Research Report on Noise-Shaped One-Bit Coefficients in Discrete Polynomial Fourier Extension

# Research Report on Noise-Shaped One-Bit Coefficients in Discrete Polynomial Fourier Extension Detailed Derivations, Rigorous Proofs, Boundary-Corrected Reconstruction, and Higher-Order Extensions
Source: [https://arxiv.org/html/2607.24868](https://arxiv.org/html/2607.24868)
\(July 2026\)

## Prefatory Note

This report grew from personal academic interest in the interaction between sigma\-delta quantization, discrete finite differences, and polynomial Fourier extension\. Its purpose is to develop the underlying identities and estimates in a detailed form that supports further mathematical study\.

## Abstract

This research report develops a rigorous analytical framework for one\-bit noise\-shaped coefficients in discrete polynomial Fourier extension\. The central object is a finite coefficient record whose quantization error admits a discrete\-difference representation\. For first\-order sigma\-delta quantization,

uk−qk=vk−vk−1,‖v‖ℓ∞≤V,u\_\{k\}\-q\_\{k\}=v\_\{k\}\-v\_\{k\-1\},\\qquad\\\|v\\\|\_\{\\ell^\{\\infty\}\}\\leq V,and the bounded state converts the quantization error into a controlled functional on weight sequences of finite discrete variation\. A complete finite\-record summation\-by\-parts argument gives

\|1N​∑k=1N\(uk−qk\)​wk\|≤VN​\(\|wN\|\+∑k=1N−1\|wk\+1−wk\|\)\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(u\_\{k\}\-q\_\{k\}\)w\_\{k\}\\right\|\\leq\\frac\{V\}\{N\}\\left\(\|w\_\{N\}\|\+\\sum\_\{k=1\}^\{N\-1\}\|w\_\{k\+1\}\-w\_\{k\}\|\\right\)when the initial state is zero\. This estimate is then applied to absolutely continuous and bounded\-variation phases, with particular attention to the parabolic phase

ϕx,t​\(ξ\)=x​ξ\+t​ξ2\.\\phi\_\{x,t\}\(\\xi\)=x\\xi\+t\\xi^\{2\}\.The resulting local estimate is

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤VN​\[1\+2​π​∫01\|x\+2​t​ξ\|​dξ\]\.\|\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\|\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\int\_\{0\}^\{1\}\|x\+2t\\xi\|\\,\\mathrm\{d\}\\xi\\right\]\.The phase\-variation factor is evaluated explicitly, compact\-set and localLpL^\{p\}consequences are derived, and a zero\-input trajectory proves that the uniform first\-order rate cannot generally be improved beyondN−1N^\{\-1\}without additional conditions\.

The report also develops the exact finite\-interval identity for errors of the forme=Δr​ve=\\Delta^\{r\}v\. All initial and terminal boundary traces are retained\. Under endpoint compatibility, repeated summation by parts transfersrrdifferences from the state to the sampled phase weight and yields anO​\(N−r\)O\(N^\{\-r\}\)compact\-set estimate\. A complementary boundary\-corrected reconstruction formula removes all finite\-record traces by using a fixed number of state\-trace values, so the same interior rate is recovered without imposing terminal compatibility\. Fractional smoothness is also treated:Cr−1,αC^\{r\-1,\\alpha\}sampled weights yield the intermediate rateO​\(N−\(r−1\+α\)\)O\(N^\{\-\(r\-1\+\\alpha\)\}\)\. The analysis is extended to polynomial phases, moment\-curve extensions, multidimensional divergence\-form errors, anisotropic growing regions, residual error models, blockwise reset, smooth termination, and oscillatory transfer estimates under explicit state hypotheses\. Numerical calculations verify indexing, constants, and predicted rates\.

The presentation follows the format of a detailed mathematical research report\. External results are cited at their substantive use, and each transfer estimate states the additional state assumptions required by its proof\.

Keywords:sigma\-delta quantization; one\-bit coefficients; noise shaping; discrete Fourier extension; polynomial phase; parabolic extension; summation by parts; total variation; endpoint compatibility; exponential sums; restriction theory\.

## Executive Summary

The report addresses a specific mathematical question\. A stable one\-bit sigma\-delta quantizer produces coefficients in\{−1,\+1\}\\\{\-1,\+1\\\}, but the pointwise quantization error is not small\. The useful structure is instead the identityu−q=Δ​vu\-q=\\Delta v, where the statevvremains uniformly bounded\. The main task is to determine how this discrete\-difference structure affects polynomial Fourier extension sums\.

The analysis produces six groups of rigorous results\.

1. Result Group 1\.First\-order stability and discrepancy\.The greedy sign quantizer is stable for inputs in\[−1,1\]\[\-1,1\]\. Every interval sum of the quantization error is controlled by the state bound, independently of the interval length\.
2. Result Group 2\.Weighted variation and parabolic extension\.Discrete summation by parts converts the shaped error into a boundary contribution and a weighted state sum\. The resulting norm is the discrete total variation of the sampled phase\. For the parabolic phase, the variation is governed byJ​\(x,t\)=∫01\|x\+2​t​ξ\|​dξJ\(x,t\)=\\int\_\{0\}^\{1\}\|x\+2t\\xi\|\\,\\mathrm\{d\}\\xi, whose exact piecewise formula is derived\.
3. Result Group 3\.Sharpness and boundary structure\.The zero\-input orbit gives an exact1/N1/Nerror at the origin for odd record lengths\. This example identifies the terminal state as the leading first\-order obstruction and proves the optimality of the uniformN−1N^\{\-1\}scale under the stated assumptions\.
4. Result Group 4\.Higher\-order finite\-record theory\.Fore=Δr​ve=\\Delta^\{r\}v, repeated summation by parts produces interior forward differences of the weight and a complete hierarchy of boundary traces\. High\-order decay follows when the relevant traces vanish or are sufficiently small\. Under exact endpoint compatibility and smooth phase sampling, the normalised error isO​\(N−r\)O\(N^\{\-r\}\)on compact parameter sets\.
5. Result Group 5\.Boundary correction and fractional regularity\.The full boundary functional can be added to the quantized extension as finite\-dimensional side information\. The corrected reconstruction has an exact pure\-interior error and therefore achieves the high\-order rate without terminal reset\. For weights inCr−1,αC^\{r\-1,\\alpha\}, the proved rate isO​\(N−\(r−1\+α\)\)O\(N^\{\-\(r\-1\+\\alpha\)\}\)\.
6. Result Group 6\.Extensions and verification\.The same mechanism applies to polynomial phases, moment curves, multidimensional divergence\-form shaping, growing observation regions, residual errors, leaky shaping, block processing, and smooth termination\. Orthogonality, exponential\-sum theory, restriction estimates, and decoupling are examined as tools for scales where total variation no longer provides decay\. Numerical examples verify the exact first\-order sharpness and the second\-order endpoint\-compatible scaling\.

The central contribution is a quantitative approximation theory for unquantized and one\-bit polynomial Fourier extensions under explicit stability and boundary hypotheses\. At critical scales, the framework connects the extension error to realisable state trajectories, oscillatory cancellation, and terminal control\.

###### Contents

1. [Prefatory Note](https://arxiv.org/html/2607.24868#Chx1)
2. [Abstract](https://arxiv.org/html/2607.24868#Chx2)
3. [Executive Summary](https://arxiv.org/html/2607.24868#Chx3)
4. [List of Abbreviations](https://arxiv.org/html/2607.24868#Chx4)
5. [Notation and Reading Guide](https://arxiv.org/html/2607.24868#Chx5)
6. [1Introduction and Research Objectives](https://arxiv.org/html/2607.24868#Ch1)1. [1\.1Research problem and motivation](https://arxiv.org/html/2607.24868#Ch1.S1)1. [1\.1\.1The mathematical problem](https://arxiv.org/html/2607.24868#Ch1.S1.SS1) 2. [1\.1\.2Relation to discrete restriction theory](https://arxiv.org/html/2607.24868#Ch1.S1.SS2) 3. [1\.1\.3Relation to finite frames and compressed sensing](https://arxiv.org/html/2607.24868#Ch1.S1.SS3) 4. [1\.1\.4Contributions](https://arxiv.org/html/2607.24868#Ch1.S1.SS4) 5. [1\.1\.5Position of the results](https://arxiv.org/html/2607.24868#Ch1.S1.SS5) 6. [1\.1\.6Organisation](https://arxiv.org/html/2607.24868#Ch1.S1.SS6) 2. [1\.2Quantization context and stable noise shaping](https://arxiv.org/html/2607.24868#Ch1.S2)1. [1\.2\.1Memoryless quantization and feedback quantization](https://arxiv.org/html/2607.24868#Ch1.S2.SS1) 2. [1\.2\.2The engineering transfer\-function picture](https://arxiv.org/html/2607.24868#Ch1.S2.SS2) 3. [1\.2\.3First\-order and higher\-order state equations](https://arxiv.org/html/2607.24868#Ch1.S2.SS3) 4. [1\.2\.4Oversampling and normalisation](https://arxiv.org/html/2607.24868#Ch1.S2.SS4) 5. [1\.2\.5Finite frames and the role of endpoint geometry](https://arxiv.org/html/2607.24868#Ch1.S2.SS5) 6. [1\.2\.6Robustness and imperfect quantizers](https://arxiv.org/html/2607.24868#Ch1.S2.SS6) 7. [1\.2\.7What stability must mean in this manuscript](https://arxiv.org/html/2607.24868#Ch1.S2.SS7)
7. [2Literature Review and Theoretical Positioning](https://arxiv.org/html/2607.24868#Ch2)1. [2\.1Exponential sums, Weyl differencing, and mean values](https://arxiv.org/html/2607.24868#Ch2.S1)1. [2\.1\.1Purpose of this literature chapter](https://arxiv.org/html/2607.24868#Ch2.S1.SS1) 2. [2\.1\.2Finite differences, moduli of smoothness, and fractional regularity](https://arxiv.org/html/2607.24868#Ch2.S1.SS2) 3. [2\.1\.3Basic notation](https://arxiv.org/html/2607.24868#Ch2.S1.SS3) 4. [2\.1\.4Abel summation and bounded partial sums](https://arxiv.org/html/2607.24868#Ch2.S1.SS4) 5. [2\.1\.5Van der Corput differencing](https://arxiv.org/html/2607.24868#Ch2.S1.SS5) 6. [2\.1\.6Weyl’s inequality](https://arxiv.org/html/2607.24868#Ch2.S1.SS6) 7. [2\.1\.7The Vinogradov mean value](https://arxiv.org/html/2607.24868#Ch2.S1.SS7) 8. [2\.1\.8Weighted mean values](https://arxiv.org/html/2607.24868#Ch2.S1.SS8) 9. [2\.1\.9Major arcs, minor arcs, and shaped coefficients](https://arxiv.org/html/2607.24868#Ch2.S1.SS9) 10. [2\.1\.10Completion and short intervals](https://arxiv.org/html/2607.24868#Ch2.S1.SS10) 11. [2\.1\.11A possible hybrid inequality](https://arxiv.org/html/2607.24868#Ch2.S1.SS11) 12. [2\.1\.12What additional state hypotheses would help](https://arxiv.org/html/2607.24868#Ch2.S1.SS12) 13. [2\.1\.13Summary of the number\-theoretic route](https://arxiv.org/html/2607.24868#Ch2.S1.SS13) 2. [2\.2Restriction, Strichartz estimates, and decoupling](https://arxiv.org/html/2607.24868#Ch2.S2)1. [2\.2\.1Continuous restriction as the geometric model](https://arxiv.org/html/2607.24868#Ch2.S2.SS1) 2. [2\.2\.2The Stein–Tomas mechanism](https://arxiv.org/html/2607.24868#Ch2.S2.SS2) 3. [2\.2\.3Strichartz estimates](https://arxiv.org/html/2607.24868#Ch2.S2.SS3) 4. [2\.2\.4Discrete restriction for the parabola](https://arxiv.org/html/2607.24868#Ch2.S2.SS4) 5. [2\.2\.5Decoupling for the moment curve](https://arxiv.org/html/2607.24868#Ch2.S2.SS5) 6. [2\.2\.6What decoupling separates](https://arxiv.org/html/2607.24868#Ch2.S2.SS6) 7. [2\.2\.7Boundary terms under frequency localisation](https://arxiv.org/html/2607.24868#Ch2.S2.SS7) 8. [2\.2\.8A model dyadic decomposition](https://arxiv.org/html/2607.24868#Ch2.S2.SS8) 9. [2\.2\.9Multilinear restriction and transversality](https://arxiv.org/html/2607.24868#Ch2.S2.SS9) 10. [2\.2\.10A negative Sobolev viewpoint](https://arxiv.org/html/2607.24868#Ch2.S2.SS10) 11. [2\.2\.11Periodic versus Euclidean regions](https://arxiv.org/html/2607.24868#Ch2.S2.SS11) 12. [2\.2\.12The realistic role of decoupling in this project](https://arxiv.org/html/2607.24868#Ch2.S2.SS12)
8. [3Analytical Methodology and Discrete Mathematical Framework](https://arxiv.org/html/2607.24868#Ch3)1. [3\.1Discrete calculus, variation, and finite summation](https://arxiv.org/html/2607.24868#Ch3.S1)1. [3\.1\.1Backward and forward differences](https://arxiv.org/html/2607.24868#Ch3.S1.SS1) 2. [3\.1\.2Discrete summation by parts](https://arxiv.org/html/2607.24868#Ch3.S1.SS2) 3. [3\.1\.3Discrete total variation](https://arxiv.org/html/2607.24868#Ch3.S1.SS3) 4. [3\.1\.4Continuous total variation](https://arxiv.org/html/2607.24868#Ch3.S1.SS4) 5. [3\.1\.5Sampling a BV phase](https://arxiv.org/html/2607.24868#Ch3.S1.SS5) 6. [3\.1\.6Riemann\-sum scaling](https://arxiv.org/html/2607.24868#Ch3.S1.SS6) 7. [3\.1\.7A useful integral formula for finite differences](https://arxiv.org/html/2607.24868#Ch3.S1.SS7) 8. [3\.1\.8Why exact indexing matters](https://arxiv.org/html/2607.24868#Ch3.S1.SS8)
9. [4First\-Order One\-Bit Noise\-Shaping Theory](https://arxiv.org/html/2607.24868#Ch4)1. [4\.1Greedy one\-bit stability and discrepancy](https://arxiv.org/html/2607.24868#Ch4.S1)1. [4\.1\.1The greedy sign quantizer](https://arxiv.org/html/2607.24868#Ch4.S1.SS1) 2. [4\.1\.2Invariant interval](https://arxiv.org/html/2607.24868#Ch4.S1.SS2) 3. [4\.1\.3Strict input margins](https://arxiv.org/html/2607.24868#Ch4.S1.SS3) 4. [4\.1\.4Interval discrepancy](https://arxiv.org/html/2607.24868#Ch4.S1.SS4) 5. [4\.1\.5Prefix sums and equivalence of formulations](https://arxiv.org/html/2607.24868#Ch4.S1.SS5) 6. [4\.1\.6The zero\-input orbit](https://arxiv.org/html/2607.24868#Ch4.S1.SS6) 7. [4\.1\.7Complex inputs and larger alphabets](https://arxiv.org/html/2607.24868#Ch4.S1.SS7) 2. [4\.2Weighted variation estimates](https://arxiv.org/html/2607.24868#Ch4.S2)1. [4\.2\.1The general theorem](https://arxiv.org/html/2607.24868#Ch4.S2.SS1) 2. [4\.2\.2Optimality of the variation form](https://arxiv.org/html/2607.24868#Ch4.S2.SS2) 3. [4\.2\.3Absolutely continuous phases](https://arxiv.org/html/2607.24868#Ch4.S2.SS3) 4. [4\.2\.4Bounded\-variation phases](https://arxiv.org/html/2607.24868#Ch4.S2.SS4) 5. [4\.2\.5Lipschitz and Hölder corollaries](https://arxiv.org/html/2607.24868#Ch4.S2.SS5) 6. [4\.2\.6Vector\-valued and operator\-valued weights](https://arxiv.org/html/2607.24868#Ch4.S2.SS6) 7. [4\.2\.7A duality interpretation](https://arxiv.org/html/2607.24868#Ch4.S2.SS7)
10. [5Detailed Derivations and Proved Results for Parabolic Fourier Extension](https://arxiv.org/html/2607.24868#Ch5)1. [5\.1Parabolic Fourier extension and the phase\-variation factor](https://arxiv.org/html/2607.24868#Ch5.S1)1. [5\.1\.1Definition and PDE interpretation](https://arxiv.org/html/2607.24868#Ch5.S1.SS1) 2. [5\.1\.2The variation factor](https://arxiv.org/html/2607.24868#Ch5.S1.SS2) 3. [5\.1\.3Exact evaluation ofJ​\(x,t\)J\(x,t\)](https://arxiv.org/html/2607.24868#Ch5.S1.SS3) 4. [5\.1\.4Compact\-set convergence](https://arxiv.org/html/2607.24868#Ch5.S1.SS4) 5. [5\.1\.5Rectangular localLpL^\{p\}bounds](https://arxiv.org/html/2607.24868#Ch5.S1.SS5) 6. [5\.1\.6An exact adjacent\-phase estimate](https://arxiv.org/html/2607.24868#Ch5.S1.SS6) 7. [5\.1\.7Geometric interpretation](https://arxiv.org/html/2607.24868#Ch5.S1.SS7) 2. [5\.2Complete derivation of the parabolic estimate](https://arxiv.org/html/2607.24868#Ch5.S2)1. [5\.2\.1Dependence of the estimate on the assumptions](https://arxiv.org/html/2607.24868#Ch5.S2.SS1) 2. [5\.2\.2Exact and simplified constants](https://arxiv.org/html/2607.24868#Ch5.S2.SS2) 3. [5\.3Sharpness, boundary obstruction, and limits of first\-order control](https://arxiv.org/html/2607.24868#Ch5.S3)1. [5\.3\.1Uniform optimality ofN−1N^\{\-1\}](https://arxiv.org/html/2607.24868#Ch5.S3.SS1) 2. [5\.3\.2The quantifiers in the sharpness statement](https://arxiv.org/html/2607.24868#Ch5.S3.SS2) 3. [5\.3\.3The terminal state as the leading obstruction](https://arxiv.org/html/2607.24868#Ch5.S3.SS3) 4. [5\.3\.4Special inputs with improved behaviour](https://arxiv.org/html/2607.24868#Ch5.S3.SS4) 5. [5\.3\.5Why global uniformity fails on large regions](https://arxiv.org/html/2607.24868#Ch5.S3.SS5) 6. [5\.3\.6No automatic globalL2L^\{2\}gain](https://arxiv.org/html/2607.24868#Ch5.S3.SS6) 7. [5\.3\.7The danger of an invalid high\-order argument](https://arxiv.org/html/2607.24868#Ch5.S3.SS7) 8. [5\.3\.8A hierarchy of possible improvements](https://arxiv.org/html/2607.24868#Ch5.S3.SS8)
11. [6Higher\-Order Finite\-Record Derivations and Endpoint\-Compatible Proofs](https://arxiv.org/html/2607.24868#Ch6)1. [6\.1Exact higher\-order summation on a finite record](https://arxiv.org/html/2607.24868#Ch6.S1)1. [6\.1\.1Why the finite interval must be treated explicitly](https://arxiv.org/html/2607.24868#Ch6.S1.SS1) 2. [6\.1\.2Difference conventions and boundary traces](https://arxiv.org/html/2607.24868#Ch6.S1.SS2) 3. [6\.1\.3The exact finite\-interval identity](https://arxiv.org/html/2607.24868#Ch6.S1.SS3) 4. [6\.1\.4Low\-order cases written in full](https://arxiv.org/html/2607.24868#Ch6.S1.SS4) 5. [6\.1\.5A full deterministic estimate](https://arxiv.org/html/2607.24868#Ch6.S1.SS5) 6. [6\.1\.6Boundary\-trace correction as a reconstruction principle](https://arxiv.org/html/2607.24868#Ch6.S1.SS6) 7. [6\.1\.7Zero padding and terminal traces](https://arxiv.org/html/2607.24868#Ch6.S1.SS7) 8. [6\.1\.8Matrix formulation](https://arxiv.org/html/2607.24868#Ch6.S1.SS8) 9. [6\.1\.9Boundary verification checklist](https://arxiv.org/html/2607.24868#Ch6.S1.SS9) 2. [6\.2Endpoint\-compatible high\-order extension theory](https://arxiv.org/html/2607.24868#Ch6.S2)1. [6\.2\.1Strong and weak compatibility conditions](https://arxiv.org/html/2607.24868#Ch6.S2.SS1) 2. [6\.2\.2Forward differences of sampled smooth functions](https://arxiv.org/html/2607.24868#Ch6.S2.SS2) 3. [6\.2\.3Fractional smoothness and intermediate rates](https://arxiv.org/html/2607.24868#Ch6.S2.SS3) 4. [6\.2\.4The high\-order compact\-set theorem](https://arxiv.org/html/2607.24868#Ch6.S2.SS4) 5. [6\.2\.5Oscillatory weights and Bell\-polynomial bounds](https://arxiv.org/html/2607.24868#Ch6.S2.SS5) 6. [6\.2\.6Second\-order parabolic estimate](https://arxiv.org/html/2607.24868#Ch6.S2.SS6) 7. [6\.2\.7Approximate endpoint compatibility](https://arxiv.org/html/2607.24868#Ch6.S2.SS7) 8. [6\.2\.8Interpretation for quantizer design](https://arxiv.org/html/2607.24868#Ch6.S2.SS8)
12. [7Polynomial, Multidimensional, and Growing\-Region Extensions](https://arxiv.org/html/2607.24868#Ch7)1. [7\.1Polynomial phases and moment\-curve extensions](https://arxiv.org/html/2607.24868#Ch7.S1)1. [7\.1\.1The polynomial extension operator](https://arxiv.org/html/2607.24868#Ch7.S1.SS1) 2. [7\.1\.2First\-order estimates](https://arxiv.org/html/2607.24868#Ch7.S1.SS2) 3. [7\.1\.3High\-order estimates under endpoint compatibility](https://arxiv.org/html/2607.24868#Ch7.S1.SS3) 4. [7\.1\.4Explicit second\- and third\-order formulae](https://arxiv.org/html/2607.24868#Ch7.S1.SS4) 5. [7\.1\.5Anisotropic parameter boxes](https://arxiv.org/html/2607.24868#Ch7.S1.SS5) 6. [7\.1\.6Stationary points and local improvement](https://arxiv.org/html/2607.24868#Ch7.S1.SS6) 7. [7\.1\.7Coefficient normalisation and comparison with restriction theory](https://arxiv.org/html/2607.24868#Ch7.S1.SS7) 8. [7\.1\.8A new family of structured restriction questions](https://arxiv.org/html/2607.24868#Ch7.S1.SS8) 2. [7\.2Multidimensional noise shaping and grid extensions](https://arxiv.org/html/2607.24868#Ch7.S2)1. [7\.2\.1From a sequence to a lattice array](https://arxiv.org/html/2607.24868#Ch7.S2.SS1) 2. [7\.2\.2Divergence\-form shaped errors](https://arxiv.org/html/2607.24868#Ch7.S2.SS2) 3. [7\.2\.3Discrete divergence theorem](https://arxiv.org/html/2607.24868#Ch7.S2.SS3) 4. [7\.2\.4Smooth multidimensional phases](https://arxiv.org/html/2607.24868#Ch7.S2.SS4) 5. [7\.2\.5A two\-dimensional polynomial example](https://arxiv.org/html/2607.24868#Ch7.S2.SS5) 6. [7\.2\.6Directional and weighted schemes](https://arxiv.org/html/2607.24868#Ch7.S2.SS6) 7. [7\.2\.7Higher\-order multi\-index shaping](https://arxiv.org/html/2607.24868#Ch7.S2.SS7) 8. [7\.2\.8Scan order and causality](https://arxiv.org/html/2607.24868#Ch7.S2.SS8) 9. [7\.2\.9Dimension\-free rate and dimension\-dependent constants](https://arxiv.org/html/2607.24868#Ch7.S2.SS9) 10. [7\.2\.10Open multidimensional questions](https://arxiv.org/html/2607.24868#Ch7.S2.SS10) 3. [7\.3Growing observation regions and scale transitions](https://arxiv.org/html/2607.24868#Ch7.S3)1. [7\.3\.1Why fixed compact sets are only the first scale](https://arxiv.org/html/2607.24868#Ch7.S3.SS1) 2. [7\.3\.2Isotropic parabolic boxes](https://arxiv.org/html/2607.24868#Ch7.S3.SS2) 3. [7\.3\.3Anisotropic parabolic boxes](https://arxiv.org/html/2607.24868#Ch7.S3.SS3) 4. [7\.3\.4Exact sine variation](https://arxiv.org/html/2607.24868#Ch7.S3.SS4) 5. [7\.3\.5Higher\-order endpoint\-compatible scaling](https://arxiv.org/html/2607.24868#Ch7.S3.SS5) 6. [7\.3\.6A boundary\-layer refinement](https://arxiv.org/html/2607.24868#Ch7.S3.SS6) 7. [7\.3\.7Volume factors in localLpL^\{p\}estimates](https://arxiv.org/html/2607.24868#Ch7.S3.SS7) 8. [7\.3\.8Physical and torus scalings](https://arxiv.org/html/2607.24868#Ch7.S3.SS8) 9. [7\.3\.9A scale diagram](https://arxiv.org/html/2607.24868#Ch7.S3.SS9) 10. [7\.3\.10Interpretation of the scale transition](https://arxiv.org/html/2607.24868#Ch7.S3.SS10)
13. [8LocalLpL^\{p\}Analysis and Oscillatory Transfer](https://arxiv.org/html/2607.24868#Ch8)1. [8\.1Exact orthogonality and baseline estimates](https://arxiv.org/html/2607.24868#Ch8.S1)1. [8\.1\.1Orthogonality in the linear parameter](https://arxiv.org/html/2607.24868#Ch8.S1.SS1) 2. [8\.1\.2Two\-parameter torus normalisation](https://arxiv.org/html/2607.24868#Ch8.S1.SS2) 3. [8\.1\.3The fourth moment and additive energy](https://arxiv.org/html/2607.24868#Ch8.S1.SS3) 4. [8\.1\.4The sixth moment and quadratic discrete restriction](https://arxiv.org/html/2607.24868#Ch8.S1.SS4) 2. [8\.2Interpolation and localLpL^\{p\}consequences](https://arxiv.org/html/2607.24868#Ch8.S2)1. [8\.2\.1Interpolation on the full parabolic cell](https://arxiv.org/html/2607.24868#Ch8.S2.SS1) 2. [8\.2\.2Fixed compact sets](https://arxiv.org/html/2607.24868#Ch8.S2.SS2) 3. [8\.2\.3Growing rectangles and the deterministic threshold](https://arxiv.org/html/2607.24868#Ch8.S2.SS3) 3. [8\.3LocalL2L^\{2\}analysis by oscillatory kernels](https://arxiv.org/html/2607.24868#Ch8.S3)1. [8\.3\.1The exact kernel identity](https://arxiv.org/html/2607.24868#Ch8.S3.SS1) 2. [8\.3\.2A deterministic localL2L^\{2\}bound](https://arxiv.org/html/2607.24868#Ch8.S3.SS2) 3. [8\.3\.3Double summation by parts for first\-order shaped errors](https://arxiv.org/html/2607.24868#Ch8.S3.SS3) 4. [8\.4Moment cancellation and endpoint\-compatible shaping](https://arxiv.org/html/2607.24868#Ch8.S4)1. [8\.4\.1Exact annihilation of low\-degree polynomial tests](https://arxiv.org/html/2607.24868#Ch8.S4.SS1) 2. [8\.4\.2Taylor subtraction for smooth weights](https://arxiv.org/html/2607.24868#Ch8.S4.SS2) 3. [8\.4\.3Polynomial\-phase weights](https://arxiv.org/html/2607.24868#Ch8.S4.SS3) 5. [8\.5Exact transfer formulas for noise\-shaped states](https://arxiv.org/html/2607.24868#Ch8.S5)1. [8\.5\.1First\-order phase factorisation](https://arxiv.org/html/2607.24868#Ch8.S5.SS1) 2. [8\.5\.2A deterministic transfer theorem](https://arxiv.org/html/2607.24868#Ch8.S5.SS2) 3. [8\.5\.3High\-order transfer](https://arxiv.org/html/2607.24868#Ch8.S5.SS3) 6. [8\.6Random and weakly correlated state models](https://arxiv.org/html/2607.24868#Ch8.S6)1. [8\.6\.1Independent centred states](https://arxiv.org/html/2607.24868#Ch8.S6.SS1) 2. [8\.6\.2Summable covariance](https://arxiv.org/html/2607.24868#Ch8.S6.SS2) 3. [8\.6\.3High\-order random states](https://arxiv.org/html/2607.24868#Ch8.S6.SS3) 7. [8\.7Correlation estimates and van der Corput transfer](https://arxiv.org/html/2607.24868#Ch8.S7)1. [8\.7\.1A finite van der Corput inequality](https://arxiv.org/html/2607.24868#Ch8.S7.SS1) 2. [8\.7\.2State\-phase correlations](https://arxiv.org/html/2607.24868#Ch8.S7.SS2) 8. [8\.8Dyadic block decompositions and square\-function transfer](https://arxiv.org/html/2607.24868#Ch8.S8)1. [8\.8\.1Block decomposition](https://arxiv.org/html/2607.24868#Ch8.S8.SS1) 2. [8\.8\.2A deterministic block\-energy baseline](https://arxiv.org/html/2607.24868#Ch8.S8.SS2) 9. [8\.9Frequency\-side interpretation of discrete differences](https://arxiv.org/html/2607.24868#Ch8.S9)1. [8\.9\.1Periodic discrete Fourier transform](https://arxiv.org/html/2607.24868#Ch8.S9.SS1) 2. [8\.9\.2A periodic Sobolev estimate](https://arxiv.org/html/2607.24868#Ch8.S9.SS2) 10. [8\.10Commutator identities for chirp modulation](https://arxiv.org/html/2607.24868#Ch8.S10)1. [8\.10\.1A discrete product rule](https://arxiv.org/html/2607.24868#Ch8.S10.SS1) 11. [8\.11Sharp obstructions and consistency tests](https://arxiv.org/html/2607.24868#Ch8.S11)1. [8\.11\.1Zero\-frequency boundary obstruction](https://arxiv.org/html/2607.24868#Ch8.S11.SS1) 2. [8\.11\.2Full\-periodL2L^\{2\}obstruction](https://arxiv.org/html/2607.24868#Ch8.S11.SS2) 3. [8\.11\.3Random bounded states](https://arxiv.org/html/2607.24868#Ch8.S11.SS3) 4. [8\.11\.4Interior\-supported states](https://arxiv.org/html/2607.24868#Ch8.S11.SS4) 5. [8\.11\.5Periodic state orbits](https://arxiv.org/html/2607.24868#Ch8.S11.SS5) 12. [8\.12A consolidated hierarchy of proved estimates](https://arxiv.org/html/2607.24868#Ch8.S12)1. [8\.12\.1Arbitrary coefficients](https://arxiv.org/html/2607.24868#Ch8.S12.SS1) 2. [8\.12\.2First\-order bounded states](https://arxiv.org/html/2607.24868#Ch8.S12.SS2) 3. [8\.12\.3Endpoint\-compatible high\-order states](https://arxiv.org/html/2607.24868#Ch8.S12.SS3) 13. [8\.13Critical\-scale research target](https://arxiv.org/html/2607.24868#Ch8.S13) 14. [8\.14Chapter conclusion](https://arxiv.org/html/2607.24868#Ch8.S14)
14. [9Robustness, Boundary Engineering, and Verification](https://arxiv.org/html/2607.24868#Ch9)1. [9\.1Robustness to residual error, leakage, and model mismatch](https://arxiv.org/html/2607.24868#Ch9.S1)1. [9\.1\.1Exact identities and physical implementations](https://arxiv.org/html/2607.24868#Ch9.S1.SS1) 2. [9\.1\.2Additive residuals](https://arxiv.org/html/2607.24868#Ch9.S1.SS2) 3. [9\.1\.3Residuals with their own noise\-shaping order](https://arxiv.org/html/2607.24868#Ch9.S1.SS3) 4. [9\.1\.4Leaky first\-order shaping](https://arxiv.org/html/2607.24868#Ch9.S1.SS4) 5. [9\.1\.5Perturbed weights and sampling jitter](https://arxiv.org/html/2607.24868#Ch9.S1.SS5) 6. [9\.1\.6Coefficient saturation and overload events](https://arxiv.org/html/2607.24868#Ch9.S1.SS6) 7. [9\.1\.7Approximate high\-order shaping](https://arxiv.org/html/2607.24868#Ch9.S1.SS7) 8. [9\.1\.8Dither and probabilistic modelling](https://arxiv.org/html/2607.24868#Ch9.S1.SS8) 9. [9\.1\.9A robustness checklist](https://arxiv.org/html/2607.24868#Ch9.S1.SS9) 2. [9\.2Blockwise reset, smooth termination, and boundary engineering](https://arxiv.org/html/2607.24868#Ch9.S2)1. [9\.2\.1The purpose of block processing](https://arxiv.org/html/2607.24868#Ch9.S2.SS1) 2. [9\.2\.2First\-order block identity](https://arxiv.org/html/2607.24868#Ch9.S2.SS2) 3. [9\.2\.3Choosing the block length](https://arxiv.org/html/2607.24868#Ch9.S2.SS3) 4. [9\.2\.4Exact terminal reset](https://arxiv.org/html/2607.24868#Ch9.S2.SS4) 5. [9\.2\.5High\-order block compatibility](https://arxiv.org/html/2607.24868#Ch9.S2.SS5) 6. [9\.2\.6Imperfect block termination](https://arxiv.org/html/2607.24868#Ch9.S2.SS6) 7. [9\.2\.7Windowed reconstruction as an alternative](https://arxiv.org/html/2607.24868#Ch9.S2.SS7) 8. [9\.2\.8A discrete polynomial window](https://arxiv.org/html/2607.24868#Ch9.S2.SS8) 9. [9\.2\.9State flushing by an appended tail](https://arxiv.org/html/2607.24868#Ch9.S2.SS9) 10. [9\.2\.10Parallel implementation and numerical experiments](https://arxiv.org/html/2607.24868#Ch9.S2.SS10) 11. [9\.2\.11Design principle](https://arxiv.org/html/2607.24868#Ch9.S2.SS11) 3. [9\.3Verification of the proved finite\-record results](https://arxiv.org/html/2607.24868#Ch9.S3)1. [9\.3\.1Computational verification protocol](https://arxiv.org/html/2607.24868#Ch9.S3.SS1) 2. [9\.3\.2Core experiment](https://arxiv.org/html/2607.24868#Ch9.S3.SS2) 3. [9\.3\.3Convergence\-order estimation](https://arxiv.org/html/2607.24868#Ch9.S3.SS3) 4. [9\.3\.4Testing the closed form ofJJ](https://arxiv.org/html/2607.24868#Ch9.S3.SS4) 5. [9\.3\.5Exact sine variation](https://arxiv.org/html/2607.24868#Ch9.S3.SS5) 6. [9\.3\.6State\-phase alignment](https://arxiv.org/html/2607.24868#Ch9.S3.SS6) 7. [9\.3\.7Growing regions](https://arxiv.org/html/2607.24868#Ch9.S3.SS7) 8. [9\.3\.8High\-order synthetic validation](https://arxiv.org/html/2607.24868#Ch9.S3.SS8) 9. [9\.3\.9Monte Carlo decorrelation experiments](https://arxiv.org/html/2607.24868#Ch9.S3.SS9) 10. [9\.3\.10Reporting standards](https://arxiv.org/html/2607.24868#Ch9.S3.SS10)
15. [10Consolidated Derivations, Main Results, and Conclusions](https://arxiv.org/html/2607.24868#Ch10)1. [10\.1Purpose of the consolidated chapter](https://arxiv.org/html/2607.24868#Ch10.S1) 2. [10\.2Master notation and standing assumptions](https://arxiv.org/html/2607.24868#Ch10.S2) 3. [10\.3Complete first\-order derivation](https://arxiv.org/html/2607.24868#Ch10.S3)1. [10\.3\.1Stability of the greedy recursion](https://arxiv.org/html/2607.24868#Ch10.S3.SS1) 2. [10\.3\.2Discrepancy identity](https://arxiv.org/html/2607.24868#Ch10.S3.SS2) 3. [10\.3\.3Weighted variation theorem](https://arxiv.org/html/2607.24868#Ch10.S3.SS3) 4. [10\.3\.4Absolutely continuous and BV phases](https://arxiv.org/html/2607.24868#Ch10.S3.SS4) 4. [10\.4Parabolic result catalogue](https://arxiv.org/html/2607.24868#Ch10.S4) 5. [10\.5Sharpness of the first\-order rate](https://arxiv.org/html/2607.24868#Ch10.S5) 6. [10\.6Higher\-order finite\-record derivation](https://arxiv.org/html/2607.24868#Ch10.S6)1. [10\.6\.1Why boundary traces appear](https://arxiv.org/html/2607.24868#Ch10.S6.SS1) 2. [10\.6\.2Endpoint\-compatible rate](https://arxiv.org/html/2607.24868#Ch10.S6.SS2) 3. [10\.6\.3Approximate compatibility](https://arxiv.org/html/2607.24868#Ch10.S6.SS3) 7. [10\.7Boundary Correction and Fractional Rates](https://arxiv.org/html/2607.24868#Ch10.S7) 8. [10\.8Polynomial and multidimensional consequences](https://arxiv.org/html/2607.24868#Ch10.S8) 9. [10\.9Growing regions and scale transition](https://arxiv.org/html/2607.24868#Ch10.S9) 10. [10\.10LocalLpL^\{p\}, orthogonality, and external harmonic analysis](https://arxiv.org/html/2607.24868#Ch10.S10) 11. [10\.11Robustness decomposition](https://arxiv.org/html/2607.24868#Ch10.S11) 12. [10\.12Boundary engineering and block processing](https://arxiv.org/html/2607.24868#Ch10.S12) 13. [10\.13Numerical verification of exact statements](https://arxiv.org/html/2607.24868#Ch10.S13) 14. [10\.14Detailed treatment of nonzero initial states](https://arxiv.org/html/2607.24868#Ch10.S14) 15. [10\.15Worked second\-order finite\-record derivation](https://arxiv.org/html/2607.24868#Ch10.S15)1. [10\.15\.1Second\-order parabolic constant](https://arxiv.org/html/2607.24868#Ch10.S15.SS1) 16. [10\.16Detailed proof of the piecewise formula for the parabolic variation](https://arxiv.org/html/2607.24868#Ch10.S16) 17. [10\.17Constant dependence and uniformity statements](https://arxiv.org/html/2607.24868#Ch10.S17) 18. [10\.18Assumption\-to\-conclusion map](https://arxiv.org/html/2607.24868#Ch10.S18) 19. [10\.19Separation between proved approximation and restriction estimates](https://arxiv.org/html/2607.24868#Ch10.S19) 20. [10\.20Summary of principal results](https://arxiv.org/html/2607.24868#Ch10.S20) 21. [10\.21Scope of the established results](https://arxiv.org/html/2607.24868#Ch10.S21) 22. [10\.22Conclusions](https://arxiv.org/html/2607.24868#Ch10.S22)
16. [AExpanded Discrete Identities](https://arxiv.org/html/2607.24868#A1)1. [A\.1Binomial formulas](https://arxiv.org/html/2607.24868#A1.S1) 2. [A\.2Low\-order expansions](https://arxiv.org/html/2607.24868#A1.S2) 3. [A\.3Verification of the second\-order finite formula](https://arxiv.org/html/2607.24868#A1.S3) 4. [A\.4Fourth\-order boundary table](https://arxiv.org/html/2607.24868#A1.S4) 5. [A\.5A two\-dimensional second\-order example](https://arxiv.org/html/2607.24868#A1.S5) 6. [A\.6Discrete product rules](https://arxiv.org/html/2607.24868#A1.S6) 7. [A\.7Periodic differences](https://arxiv.org/html/2607.24868#A1.S7) 8. [A\.8Zero extension and boundary impulses](https://arxiv.org/html/2607.24868#A1.S8) 9. [A\.9Norm bounds for state differences](https://arxiv.org/html/2607.24868#A1.S9)
17. [BDerivative Bounds for Oscillatory Weights](https://arxiv.org/html/2607.24868#A2)1. [B\.1Faà di Bruno and Bell polynomials](https://arxiv.org/html/2607.24868#A2.S1) 2. [B\.2Low\-order phase bounds](https://arxiv.org/html/2607.24868#A2.S2) 3. [B\.3Quadratic phase formulas](https://arxiv.org/html/2607.24868#A2.S3) 4. [B\.4Polynomial phases](https://arxiv.org/html/2607.24868#A2.S4) 5. [B\.5Finite differences versus derivatives](https://arxiv.org/html/2607.24868#A2.S5) 6. [B\.6Verification of the displayed third derivative](https://arxiv.org/html/2607.24868#A2.S6)
18. [CBackground on Bounded Variation](https://arxiv.org/html/2607.24868#A3)1. [C\.1Definition](https://arxiv.org/html/2607.24868#A3.S1) 2. [C\.2Absolutely continuous functions](https://arxiv.org/html/2607.24868#A3.S2) 3. [C\.3Composition with a Lipschitz map](https://arxiv.org/html/2607.24868#A3.S3) 4. [C\.4Choice of representative](https://arxiv.org/html/2607.24868#A3.S4) 5. [C\.5Jordan decomposition](https://arxiv.org/html/2607.24868#A3.S5) 6. [C\.6Complex BV functions](https://arxiv.org/html/2607.24868#A3.S6) 7. [C\.7Discrete sampling inequality](https://arxiv.org/html/2607.24868#A3.S7) 8. [C\.8Higher\-dimensional BV](https://arxiv.org/html/2607.24868#A3.S8)
19. [DReference Map and Theoretical Sources](https://arxiv.org/html/2607.24868#A4)
20. [EReproducible Code](https://arxiv.org/html/2607.24868#A5)1. [E\.1Python implementation of the greedy quantizer](https://arxiv.org/html/2607.24868#A5.S1) 2. [E\.2Parabolic extension and theoretical bound](https://arxiv.org/html/2607.24868#A5.S2) 3. [E\.3Identity check](https://arxiv.org/html/2607.24868#A5.S3) 4. [E\.4Synthetic high\-order identity test](https://arxiv.org/html/2607.24868#A5.S4) 5. [E\.5MATLAB\-style implementation](https://arxiv.org/html/2607.24868#A5.S5) 6. [E\.6Convergence\-order script outline](https://arxiv.org/html/2607.24868#A5.S6) 7. [E\.7Numerical verification notes](https://arxiv.org/html/2607.24868#A5.S7)
21. [FGlossary of Terms](https://arxiv.org/html/2607.24868#A6)1. [F\.1Symbols most likely to be confused](https://arxiv.org/html/2607.24868#A6.S1)
22. [References](https://arxiv.org/html/2607.24868#bib)

###### List of Figures

1. [9\.1Computed first\-order error and the deterministic variation bound at a fixed parabolic parameter\. The bound has the guaranteedN−1N^\{\-1\}scale, while the selected input exhibits additional cancellation\.](https://arxiv.org/html/2607.24868#Ch9.F1)
2. [9\.2Exact sharpness experiment for the zero input and odd record lengths\. The computed values coincide withN−1N^\{\-1\}\.](https://arxiv.org/html/2607.24868#Ch9.F2)
3. [9\.3The phase\-variation factorJ​\(x,t\)=∫01\|x\+2​t​ξ\|​dξJ\(x,t\)=\\int\_\{0\}^\{1\}\\left\\lvert x\+2t\\xi\\right\\rvert\\,\\,\\mathrm\{d\}\\xion a bounded parameter square\. The change in formula occurs where the derivative crosses zero inside the sampling interval\.](https://arxiv.org/html/2607.24868#Ch9.F3)
4. [9\.4Synthetic second\-order validation with exact endpoint compatibility\. The fitted exponent is approximately−1\.991\-1\.991\. This experiment verifies the analytic scaling but does not by itself construct a one\-bit second\-order quantizer with terminal control\.](https://arxiv.org/html/2607.24868#Ch9.F4)

###### List of Tables

1. [List of Abbreviations](https://arxiv.org/html/2607.24868#Chx4.T1)
2. [Notation and Reading Guide](https://arxiv.org/html/2607.24868#Chx5.T2)
3. [7\.3\.9A scale diagram](https://arxiv.org/html/2607.24868#Ch7.T1)
4. [10\.1Role of the principal assumptions in the proved estimates](https://arxiv.org/html/2607.24868#Ch10.T1)
5. [10\.2Principal mathematical results and analytical settings](https://arxiv.org/html/2607.24868#Ch10.T2)
6. [A\.1Boundary hierarchy for order four](https://arxiv.org/html/2607.24868#A1.T1)
7. [DReference Map and Theoretical Sources](https://arxiv.org/html/2607.24868#A4.T1)

## List of Abbreviations

ADCanalogue\-to\-digital converterACabsolutely continuousBVbounded variationBCRboundary\-corrected reconstructionCScompressed sensingDFRdiscrete Fourier restrictionDFTdiscrete Fourier transformDSPdigital signal processingENOBeffective number of bitsFFTfast Fourier transformLHSleft\-hand sideMSEmean squared errorNTFnoise transfer functionOSRoversampling ratioPCMpulse\-code modulationPDEpartial differential equationRHSright\-hand sideRIPrestricted isometry propertySNRsignal\-to\-noise ratioSNDRsignal\-to\-noise\-and\-distortion ratioSTFsignal transfer functionVMVTVinogradov mean value theorem
## Notation and Reading Guide

NNnumber of samples or coefficientsu=\(uk\)u=\(u\_\{k\}\)unquantized input sequenceq=\(qk\)q=\(q\_\{k\}\)quantized output sequence, usually one\-bite=u−qe=u\-qquantization error sequencevvinternal state sequenceΔ​vk=vk−vk−1\\Delta v\_\{k\}=v\_\{k\}\-v\_\{k\-1\}backward finite differenceΔ\+​wk=wk\+1−wk\\Delta\_\{\+\}w\_\{k\}=w\_\{k\+1\}\-w\_\{k\}forward finite differenceℰN\\mathcal\{E\}\_\{N\}normalized discrete extension operatorTVN⁡\(w\)\\operatorname\{TV\}\_\{N\}\(w\)discrete total variation of a weight sequenceVar\[0,1\]⁡\(ϕ\)\\operatorname\{Var\}\_\{\[0,1\]\}\(\\phi\)total variation of a scalar phaseA≲BA\\lesssim BA≤C​BA\\leq CBfor a constant independent of the main asymptotic parameter𝕋=ℝ/ℤ\\mathbb\{T\}=\\mathbb\{R\}/\\mathbb\{Z\}one\-dimensional toruse​\(z\)e\(z\)shorthand forexp⁡\(2​π​i​z\)\\exp\(2\\pi\\mathrm\{i\}z\)when used in number\-theoretic sectionsThe core first\-order argument is contained in Chapters 3–5\. The exact higher\-order finite\-record theory appears in Chapter 6\. Polynomial, multidimensional, and growing\-region extensions are collected in Chapter 7\. External harmonic\-analysis tools and conditional transfer statements are separated in Chapters 2 and 8\.

## Chapter 1Introduction and Research Objectives

### 1\.1Research problem and motivation

#### 1\.1\.1The mathematical problem

One\-bit sigma\-delta quantization converts an analogue or finely represented sequence into a binary sequence while feeding the accumulated error back into the quantizer\. The output symbols are coarse, but the error is organised\. This distinction is the reason sigma\-delta methods can outperform memoryless scalar quantization under oversampling\. The engineering origins of delta\-sigma modulation go back to early feedback coding systems, while the spectral and probabilistic analysis of quantization noise developed through the work of Bennett, Gray, Candy, and many others\[[7](https://arxiv.org/html/2607.24868#bib.bib1),[58](https://arxiv.org/html/2607.24868#bib.bib2),[20](https://arxiv.org/html/2607.24868#bib.bib3),[43](https://arxiv.org/html/2607.24868#bib.bib5),[44](https://arxiv.org/html/2607.24868#bib.bib4)\]\. A rigorous modern mathematical theory was established through stable high\-order constructions, invariant\-region arguments, exponential accuracy, and robustness analysis\[[45](https://arxiv.org/html/2607.24868#bib.bib6),[25](https://arxiv.org/html/2607.24868#bib.bib7),[48](https://arxiv.org/html/2607.24868#bib.bib8),[49](https://arxiv.org/html/2607.24868#bib.bib9),[24](https://arxiv.org/html/2607.24868#bib.bib10),[27](https://arxiv.org/html/2607.24868#bib.bib12),[50](https://arxiv.org/html/2607.24868#bib.bib14)\]\.

The present manuscript studies a specific interface between noise shaping and harmonic analysis\. Let

ℰN,d​a​\(x\)=1N​∑k=1Nak​e​\(∑j=1dxj​\(k/N\)j\),e​\(z\)=exp⁡\(2​π​i​z\)\.\\mathcal\{E\}\_\{N,d\}a\(x\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}a\_\{k\}e\\\!\\left\(\\sum\_\{j=1\}^\{d\}x\_\{j\}\(k/N\)^\{j\}\\right\),\\qquad e\(z\)=\\exp\(2\\pi\\mathrm\{i\}z\)\.Ford=2d=2, this is a discrete parabolic extension sum\. Such sums are closely related to periodic Schrödinger evolution, discrete restriction estimates, Weyl sums, and moment\-curve decoupling\[[15](https://arxiv.org/html/2607.24868#bib.bib41),[56](https://arxiv.org/html/2607.24868#bib.bib43),[13](https://arxiv.org/html/2607.24868#bib.bib45),[12](https://arxiv.org/html/2607.24868#bib.bib46),[28](https://arxiv.org/html/2607.24868#bib.bib55)\]\. The question is whether replacinguku\_\{k\}by a one\-bit sequenceqkq\_\{k\}can preserve the extension sum in a quantitatively controlled manner\.

A direct estimate gives

\|ℰN,d​u​\(x\)−ℰN,d​q​\(x\)\|≤1N​∑k=1N\|uk−qk\|\.\\left\\lvert\\mathcal\{E\}\_\{N,d\}u\(x\)\-\\mathcal\{E\}\_\{N,d\}q\(x\)\\right\\rvert\\leq\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\\left\\lvert u\_\{k\}\-q\_\{k\}\\right\\rvert\.This bound is usually of order one\. It does not use noise shaping and therefore cannot reveal oversampling gain\. The central observation is that a stable first\-order sigma\-delta scheme produces

ek=uk−qk=Δ​vk,‖v‖ℓ∞≤V\.e\_\{k\}=u\_\{k\}\-q\_\{k\}=\\Delta v\_\{k\},\\qquad\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\.Discrete summation by parts then transfersΔ\\Deltafromvvto the oscillatory weight\. If the phase changes slowly across adjacent indices, the resulting weight difference is of orderN−1N^\{\-1\}\. This produces a normalized error of orderN−1N^\{\-1\}on fixed parameter sets\.

#### 1\.1\.2Relation to discrete restriction theory

Three related analytical quantities organise the discussion\. The approximation problem concerns

ℰN​u−ℰN​q\.\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\.The extension\-size problem concerns

‖ℰN​q‖Lp​\(Ω\)\.\\left\\lVert\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{p\}\(\\Omega\)\}\.Discrete restriction theory studies scale\-sensitive bounds that are uniform over coefficient sequences in an appropriate norm\. The first\-order argument establishes a quantitative approximation theorem, while the later oscillatory sections describe the additional state information relevant to extension\-size and restriction estimates\.

The distinction is structural\. The total\-variation proof takes absolute values of adjacent phase increments\. Once this is done, true oscillatory cancellation has been discarded\. On a fixed compact region this loss is acceptable because each increment is already small\. On a region whose diameter is comparable withNN, the adjacent phase increment is no longer small\. The estimate then becomes order one\. Critical\-scale results require methods designed to preserve and exploit oscillation, including level\-set arguments, bilinear restriction, multilinear Kakeya estimates, polynomial partitioning, decoupling, and Vinogradov mean value estimates\[[72](https://arxiv.org/html/2607.24868#bib.bib49),[6](https://arxiv.org/html/2607.24868#bib.bib48),[14](https://arxiv.org/html/2607.24868#bib.bib47),[53](https://arxiv.org/html/2607.24868#bib.bib52),[54](https://arxiv.org/html/2607.24868#bib.bib53),[13](https://arxiv.org/html/2607.24868#bib.bib45),[12](https://arxiv.org/html/2607.24868#bib.bib46)\]\.

#### 1\.1\.3Relation to finite frames and compressed sensing

Noise shaping has already been extended far beyond classical bandlimited sampling\. Finite\-frame quantization showed that the ordering and variation of frame vectors influence reconstruction accuracy\[[5](https://arxiv.org/html/2607.24868#bib.bib17),[4](https://arxiv.org/html/2607.24868#bib.bib18),[10](https://arxiv.org/html/2607.24868#bib.bib19)\]\. Higher\-order accuracy requires suitable dual frames or smooth endpoint termination\[[9](https://arxiv.org/html/2607.24868#bib.bib20),[8](https://arxiv.org/html/2607.24868#bib.bib21)\]\. Random frames and compressed sensing provide another major branch, in which Sobolev duals and convex recovery methods convert high\-order noise shaping into polynomial or root\-exponential reconstruction accuracy\[[46](https://arxiv.org/html/2607.24868#bib.bib22),[62](https://arxiv.org/html/2607.24868#bib.bib23),[68](https://arxiv.org/html/2607.24868#bib.bib24),[35](https://arxiv.org/html/2607.24868#bib.bib25)\]\. These results are conceptually close to the present work because they also transfer a finite\-difference operator onto a reconstruction object\.

There is, however, an important difference\. In finite\-frame reconstruction, one may choose a dual frame adapted to the difference operator\. In a Fourier extension problem, the test weights are imposed by the phase\. This makes endpoint terms and phase increments unavoidable\. The manuscript therefore places greater emphasis on exact finite\-interval identities\.

#### 1\.1\.4Contributions

The rigorous contributions are organised as follows\.

1. \(i\)A complete invariant\-interval proof is given for the greedy first\-order one\-bit recursion, together with interval discrepancy bounds\.
2. \(ii\)A weighted summation theorem is proved for arbitrary complex weights, with both initial and terminal boundary terms retained\.
3. \(iii\)The phase estimate is extended from continuously differentiable phases to absolutely continuous and bounded\-variation phases by means of standard BV theory\[[1](https://arxiv.org/html/2607.24868#bib.bib72),[33](https://arxiv.org/html/2607.24868#bib.bib73)\]\.
4. \(iv\)A sharp variation factor for the parabolic phase is derived and evaluated explicitly\.
5. \(v\)Uniform optimality of the orderN−1N^\{\-1\}is proved in the absence of additional endpoint conditions\.
6. \(vi\)An exactrrth\-order finite\-interval summation formula is established\. The formula displays every boundary term and prevents an invalid automatic inference ofO​\(N−r\)O\(N^\{\-r\}\)decay\.
7. \(vii\)Under explicit endpoint compatibility, anO​\(N−r\)O\(N^\{\-r\}\)error estimate is proved for smooth phases\.
8. \(viii\)A boundary\-trace corrected reconstruction formula is established\. Exact trace metadata removes every finite\-record boundary term and recovers theO​\(N−r\)O\(N^\{\-r\}\)interior rate without modifying the original coefficient record\.
9. \(ix\)Fractional regularity is treated through finite differences: if the sampled weight belongs toCr−1,αC^\{r\-1,\\alpha\}, the corrected or endpoint\-compatible rate isO​\(N−\(r−1\+α\)\)O\(N^\{\-\(r\-1\+\\alpha\)\}\)\.
10. \(x\)The first\-order theory is extended to polynomial moment curves and multidimensional divergence\-form noise shaping\.
11. \(xi\)The subcritical growing\-region regime is identified, and a precise scale\-transition estimate is proved\.
12. \(xii\)Orthogonality obstructions are derived, showing why globalL2L^\{2\}decay cannot follow from bounded\-state noise shaping alone\.

#### 1\.1\.5Position of the results

The higher\-order finite\-interval identity, the endpoint\-compatible phase theorem, the boundary\-trace corrected reconstruction, the polynomial\-phase extension, and the divergence\-form grid estimate follow from exact finite\-difference identities developed in the report\. Chapter[8](https://arxiv.org/html/2607.24868#Ch8)adds transfer theorems under explicit state\-sum, correlation, covariance, or block\-square hypotheses\. The final chapter consolidates the derivations, principal estimates, and their relation to discrete Fourier restriction and periodic Schrödinger analysis\.

#### 1\.1\.6Organisation

Chapter[2](https://arxiv.org/html/2607.24868#Ch2)positions the problem within sigma\-delta quantization, finite\-frame theory, approximation theory, and harmonic analysis\. Chapter[3](https://arxiv.org/html/2607.24868#Ch3)fixes the analytical conventions\. Chapters[4](https://arxiv.org/html/2607.24868#Ch4)and[5](https://arxiv.org/html/2607.24868#Ch5)contain the complete first\-order and parabolic proofs\. Chapter[6](https://arxiv.org/html/2607.24868#Ch6)develops finite\-record higher\-order identities, endpoint\-compatible estimates, and boundary\-trace corrected reconstruction\. Chapters[7](https://arxiv.org/html/2607.24868#Ch7)–[9](https://arxiv.org/html/2607.24868#Ch9)treat polynomial, multidimensional, oscillatory, and implementation\-sensitive extensions\. Chapter[10](https://arxiv.org/html/2607.24868#Ch10)consolidates the detailed derivations and principal results\. The appendices contain expanded identities, derivative estimates, BV background, a reference map, reproducible code, and a glossary\.

### 1\.2Quantization context and stable noise shaping

#### 1\.2\.1Memoryless quantization and feedback quantization

A memoryless scalar quantizer maps each input sample to a nearby alphabet point independently of previous samples\. Pulse\-code modulation is the standard example\. If the quantization step isδ\\delta, one often begins with a local error estimate of sizeδ/2\\delta/2\. Such an estimate is useful but does not improve when the same signal is oversampled unless the reconstruction method introduces further averaging\. Classical models of quantization noise and their limitations are discussed in\[[7](https://arxiv.org/html/2607.24868#bib.bib1),[44](https://arxiv.org/html/2607.24868#bib.bib4),[39](https://arxiv.org/html/2607.24868#bib.bib28)\]\.

Sigma\-delta quantization changes the problem by introducing memory\. The current quantization decision depends on an internal state that stores previous error\. In the simplest recursion,

qk=Q​\(uk\+vk−1\),vk=vk−1\+uk−qk\.q\_\{k\}=Q\(u\_\{k\}\+v\_\{k\-1\}\),\\qquad v\_\{k\}=v\_\{k\-1\}\+u\_\{k\}\-q\_\{k\}\.Therefore,

uk−qk=vk−vk−1\.u\_\{k\}\-q\_\{k\}=v\_\{k\}\-v\_\{k\-1\}\.The error is a discrete derivative\. In the frequency domain, a derivative corresponds to multiplication by1−e−2​π​i​ω1\-e^\{\-2\\pi\\mathrm\{i\}\\omega\}, which is small near zero frequency\. This is the elementary noise\-shaping mechanism\.

#### 1\.2\.2The engineering transfer\-function picture

In a linearised sigma\-delta model, one writes the quantizer output as the quantizer input plus an additive noise source\. The output then separates into a signal transfer function and a noise transfer function\. A first\-order loop gives an idealised noise transfer factor1−z−11\-z^\{\-1\}\. Higher\-order designs aim for\(1−z−1\)r\(1\-z^\{\-1\}\)^\{r\}or a stable approximation to it\. This circuit\-level language is useful, but it does not by itself prove nonlinear stability\. The rigorous mathematical theory replaces the additive\-noise assumption with an exact state recursion and an invariant\-set argument\[[45](https://arxiv.org/html/2607.24868#bib.bib6),[25](https://arxiv.org/html/2607.24868#bib.bib7),[50](https://arxiv.org/html/2607.24868#bib.bib14)\]\.

The difference between the two viewpoints should be kept explicit\. Transfer\-function calculations describe the desired spectral shape\. Stability analysis proves that the nonlinear state remains bounded for all admissible inputs\. Without boundedness, the formal identitye=Δr​ve=\\Delta^\{r\}vis not enough to control reconstruction error\.

#### 1\.2\.3First\-order and higher\-order state equations

For a first\-order scheme, the state equation is

For anrrth\-order scheme, one often writes

This notation compresses a larger state\-space system\. In implementations, the modulator may contain several integrator states, and the relationship between those states and the scalar sequencevvdepends on the architecture\. The analysis in this manuscript begins only after a valid identity of the forme=Δr​ve=\\Delta^\{r\}vhas been established\.

Stable high\-order one\-bit schemes are nontrivial\. The existence of stable families of arbitrary order and the possibility of exponential reconstruction accuracy were major achievements\[[25](https://arxiv.org/html/2607.24868#bib.bib7),[48](https://arxiv.org/html/2607.24868#bib.bib8),[49](https://arxiv.org/html/2607.24868#bib.bib9),[27](https://arxiv.org/html/2607.24868#bib.bib12)\]\. The present manuscript does not reproduce those constructions\. Instead, it assumes a bounded state when treating general order and focuses on finite\-record Fourier testing\.

#### 1\.2\.4Oversampling and normalisation

The extension operator is normalised by1/N1/N\. This choice corresponds to a Riemann\-sum scale rather than an unnormalised Weyl sum\. The distinction matters\. If

SN​\(x,t\)=∑k=1Nak​e​\(x​kN\+t​k2N2\),S\_\{N\}\(x,t\)=\\sum\_\{k=1\}^\{N\}a\_\{k\}e\\\!\\left\(x\\frac\{k\}\{N\}\+t\\frac\{k^\{2\}\}\{N^\{2\}\}\\right\),thenSN=N​ℰNS\_\{N\}=N\\mathcal\{E\}\_\{N\}\. AnO​\(N−1\)O\(N^\{\-1\}\)bound forℰN​u−ℰN​q\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}qis equivalent to anO​\(1\)O\(1\)bound for the unnormalised difference\. Neither statement should be confused with square\-root cancellation in a classical unnormalised exponential sum\.

Oversampling appears here through the small mesh sizeh=1/Nh=1/N\. A slowly varying phase satisfies

wk\+1−wk=O​\(h\)\.w\_\{k\+1\}\-w\_\{k\}=O\(h\)\.The state difference is transferred to this increment\. This is the exact analogue of moving a derivative onto a smooth reconstruction kernel in bandlimited sigma\-delta analysis\[[25](https://arxiv.org/html/2607.24868#bib.bib7),[26](https://arxiv.org/html/2607.24868#bib.bib15)\]\.

#### 1\.2\.5Finite frames and the role of endpoint geometry

Finite\-frame quantization provides a useful comparison\. Supposexxis represented by frame coefficients⟨x,fk⟩\\left\\langle x,f\_\{k\}\\right\\rangleand these coefficients are quantized\. Summation by parts transfers differences to the frame pathfkf\_\{k\}\. Error bounds then depend on frame variation\. Higher\-order accuracy requires higher finite differences of the frame vectors to be small and, crucially, requires endpoint terms to vanish or be controlled\[[5](https://arxiv.org/html/2607.24868#bib.bib17),[4](https://arxiv.org/html/2607.24868#bib.bib18),[10](https://arxiv.org/html/2607.24868#bib.bib19),[9](https://arxiv.org/html/2607.24868#bib.bib20)\]\.

The same phenomenon occurs in the Fourier extension problem\. A high\-order difference identity produces interior terms involvingΔ\+r​w\\Delta\_\{\+\}^\{r\}w, but it also produces lower\-order boundary terms\. If those terms are ignored, the conclusion is generally false\. This observation is one of the main organising principles of the manuscript\.

#### 1\.2\.6Robustness and imperfect quantizers

Real quantizers have threshold errors, saturation limits, circuit noise, and finite settling\. The mathematical literature includes robustness results for imperfect quantizers and decimation procedures\[[24](https://arxiv.org/html/2607.24868#bib.bib10),[26](https://arxiv.org/html/2607.24868#bib.bib15)\]\. In the present setting, a convenient perturbed model is

uk−qk=Δ​vk\+ηk,u\_\{k\}\-q\_\{k\}=\\Delta v\_\{k\}\+\\eta\_\{k\},whereη\\etacollects unshaped error\. The shaped part receives a variation gain, while the residual part contributes through its averageℓ1\\ell^\{1\}size or another norm appropriate to the application\. The robustness decomposition is developed in Chapter[9](https://arxiv.org/html/2607.24868#Ch9)\.

#### 1\.2\.7What stability must mean in this manuscript

The phrase “stable quantizer” will always mean that the relevant state sequence satisfies a uniform bound independent of the record lengthNN\. For first order,

supN≥1sup1≤k≤N\|vk\|≤V\.\\sup\_\{N\\geq 1\}\\sup\_\{1\\leq k\\leq N\}\\left\\lvert v\_\{k\}\\right\\rvert\\leq V\.For higher order, the exact state variable included ine=Δr​ve=\\Delta^\{r\}vmust obey the same type of bound\. A simulation showing bounded states for one input is evidence, not a stability proof\. Conversely, a mathematical bound may be conservative relative to circuit behaviour\. These two statements are compatible\.

## Chapter 2Literature Review and Theoretical Positioning

This chapter places the report within four established areas: mathematical sigma\-delta quantization, frame and compressed\-sensing quantization, exponential\-sum theory, and Fourier restriction with decoupling\. The purpose is not to reproduce the literature exhaustively\. The purpose is to identify the exact theorem used at each stage and to distinguish those external theorems from the finite\-record results derived later\.

### 2\.1Exponential sums, Weyl differencing, and mean values

#### 2\.1\.1Purpose of this literature chapter

The total\-variation method takes absolute values of adjacent phase differences\. It is therefore effective when those differences are uniformly small, but it cannot capture cancellation among terms whose phases rotate substantially\. Classical exponential\-sum theory was developed to quantify precisely this cancellation\. The main tools include Weyl differencing, van der Corput estimates, the Hardy–Littlewood circle method, Vinogradov mean values, efficient congruencing, and decoupling\[[79](https://arxiv.org/html/2607.24868#bib.bib56),[78](https://arxiv.org/html/2607.24868#bib.bib57),[57](https://arxiv.org/html/2607.24868#bib.bib58),[42](https://arxiv.org/html/2607.24868#bib.bib60),[77](https://arxiv.org/html/2607.24868#bib.bib59),[66](https://arxiv.org/html/2607.24868#bib.bib61),[59](https://arxiv.org/html/2607.24868#bib.bib62),[81](https://arxiv.org/html/2607.24868#bib.bib63),[12](https://arxiv.org/html/2607.24868#bib.bib46)\]\.

This chapter does not reproduce the deep proofs of these theorems\. It states the forms relevant to structured one\-bit coefficients, explains their normalisations, and identifies which parts of the classical machinery do not immediately apply when the coefficient sequence isΔr​v\\Delta^\{r\}v\.

#### 2\.1\.2Finite differences, moduli of smoothness, and fractional regularity

The high\-order interior term is naturally measured by a modulus of smoothness rather than by classical derivatives alone\. Constructive approximation theory treats finite differences as intrinsic regularity measurements\. Hölder, Lipschitz, Besov, and Sobolev scales can be characterised by the rate at which suitable finite differences vanish\[[29](https://arxiv.org/html/2607.24868#bib.bib83),[30](https://arxiv.org/html/2607.24868#bib.bib84),[76](https://arxiv.org/html/2607.24868#bib.bib85)\]\. This viewpoint is especially appropriate for a sampled Fourier weight because the finite\-record summation identity producesΔ\+r​w\\Delta\_\{\+\}^\{r\}wexactly\. A continuous derivative estimate is introduced only after the discrete identity has been established\.

ForW∈Cr−1,αW\\in C^\{r\-1,\\alpha\}, the elementary estimate

\|Δhr​W​\(x\)\|≤hr−1\+α​\[W\(r−1\)\]C0,α\|\\Delta\_\{h\}^\{r\}W\(x\)\|\\leq h^\{r\-1\+\\alpha\}\[W^\{\(r\-1\)\}\]\_\{C^\{0,\\alpha\}\}provides an intermediate scale between integer ordersr−1r\-1andrr\. Chapter[6](https://arxiv.org/html/2607.24868#Ch6)proves this estimate directly and inserts it into the finite\-record formula\. The resulting theorem replaces a binary distinction between “smooth” and “nonsmooth” weights by a regularity\-sensitive decay law\.

#### 2\.1\.3Basic notation

For a real polynomial

P​\(k\)=αd​kd\+⋯\+α1​k\+α0,P\(k\)=\\alpha\_\{d\}k^\{d\}\+\\cdots\+\\alpha\_\{1\}k\+\\alpha\_\{0\},define the weighted exponential sum

SP​\(a;N\)=∑k=1Nak​e​\(P​\(k\)\)\.S\_\{P\}\(a;N\)=\\sum\_\{k=1\}^\{N\}a\_\{k\}e\(P\(k\)\)\.\(2\.1\)The unweighted case hasak=1a\_\{k\}=1\. The quantization\-error sum considered here is weighted, withak=eka\_\{k\}=e\_\{k\}orak=vka\_\{k\}=v\_\{k\}after summation by parts\.

The trivial estimate is

\|SP​\(a;N\)\|≤‖a‖ℓ1\.\\left\\lvert S\_\{P\}\(a;N\)\\right\\rvert\\leq\\left\\lVert a\\right\\rVert\_\{\\ell^\{1\}\}\.If\|ak\|≤1\\left\\lvert a\_\{k\}\\right\\rvert\\leq 1, this isNN\. Any boundo​\(N\)o\(N\)represents cancellation\.

#### 2\.1\.4Abel summation and bounded partial sums

Before using curvature, one can exploit bounded partial sums of the coefficients\. Let

A​\(m\)=∑k=1mak\.A\(m\)=\\sum\_\{k=1\}^\{m\}a\_\{k\}\.Abel summation gives

∑k=1Nak​bk=A​\(N\)​bN−∑k=1N−1A​\(k\)​\(bk\+1−bk\)\.\\sum\_\{k=1\}^\{N\}a\_\{k\}b\_\{k\}=A\(N\)b\_\{N\}\-\\sum\_\{k=1\}^\{N\-1\}A\(k\)\(b\_\{k\+1\}\-b\_\{k\}\)\.\(2\.2\)This is the first\-order weighted theorem withA=vA=vwhena=e=Δ​va=e=\\Delta vandv0=0v\_\{0\}=0\.

Ifsupm\|A​\(m\)\|≤V\\sup\_\{m\}\\left\\lvert A\(m\)\\right\\rvert\\leq V, then

\|SP​\(a;N\)\|≤V​\(1\+∑k=1N−1\|e​\(P​\(k\+1\)\)−e​\(P​\(k\)\)\|\)\.\\left\\lvert S\_\{P\}\(a;N\)\\right\\rvert\\leq V\\left\(1\+\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert e\(P\(k\+1\)\)\-e\(P\(k\)\)\\right\\rvert\\right\)\.The right\-hand side is useful only while the phase increments are small or arithmetically resonant\. Classical Weyl estimates instead exploit cancellation inside the sum without taking all these absolute values\.

#### 2\.1\.5Van der Corput differencing

A fundamental identity compares a sum with shifted correlations\. Let

S=∑k=1Nak\.S=\\sum\_\{k=1\}^\{N\}a\_\{k\}\.For1≤H≤N1\\leq H\\leq N, a standard van der Corput inequality has the schematic form

\|S\|2≲N\+HH​\[∑k=1N\|ak\|2\+2​∑h=1H\(1−hH\)​\|∑k=1N−hak\+h​ak¯\|\]\.\\left\\lvert S\\right\\rvert^\{2\}\\lesssim\\frac\{N\+H\}\{H\}\\left\[\\sum\_\{k=1\}^\{N\}\\left\\lvert a\_\{k\}\\right\\rvert^\{2\}\+2\\sum\_\{h=1\}^\{H\}\\left\(1\-\\frac\{h\}\{H\}\\right\)\\left\\lvert\\sum\_\{k=1\}^\{N\-h\}a\_\{k\+h\}\\overline\{a\_\{k\}\}\\right\\rvert\\right\]\.\(2\.3\)Precise variants and normalisations are given in standard texts\[[42](https://arxiv.org/html/2607.24868#bib.bib60),[66](https://arxiv.org/html/2607.24868#bib.bib61),[59](https://arxiv.org/html/2607.24868#bib.bib62)\]\.

Forak=e​\(P​\(k\)\)a\_\{k\}=e\(P\(k\)\), the correlation phase is

P​\(k\+h\)−P​\(k\),P\(k\+h\)\-P\(k\),a polynomial of degreed−1d\-1\. Repeated differencing reduces the degree\. This is the heart of Weyl’s method\.

For shaped coefficients, take

ak=\(Δr​v\)k​e​\(P​\(k\)\)\.a\_\{k\}=\(\\Delta^\{r\}v\)\_\{k\}e\(P\(k\)\)\.The correlation contains products

\(Δr​v\)k\+h​\(Δr​v\)k¯​e​\(P​\(k\+h\)−P​\(k\)\)\.\(\\Delta^\{r\}v\)\_\{k\+h\}\\overline\{\(\\Delta^\{r\}v\)\_\{k\}\}e\(P\(k\+h\)\-P\(k\)\)\.Boundedness ofvvdoes not directly control the partial sums of these products\. Thus van der Corput differencing preserves phase cancellation but complicates the coefficient structure\. A new theory must track both effects simultaneously\.

#### 2\.1\.6Weyl’s inequality

For an unweighted polynomial sum, Weyl’s inequality bounds the sum in terms of a rational approximation to the leading coefficient\. One standard form states that if

\|αd−aq\|≤q−2,\(a,q\)=1,\\left\|\\alpha\_\{d\}\-\\frac\{a\}\{q\}\\right\|\\leq q^\{\-2\},\\qquad\(a,q\)=1,then, for everyε\>0\\varepsilon\>0,

\|∑k=1Ne​\(P​\(k\)\)\|≤Cd,ε​N1\+ε​\(q−1\+N−1\+q​N−d\)21−d\.\\left\\lvert\\sum\_\{k=1\}^\{N\}e\(P\(k\)\)\\right\\rvert\\leq C\_\{d,\\varepsilon\}N^\{1\+\\varepsilon\}\\left\(q^\{\-1\}\+N^\{\-1\}\+qN^\{\-d\}\\right\)^\{2^\{1\-d\}\}\.\(2\.4\)The exact exponent and refinements depend on the version used\. The classical source is Weyl’s equidistribution paper, and modern treatments appear in Vaughan and Iwaniec–Kowalski\[[79](https://arxiv.org/html/2607.24868#bib.bib56),[77](https://arxiv.org/html/2607.24868#bib.bib59),[59](https://arxiv.org/html/2607.24868#bib.bib62)\]\.

The estimate divides parameter space into major arcs, where a coefficient has a good rational approximation with small denominator, and minor arcs, where Weyl cancellation is strong\. In the parabolic extension, the relevant leading coefficient ist/N2t/N^\{2\}in the unscaled integer polynomial\. A critical\-scale analysis must therefore respect the arithmetic oft/N2t/N^\{2\}\.

#### 2\.1\.7The Vinogradov mean value

For degreedd, define

fd​\(α;N\)=∑k=1Ne​\(α1​k\+⋯\+αd​kd\)f\_\{d\}\(\\alpha;N\)=\\sum\_\{k=1\}^\{N\}e\(\\alpha\_\{1\}k\+\\cdots\+\\alpha\_\{d\}k^\{d\}\)and

Js,d​\(N\)=∫𝕋d\|fd​\(α;N\)\|2​s​dα\.J\_\{s,d\}\(N\)=\\int\_\{\\mathbb\{T\}^\{d\}\}\\left\\lvert f\_\{d\}\(\\alpha;N\)\\right\\rvert^\{2s\}\\,\\mathrm\{d\}\\alpha\.\(2\.5\)By orthogonality,Js,d​\(N\)J\_\{s,d\}\(N\)counts integer solutions of

x1j\+⋯\+xsj=y1j\+⋯\+ysj,1≤j≤d,x\_\{1\}^\{j\}\+\\cdots\+x\_\{s\}^\{j\}=y\_\{1\}^\{j\}\+\\cdots\+y\_\{s\}^\{j\},\\qquad 1\\leq j\\leq d,\(2\.6\)with all variables in\{1,…,N\}\\\{1,\\ldots,N\\\}\.

The main conjecture, now a theorem, states that for everyε\>0\\varepsilon\>0,

Js,d​\(N\)≤Cs,d,ε​\(Ns\+ε\+N2​s−d​\(d\+1\)/2\+ε\)\.J\_\{s,d\}\(N\)\\leq C\_\{s,d,\\varepsilon\}\\left\(N^\{s\+\\varepsilon\}\+N^\{2s\-d\(d\+1\)/2\+\\varepsilon\}\\right\)\.\(2\.7\)Efficient congruencing produced major advances and resolved the cubic case; decoupling resolved degrees above three, with later work completing and simplifying several aspects\[[81](https://arxiv.org/html/2607.24868#bib.bib63),[36](https://arxiv.org/html/2607.24868#bib.bib64),[82](https://arxiv.org/html/2607.24868#bib.bib65),[83](https://arxiv.org/html/2607.24868#bib.bib66),[12](https://arxiv.org/html/2607.24868#bib.bib46),[84](https://arxiv.org/html/2607.24868#bib.bib67),[51](https://arxiv.org/html/2607.24868#bib.bib70)\]\.

The cited sources provide the proofs of \([2\.7](https://arxiv.org/html/2607.24868#Ch2.E7)\) and trace the historical development of efficient congruencing and decoupling methods\.

#### 2\.1\.8Weighted mean values

Decoupling yields a weighted version at the critical exponentp=d​\(d\+1\)p=d\(d\+1\)\. For a complex sequencea=\(ak\)a=\(a\_\{k\}\),

‖∑k=1Nak​e​\(α1​k\+⋯\+αd​kd\)‖Ld​\(d\+1\)​\(𝕋d\)≤Cd,ε​Nε​‖a‖ℓ2\.\\left\\lVert\\sum\_\{k=1\}^\{N\}a\_\{k\}e\(\\alpha\_\{1\}k\+\\cdots\+\\alpha\_\{d\}k^\{d\}\)\\right\\rVert\_\{L^\{d\(d\+1\)\}\(\\mathbb\{T\}^\{d\}\)\}\\leq C\_\{d,\\varepsilon\}N^\{\\varepsilon\}\\left\\lVert a\\right\\rVert\_\{\\ell^\{2\}\}\.\(2\.8\)This form is particularly relevant because the coefficients need not be equal\. For a bounded shaped error, however,

‖e‖ℓ2≲N1/2,\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\\lesssim N^\{1/2\},so the theorem does not by itself yield a gain over the natural unit\-coefficient scale\.

To improve the estimate usinge=Δr​ve=\\Delta^\{r\}v, one needs a weighted theorem whose right\-hand side measures a negative discrete Sobolev norm, bounded partial sums, endpoint\-compatible moments, or another norm adapted to the difference structure\.

#### 2\.1\.9Major arcs, minor arcs, and shaped coefficients

The circle method decomposes the torus into regions near rational points and their complement\. On minor arcs, one seeks strong cancellation in polynomial sums\. On major arcs, one approximates the sum by a product of a complete exponential sum and a continuous oscillatory integral\.

For shaped coefficients, each region presents a different issue\.

1. \(i\)On minor arcs, summation by parts may combine bounded partial sums with a Weyl estimate for short phase increments\.
2. \(ii\)On major arcs, the binary or difference structure changes the complete sum and may create additional cancellation or new resonances\.
3. \(iii\)Boundary traces contribute low\-frequency terms that are naturally concentrated near the origin, which is a principal major arc\.

A useful research strategy is therefore to remove boundary traces first, then study minor and major arcs separately\.

#### 2\.1\.10Completion and short intervals

Classical estimates often treat a complete interval1≤k≤N1\\leq k\\leq N\. Blockwise reset and localisation lead to shorter intervals\. Completion methods embed an incomplete sum into a complete periodic sum at the cost of logarithmic or arithmetic factors\. For coefficients with bounded partial sums, completion must be performed carefully because extending the interval can create new boundary terms\.

If a state is compactly supported away from the endpoints, then the shaped error also has a short transition region\. Such states may be suitable test cases for combining finite\-difference cancellation with classical incomplete\-sum estimates\.

#### 2\.1\.11A possible hybrid inequality

Supposee=Δ​ve=\\Delta v,v0=vN=0v\_\{0\}=v\_\{N\}=0\. Then

SP​\(e;N\)=−∑k=1N−1vk​e​\(P​\(k\)\)​\[e​\(P​\(k\+1\)−P​\(k\)\)−1\]\.S\_\{P\}\(e;N\)=\-\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}e\(P\(k\)\)\\left\[e\(P\(k\+1\)\-P\(k\)\)\-1\\right\]\.Define

mk=e​\(P​\(k\+1\)−P​\(k\)\)−1\.m\_\{k\}=e\(P\(k\+1\)\-P\(k\)\)\-1\.The sum is now an exponential sum with bounded coefficientsvk​mkv\_\{k\}m\_\{k\}\. On subcritical regions,mkm\_\{k\}is uniformly small and total variation is sufficient\. On critical regions,mkm\_\{k\}is not small but has its own polynomial phase dependence\. One may attempt to apply Weyl or mean\-value estimates to the productmk​e​\(P​\(k\)\)m\_\{k\}e\(P\(k\)\), which is a difference of two polynomial phases:

mk​e​\(P​\(k\)\)=e​\(P​\(k\+1\)\)−e​\(P​\(k\)\)\.m\_\{k\}e\(P\(k\)\)=e\(P\(k\+1\)\)\-e\(P\(k\)\)\.This identity returns to the original difference, but it suggests a multi\-phase decomposition rather than an absolute\-value bound\.

For orderrr, repeated summation produces a finite linear combination of shifted phases

Δ\+r​e​\(P​\(k\)\)=∑j=0r\(−1\)r−j​\(rj\)​e​\(P​\(k\+j\)\)\.\\Delta\_\{\+\}^\{r\}e\(P\(k\)\)=\\sum\_\{j=0\}^\{r\}\(\-1\)^\{r\-j\}\\binom\{r\}\{j\}e\(P\(k\+j\)\)\.Each term is a classical polynomial exponential sum with a shifted polynomial\. Endpoint\-compatible shaping therefore converts the problem into a controlled linear combination of classical sums with bounded state coefficients\. The remaining difficulty is that the coefficientsvkv\_\{k\}are arbitrary bounded sequences, and classical Weyl cancellation can be destroyed by adversarial weights\.

#### 2\.1\.12What additional state hypotheses would help

Several stronger assumptions are compatible with practical quantization models and could make exponential\-sum estimates possible\.

1. \(a\)vvhas bounded variation or a bounded number of jumps\.
2. \(b\)vvis periodic with a period much shorter thanNN\.
3. \(c\)vvis generated by an ergodic map with quantitative mixing\.
4. \(d\)vvhas a sparse or high\-frequency discrete Fourier spectrum\.
5. \(e\)vvis random after dither and satisfies concentration estimates\.

Ergodic aspects of sigma\-delta state dynamics have been studied for specific recursions\[[47](https://arxiv.org/html/2607.24868#bib.bib11)\]\. A future theorem could combine such dynamical information with number\-theoretic phase cancellation\.

#### 2\.1\.13Summary of the number\-theoretic route

The key lesson is not that a particular Weyl inequality immediately solves the structured extension problem\. It is that critical\-scale cancellation is controlled by rational approximation, differencing, and mean values\. The difference structure of sigma\-delta error must be incorporated into these mechanisms rather than estimated before they are used\.

A realistic first target is a minor\-arc estimate for endpoint\-compatible first\- or second\-order shaped errors under an additional regularity condition on the state\. A second target is a mean\-value inequality in which‖a‖ℓ2\\left\\lVert a\\right\\rVert\_\{\\ell^\{2\}\}is replaced or supplemented by bounded partial sums or vanishing discrete moments\.

### 2\.2Restriction, Strichartz estimates, and decoupling

#### 2\.2\.1Continuous restriction as the geometric model

Fourier restriction asks whether the Fourier transform of anLpL^\{p\}function can be meaningfully restricted to a curved set of measure zero\. The dual extension operator for a parametrised surfaceΣ\\Sigmahas the form

E​f​\(x\)=∫Σf​\(ξ\)​e2​π​i​x⋅ξ​dσ​\(ξ\)\.Ef\(x\)=\\int\_\{\\Sigma\}f\(\\xi\)e^\{2\\pi\\mathrm\{i\}x\\cdot\\xi\}\\,\\mathrm\{d\}\\sigma\(\\xi\)\.Curvature creates oscillation and hence integrability beyond the trivialL1→L∞L^\{1\}\\to L^\{\\infty\}estimate\. Foundational results include the restriction work of Fefferman, Tomas, and Strichartz\[[34](https://arxiv.org/html/2607.24868#bib.bib37),[75](https://arxiv.org/html/2607.24868#bib.bib38),[71](https://arxiv.org/html/2607.24868#bib.bib39)\]\. Modern developments use bilinear and multilinear restriction, Kakeya geometry, polynomial partitioning, and decoupling\[[72](https://arxiv.org/html/2607.24868#bib.bib49),[80](https://arxiv.org/html/2607.24868#bib.bib51),[73](https://arxiv.org/html/2607.24868#bib.bib50),[6](https://arxiv.org/html/2607.24868#bib.bib48),[14](https://arxiv.org/html/2607.24868#bib.bib47),[53](https://arxiv.org/html/2607.24868#bib.bib52),[54](https://arxiv.org/html/2607.24868#bib.bib53)\]\.

The discrete operator in this report is a Riemann\-sum analogue of extension from the moment curve\. The analogy is valuable, but the normalisation and parameter scales must be tracked carefully\.

#### 2\.2\.2The Stein–Tomas mechanism

For a compact hypersurface with nonvanishing curvature, the Stein–Tomas theorem gives anL2L^\{2\}extension estimate in a nontrivial range\. Its proof uses theT​T∗TT^\{\*\}method and decay of the Fourier transform of surface measure\. The original sources are Tomas and the subsequent restriction theory developed by Stein and Strichartz\[[75](https://arxiv.org/html/2607.24868#bib.bib38),[71](https://arxiv.org/html/2607.24868#bib.bib39),[70](https://arxiv.org/html/2607.24868#bib.bib76)\]\.

The important structural point is that anL2L^\{2\}coefficient norm is converted into an ambientLpL^\{p\}norm through curvature\. Standard discrete restriction inherits the same coefficient norm\. Since a one\-bit error hasℓ2\\ell^\{2\}size of orderN1/2N^\{1/2\}, a direct application does not distinguish shaped and unshaped errors\.

#### 2\.2\.3Strichartz estimates

For the Schrödinger equation

i​∂tu\+Δ​u=0,u​\(0\)=f,\\mathrm\{i\}\\partial\_\{t\}u\+\\Delta u=0,\\qquad u\(0\)=f,the solution is an extension operator over the paraboloid\. Strichartz estimates control mixed space\-time norms of the solution\. The endpoint theory was established in a general dispersive framework by Keel and Tao\[[60](https://arxiv.org/html/2607.24868#bib.bib40)\]; standard expositions appear in Cazenave and Tao\[[22](https://arxiv.org/html/2607.24868#bib.bib79),[74](https://arxiv.org/html/2607.24868#bib.bib78)\]\.

On a torus, the spectrum is discrete and Euclidean scaling is modified by arithmetic effects\. Bourgain’s work on periodic nonlinear evolution equations introduced decisive discrete restriction and periodic Strichartz estimates\[[15](https://arxiv.org/html/2607.24868#bib.bib41),[16](https://arxiv.org/html/2607.24868#bib.bib42)\]\. Later work connected periodic estimates to decoupling and scale\-invariant bounds\[[13](https://arxiv.org/html/2607.24868#bib.bib45),[61](https://arxiv.org/html/2607.24868#bib.bib54),[28](https://arxiv.org/html/2607.24868#bib.bib55)\]\.

For the one\-dimensional periodic Schrödinger flow,

u​\(x,t\)=∑k∈ℤak​e​\(k​x\+k2​t\)\.u\(x,t\)=\\sum\_\{k\\in\\mathbb\{Z\}\}a\_\{k\}e\(kx\+k^\{2\}t\)\.The finite sum over1≤k≤N1\\leq k\\leq Nis exactly the unscaled quadratic extension that appears after the change of variables in Chapter[8](https://arxiv.org/html/2607.24868#Ch8)\.

#### 2\.2\.4Discrete restriction for the parabola

A representative critical estimate is

‖∑k=1Nak​e​\(k​x\+k2​t\)‖L6​\(𝕋2\)≤Cε​Nε​‖a‖ℓ2\.\\left\\lVert\\sum\_\{k=1\}^\{N\}a\_\{k\}e\(kx\+k^\{2\}t\)\\right\\rVert\_\{L^\{6\}\(\\mathbb\{T\}^\{2\}\)\}\\leq C\_\{\\varepsilon\}N^\{\\varepsilon\}\\left\\lVert a\\right\\rVert\_\{\\ell^\{2\}\}\.\(2\.9\)Forak=1a\_\{k\}=1, the sixth power counts solutions of

k1\+k2\+k3\\displaystyle k\_\{1\}\+k\_\{2\}\+k\_\{3\}=k4\+k5\+k6,\\displaystyle=k\_\{4\}\+k\_\{5\}\+k\_\{6\},k12\+k22\+k32\\displaystyle k\_\{1\}^\{2\}\+k\_\{2\}^\{2\}\+k\_\{3\}^\{2\}=k42\+k52\+k62\.\\displaystyle=k\_\{4\}^\{2\}\+k\_\{5\}^\{2\}\+k\_\{6\}^\{2\}\.The estimate is a quadratic Vinogradov mean value theorem\. Discrete restriction associated with Schrödinger equations has also been studied directly by Hu and Li\[[56](https://arxiv.org/html/2607.24868#bib.bib43)\]and in a general formulation by Lai and Ding\[[65](https://arxiv.org/html/2607.24868#bib.bib44)\]\.

#### 2\.2\.5Decoupling for the moment curve

Let

EI​g​\(x\)=∫Ig​\(ξ\)​e​\(x1​ξ\+x2​ξ2\+⋯\+xd​ξd\)​dξ\.E\_\{I\}g\(x\)=\\int\_\{I\}g\(\\xi\)e\\\!\\left\(x\_\{1\}\\xi\+x\_\{2\}\\xi^\{2\}\+\\cdots\+x\_\{d\}\\xi^\{d\}\\right\)\\,\\mathrm\{d\}\\xi\.Partition\[0,1\]\[0,1\]into intervalsJJof lengthδ\\delta\. The criticalℓ2\\ell^\{2\}decoupling theorem for the moment curve states, schematically, that for

and for a ballBBof radiusδ−d\\delta^\{\-d\},

‖E\[0,1\]​g‖Lp​\(wB\)≤Cd,ε​δ−ε​\(∑J‖EJ​g‖Lp​\(wB\)2\)1/2\.\\left\\lVert E\_\{\[0,1\]\}g\\right\\rVert\_\{L^\{p\}\(w\_\{B\}\)\}\\leq C\_\{d,\\varepsilon\}\\delta^\{\-\\varepsilon\}\\left\(\\sum\_\{J\}\\left\\lVert E\_\{J\}g\\right\\rVert\_\{L^\{p\}\(w\_\{B\}\)\}^\{2\}\\right\)^\{1/2\}\.\(2\.10\)Bourgain and Demeter proved the generalℓ2\\ell^\{2\}decoupling conjecture, and Bourgain, Demeter, and Guth obtained the sharp moment\-curve consequence used in Vinogradov’s mean value theorem\[[13](https://arxiv.org/html/2607.24868#bib.bib45),[12](https://arxiv.org/html/2607.24868#bib.bib46)\]\. A later proof by Guo, Li, Yung, and Zorin\-Kranich gave a shorter route for the moment curve\[[51](https://arxiv.org/html/2607.24868#bib.bib70)\]\. Demeter’s monograph provides a detailed account\[[28](https://arxiv.org/html/2607.24868#bib.bib55)\]\.

The subsequent analysis uses the stated consequences of the decoupling theorem, with the full proof and broader formulation available in the cited sources\.

#### 2\.2\.6What decoupling separates

Decoupling separates contributions from short frequency intervals in anℓ2\\ell^\{2\}manner\. It is effective because pieces of a curved surface point in different directions\. For a discrete sum, one may thicken each lattice frequency into a short interval and transfer the continuous estimate to the discrete setting\.

Noise shaping creates a different type of structure\. It correlates neighbouring coefficients through finite differences\. This is local in the frequency index, whereas decoupling separates frequency intervals\. The two structures are compatible in principle: finite differences act within or across adjacent intervals, while decoupling controls how the intervals recombine\.

A naive approach loses the shaping\. If one applies decoupling directly with coefficient sequenceee, the right\-hand side contains‖e‖ℓ2\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}or localℓ2\\ell^\{2\}norms\. These can be as large fore=Δ​ve=\\Delta vas for arbitrary bounded coefficients\. The finite\-difference representation must therefore be inserted before or during the decoupling decomposition\.

#### 2\.2\.7Boundary terms under frequency localisation

LetψI​\(k\)\\psi\_\{I\}\(k\)be a smooth cutoff to a frequency intervalII\. Then

∑ek​ψI​\(k\)​wk=∑\(Δ​v\)k​ψI​\(k\)​wk\.\\sum e\_\{k\}\\psi\_\{I\}\(k\)w\_\{k\}=\\sum\(\\Delta v\)\_\{k\}\\psi\_\{I\}\(k\)w\_\{k\}\.Summation by parts moves the difference onto the product:

Δ\+​\(ψI​w\)k=ψI​\(k\+1\)​Δ\+​wk\+wk​Δ\+​ψI​\(k\)\.\\Delta\_\{\+\}\(\\psi\_\{I\}w\)\_\{k\}=\\psi\_\{I\}\(k\+1\)\\Delta\_\{\+\}w\_\{k\}\+w\_\{k\}\\Delta\_\{\+\}\\psi\_\{I\}\(k\)\.The first term contains the small phase increment in a subcritical regime\. The second is a localisation cost\. IfIIhas lengthMM, a smooth cutoff satisfies

\|Δ\+​ψI\|≲M−1\.\\left\\lvert\\Delta\_\{\+\}\\psi\_\{I\}\\right\\rvert\\lesssim M^\{\-1\}\.Summed overMMindices, this cost isO​\(1\)O\(1\)\. After the global1/N1/Nnormalisation, each interval contributesO​\(N−1\)O\(N^\{\-1\}\)before recombination\.

Decoupling may recombineN/MN/Mintervals inℓ2\\ell^\{2\}, producing a factor\(N/M\)1/2\(N/M\)^\{1/2\}rather thanN/MN/M\. This heuristic suggests a possible gain over global total variation, but a rigorous theorem must control the dependence ofΔ\+​w\\Delta\_\{\+\}won the observation point and the boundary traces created at each interval\.

#### 2\.2\.8A model dyadic decomposition

Partition\{1,…,N\}\\\{1,\\ldots,N\\\}into intervalsIIof lengthMM\. Write

ℰN​e=∑IℰN​\(e​ψI\)\.\\mathcal\{E\}\_\{N\}e=\\sum\_\{I\}\\mathcal\{E\}\_\{N\}\(e\\psi\_\{I\}\)\.After summation by parts insideII, each piece has three components:

1. \(i\)an initial boundary term;
2. \(ii\)a terminal boundary term;
3. \(iii\)an interior term involvingvk​Δ\+​\(ψI​w\)kv\_\{k\}\\Delta\_\{\+\}\(\\psi\_\{I\}w\)\_\{k\}\.

Ifvvis globally bounded but not locally reset, the artificial boundaries at interval endpoints do not vanish\. Smooth cutoffs spread these boundaries into the derivative term, but do not remove their total mass\. A successful decoupling argument must show that the contributions of many interval boundaries combine orthogonally or cancel\.

This observation parallels smooth frame\-path termination\. Hard partitioning creates large boundary errors; smooth localisation trades them for controlled derivatives\.

#### 2\.2\.9Multilinear restriction and transversality

Bilinear and multilinear restriction estimates improve linear estimates when frequency pieces are transverse\. Foundational results include the bilinear approach of Tao, Vargas, and Vega, Wolff’s cone estimate, Tao’s paraboloid estimate, and the multilinear restriction theorem of Bennett, Carbery, and Tao\[[72](https://arxiv.org/html/2607.24868#bib.bib49),[80](https://arxiv.org/html/2607.24868#bib.bib51),[73](https://arxiv.org/html/2607.24868#bib.bib50),[6](https://arxiv.org/html/2607.24868#bib.bib48)\]\. Bourgain and Guth developed a broad\-narrow method that converts multilinear information into linear estimates\[[14](https://arxiv.org/html/2607.24868#bib.bib47)\]\. Guth’s polynomial partitioning method produced further advances\[[53](https://arxiv.org/html/2607.24868#bib.bib52),[54](https://arxiv.org/html/2607.24868#bib.bib53)\]\.

For a one\-dimensional moment curve, separated frequency intervals have transverse tangent directions after lifting to the appropriate ambient dimension\. A shaped coefficient sequence couples neighbours, but intervals separated by more than the shaping order remain algebraically independent at the level of the finite difference\. This may allow multilinear estimates to be applied after grouping adjacent intervals into packets\.

#### 2\.2\.10A negative Sobolev viewpoint

The identitye=Δr​ve=\\Delta^\{r\}vsays thateeis small in a discrete negative Sobolev norm\. Standard extension estimates control the operator fromℓ2\\ell^\{2\}toLpL^\{p\}\. A natural target is an estimate of the form

‖ℰN​e‖Lp​\(Ω\)≤AN​‖DN−r​e‖ℓ∞\.\\left\\lVert\\mathcal\{E\}\_\{N\}e\\right\\rVert\_\{L^\{p\}\(\\Omega\)\}\\leq A\_\{N\}\\left\\lVert D\_\{N\}^\{\-r\}e\\right\\rVert\_\{\\ell^\{\\infty\}\}\.\(2\.11\)This estimate is intended for a class of errors satisfying compatible boundary conditions\. In frame quantization, Sobolev duals are designed precisely to exploit the inverse difference operator\[[8](https://arxiv.org/html/2607.24868#bib.bib21),[46](https://arxiv.org/html/2607.24868#bib.bib22)\]\. In curved extension, no fixed linear dual is available because the observation point varies continuously\.

One possibility is to treatℰN​DNr\\mathcal\{E\}\_\{N\}D\_\{N\}^\{r\}as a family of differentiated extension operators and estimate its operator norm fromℓ∞\\ell^\{\\infty\}toLpL^\{p\}\. The adjoint difference produces observation\-dependent multipliers\. Decoupling for such variable discrete multipliers is not an immediate consequence of the standard theorem\.

#### 2\.2\.11Periodic versus Euclidean regions

Euclidean decoupling is stated on large balls whose radius is linked to the frequency scale\. Periodic discrete restriction is stated on a torus\. The variables\(x,t\)\(x,t\)used here employ a normalised phase, so the corresponding physical region is anisotropic\. A correct transfer must include:

1. \(a\)the change of variables from\(x,t\)\(x,t\)to torus variables;
2. \(b\)the Jacobian of the change;
3. \(c\)the1/N1/Ncoefficient normalisation;
4. \(d\)the coefficient norm after finite differences;
5. \(e\)the location and size of boundary terms\.

Many apparent gains disappear when one of these factors is omitted\. Chapter[8](https://arxiv.org/html/2607.24868#Ch8)formulates transfer principles with all normalisations displayed\.

#### 2\.2\.12The realistic role of decoupling in this project

Decoupling is not needed for the compact\-setO​\(N−1\)O\(N^\{\-1\}\)theorem\. It becomes relevant only when the observation region grows to a scale on which adjacent phase variation is not small\. At that point, the aim is not to reproduce a standard restriction estimate\. The aim is to prove that the special coefficient class generated by stable sigma\-delta quantization obeys a stronger estimate than arbitrary coefficients of the sameℓ2\\ell^\{2\}size\.

A plausible first result would be localised and conditional\. For example, one might assume endpoint compatibility and a bounded\-variation state, then prove an improvedL6L^\{6\}estimate on a subregion of the full parabolic torus\. Establishing such a theorem would already create a genuine bridge between noise shaping and restriction theory\.

## Chapter 3Analytical Methodology and Discrete Mathematical Framework

The analytical method is based on exact finite\-record identities\. Infinite\-sequence notation is avoided whenever it would conceal a boundary contribution\. Backward differences describe the quantization error, forward differences act on sampled weights, and total variation measures the cost of transferring a difference from the state to the weight\.

### 3\.1Discrete calculus, variation, and finite summation

#### 3\.1\.1Backward and forward differences

For a sequencea=\(ak\)a=\(a\_\{k\}\), define the backward difference

Δ​ak=ak−ak−1\\Delta a\_\{k\}=a\_\{k\}\-a\_\{k\-1\}and the forward difference

Δ\+​ak=ak\+1−ak\.\\Delta\_\{\+\}a\_\{k\}=a\_\{k\+1\}\-a\_\{k\}\.The two operators differ only by an index shift, but keeping both notations is helpful on finite intervals\. The quantization identity naturally uses the backward difference, whereas summation by parts transfers it to a forward difference of the weight\.

Repeated differences are defined recursively\. For example,

Δ2​ak=ak−2​ak−1\+ak−2,Δ\+2​ak=ak\+2−2​ak\+1\+ak\.\\Delta^\{2\}a\_\{k\}=a\_\{k\}\-2a\_\{k\-1\}\+a\_\{k\-2\},\\qquad\\Delta\_\{\+\}^\{2\}a\_\{k\}=a\_\{k\+2\}\-2a\_\{k\+1\}\+a\_\{k\}\.The binomial formula gives

Δr​ak=∑j=0r\(−1\)j​\(rj\)​ak−j,Δ\+r​ak=∑j=0r\(−1\)r−j​\(rj\)​ak\+j\.\\Delta^\{r\}a\_\{k\}=\\sum\_\{j=0\}^\{r\}\(\-1\)^\{j\}\\binom\{r\}\{j\}a\_\{k\-j\},\\qquad\\Delta\_\{\+\}^\{r\}a\_\{k\}=\\sum\_\{j=0\}^\{r\}\(\-1\)^\{r\-j\}\\binom\{r\}\{j\}a\_\{k\+j\}\.These identities are elementary, but they should be written explicitly because boundary indices become decisive in finite records\.

#### 3\.1\.2Discrete summation by parts

The finite analogue of integration by parts is also called Abel summation\. It is standard in Fourier analysis and analytic number theory\[[2](https://arxiv.org/html/2607.24868#bib.bib80),[85](https://arxiv.org/html/2607.24868#bib.bib74),[69](https://arxiv.org/html/2607.24868#bib.bib77)\]\. The form used below is adapted to noise shaping\.

###### Lemma 3\.1\(Finite summation by parts\)\.

Letv0,…,vN∈ℂv\_\{0\},\\ldots,v\_\{N\}\\in\\mathbb\{C\}andw1,…,wN∈ℂw\_\{1\},\\ldots,w\_\{N\}\\in\\mathbb\{C\}\. Then

∑k=1N\(Δ​vk\)​wk=vN​wN−v0​w1−∑k=1N−1vk​Δ\+​wk\.\\sum\_\{k=1\}^\{N\}\(\\Delta v\_\{k\}\)w\_\{k\}=v\_\{N\}w\_\{N\}\-v\_\{0\}w\_\{1\}\-\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\\Delta\_\{\+\}w\_\{k\}\.\(3\.1\)

###### Proof\.

Expand the left\-hand side:

∑k=1Nvk​wk−∑k=1Nvk−1​wk\.\\sum\_\{k=1\}^\{N\}v\_\{k\}w\_\{k\}\-\\sum\_\{k=1\}^\{N\}v\_\{k\-1\}w\_\{k\}\.In the second sum, replacekkbyk\+1k\+1in the interior terms\. This yields

vN​wN−v0​w1\+∑k=1N−1vk​\(wk−wk\+1\),v\_\{N\}w\_\{N\}\-v\_\{0\}w\_\{1\}\+\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(w\_\{k\}\-w\_\{k\+1\}\),which is exactly \([3\.1](https://arxiv.org/html/2607.24868#Ch3.E1)\)\. ∎

The identity has three parts\. The terminal state producesvN​wNv\_\{N\}w\_\{N\}\. The initial state produces−v0​w1\-v\_\{0\}w\_\{1\}\. The interior produces a weighted forward difference\. Every later estimate is a consequence of deciding how these three parts are controlled\.

#### 3\.1\.3Discrete total variation

For a finite complex sequencew=\(wk\)k=1Nw=\(w\_\{k\}\)\_\{k=1\}^\{N\}, define

TVN⁡\(w\)=∑k=1N−1\|wk\+1−wk\|\.\\operatorname\{TV\}\_\{N\}\(w\)=\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert w\_\{k\+1\}\-w\_\{k\}\\right\\rvert\.This is the variation of the polygonal path through the pointsw1,…,wNw\_\{1\},\\ldots,w\_\{N\}in the complex plane\. It is not the variation of the real and imaginary parts separately, although the two notions are comparable\. If\|wk\|=1\\left\\lvert w\_\{k\}\\right\\rvert=1, thenTVN⁡\(w\)\\operatorname\{TV\}\_\{N\}\(w\)measures the total chordal motion of the sampled points on the unit circle\.

The elementary inequality

\|ei​α−ei​β\|=2​\|sin⁡\(\(α−β\)/2\)\|≤\|α−β\|\\left\\lvert e^\{\\mathrm\{i\}\\alpha\}\-e^\{\\mathrm\{i\}\\beta\}\\right\\rvert=2\\left\\lvert\\sin\(\(\\alpha\-\\beta\)/2\)\\right\\rvert\\leq\\left\\lvert\\alpha\-\\beta\\right\\rvertconverts phase variation into weight variation\. The constant one in this inequality is sharp nearα=β\\alpha=\\beta\.

#### 3\.1\.4Continuous total variation

A real\-valued functionϕ\\phion\[0,1\]\[0,1\]has bounded variation if

Var\[0,1\]⁡\(ϕ\):=sup0=t0<⋯<tm=1∑j=0m−1\|ϕ​\(tj\+1\)−ϕ​\(tj\)\|<∞\.\\operatorname\{Var\}\_\{\[0,1\]\}\(\\phi\):=\\sup\_\{0=t\_\{0\}<\\cdots<t\_\{m\}=1\}\\sum\_\{j=0\}^\{m\-1\}\\left\\lvert\\phi\(t\_\{j\+1\}\)\-\\phi\(t\_\{j\}\)\\right\\rvert<\\infty\.The standard theory of BV functions includes decomposition into absolutely continuous, jump, and Cantor parts\[[1](https://arxiv.org/html/2607.24868#bib.bib72),[33](https://arxiv.org/html/2607.24868#bib.bib73)\]\. The first\-order Fourier estimate needs only the defining variation inequality\. It does not require differentiability\.

Ifϕ\\phiis absolutely continuous, then

Var\[0,1\]⁡\(ϕ\)=∫01\|ϕ′​\(ξ\)\|​dξ\.\\operatorname\{Var\}\_\{\[0,1\]\}\(\\phi\)=\\int\_\{0\}^\{1\}\\left\\lvert\\phi^\{\\prime\}\(\\xi\)\\right\\rvert\\,\\mathrm\{d\}\\xi\.The equality is a standard theorem\. In the parabolic case,ϕ\\phiis smooth, so no measure\-theoretic subtlety arises\. The BV formulation is included because it reveals the actual regularity threshold of the first\-order argument\.

#### 3\.1\.5Sampling a BV phase

Letξk=k/N\\xi\_\{k\}=k/Nand set

wk=e2​π​i​ϕ​\(ξk\)\.w\_\{k\}=e^\{2\\pi\\mathrm\{i\}\\phi\(\\xi\_\{k\}\)\}\.Then

TVN⁡\(w\)\\displaystyle\\operatorname\{TV\}\_\{N\}\(w\)≤2​π​∑k=1N−1\|ϕ​\(ξk\+1\)−ϕ​\(ξk\)\|\\displaystyle\\leq 2\\pi\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert\\phi\(\\xi\_\{k\+1\}\)\-\\phi\(\\xi\_\{k\}\)\\right\\rvert≤2​π​Var\[0,1\]⁡\(ϕ\)\.\\displaystyle\\leq 2\\pi\\operatorname\{Var\}\_\{\[0,1\]\}\(\\phi\)\.This inequality remains valid whenϕ\\phihas jumps, provided a representative has been chosen at the sample points\. Since only finitely many values are used, changing a representative on a null set may change the discrete sum\. For applications involving discontinuous phases, the representative must therefore be specified\. In the smooth polynomial cases considered later, this issue does not occur\.

#### 3\.1\.6Riemann\-sum scaling

Supposef∈C1​\(\[0,1\]\)f\\in C^\{1\}\(\[0,1\]\)\. The mean\-value theorem gives

\|f​\(\(k\+1\)/N\)−f​\(k/N\)\|≤1N​‖f′‖L∞\.\\left\\lvert f\(\(k\+1\)/N\)\-f\(k/N\)\\right\\rvert\\leq\\frac\{1\}\{N\}\\left\\lVert f^\{\\prime\}\\right\\rVert\_\{L^\{\\infty\}\}\.Summing overkkgives a bound independent ofNN\. Therefore, first\-order noise shaping and a normalisation1/N1/Nyield anN−1N^\{\-1\}estimate\.

Forrrth differences, Taylor expansion suggests

\|Δ\+r​f​\(k/N\)\|≲rN−r​‖f\(r\)‖L∞\.\\left\\lvert\\Delta\_\{\+\}^\{r\}f\(k/N\)\\right\\rvert\\lesssim\_\{r\}N^\{\-r\}\\left\\lVert f^\{\(r\)\}\\right\\rVert\_\{L^\{\\infty\}\}\.There areO​\(N\)O\(N\)interior terms, so their sum isO​\(N1−r\)O\(N^\{1\-r\}\)\. After the external factor1/N1/N, the contribution isO​\(N−r\)O\(N^\{\-r\}\)\. This heuristic is correct, but only after the boundary terms from repeated summation by parts have been removed or controlled\.

#### 3\.1\.7A useful integral formula for finite differences

###### Lemma 3\.2\(Integral representation\)\.

Letf∈Cr​\(\[0,1\]\)f\\in C^\{r\}\(\[0,1\]\), leth\>0h\>0, and assumex\+r​h≤1x\+rh\\leq 1\. Then

Δhr​f​\(x\)=∫\[0,h\]rf\(r\)​\(x\+s1\+⋯\+sr\)​ds1​⋯​dsr,\\Delta\_\{h\}^\{r\}f\(x\)=\\int\_\{\[0,h\]^\{r\}\}f^\{\(r\)\}\(x\+s\_\{1\}\+\\cdots\+s\_\{r\}\)\\,\\mathrm\{d\}s\_\{1\}\\cdots\\,\\mathrm\{d\}s\_\{r\},whereΔh​f​\(x\)=f​\(x\+h\)−f​\(x\)\\Delta\_\{h\}f\(x\)=f\(x\+h\)\-f\(x\)\.

###### Proof\.

Forr=1r=1, this is the fundamental theorem of calculus\. Assume the formula forrr\. Then

Δhr\+1​f​\(x\)\\displaystyle\\Delta\_\{h\}^\{r\+1\}f\(x\)=Δhr​f​\(x\+h\)−Δhr​f​\(x\)\\displaystyle=\\Delta\_\{h\}^\{r\}f\(x\+h\)\-\\Delta\_\{h\}^\{r\}f\(x\)=∫\[0,h\]r\[f\(r\)​\(x\+h\+s1\+⋯\+sr\)−f\(r\)​\(x\+s1\+⋯\+sr\)\]​ds\\displaystyle=\\int\_\{\[0,h\]^\{r\}\}\\left\[f^\{\(r\)\}\(x\+h\+s\_\{1\}\+\\cdots\+s\_\{r\}\)\-f^\{\(r\)\}\(x\+s\_\{1\}\+\\cdots\+s\_\{r\}\)\\right\]\\,\\mathrm\{d\}s=∫\[0,h\]r\+1f\(r\+1\)​\(x\+s1\+⋯\+sr\+1\)​ds,\\displaystyle=\\int\_\{\[0,h\]^\{r\+1\}\}f^\{\(r\+1\)\}\(x\+s\_\{1\}\+\\cdots\+s\_\{r\+1\}\)\\,\\mathrm\{d\}s,by another application of the fundamental theorem of calculus\. ∎

As an immediate consequence,

\|Δhr​f​\(x\)\|≤hr​‖f\(r\)‖L∞\.\\left\\lvert\\Delta\_\{h\}^\{r\}f\(x\)\\right\\rvert\\leq h^\{r\}\\left\\lVert f^\{\(r\)\}\\right\\rVert\_\{L^\{\\infty\}\}\.\(3\.2\)This estimate will be used repeatedly in the higher\-order theory\.

#### 3\.1\.8Why exact indexing matters

It is tempting to write symbolically

⟨Δr​v,w⟩=\(−1\)r​⟨v,Δ\+r​w⟩\\left\\langle\\Delta^\{r\}v,w\\right\\rangle=\(\-1\)^\{r\}\\left\\langle v,\\Delta\_\{\+\}^\{r\}w\\right\\rangleand ignore the endpoints\. This identity is correct on a bi\-infinite sequence when all terms decay sufficiently, or on a periodic sequence with compatible boundary conditions\. It is not correct on a finite interval without correction terms\. The boundary terms are not a technical nuisance\. They determine the leading asymptotic order\.

The frame\-quantization literature reached the same conclusion from a geometric perspective\. Smooth frame\-path termination was introduced precisely to suppress the endpoint contributions that obstruct high\-order convergence\[[9](https://arxiv.org/html/2607.24868#bib.bib20)\]\. The exact finite formula is derived in Chapter[6](https://arxiv.org/html/2607.24868#Ch6)\.

## Chapter 4First\-Order One\-Bit Noise\-Shaping Theory

This chapter develops the complete first\-order theory required by the later Fourier analysis\. Stability is proved directly from an invariant interval\. Discrepancy bounds follow by telescoping\. The weighted variation theorem is then established for arbitrary complex weights, followed by absolutely continuous, bounded\-variation, Hölder, vector\-valued, and operator\-valued consequences\.

### 4\.1Greedy one\-bit stability and discrepancy

#### 4\.1\.1The greedy sign quantizer

Define

Q​\(y\)=\{\+1,y≥0,−1,y<0\.Q\(y\)=\\begin\{cases\}\+1,&y\\geq 0,\\\\ \-1,&y<0\.\\end\{cases\}Given a real input sequenceu=\(uk\)k=1Nu=\(u\_\{k\}\)\_\{k=1\}^\{N\}, the greedy first\-order recursion is

qk=Q​\(uk\+vk−1\),vk=vk−1\+uk−qk,v0=0\.q\_\{k\}=Q\(u\_\{k\}\+v\_\{k\-1\}\),\\qquad v\_\{k\}=v\_\{k\-1\}\+u\_\{k\}\-q\_\{k\},\\qquad v\_\{0\}=0\.\(4\.1\)The conventionQ​\(0\)=1Q\(0\)=1is fixed\. Another convention at zero changes some special trajectories but does not affect the main variation theorem\.

The error identity follows immediately:

ek:=uk−qk=vk−vk−1=Δ​vk\.e\_\{k\}:=u\_\{k\}\-q\_\{k\}=v\_\{k\}\-v\_\{k\-1\}=\\Delta v\_\{k\}\.\(4\.2\)

#### 4\.1\.2Invariant interval

###### Theorem 4\.1\(Stability of the greedy one\-bit scheme\)\.

Assume

\|uk\|≤1,k=1,…,N\.\\left\\lvert u\_\{k\}\\right\\rvert\\leq 1,\\qquad k=1,\\ldots,N\.Then the recursion \([4\.1](https://arxiv.org/html/2607.24868#Ch4.E1)\) satisfies

\|vk\|≤1,k=0,1,…,N\.\\left\\lvert v\_\{k\}\\right\\rvert\\leq 1,\\qquad k=0,1,\\ldots,N\.Thus the scheme is stable withV=1V=1\.

###### Proof\.

The interval\[−1,1\]\[\-1,1\]is shown to be forward invariant\. The initial statev0=0v\_\{0\}=0lies in the interval\. Supposevk−1∈\[−1,1\]v\_\{k\-1\}\\in\[\-1,1\]and define

yk=uk\+vk−1\.y\_\{k\}=u\_\{k\}\+v\_\{k\-1\}\.Since both terms lie in\[−1,1\]\[\-1,1\], one hasyk∈\[−2,2\]y\_\{k\}\\in\[\-2,2\]\.

Ifyk≥0y\_\{k\}\\geq 0, thenqk=1q\_\{k\}=1and

The condition0≤yk≤20\\leq y\_\{k\}\\leq 2gives−1≤vk≤1\-1\\leq v\_\{k\}\\leq 1\.

Ifyk<0y\_\{k\}<0, thenqk=−1q\_\{k\}=\-1and

The condition−2≤yk<0\-2\\leq y\_\{k\}<0gives−1≤vk<1\-1\\leq v\_\{k\}<1\.

Thereforevk∈\[−1,1\]v\_\{k\}\\in\[\-1,1\]in both cases\. Induction completes the proof\. ∎

This invariant\-region argument is a basic example of nonlinear stability\. More elaborate sigma\-delta systems require higher\-dimensional invariant sets or carefully designed feedback filters\[[45](https://arxiv.org/html/2607.24868#bib.bib6),[25](https://arxiv.org/html/2607.24868#bib.bib7),[47](https://arxiv.org/html/2607.24868#bib.bib11)\]\.

#### 4\.1\.3Strict input margins

Suppose\|uk\|≤1−μ\\left\\lvert u\_\{k\}\\right\\rvert\\leq 1\-\\mufor someμ\>0\\mu\>0\. The same proof givesvk∈\[−1,1\]v\_\{k\}\\in\[\-1,1\]\. The strict margin does not automatically shrink the invariant interval for every trajectory because the state can approach the endpoints after a sequence of inputs\. It does, however, provide overload margin for perturbed thresholds and finite analogue errors\. A robustness theorem must state the perturbation model explicitly\.

#### 4\.1\.4Interval discrepancy

###### Proposition 4\.2\(Uniform discrepancy on intervals\)\.

Assumeek=Δ​vke\_\{k\}=\\Delta v\_\{k\}and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\. Then, for every1≤m≤n≤N1\\leq m\\leq n\\leq N,

\|∑k=mnek\|=\|vn−vm−1\|≤2​V\.\\left\\lvert\\sum\_\{k=m\}^\{n\}e\_\{k\}\\right\\rvert=\\left\\lvert v\_\{n\}\-v\_\{m\-1\}\\right\\rvert\\leq 2V\.Ifm=1m=1andv0=0v\_\{0\}=0, then

\|∑k=1nek\|≤V\.\\left\\lvert\\sum\_\{k=1\}^\{n\}e\_\{k\}\\right\\rvert\\leq V\.

###### Proof\.

The sum telescopes:

∑k=mnek=∑k=mn\(vk−vk−1\)=vn−vm−1\.\\sum\_\{k=m\}^\{n\}e\_\{k\}=\\sum\_\{k=m\}^\{n\}\(v\_\{k\}\-v\_\{k\-1\}\)=v\_\{n\}\-v\_\{m\-1\}\.The bounds follow from the triangle inequality\. ∎

The proposition says that the quantization error has uniformly bounded discrepancy on every interval\. Individual errors may be large, but their cumulative sum cannot drift linearly with the interval length\. This property is stronger than a pointwise bound and is the discrete cancellation used later\.

#### 4\.1\.5Prefix sums and equivalence of formulations

Let

En=∑k=1nek\.E\_\{n\}=\\sum\_\{k=1\}^\{n\}e\_\{k\}\.Whenv0=0v\_\{0\}=0, one hasEn=vnE\_\{n\}=v\_\{n\}\. Therefore first\-order noise shaping is equivalent to bounded prefix sums of the error\. Conversely, if an error sequence has bounded prefix sums, definingvn=Env\_\{n\}=E\_\{n\}givesen=Δ​vne\_\{n\}=\\Delta v\_\{n\}\.

This equivalence is useful because discrepancy theory often begins from partial sums, while sigma\-delta theory begins from a state recursion\. The two languages describe the same first\-order structure\.

#### 4\.1\.6The zero\-input orbit

The zero input is an instructive trajectory\. Letuk=0u\_\{k\}=0for allkkandv0=0v\_\{0\}=0\. Then

q1=1,v1=−1,q\_\{1\}=1,\\qquad v\_\{1\}=\-1,and

q2=−1,v2=0\.q\_\{2\}=\-1,\\qquad v\_\{2\}=0\.The pattern repeats:

qk=\(−1\)k\+1,vk=\{−1,k​odd,0,k​even\.q\_\{k\}=\(\-1\)^\{k\+1\},\\qquad v\_\{k\}=\\begin\{cases\}\-1,&k\\text\{ odd\},\\\\ 0,&k\\text\{ even\}\.\\end\{cases\}This sequence is stable, but its terminal state is nonzero for oddNN\. It provides the sharpness example in Chapter[5](https://arxiv.org/html/2607.24868#Ch5)\.

#### 4\.1\.7Complex inputs and larger alphabets

The greedy sign rule is real\. Complex Fourier coefficients may nevertheless be tested because the weights are complex while the coefficients remain real\. A genuinely complex quantizer would require a planar alphabet and an invariant set inℂ\\mathbb\{C\}\. The weighted variation theorem itself is indifferent to how the bounded state was generated\. It applies to complexvvand complexeeas soon ase=Δ​ve=\\Delta vand‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq Vhold\.

Similarly, a multibit alphabet changes the stability region but not the summation argument\. The separation between quantizer design and Fourier testing is therefore useful: stability is established in the quantizer model, and the extension estimate uses only the resulting difference identity\.

### 4\.2Weighted variation estimates

#### 4\.2\.1The general theorem

###### Theorem 4\.3\(Weighted variation with full boundary terms\)\.

Letek=Δ​vke\_\{k\}=\\Delta v\_\{k\}fork=1,…,Nk=1,\\ldots,N, and assume

‖v‖ℓ∞≤V\.\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\.For arbitrary complex weightsw1,…,wNw\_\{1\},\\ldots,w\_\{N\},

\|1N​∑k=1Nek​wk\|≤VN​\(\|wN\|\+\|w1\|\+TVN⁡\(w\)\)\.\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\(\\left\\lvert w\_\{N\}\\right\\rvert\+\\left\\lvert w\_\{1\}\\right\\rvert\+\\operatorname\{TV\}\_\{N\}\(w\)\\right\)\.\(4\.3\)Ifv0=0v\_\{0\}=0, the term\|w1\|\\left\\lvert w\_\{1\}\\right\\rvertmay be removed:

\|1N​∑k=1Nek​wk\|≤VN​\(\|wN\|\+TVN⁡\(w\)\)\.\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\(\\left\\lvert w\_\{N\}\\right\\rvert\+\\operatorname\{TV\}\_\{N\}\(w\)\\right\)\.\(4\.4\)

###### Proof\.

By[Lemma˜3\.1](https://arxiv.org/html/2607.24868#Ch3.Thmtheorem1),

∑k=1Nek​wk=vN​wN−v0​w1−∑k=1N−1vk​Δ\+​wk\.\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}=v\_\{N\}w\_\{N\}\-v\_\{0\}w\_\{1\}\-\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\\Delta\_\{\+\}w\_\{k\}\.Taking absolute values gives

\|∑k=1Nek​wk\|\\displaystyle\\left\\lvert\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rvert≤\|vN\|​\|wN\|\+\|v0\|​\|w1\|\+∑k=1N−1\|vk\|​\|Δ\+​wk\|\\displaystyle\\leq\\left\\lvert v\_\{N\}\\right\\rvert\\left\\lvert w\_\{N\}\\right\\rvert\+\\left\\lvert v\_\{0\}\\right\\rvert\\left\\lvert w\_\{1\}\\right\\rvert\+\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert v\_\{k\}\\right\\rvert\\left\\lvert\\Delta\_\{\+\}w\_\{k\}\\right\\rvert≤V​\(\|wN\|\+\|w1\|\+TVN⁡\(w\)\)\.\\displaystyle\\leq V\\left\(\\left\\lvert w\_\{N\}\\right\\rvert\+\\left\\lvert w\_\{1\}\\right\\rvert\+\\operatorname\{TV\}\_\{N\}\(w\)\\right\)\.Divide byNN\. Ifv0=0v\_\{0\}=0, the initial term vanishes\. ∎

The theorem is deterministic and contains no oscillatory estimate\. It applies to every complex weight sequence\. Its usefulness depends on finding weights with bounded total variation\.

#### 4\.2\.2Optimality of the variation form

The right\-hand side cannot be replaced by a bound involving onlymaxk⁡\|wk\|\\max\_\{k\}\\left\\lvert w\_\{k\}\\right\\rvertunder the same hypotheses\. Indeed, a rapidly alternating weight may align with an alternating error sequence\. The variation term measures this possible alignment\.

Conversely, the theorem is not always sharp in its constant because the triangle inequality discards cancellation among the interior terms\. The purpose of later oscillatory analysis is precisely to improve on this absolute\-value step in regimes whereTVN⁡\(w\)\\operatorname\{TV\}\_\{N\}\(w\)is large\.

#### 4\.2\.3Absolutely continuous phases

For a real phaseϕ\\phion\[0,1\]\[0,1\], define

ℰNϕ​a=1N​∑k=1Nak​e2​π​i​ϕ​\(k/N\)\.\\mathcal\{E\}\_\{N\}^\{\\phi\}a=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}a\_\{k\}e^\{2\\pi\\mathrm\{i\}\\phi\(k/N\)\}\.
###### Theorem 4\.4\(Absolutely continuous phase estimate\)\.

Assumeϕ\\phiis absolutely continuous,ek=Δ​vke\_\{k\}=\\Delta v\_\{k\},v0=0v\_\{0\}=0, and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\. Then

\|ℰNϕ​u−ℰNϕ​q\|≤VN​\[1\+2​π​∫01\|ϕ′​\(ξ\)\|​dξ\]\.\\left\\lvert\\mathcal\{E\}\_\{N\}^\{\\phi\}u\-\\mathcal\{E\}\_\{N\}^\{\\phi\}q\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\int\_\{0\}^\{1\}\\left\\lvert\\phi^\{\\prime\}\(\\xi\)\\right\\rvert\\,\\mathrm\{d\}\\xi\\right\]\.\(4\.5\)

###### Proof\.

Setwk=e2​π​i​ϕ​\(k/N\)w\_\{k\}=e^\{2\\pi\\mathrm\{i\}\\phi\(k/N\)\}\. Since\|wN\|=1\\left\\lvert w\_\{N\}\\right\\rvert=1and

\|ei​α−ei​β\|≤\|α−β\|,\\left\\lvert e^\{\\mathrm\{i\}\\alpha\}\-e^\{\\mathrm\{i\}\\beta\}\\right\\rvert\\leq\\left\\lvert\\alpha\-\\beta\\right\\rvert,one has

\|wk\+1−wk\|\\displaystyle\\left\\lvert w\_\{k\+1\}\-w\_\{k\}\\right\\rvert≤2​π​\|ϕ​\(\(k\+1\)/N\)−ϕ​\(k/N\)\|\\displaystyle\\leq 2\\pi\\left\\lvert\\phi\(\(k\+1\)/N\)\-\\phi\(k/N\)\\right\\rvert≤2​π​∫k/N\(k\+1\)/N\|ϕ′​\(ξ\)\|​dξ\.\\displaystyle\\leq 2\\pi\\int\_\{k/N\}^\{\(k\+1\)/N\}\\left\\lvert\\phi^\{\\prime\}\(\\xi\)\\right\\rvert\\,\\mathrm\{d\}\\xi\.Summation gives

TVN⁡\(w\)≤2​π​∫1/N1\|ϕ′​\(ξ\)\|​dξ≤2​π​∫01\|ϕ′​\(ξ\)\|​dξ\.\\operatorname\{TV\}\_\{N\}\(w\)\\leq 2\\pi\\int\_\{1/N\}^\{1\}\\left\\lvert\\phi^\{\\prime\}\(\\xi\)\\right\\rvert\\,\\mathrm\{d\}\\xi\\leq 2\\pi\\int\_\{0\}^\{1\}\\left\\lvert\\phi^\{\\prime\}\(\\xi\)\\right\\rvert\\,\\mathrm\{d\}\\xi\.Apply \([4\.4](https://arxiv.org/html/2607.24868#Ch4.E4)\)\. ∎

#### 4\.2\.4Bounded\-variation phases

###### Theorem 4\.5\(BV phase estimate\)\.

Assumeϕ∈B​V​\(\[0,1\]\)\\phi\\in BV\(\[0,1\]\),ek=Δ​vke\_\{k\}=\\Delta v\_\{k\},v0=0v\_\{0\}=0, and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\. Then

\|ℰNϕ​u−ℰNϕ​q\|≤VN​\[1\+2​π​Var\[0,1\]⁡\(ϕ\)\]\.\\left\\lvert\\mathcal\{E\}\_\{N\}^\{\\phi\}u\-\\mathcal\{E\}\_\{N\}^\{\\phi\}q\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\operatorname\{Var\}\_\{\[0,1\]\}\(\\phi\)\\right\]\.

###### Proof\.

For each adjacent pair,

\|wk\+1−wk\|≤2​π​\|ϕ​\(\(k\+1\)/N\)−ϕ​\(k/N\)\|\.\\left\\lvert w\_\{k\+1\}\-w\_\{k\}\\right\\rvert\\leq 2\\pi\\left\\lvert\\phi\(\(k\+1\)/N\)\-\\phi\(k/N\)\\right\\rvert\.The sum of the phase increments over the grid is bounded by the total variation ofϕ\\phi\. Apply[Theorem˜4\.3](https://arxiv.org/html/2607.24868#Ch4.Thmtheorem3)withv0=0v\_\{0\}=0\. ∎

The BV theorem reveals that differentiability is not the essential assumption\. The essential assumption is finite phase variation on the sampled interval\.

#### 4\.2\.5Lipschitz and Hölder corollaries

Ifϕ\\phiis Lipschitz with constantLL, then

Var\[0,1\]⁡\(ϕ\)≤L,\\operatorname\{Var\}\_\{\[0,1\]\}\(\\phi\)\\leq L,so

\|ℰNϕ​u−ℰNϕ​q\|≤VN​\(1\+2​π​L\)\.\\left\\lvert\\mathcal\{E\}\_\{N\}^\{\\phi\}u\-\\mathcal\{E\}\_\{N\}^\{\\phi\}q\\right\\rvert\\leq\\frac\{V\}\{N\}\(1\+2\\pi L\)\.A merely Hölder phase with exponentγ<1\\gamma<1may have grid variation of orderN1−γN^\{1\-\\gamma\}\. The same method then gives onlyN−γN^\{\-\\gamma\}decay:

TVN⁡\(w\)≲N1−γ\.\\operatorname\{TV\}\_\{N\}\(w\)\\lesssim N^\{1\-\\gamma\}\.This observation suggests a regularity scale\. First\-order noise shaping gives one power of the mesh size, but rough weights can consume part of that gain\.

###### Proposition 4\.6\(Hölder phase estimate\)\.

Assume

\|ϕ​\(x\)−ϕ​\(y\)\|≤L​\|x−y\|γ,0<γ≤1\.\\left\\lvert\\phi\(x\)\-\\phi\(y\)\\right\\rvert\\leq L\\left\\lvert x\-y\\right\\rvert^\{\\gamma\},\\qquad 0<\\gamma\\leq 1\.Then

\|ℰNϕ​u−ℰNϕ​q\|≤VN\+2​π​V​L​N−γ\.\\left\\lvert\\mathcal\{E\}\_\{N\}^\{\\phi\}u\-\\mathcal\{E\}\_\{N\}^\{\\phi\}q\\right\\rvert\\leq\\frac\{V\}\{N\}\+2\\pi VLN^\{\-\\gamma\}\.

###### Proof\.

Each adjacent phase increment is at mostL​N−γLN^\{\-\\gamma\}\. There areN−1N\-1increments, so

TVN⁡\(w\)≤2​π​L​\(N−1\)​N−γ\.\\operatorname\{TV\}\_\{N\}\(w\)\\leq 2\\pi L\(N\-1\)N^\{\-\\gamma\}\.Substitute into \([4\.4](https://arxiv.org/html/2607.24868#Ch4.E4)\)\. ∎

#### 4\.2\.6Vector\-valued and operator\-valued weights

The same proof works in a normed space ifwkw\_\{k\}is replaced by a bounded linear functional or operator and absolute values are replaced by operator norms\. LetXXbe a Banach space andwk∈Xw\_\{k\}\\in X\. Then

‖1N​∑k=1Nek​wk‖X≤VN​\(‖wN‖X\+∑k=1N−1‖wk\+1−wk‖X\)\\left\\lVert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rVert\_\{X\}\\leq\\frac\{V\}\{N\}\\left\(\\left\\lVert w\_\{N\}\\right\\rVert\_\{X\}\+\\sum\_\{k=1\}^\{N\-1\}\\left\\lVert w\_\{k\+1\}\-w\_\{k\}\\right\\rVert\_\{X\}\\right\)whenv0=0v\_\{0\}=0\. This form may be useful for vector\-valued extensions, simultaneous testing at several parameter points, or operator\-valued reconstruction kernels\.

###### Proposition 4\.7\(Exact variation duality for complex states\)\.

Letw1,…,wN∈ℂw\_\{1\},\\ldots,w\_\{N\}\\in\\mathbb\{C\}and letV\>0V\>0\. Among all complex state sequences satisfyingv0=0v\_\{0\}=0and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V, one has

supv0=0‖v‖ℓ∞≤V\|∑k=1N\(Δ​vk\)​wk\|=V​\(\|wN\|\+∑k=1N−1\|wk\+1−wk\|\)\.\\sup\_\{\\begin\{subarray\}\{c\}v\_\{0\}=0\\\\ \\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\\end\{subarray\}\}\\left\\lvert\\sum\_\{k=1\}^\{N\}\(\\Delta v\_\{k\}\)w\_\{k\}\\right\\rvert=V\\left\(\\left\\lvert w\_\{N\}\\right\\rvert\+\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert w\_\{k\+1\}\-w\_\{k\}\\right\\rvert\\right\)\.\(4\.6\)Consequently, the constant in the weighted variation theorem cannot be reduced when the admissible class contains every bounded complex state\.

###### Proof\.

Summation by parts gives

∑k=1N\(Δ​vk\)​wk=vN​wN\+∑k=1N−1vk​\(wk−wk\+1\)\.\\sum\_\{k=1\}^\{N\}\(\\Delta v\_\{k\}\)w\_\{k\}=v\_\{N\}w\_\{N\}\+\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(w\_\{k\}\-w\_\{k\+1\}\)\.The triangle inequality proves the upper bound\. To attain it, choose each state value independently so that its product with the corresponding coefficient is a nonnegative real number\. More precisely, set

vN=V​exp⁡\(−i​arg⁡wN\)v\_\{N\}=V\\exp\\\!\\bigl\(\-\\mathrm\{i\}\\arg w\_\{N\}\\bigr\)whenwN≠0w\_\{N\}\\neq 0, and for1≤k≤N−11\\leq k\\leq N\-1set

vk=V​exp⁡\(−i​arg⁡\(wk−wk\+1\)\)v\_\{k\}=V\\exp\\\!\\bigl\(\-\\mathrm\{i\}\\arg\(w\_\{k\}\-w\_\{k\+1\}\)\\bigr\)whenwk≠wk\+1w\_\{k\}\\neq w\_\{k\+1\}\. Values attached to zero coefficients may be chosen arbitrarily in the closed disc of radiusVV\. Every nonzero summand is then equal to the modulus of its coefficient multiplied byVV, which proves equality\. ∎

#### 4\.2\.7A duality interpretation

The space of finite sequences whose prefix sums are bounded is the discrete analogue of a negative Sobolev space\. The weighted theorem says thate=Δ​ve=\\Delta vacts continuously on weights with bounded variation\. Symbolically,

\|⟨e,w⟩\|≲‖v‖ℓ∞​‖w‖B​Vd\.\\left\\lvert\\left\\langle e,w\\right\\rangle\\right\\rvert\\lesssim\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\left\\lVert w\\right\\rVert\_\{BV\_\{d\}\}\.This is a finite\-dimensional duality between discrete derivatives of bounded sequences and discrete BV weights\. The viewpoint parallels continuous distribution theory, where derivatives of bounded functions act on test functions through integration by parts\.

## Chapter 5Detailed Derivations and Proved Results for Parabolic Fourier Extension

### 5\.1Parabolic Fourier extension and the phase\-variation factor

#### 5\.1\.1Definition and PDE interpretation

Define

ℰN​a​\(x,t\)=1N​∑k=1Nak​e​\(x​kN\+t​k2N2\)\.\\mathcal\{E\}\_\{N\}a\(x,t\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}a\_\{k\}e\\\!\\left\(x\\frac\{k\}\{N\}\+t\\frac\{k^\{2\}\}\{N^\{2\}\}\\right\)\.\(5\.1\)The variablesxxandttmay be viewed as space and time parameters for a periodic Schrödinger\-type evolution after a suitable rescaling\. Discrete restriction estimates for parabolic phases are closely connected with periodic Strichartz inequalities\[[15](https://arxiv.org/html/2607.24868#bib.bib41),[56](https://arxiv.org/html/2607.24868#bib.bib43),[61](https://arxiv.org/html/2607.24868#bib.bib54),[13](https://arxiv.org/html/2607.24868#bib.bib45)\]\.

The present normalisation differs from the most common PDE normalisation\. The frequency index isk/Nk/Nrather thankk, and the sum is divided byNN\. This scaling is natural for comparing a sequence with a Riemann integral on\[0,1\]\[0,1\]\.

#### 5\.1\.2The variation factor

Let

ϕx,t​\(ξ\)=x​ξ\+t​ξ2,J​\(x,t\)=∫01\|x\+2​t​ξ\|​dξ\.\\phi\_\{x,t\}\(\\xi\)=x\\xi\+t\\xi^\{2\},\\qquad J\(x,t\)=\\int\_\{0\}^\{1\}\\left\\lvert x\+2t\\xi\\right\\rvert\\,\\mathrm\{d\}\\xi\.Sinceϕx,t′​\(ξ\)=x\+2​t​ξ\\phi^\{\\prime\}\_\{x,t\}\(\\xi\)=x\+2t\\xi, the absolutely continuous phase theorem applies directly\.

###### Theorem 5\.1\(Parabolic quantization error\)\.

Assumeek=uk−qk=Δ​vke\_\{k\}=u\_\{k\}\-q\_\{k\}=\\Delta v\_\{k\},v0=0v\_\{0\}=0, and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\. Then

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤VN​\[1\+2​π​J​\(x,t\)\]\.\\left\\lvert\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi J\(x,t\)\\right\]\.\(5\.2\)Moreover,

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤VN​\[1\+2​π​\(\|x\|\+\|t\|\)\]\.\\left\\lvert\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\bigl\(\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert\\bigr\)\\right\]\.\(5\.3\)

###### Proof\.

Apply[Theorem˜4\.4](https://arxiv.org/html/2607.24868#Ch4.Thmtheorem4)toϕx,t\\phi\_\{x,t\}\. For the simpler bound, use

\|x\+2​t​ξ\|≤\|x\|\+2​\|t\|​ξ\\left\\lvert x\+2t\\xi\\right\\rvert\\leq\\left\\lvert x\\right\\rvert\+2\\left\\lvert t\\right\\rvert\\xiand integrate over\[0,1\]\[0,1\]\. ∎

#### 5\.1\.3Exact evaluation ofJ​\(x,t\)J\(x,t\)

###### Proposition 5\.2\(Closed form\)\.

The variation factor is given by the following cases\.

Ift=0t=0, then

J​\(x,0\)=\|x\|\.J\(x,0\)=\\left\\lvert x\\right\\rvert\.Ift≠0t\\neq 0andx​\(x\+2​t\)≥0x\(x\+2t\)\\geq 0, then

J​\(x,t\)=\|x\+t\|\.J\(x,t\)=\\left\\lvert x\+t\\right\\rvert\.Ift≠0t\\neq 0andx​\(x\+2​t\)<0x\(x\+2t\)<0, then

J​\(x,t\)=x2\+\(x\+2​t\)24​\|t\|\.J\(x,t\)=\\frac\{x^\{2\}\+\(x\+2t\)^\{2\}\}\{4\\left\\lvert t\\right\\rvert\}\.

###### Proof\.

Whent=0t=0, the integrand is constant\. Supposet≠0t\\neq 0\. The affine functiong​\(ξ\)=x\+2​t​ξg\(\\xi\)=x\+2t\\xihas endpoint valuesxxandx\+2​tx\+2t\.

If the endpoint values have the same sign, thenggdoes not change sign on\[0,1\]\[0,1\]\. Hence

J​\(x,t\)=\|∫01g​\(ξ\)​dξ\|=\|x\+t\|\.J\(x,t\)=\\left\\lvert\\int\_\{0\}^\{1\}g\(\\xi\)\\,\\mathrm\{d\}\\xi\\right\\rvert=\\left\\lvert x\+t\\right\\rvert\.
If the endpoint values have opposite signs, the unique zero is

ξ0=−x2​t∈\(0,1\)\.\\xi\_\{0\}=\-\\frac\{x\}\{2t\}\\in\(0,1\)\.The graph of\|g\|\\left\\lvert g\\right\\rvertconsists of two triangles\. The first has baseξ0\\xi\_\{0\}and height\|x\|\\left\\lvert x\\right\\rvert; the second has base1−ξ01\-\\xi\_\{0\}and height\|x\+2​t\|\\left\\lvert x\+2t\\right\\rvert\. Therefore

J​\(x,t\)=12​ξ0​\|x\|\+12​\(1−ξ0\)​\|x\+2​t\|\.J\(x,t\)=\\frac\{1\}\{2\}\\xi\_\{0\}\\left\\lvert x\\right\\rvert\+\\frac\{1\}\{2\}\(1\-\\xi\_\{0\}\)\\left\\lvert x\+2t\\right\\rvert\.Substitutingξ0=−x/\(2​t\)\\xi\_\{0\}=\-x/\(2t\)and simplifying gives the stated formula\. ∎

The closed form is often substantially smaller than\|x\|\+\|t\|\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert\. For example, ifx=−tx=\-t, the phase derivative changes sign symmetrically and

J​\(−t,t\)=\|t\|2\.J\(\-t,t\)=\\frac\{\\left\\lvert t\\right\\rvert\}\{2\}\.The simple triangle bound would give2​\|t\|2\\left\\lvert t\\right\\rvert\.

#### 5\.1\.4Compact\-set convergence

###### Corollary 5\.3\(Uniform convergence on compact sets\)\.

LetK⊂ℝ2K\\subset\\mathbb\{R\}^\{2\}be compact and define

MK=sup\(x,t\)∈KJ​\(x,t\)\.M\_\{K\}=\\sup\_\{\(x,t\)\\in K\}J\(x,t\)\.Then

‖ℰN​u−ℰN​q‖L∞​\(K\)≤VN​\(1\+2​π​MK\)\.\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{\\infty\}\(K\)\}\\leq\\frac\{V\}\{N\}\(1\+2\\pi M\_\{K\}\)\.In particular,

‖ℰN​u−ℰN​q‖L∞​\(K\)=OK​\(N−1\)\.\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{\\infty\}\(K\)\}=O\_\{K\}\(N^\{\-1\}\)\.

###### Proof\.

Take the supremum in \([5\.2](https://arxiv.org/html/2607.24868#Ch5.E2)\)\. Continuity ofJJand compactness ofKKimplyMK<∞M\_\{K\}<\\infty\. ∎

#### 5\.1\.5Rectangular localLpL^\{p\}bounds

Let

ΩR=\[−R,R\]2\.\\Omega\_\{R\}=\[\-R,R\]^\{2\}\.For\(x,t\)∈ΩR\(x,t\)\\in\\Omega\_\{R\}, one has\|x\|\+\|t\|≤2​R\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert\\leq 2R\. Therefore

‖ℰN​u−ℰN​q‖L∞​\(ΩR\)≤VN​\(1\+4​π​R\)\.\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\_\{R\}\)\}\\leq\\frac\{V\}\{N\}\(1\+4\\pi R\)\.Using\|ΩR\|=4​R2\\left\\lvert\\Omega\_\{R\}\\right\\rvert=4R^\{2\}gives the following result\.

###### Corollary 5\.4\(LocalLpL^\{p\}estimate\)\.

For1≤p<∞1\\leq p<\\infty,

‖ℰN​u−ℰN​q‖Lp​\(ΩR\)≤\(4​R2\)1/p​VN​\(1\+4​π​R\)\.\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{p\}\(\\Omega\_\{R\}\)\}\\leq\(4R^\{2\}\)^\{1/p\}\\frac\{V\}\{N\}\(1\+4\\pi R\)\.Forp=∞p=\\infty, the factor\(4​R2\)1/p\(4R^\{2\}\)^\{1/p\}is omitted\.

This estimate is local and deterministic\. It does not use a restriction theorem\. The factorR2/pR^\{2/p\}comes only from the measure of the region\.

#### 5\.1\.6An exact adjacent\-phase estimate

The inequality\|ei​α−ei​β\|≤\|α−β\|\\left\\lvert e^\{\\mathrm\{i\}\\alpha\}\-e^\{\\mathrm\{i\}\\beta\}\\right\\rvert\\leq\\left\\lvert\\alpha\-\\beta\\right\\rvertis convenient but may be loose when phase increments are not small\. An exact identity is

\|wk\+1−wk\|=2​\|sin⁡\(π​δk\)\|,\\left\\lvert w\_\{k\+1\}\-w\_\{k\}\\right\\rvert=2\\left\\lvert\\sin\(\\pi\\delta\_\{k\}\)\\right\\rvert,where

δk=xN\+t​\(2​k\+1\)N2\.\\delta\_\{k\}=\\frac\{x\}\{N\}\+\\frac\{t\(2k\+1\)\}\{N^\{2\}\}\.Therefore the weighted theorem also gives

\|ℰN​u−ℰN​q\|≤VN​\[1\+2​∑k=1N−1\|sin⁡\(π​δk\)\|\]\.\\left\\lvert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\[1\+2\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert\\sin\(\\pi\\delta\_\{k\}\)\\right\\rvert\\right\]\.\(5\.4\)This bound is exact at the variation step\. It may improve the simple derivative estimate when some increments are close to integers\. It still takes absolute values and therefore does not exploit cancellation between differentkk\.

#### 5\.1\.7Geometric interpretation

The sampled weights trace a polygonal path on the unit circle\. The quantityJ​\(x,t\)J\(x,t\)is the total variation of the continuous phase, whileTVN⁡\(w\)\\operatorname\{TV\}\_\{N\}\(w\)is the chord length of the sampled path\. The estimate says that a first\-order shaped error acts weakly on a slowly moving point of the unit circle\. The faster the phase winds, the weaker this deterministic conclusion becomes\.

### 5\.2Complete derivation of the parabolic estimate

The complete argument is recorded here in a single chain so that every hypothesis can be traced to the final estimate\. Let

ek=uk−qk=vk−vk−1,v0=0,e\_\{k\}=u\_\{k\}\-q\_\{k\}=v\_\{k\}\-v\_\{k\-1\},\\qquad v\_\{0\}=0,and suppose that‖v‖ℓ∞≤V\\\|v\\\|\_\{\\ell^\{\\infty\}\}\\leq V\. Define

wk​\(x,t\)=exp⁡\(2​π​i​\(x​kN\+t​k2N2\)\)\.w\_\{k\}\(x,t\)=\\exp\\\!\\left\(2\\pi\\mathrm\{i\}\\left\(x\\frac\{k\}\{N\}\+t\\frac\{k^\{2\}\}\{N^\{2\}\}\\right\)\\right\)\.The extension difference is

ℰN​u​\(x,t\)−ℰN​q​\(x,t\)=1N​∑k=1Nek​wk​\(x,t\)\.\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\(x,t\)\.Discrete summation by parts gives

∑k=1Nek​wk\\displaystyle\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}=∑k=1N\(vk−vk−1\)​wk\\displaystyle=\\sum\_\{k=1\}^\{N\}\(v\_\{k\}\-v\_\{k\-1\}\)w\_\{k\}\(5\.5\)=vN​wN−v0​w1\+∑k=1N−1vk​\(wk−wk\+1\)\.\\displaystyle=v\_\{N\}w\_\{N\}\-v\_\{0\}w\_\{1\}\+\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(w\_\{k\}\-w\_\{k\+1\}\)\.\(5\.6\)The initial contribution vanishes becausev0=0v\_\{0\}=0\. Since\|wN\|=1\|w\_\{N\}\|=1,

\|ℰN​u−ℰN​q\|\\displaystyle\\left\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\|≤VN​\(1\+∑k=1N−1\|wk\+1−wk\|\)\.\\displaystyle\\leq\\frac\{V\}\{N\}\\left\(1\+\\sum\_\{k=1\}^\{N\-1\}\|w\_\{k\+1\}\-w\_\{k\}\|\\right\)\.\(5\.7\)The phase increment is

δk\\displaystyle\\delta\_\{k\}=x​1N\+t​\(k\+1\)2−k2N2\\displaystyle=x\\frac\{1\}\{N\}\+t\\frac\{\(k\+1\)^\{2\}\-k^\{2\}\}\{N^\{2\}\}\(5\.8\)=xN\+t​\(2​k\+1\)N2\.\\displaystyle=\\frac\{x\}\{N\}\+\\frac\{t\(2k\+1\)\}\{N^\{2\}\}\.\(5\.9\)Therefore,

\|wk\+1−wk\|=2​\|sin⁡\(π​δk\)\|≤2​π​\|δk\|\.\|w\_\{k\+1\}\-w\_\{k\}\|=2\|\\sin\(\\pi\\delta\_\{k\}\)\|\\leq 2\\pi\|\\delta\_\{k\}\|\.A second derivation uses the sampled phaseϕx,t​\(ξ\)=x​ξ\+t​ξ2\\phi\_\{x,t\}\(\\xi\)=x\\xi\+t\\xi^\{2\}\. The fundamental theorem of calculus gives

\|ϕx,t​\(ξk\+1\)−ϕx,t​\(ξk\)\|≤∫ξkξk\+1\|x\+2​t​ξ\|​dξ\.\|\\phi\_\{x,t\}\(\\xi\_\{k\+1\}\)\-\\phi\_\{x,t\}\(\\xi\_\{k\}\)\|\\leq\\int\_\{\\xi\_\{k\}\}^\{\\xi\_\{k\+1\}\}\|x\+2t\\xi\|\\,\\mathrm\{d\}\\xi\.Summation over the adjacent intervals yields

∑k=1N−1\|wk\+1−wk\|≤2​π​∫1/N1\|x\+2​t​ξ\|​dξ≤2​π​J​\(x,t\),\\sum\_\{k=1\}^\{N\-1\}\|w\_\{k\+1\}\-w\_\{k\}\|\\leq 2\\pi\\int\_\{1/N\}^\{1\}\|x\+2t\\xi\|\\,\\mathrm\{d\}\\xi\\leq 2\\pi J\(x,t\),where

J​\(x,t\)=∫01\|x\+2​t​ξ\|​dξ\.J\(x,t\)=\\int\_\{0\}^\{1\}\|x\+2t\\xi\|\\,\\mathrm\{d\}\\xi\.Substitution into \([5\.7](https://arxiv.org/html/2607.24868#Ch5.E7)\) proves

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤VN​\[1\+2​π​J​\(x,t\)\]\.\\left\|\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\\right\|\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi J\(x,t\)\\right\]\.Every term has a distinct origin\. The factorN−1N^\{\-1\}is the normalisation of the extension operator\. The constantVVis the state bound\. The number11is the terminal\-state contribution\. The functionJJmeasures the continuous total variation of the phase\. No cancellation beyond the triangle inequality is required\.

#### 5\.2\.1Dependence of the estimate on the assumptions

The difference representation is essential\. A pointwise bound\|ek\|≤C\|e\_\{k\}\|\\leq Calone gives only

\|1N​∑k=1Nek​wk\|≤C,\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\|\\leq C,which does not decay\. The state bound is also essential because the summation\-by\-parts identity contains the values ofvkv\_\{k\}\. The zero initial state removes one boundary term\. A nonzero initial state produces the additional contributionV0​\|w1\|/NV\_\{0\}\|w\_\{1\}\|/N\. The smoothness of the parabolic phase is used only to estimate its sampled variation\. Consequently, the same proof applies to every absolutely continuous phase with integrable derivative\.

#### 5\.2\.2Exact and simplified constants

The integralJ​\(x,t\)J\(x,t\)is exact as a continuous variation factor\. The estimate

J​\(x,t\)≤\|x\|\+\|t\|J\(x,t\)\\leq\|x\|\+\|t\|follows from\|x\+2​t​ξ\|≤\|x\|\+2​\|t\|​ξ\|x\+2t\\xi\|\\leq\|x\|\+2\|t\|\\xi\. The simpler bound

\|ℰN​u−ℰN​q\|≤VN​\[1\+2​π​\(\|x\|\+\|t\|\)\]\\left\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\|\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\(\|x\|\+\|t\|\)\\right\]is convenient on rectangular parameter sets\. The discrete sine expression is generally sharper:

\|ℰN​u−ℰN​q\|≤VN​\[1\+2​∑k=1N−1\|sin⁡\(π​δk\)\|\]\.\\left\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\|\\leq\\frac\{V\}\{N\}\\left\[1\+2\\sum\_\{k=1\}^\{N\-1\}\|\\sin\(\\pi\\delta\_\{k\}\)\|\\right\]\.The three versions serve different purposes\. The sine form is closest to the finite record, theJJform is geometrically transparent, and the linear form gives simple region\-dependent constants\.

### 5\.3Sharpness, boundary obstruction, and limits of first\-order control

#### 5\.3\.1Uniform optimality ofN−1N^\{\-1\}

The upper bound in[Theorem˜5\.1](https://arxiv.org/html/2607.24868#Ch5.Thmtheorem1)is of orderN−1N^\{\-1\}on fixed compact sets\. The next result proves that this order is optimal when all admissible input sequences are allowed\.

###### Proposition 5\.5\(Zero\-input lower bound\)\.

Letuk=0u\_\{k\}=0for everykk, and apply the greedy recursion withv0=0v\_\{0\}=0\. Then

qk=\(−1\)k\+1q\_\{k\}=\(\-1\)^\{k\+1\}and

vk=\{−1,k​odd,0,k​even\.v\_\{k\}=\\begin\{cases\}\-1,&k\\text\{ odd\},\\\\ 0,&k\\text\{ even\}\.\\end\{cases\}IfNNis odd, then

\|ℰN​u​\(0,0\)−ℰN​q​\(0,0\)\|=1N\.\\left\\lvert\\mathcal\{E\}\_\{N\}u\(0,0\)\-\\mathcal\{E\}\_\{N\}q\(0,0\)\\right\\rvert=\\frac\{1\}\{N\}\.Consequently, no bound of ordero​\(N−1\)o\(N^\{\-1\}\)can hold uniformly over all input sequences satisfying\|uk\|≤1\\left\\lvert u\_\{k\}\\right\\rvert\\leq 1\.

###### Proof\.

The trajectory was computed in Chapter[4](https://arxiv.org/html/2607.24868#Ch4)\. At\(x,t\)=\(0,0\)\(x,t\)=\(0,0\), all weights are one\. Hence

ℰN​u​\(0,0\)−ℰN​q​\(0,0\)\\displaystyle\\mathcal\{E\}\_\{N\}u\(0,0\)\-\\mathcal\{E\}\_\{N\}q\(0,0\)=1N​∑k=1N\(uk−qk\)\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(u\_\{k\}\-q\_\{k\}\)=1N​∑k=1NΔ​vk=vN−v0N\.\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\\Delta v\_\{k\}=\\frac\{v\_\{N\}\-v\_\{0\}\}\{N\}\.For oddNN,vN=−1v\_\{N\}=\-1andv0=0v\_\{0\}=0\. The absolute value is1/N1/N\. ∎

#### 5\.3\.2The quantifiers in the sharpness statement

The proposition does not say that every input has error exactly of orderN−1N^\{\-1\}\. Some inputs may producevN=0v\_\{N\}=0or additional cancellation\. It says that there is no sequencerN=o​\(N−1\)r\_\{N\}=o\(N^\{\-1\}\)such that

sup\|uk\|≤1\|ℰN​u​\(0,0\)−ℰN​q​\(0,0\)\|≤rN\\sup\_\{\\left\\lvert u\_\{k\}\\right\\rvert\\leq 1\}\\left\\lvert\\mathcal\{E\}\_\{N\}u\(0,0\)\-\\mathcal\{E\}\_\{N\}q\(0,0\)\\right\\rvert\\leq r\_\{N\}for all largeNN\. The distinction between a worst\-case lower bound and a pointwise asymptotic statement is essential\.

#### 5\.3\.3The terminal state as the leading obstruction

At a constant weightwk=1w\_\{k\}=1, summation by parts gives

1N​∑k=1Nek=vN−v0N\.\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}=\\frac\{v\_\{N\}\-v\_\{0\}\}\{N\}\.There is no interior variation term\. The entire error is a boundary term\. Therefore no argument based only on smoother weights can improve the order at the point\(0,0\)\(0,0\)unless the terminal state is controlled more strongly\.

This observation foreshadows the higher\-order theory\. Repeated summation by parts produces several terminal traces\. Unless they vanish, the leading trace remains of orderN−1N^\{\-1\}after normalisation, regardless of the formal order of the noise\-shaping operator\.

#### 5\.3\.4Special inputs with improved behaviour

IfvN=0v\_\{N\}=0, the constant\-weight error vanishes exactly\. For a nonconstant smooth weight, the remaining first\-order term is still generallyO​\(N−1\)O\(N^\{\-1\}\)\. Additional improvement requires more structure, such ase=Δr​ve=\\Delta^\{r\}vwith compatible endpoints\.

A useful special case is a block for which bothv0v\_\{0\}andvNv\_\{N\}vanish\. Then

∑k=1Nek​wk=−∑k=1N−1vk​Δ\+​wk\.\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}=\-\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\\Delta\_\{\+\}w\_\{k\}\.This removes the terminal amplitude term from the weighted estimate\. It improves the constant but not the generic order for first\-order shaping because∑\|Δ\+​wk\|=O​\(1\)\\sum\\left\\lvert\\Delta\_\{\+\}w\_\{k\}\\right\\rvert=O\(1\)\.

#### 5\.3\.5Why global uniformity fails on large regions

For the parabolic phase, \([5\.3](https://arxiv.org/html/2607.24868#Ch5.E3)\) gives

\|ℰN​u−ℰN​q\|≲1\+\|x\|\+\|t\|N\.\\left\\lvert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rvert\\lesssim\\frac\{1\+\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert\}\{N\}\.If\|x\|\+\|t\|∼N\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert\\sim N, the bound is order one\. This does not prove that the actual error is order one\. It proves that the total\-variation method alone cannot guarantee decay\.

The exact sine formula \([5\.4](https://arxiv.org/html/2607.24868#Ch5.E4)\) leads to the same worst\-case conclusion because each chord length is at most two and there areN−1N\-1chords\. At critical scale the sampled phase may wind around the circle many times\.

#### 5\.3\.6No automatic globalL2L^\{2\}gain

There is a stronger obstruction\. Fixttand integrate inxxover a full orthogonality interval\[0,N\]\[0,N\]\. Since the frequencies arek/Nk/N,

∫0N\|ℰN​e​\(x,t\)\|2​dx\\displaystyle\\int\_\{0\}^\{N\}\\left\\lvert\\mathcal\{E\}\_\{N\}e\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}x=1N2​∑k,ℓ=1Nek​eℓ¯​e​\(t​k2−ℓ2N2\)​∫0Ne​\(x​k−ℓN\)​dx\\displaystyle=\\frac\{1\}\{N^\{2\}\}\\sum\_\{k,\\ell=1\}^\{N\}e\_\{k\}\\overline\{e\_\{\\ell\}\}e\\\!\\left\(t\\frac\{k^\{2\}\-\\ell^\{2\}\}\{N^\{2\}\}\\right\)\\int\_\{0\}^\{N\}e\\\!\\left\(x\\frac\{k\-\\ell\}\{N\}\\right\)\\,\\mathrm\{d\}x=1N​∑k=1N\|ek\|2\.\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\\left\\lvert e\_\{k\}\\right\\rvert^\{2\}\.The phase intthas modulus one and disappears on the diagonal\. A stable first\-order scheme does not force∑\|ek\|2\\sum\\left\\lvert e\_\{k\}\\right\\rvert^\{2\}to be small\. For the zero input,ek=−qke\_\{k\}=\-q\_\{k\}and the right\-hand side equals one\.

Thus the unnormalisedLx2L^\{2\}\_\{x\}energy on a full period need not decay\. Any global theorem must use a different norm, a smaller region, extra averaging, coefficient normalisation, or additional structure\.

#### 5\.3\.7The danger of an invalid high\-order argument

A common formal argument is

e=Δr​v⟹∑ek​wk≈∑vk​Δ\+r​wk,e=\\Delta^\{r\}v\\quad\\Longrightarrow\\quad\\sum e\_\{k\}w\_\{k\}\\approx\\sum v\_\{k\}\\Delta\_\{\+\}^\{r\}w\_\{k\},followed byΔ\+r​wk=O​\(N−r\)\\Delta\_\{\+\}^\{r\}w\_\{k\}=O\(N^\{\-r\}\)\. On an infinite or periodic domain this may be legitimate\. On a finite interval it omits boundary terms\. The omitted term with no forward difference is typically of size one before the1/N1/Nnormalisation\. It therefore produces onlyO​\(N−1\)O\(N^\{\-1\}\)decay\.

The correct conclusion is not that high\-order noise shaping is useless\. The correct conclusion is that endpoint compatibility is part of the theorem\. This principle is consistent with high\-order frame quantization and smooth frame\-path termination\[[4](https://arxiv.org/html/2607.24868#bib.bib18),[9](https://arxiv.org/html/2607.24868#bib.bib20),[8](https://arxiv.org/html/2607.24868#bib.bib21)\]\.

#### 5\.3\.8A hierarchy of possible improvements

There are several ways to move beyond the first\-order worst\-case result\.

1. \(a\)Impose terminal state conditions\.
2. \(b\)Use a higher\-order stable scheme together with endpoint compatibility\.
3. \(c\)Restrict the input class, for example by smoothness or moment conditions\.
4. \(d\)Average over random dither or a probabilistic input model\.
5. \(e\)Work in a subcritical growing region\.
6. \(f\)Preserve phase cancellation through exponential\-sum or restriction estimates\.

Each mechanism addresses a different obstruction\. They should not be conflated\.

## Chapter 6Higher\-Order Finite\-Record Derivations and Endpoint\-Compatible Proofs

Higher\-order noise shaping requires a finite\-interval calculation that retains every boundary trace\. Formal repetition of the first\-order argument is insufficient\. This chapter derives the exact identity fore=Δr​ve=\\Delta^\{r\}v, gives the low\-order cases in full, and proves the endpoint\-compatibleO​\(N−r\)O\(N^\{\-r\}\)result for sampled smooth phases\.

### 6\.1Exact higher\-order summation on a finite record

#### 6\.1\.1Why the finite interval must be treated explicitly

High\-order sigma\-delta quantization is usually expressed through the formal identity

ek=uk−qk=Δr​vk,e\_\{k\}=u\_\{k\}\-q\_\{k\}=\\Delta^\{r\}v\_\{k\},\(6\.1\)wherer≥1r\\geq 1is the order and the statevvis bounded\. On an infinite sequence, or on a periodic sequence with compatible traces, one may move the difference operator fromvvto a test sequence by repeated summation by parts\. A finite data record is different\. Every application of summation by parts creates a new left boundary term and a new right boundary term\. If these terms are omitted, the conclusion can be wrong by an entire power ofNN\.

The importance of high\-order stability in bandlimited reconstruction was established by Daubechies and DeVore, and sharper constructions were developed by Güntürk and by Deift, Güntürk, and Krahmer\[[25](https://arxiv.org/html/2607.24868#bib.bib7),[48](https://arxiv.org/html/2607.24868#bib.bib8),[49](https://arxiv.org/html/2607.24868#bib.bib9),[27](https://arxiv.org/html/2607.24868#bib.bib12)\]\. Higher\-order finite\-frame quantization likewise requires careful control of how a frame path begins and ends\[[4](https://arxiv.org/html/2607.24868#bib.bib18),[9](https://arxiv.org/html/2607.24868#bib.bib20),[8](https://arxiv.org/html/2607.24868#bib.bib21)\]\. The purpose of this chapter is narrower\. The exact algebraic identity needed for a finite coefficient record is derived independently of any particular quantizer construction\.

#### 6\.1\.2Difference conventions and boundary traces

For a sequencez=\(zk\)z=\(z\_\{k\}\)indexed on a sufficiently large integer interval, define the backward and forward differences by

Δ​zk=zk−zk−1,Δ\+​zk=zk\+1−zk\.\\Delta z\_\{k\}=z\_\{k\}\-z\_\{k\-1\},\\qquad\\Delta\_\{\+\}z\_\{k\}=z\_\{k\+1\}\-z\_\{k\}\.The iterates are denoted byΔj\\Delta^\{j\}andΔ\+j\\Delta\_\{\+\}^\{j\}, withΔ0=Δ\+0=I\\Delta^\{0\}=\\Delta\_\{\+\}^\{0\}=I\.

To interpretΔr​vk\\Delta^\{r\}v\_\{k\}for1≤k≤N1\\leq k\\leq N, the state must be defined at least on

k=1−r,…,0,1,…,N\.k=1\-r,\\ldots,0,1,\\ldots,N\.This is not an artificial extension\. A causalrrth\-order recursion always carriesrrpieces of initial data, even when they are set equal to zero by convention\.

###### Definition 6\.1\(Finite\-record boundary traces\)\.

For0≤j≤r−10\\leq j\\leq r\-1, define

Lj​\(v\)\\displaystyle L\_\{j\}\(v\)=\(Δr−1−j​v\)0,\\displaystyle=\(\\Delta^\{r\-1\-j\}v\)\_\{0\},Rj,N​\(v\)\\displaystyle R\_\{j,N\}\(v\)=\(Δr−1−j​v\)N−j\.\\displaystyle=\(\\Delta^\{r\-1\-j\}v\)\_\{N\-j\}\.The valuesLj​\(v\)L\_\{j\}\(v\)are the left traces andRj,N​\(v\)R\_\{j,N\}\(v\)are the staggered right traces associated withrrfold summation by parts\.

The staggering on the right is forced by the shortening of the summation interval after each forward difference is applied to the weight\.

#### 6\.1\.3The exact finite\-interval identity

###### Theorem 6\.2\(Repeated summation by parts\)\.

Letr≥1r\\geq 1andN≥r\+1N\\geq r\+1\. Letvkv\_\{k\}be defined for1−r≤k≤N1\-r\\leq k\\leq N, and letwkw\_\{k\}be defined for1≤k≤N1\\leq k\\leq N\. Then

∑k=1N\(Δr​v\)k​wk\\displaystyle\\sum\_\{k=1\}^\{N\}\(\\Delta^\{r\}v\)\_\{k\}w\_\{k\}=∑j=0r−1\(−1\)j​\[\(Δr−1−j​v\)N−j​\(Δ\+j​w\)N−j−\(Δr−1−j​v\)0​\(Δ\+j​w\)1\]\\displaystyle=\\sum\_\{j=0\}^\{r\-1\}\(\-1\)^\{j\}\\left\[\(\\Delta^\{r\-1\-j\}v\)\_\{N\-j\}\(\\Delta\_\{\+\}^\{j\}w\)\_\{N\-j\}\-\(\\Delta^\{r\-1\-j\}v\)\_\{0\}\(\\Delta\_\{\+\}^\{j\}w\)\_\{1\}\\right\]\+\(−1\)r​∑k=1N−rvk​\(Δ\+r​w\)k\.\\displaystyle\\quad\+\(\-1\)^\{r\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\.\(6\.2\)Every boundary term in this formula is necessary in general\.

###### Proof\.

The proof proceeds by induction onrr\. The caser=1r=1is the first\-order identity

∑k=1N\(Δ​v\)k​wk=vN​wN−v0​w1−∑k=1N−1vk​\(Δ\+​w\)k\.\\sum\_\{k=1\}^\{N\}\(\\Delta v\)\_\{k\}w\_\{k\}=v\_\{N\}w\_\{N\}\-v\_\{0\}w\_\{1\}\-\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(\\Delta\_\{\+\}w\)\_\{k\}\.This agrees with \([6\.2](https://arxiv.org/html/2607.24868#Ch6.Ex5)\)\.

Assume the identity holds for orderrr\. Apply the first\-order formula toΔr​v\\Delta^\{r\}v:

∑k=1N\(Δr\+1​v\)k​wk\\displaystyle\\sum\_\{k=1\}^\{N\}\(\\Delta^\{r\+1\}v\)\_\{k\}w\_\{k\}=\(Δr​v\)N​wN−\(Δr​v\)0​w1−∑k=1N−1\(Δr​v\)k​\(Δ\+​w\)k\.\\displaystyle=\(\\Delta^\{r\}v\)\_\{N\}w\_\{N\}\-\(\\Delta^\{r\}v\)\_\{0\}w\_\{1\}\-\\sum\_\{k=1\}^\{N\-1\}\(\\Delta^\{r\}v\)\_\{k\}\(\\Delta\_\{\+\}w\)\_\{k\}\.Apply the induction hypothesis to the last sum\. Its length isN−1N\-1, its weight isΔ\+​w\\Delta\_\{\+\}w, and its difference order isrr\. Hence

∑k=1N−1\(Δr​v\)k​\(Δ\+​w\)k\\displaystyle\\sum\_\{k=1\}^\{N\-1\}\(\\Delta^\{r\}v\)\_\{k\}\(\\Delta\_\{\+\}w\)\_\{k\}=∑j=0r−1\(−1\)j\[\(Δr−1−jv\)N−1−j\(Δ\+j\+1w\)N−1−j\\displaystyle=\\sum\_\{j=0\}^\{r\-1\}\(\-1\)^\{j\}\\bigl\[\(\\Delta^\{r\-1\-j\}v\)\_\{N\-1\-j\}\(\\Delta\_\{\+\}^\{j\+1\}w\)\_\{N\-1\-j\}−\(Δr−1−jv\)0\(Δ\+j\+1w\)1\]\\displaystyle\\hskip 80\.00012pt\-\(\\Delta^\{r\-1\-j\}v\)\_\{0\}\(\\Delta\_\{\+\}^\{j\+1\}w\)\_\{1\}\\bigr\]\+\(−1\)r​∑k=1N−r−1vk​\(Δ\+r\+1​w\)k\.\\displaystyle\\quad\+\(\-1\)^\{r\}\\sum\_\{k=1\}^\{N\-r\-1\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\+1\}w\)\_\{k\}\.Insert this expression into the first\-order formula\. The two terms containingΔr​v\\Delta^\{r\}vare exactly thej=0j=0boundary contribution for orderr\+1r\+1\. Relabelj\+1j\+1as the new index in the remaining boundary terms\. The interior sign becomes\(−1\)r\+1\(\-1\)^\{r\+1\}\. This yields \([6\.2](https://arxiv.org/html/2607.24868#Ch6.Ex5)\) withrrreplaced byr\+1r\+1\. ∎

#### 6\.1\.4Low\-order cases written in full

The general formula is easier to use after the first few cases have been displayed\.

Forr=2r=2,

∑k=1N\(Δ2​v\)k​wk\\displaystyle\\sum\_\{k=1\}^\{N\}\(\\Delta^\{2\}v\)\_\{k\}w\_\{k\}=\(Δ​v\)N​wN−\(Δ​v\)0​w1−vN−1​\(Δ\+​w\)N−1\+v0​\(Δ\+​w\)1\\displaystyle=\(\\Delta v\)\_\{N\}w\_\{N\}\-\(\\Delta v\)\_\{0\}w\_\{1\}\-v\_\{N\-1\}\(\\Delta\_\{\+\}w\)\_\{N\-1\}\+v\_\{0\}\(\\Delta\_\{\+\}w\)\_\{1\}\+∑k=1N−2vk​\(Δ\+2​w\)k\.\\displaystyle\\quad\+\\sum\_\{k=1\}^\{N\-2\}v\_\{k\}\(\\Delta\_\{\+\}^\{2\}w\)\_\{k\}\.\(6\.3\)Forr=3r=3,

∑k=1N\(Δ3​v\)k​wk\\displaystyle\\sum\_\{k=1\}^\{N\}\(\\Delta^\{3\}v\)\_\{k\}w\_\{k\}=\(Δ2​v\)N​wN−\(Δ2​v\)0​w1\\displaystyle=\(\\Delta^\{2\}v\)\_\{N\}w\_\{N\}\-\(\\Delta^\{2\}v\)\_\{0\}w\_\{1\}−\(Δ​v\)N−1​\(Δ\+​w\)N−1\+\(Δ​v\)0​\(Δ\+​w\)1\\displaystyle\\quad\-\(\\Delta v\)\_\{N\-1\}\(\\Delta\_\{\+\}w\)\_\{N\-1\}\+\(\\Delta v\)\_\{0\}\(\\Delta\_\{\+\}w\)\_\{1\}\+vN−2​\(Δ\+2​w\)N−2−v0​\(Δ\+2​w\)1\\displaystyle\\quad\+v\_\{N\-2\}\(\\Delta\_\{\+\}^\{2\}w\)\_\{N\-2\}\-v\_\{0\}\(\\Delta\_\{\+\}^\{2\}w\)\_\{1\}−∑k=1N−3vk​\(Δ\+3​w\)k\.\\displaystyle\\quad\-\\sum\_\{k=1\}^\{N\-3\}v\_\{k\}\(\\Delta\_\{\+\}^\{3\}w\)\_\{k\}\.\(6\.4\)
These identities show why the informal statement “moveΔr\\Delta^\{r\}onto the weight” is incomplete\. Even whenvvis bounded, the term\(Δr−1​v\)N​wN\(\\Delta^\{r\-1\}v\)\_\{N\}w\_\{N\}can be of order one\. After the normalising factor1/N1/Nis restored, it yields onlyO​\(N−1\)O\(N^\{\-1\}\)decay\.

#### 6\.1\.5A full deterministic estimate

###### Corollary 6\.3\(Higher\-order weighted estimate with traces\)\.

Under the hypotheses of[Theorem˜6\.2](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem2),

\|1N​∑k=1N\(Δr​v\)k​wk\|\\displaystyle\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(\\Delta^\{r\}v\)\_\{k\}w\_\{k\}\\right\\rvert≤1N​∑j=0r−1\(\|Rj,N​\(v\)\|​\|\(Δ\+j​w\)N−j\|\+\|Lj​\(v\)\|​\|\(Δ\+j​w\)1\|\)\\displaystyle\\leq\\frac\{1\}\{N\}\\sum\_\{j=0\}^\{r\-1\}\\left\(\\left\\lvert R\_\{j,N\}\(v\)\\right\\rvert\\left\\lvert\(\\Delta\_\{\+\}^\{j\}w\)\_\{N\-j\}\\right\\rvert\+\\left\\lvert L\_\{j\}\(v\)\\right\\rvert\\left\\lvert\(\\Delta\_\{\+\}^\{j\}w\)\_\{1\}\\right\\rvert\\right\)\+‖v‖ℓ∞N​∑k=1N−r\|\(Δ\+r​w\)k\|\.\\displaystyle\\quad\+\\frac\{\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\}\{N\}\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\\right\\rvert\.\(6\.5\)

###### Proof\.

Take absolute values in \([6\.2](https://arxiv.org/html/2607.24868#Ch6.Ex5)\) and apply the triangle inequality only after all terms have been identified\. ∎

This corollary separates three effects\. The initialisation error is measured by the left traces\. The termination error is measured by the staggered right traces\. The interior smoothness of the test sequence is measured by therrth discrete variation\. This three\-part decomposition will be used throughout the rest of the report\.

#### 6\.1\.6Boundary\-trace correction as a reconstruction principle

Endpoint compatibility is sufficient for high\-order decay, but it is not the only analytical route\. The exact boundary functional can instead be added to the quantized extension\. This produces a corrected reconstruction whose remaining error is the pure interior term\. The correction does not alter the one\-bit coefficient record\. It requires the relevant initial and terminal traces, or sufficiently accurate estimates of those traces\.

For a statevvand a weightww, define

ℬN,r\(v,w\):=∑j=0r−1\(−1\)j\[Rj,N\(v\)\(Δ\+jw\)N−j−Lj\(v\)\(Δ\+jw\)1\]\.\\displaystyle\\mathcal\{B\}\_\{N,r\}\(v,w\):=\\sum\_\{j=0\}^\{r\-1\}\(\-1\)^\{j\}\\left\[R\_\{j,N\}\(v\)\(\\Delta\_\{\+\}^\{j\}w\)\_\{N\-j\}\-L\_\{j\}\(v\)\(\\Delta\_\{\+\}^\{j\}w\)\_\{1\}\\right\]\.\(6\.6\)The functional contains exactly the boundary terms in[Theorem˜6\.2](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem2)\.

###### Theorem 6\.4\(Boundary\-corrected finite\-record reconstruction\)\.

Letr≥1r\\geq 1, letek=uk−qk=\(Δr​v\)ke\_\{k\}=u\_\{k\}\-q\_\{k\}=\(\\Delta^\{r\}v\)\_\{k\}, and letw1,…,wN∈ℂw\_\{1\},\\ldots,w\_\{N\}\\in\\mathbb\{C\}\. Define

ℛN,r​\(q;v,w\):=1N​∑k=1Nqk​wk\+1N​ℬN,r​\(v,w\)\.\\mathcal\{R\}\_\{N,r\}\(q;v,w\):=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}q\_\{k\}w\_\{k\}\+\\frac\{1\}\{N\}\\mathcal\{B\}\_\{N,r\}\(v,w\)\.\(6\.7\)Then the following identity is exact:

1N​∑k=1Nuk​wk−ℛN,r​\(q;v,w\)=\(−1\)rN​∑k=1N−rvk​\(Δ\+r​w\)k\.\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}u\_\{k\}w\_\{k\}\-\\mathcal\{R\}\_\{N,r\}\(q;v,w\)=\\frac\{\(\-1\)^\{r\}\}\{N\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\.\(6\.8\)Consequently, if‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V, then

\|1N​∑k=1Nuk​wk−ℛN,r​\(q;v,w\)\|≤VN​∑k=1N−r\|\(Δ\+r​w\)k\|\.\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}u\_\{k\}w\_\{k\}\-\\mathcal\{R\}\_\{N,r\}\(q;v,w\)\\right\|\\leq\\frac\{V\}\{N\}\\sum\_\{k=1\}^\{N\-r\}\|\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\|\.\(6\.9\)No endpoint\-compatibility assumption is required\.

###### Proof\.

Apply[Theorem˜6\.2](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem2)toe=Δr​ve=\\Delta^\{r\}v\. Its boundary sum is preciselyℬN,r​\(v,w\)\\mathcal\{B\}\_\{N,r\}\(v,w\)\. Since

1N​∑k=1Nuk​wk=1N​∑k=1Nqk​wk\+1N​∑k=1Nek​wk,\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}u\_\{k\}w\_\{k\}=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}q\_\{k\}w\_\{k\}\+\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\},subtraction of \([6\.7](https://arxiv.org/html/2607.24868#Ch6.E7)\) leaves only the interior term in \([6\.2](https://arxiv.org/html/2607.24868#Ch6.Ex5)\)\. Taking absolute values yields \([6\.9](https://arxiv.org/html/2607.24868#Ch6.E9)\)\. ∎

###### Corollary 6\.6\(Corrected high\-order rate for sampled smooth weights\)\.

Assume the hypotheses of[Theorem˜6\.4](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem4), letwk=W​\(k/N\)w\_\{k\}=W\(k/N\), and supposeW∈Cr​\(\[0,1\]\)W\\in C^\{r\}\(\[0,1\]\)\. Then

\|1N​∑k=1Nuk​W​\(k/N\)−ℛN,r​\(q;v,w\)\|≤V​N−r​‖W\(r\)‖L∞​\(0,1\)\.\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}u\_\{k\}W\(k/N\)\-\\mathcal\{R\}\_\{N,r\}\(q;v,w\)\\right\|\\leq VN^\{\-r\}\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{L^\{\\infty\}\(0,1\)\}\.\(6\.10\)The same conclusion holds withr​‖W\(r\)‖L1r\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{L^\{1\}\}in place of theL∞L^\{\\infty\}norm whenW\(r−1\)W^\{\(r\-1\)\}is absolutely continuous andW\(r\)∈L1W^\{\(r\)\}\\in L^\{1\}\.

###### Proof\.

Combine \([6\.9](https://arxiv.org/html/2607.24868#Ch6.E9)\) with the finite\-difference estimates in[Lemmas˜6\.11](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem11)and[6\.12](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem12)\. ∎

The correction theorem separates quantizer dynamics from finite\-record reconstruction\. Endpoint\-compatible dynamics eliminate the traces inside the quantizer\. Boundary\-corrected reconstruction retains the original dynamics and cancels the traces after acquisition\. With zero initialisation, every left trace is already known to be zero, so only therrstaggered terminal traces are needed\. This side information has dimension independent ofNN, although its numerical precision must be specified in any implementation claim\.

###### Proposition 6\.7\(Perturbed trace metadata\)\.

LetL^j\\widehat\{L\}\_\{j\}andR^j\\widehat\{R\}\_\{j\}be estimates ofLj​\(v\)L\_\{j\}\(v\)andRj,N​\(v\)R\_\{j,N\}\(v\)satisfying

\|L^j−Lj​\(v\)\|≤δj,NL,\|R^j−Rj,N​\(v\)\|≤δj,NR\.\|\\widehat\{L\}\_\{j\}\-L\_\{j\}\(v\)\|\\leq\\delta^\{L\}\_\{j,N\},\\qquad\|\\widehat\{R\}\_\{j\}\-R\_\{j,N\}\(v\)\|\\leq\\delta^\{R\}\_\{j,N\}\.Letℛ^N,r\\widehat\{\\mathcal\{R\}\}\_\{N,r\}be defined by \([6\.7](https://arxiv.org/html/2607.24868#Ch6.E7)\) with the estimated traces\. Then

\|1N​∑k=1Nuk​wk−ℛ^N,r\|\\displaystyle\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}u\_\{k\}w\_\{k\}\-\\widehat\{\\mathcal\{R\}\}\_\{N,r\}\\right\|≤VN​∑k=1N−r\|\(Δ\+r​w\)k\|\\displaystyle\\leq\\frac\{V\}\{N\}\\sum\_\{k=1\}^\{N\-r\}\|\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\|\+1N​∑j=0r−1\[δj,NR​\|\(Δ\+j​w\)N−j\|\+δj,NL​\|\(Δ\+j​w\)1\|\]\.\\displaystyle\\quad\+\\frac\{1\}\{N\}\\sum\_\{j=0\}^\{r\-1\}\\left\[\\delta^\{R\}\_\{j,N\}\|\(\\Delta\_\{\+\}^\{j\}w\)\_\{N\-j\}\|\+\\delta^\{L\}\_\{j,N\}\|\(\\Delta\_\{\+\}^\{j\}w\)\_\{1\}\|\\right\]\.\(6\.11\)

###### Proof\.

Subtract the exact and estimated boundary functionals, apply[Theorem˜6\.4](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem4), and use the triangle inequality\. ∎

For a sampledCrC^\{r\}weight, the endpoint factor\|\(Δ\+j​w\)\|\|\(\\Delta\_\{\+\}^\{j\}w\)\|isO​\(N−j\)O\(N^\{\-j\}\)\. A trace error of orderN−\(r−1−j\)N^\{\-\(r\-1\-j\)\}therefore contributes at mostO​\(N−r\)O\(N^\{\-r\}\)after normalisation\. The hierarchy matches the approximate\-compatibility conditions derived later in this chapter\.

#### 6\.1\.7Zero padding and terminal traces

A common convention sets

v1−r=⋯=v−1=v0=0\.v\_\{1\-r\}=\\cdots=v\_\{\-1\}=v\_\{0\}=0\.Then every left trace vanishes\. Indeed, eachLj​\(v\)L\_\{j\}\(v\)is a linear combination of these zero values\. This removes only half of the boundary problem\. Unless the terminal state is also prepared, the right traces remain\.

Simple zero padding after the last sample does not retroactively make the right traces vanish\. If one declaresvN\+1=vN\+2=⋯=0v\_\{N\+1\}=v\_\{N\+2\}=\\cdots=0, the traces in \([6\.2](https://arxiv.org/html/2607.24868#Ch6.Ex5)\) still involvevN,vN−1,…v\_\{N\},v\_\{N\-1\},\\ldots\. A valid termination procedure must alter the final quantization steps, append a controlled tail, or design the frame path so that the relevant state components reach zero\. Smooth termination methods in finite\-frame quantization were developed precisely for this reason\[[9](https://arxiv.org/html/2607.24868#bib.bib20)\]\.

#### 6\.1\.8Matrix formulation

LetDND\_\{N\}be theN×NN\\times Nlower bidiagonal matrix with ones on the diagonal and minus ones on the first subdiagonal\. With zero initial conditions, the finite difference relation can be written as

The transpose satisfies

⟨DNr​v,w⟩=⟨v,\(DN𝖳\)r​w⟩\.\\langle D\_\{N\}^\{r\}v,w\\rangle=\\langle v,\(D\_\{N\}^\{\\mathsf\{T\}\}\)^\{r\}w\\rangle\.This matrix identity is exact, but the vectorvvin it contains a particular choice of boundary convention\. The entries of\(DN𝖳\)r​w\(D\_\{N\}^\{\\mathsf\{T\}\}\)^\{r\}wencode the interior forward differences together with the terminal rows\. The scalar formula \([6\.2](https://arxiv.org/html/2607.24868#Ch6.Ex5)\) is therefore preferable when the boundary traces must remain visible\.

Singular values of powers of the finite difference matrix play a central role in Sobolev\-dual reconstruction and quantized compressed sensing\[[8](https://arxiv.org/html/2607.24868#bib.bib21),[46](https://arxiv.org/html/2607.24868#bib.bib22),[62](https://arxiv.org/html/2607.24868#bib.bib23),[68](https://arxiv.org/html/2607.24868#bib.bib24)\]\. In the present extension problem, however, the principal issue is not inversion ofDNrD\_\{N\}^\{r\}\. It is the action ofDNr​vD\_\{N\}^\{r\}von a highly structured oscillatory vector\.

#### 6\.1\.9Boundary verification checklist

AnO​\(N−r\)O\(N^\{\-r\}\)derivation frome=Δr​ve=\\Delta^\{r\}vis verified by recording the following boundary data\.

1. \(i\)On what index set isvvdefined?
2. \(ii\)Which left traces are assumed to vanish?
3. \(iii\)Which right traces are assumed to vanish?
4. \(iv\)Is therrth forward difference of the weight bounded inℓ1\\ell^\{1\}orℓ∞\\ell^\{\\infty\}?
5. \(v\)Are the constants uniform in the observation parameters?

Failure to specify any of these points may conceal anO​\(N−1\)O\(N^\{\-1\}\)boundary contribution\. The next chapter gives sufficient endpoint conditions and derives the corresponding high\-order extension theorem\.

### 6\.2Endpoint\-compatible high\-order extension theory

#### 6\.2\.1Strong and weak compatibility conditions

The repeated summation formula identifies the exact traces that prevent high\-order decay\. There are two convenient ways to remove them\.

###### Definition 6\.8\(Trace compatibility\)\.

A state sequencevvis trace\-compatible of orderrron\{1,…,N\}\\\{1,\\ldots,N\\\}if

Lj​\(v\)=0,Rj,N​\(v\)=0,0≤j≤r−1\.L\_\{j\}\(v\)=0,\\qquad R\_\{j,N\}\(v\)=0,\\qquad 0\\leq j\\leq r\-1\.

###### Definition 6\.9\(Strong endpoint compatibility\)\.

A state sequencevvis strongly endpoint\-compatible of orderrrif

v1−r=⋯=v0=0andvN−r\+1=⋯=vN=0\.v\_\{1\-r\}=\\cdots=v\_\{0\}=0\\quad\\text\{and\}\\quad v\_\{N\-r\+1\}=\\cdots=v\_\{N\}=0\.

Strong compatibility implies trace compatibility because every difference appearing in a trace is a linear combination of values in the corresponding zero block\. The converse is not required\. Trace compatibility is the minimal algebraic condition for the boundary terms in[Theorem˜6\.2](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem2)to vanish\.

###### Proposition 6\.10\(Pure interior formula\)\.

Ifvvis trace\-compatible of orderrr, then

∑k=1N\(Δr​v\)k​wk=\(−1\)r​∑k=1N−rvk​\(Δ\+r​w\)k\.\\sum\_\{k=1\}^\{N\}\(\\Delta^\{r\}v\)\_\{k\}w\_\{k\}=\(\-1\)^\{r\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\.\(6\.12\)Consequently,

\|1N​∑k=1N\(Δr​v\)k​wk\|≤‖v‖ℓ∞N​∑k=1N−r\|\(Δ\+r​w\)k\|\.\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(\\Delta^\{r\}v\)\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\}\{N\}\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\\right\\rvert\.\(6\.13\)

###### Proof\.

All boundary contributions in \([6\.2](https://arxiv.org/html/2607.24868#Ch6.Ex5)\) vanish by hypothesis\. ∎

#### 6\.2\.2Forward differences of sampled smooth functions

Leth=N−1h=N^\{\-1\}and letW:\[0,1\]→ℂW:\[0,1\]\\to\\mathbb\{C\}\. Forx,x\+r​h∈\[0,1\]x,x\+rh\\in\[0,1\], therrth forward difference has the integral representation

Δhr​W​\(x\)=∫0h⋯​∫0hW\(r\)​\(x\+s1\+⋯\+sr\)​ds1​⋯​dsr,\\Delta\_\{h\}^\{r\}W\(x\)=\\int\_\{0\}^\{h\}\\cdots\\int\_\{0\}^\{h\}W^\{\(r\)\}\(x\+s\_\{1\}\+\\cdots\+s\_\{r\}\)\\,\\mathrm\{d\}s\_\{1\}\\cdots\\,\\mathrm\{d\}s\_\{r\},\(6\.14\)providedW\(r−1\)W^\{\(r\-1\)\}is absolutely continuous\. This formula follows by applying the fundamental theorem of calculusrrtimes\.

###### Lemma 6\.11\(Uniform derivative control\)\.

IfW∈Cr​\(\[0,1\]\)W\\in C^\{r\}\(\[0,1\]\)andwk=W​\(k/N\)w\_\{k\}=W\(k/N\), then

\|\(Δ\+r​w\)k\|≤N−r​‖W\(r\)‖L∞​\(0,1\),1≤k≤N−r\.\\left\\lvert\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\\right\\rvert\\leq N^\{\-r\}\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{L^\{\\infty\}\(0,1\)\},\\qquad 1\\leq k\\leq N\-r\.Hence

∑k=1N−r\|\(Δ\+r​w\)k\|≤N1−r​‖W\(r\)‖L∞​\(0,1\)\.\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\\right\\rvert\\leq N^\{1\-r\}\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{L^\{\\infty\}\(0,1\)\}\.\(6\.15\)

###### Proof\.

Apply \([6\.14](https://arxiv.org/html/2607.24868#Ch6.E14)\)\. The integration cube has volumehr=N−rh^\{r\}=N^\{\-r\}\. There are at mostNNterms in the sum\. ∎

A version with anL1L^\{1\}derivative is also useful\.

###### Lemma 6\.12\(L1L^\{1\}derivative control\)\.

AssumeW\(r−1\)W^\{\(r\-1\)\}is absolutely continuous\. Then

∑k=1N−r\|\(Δ\+r​w\)k\|≤r​N1−r​‖W\(r\)‖L1​\(0,1\)\.\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\\right\\rvert\\leq rN^\{1\-r\}\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{L^\{1\}\(0,1\)\}\.\(6\.16\)

###### Proof\.

Use \([6\.14](https://arxiv.org/html/2607.24868#Ch6.E14)\), sum overkk, and apply Tonelli’s theorem\. For each fixed\(s1,…,sr\)\(s\_\{1\},\\ldots,s\_\{r\}\), the intervals

\[k/N\+s1\+⋯\+sr,\(k\+1\)/N\+s1\+⋯\+sr\]\[k/N\+s\_\{1\}\+\\cdots\+s\_\{r\},\\,\(k\+1\)/N\+s\_\{1\}\+\\cdots\+s\_\{r\}\]have overlap bounded byrrafter therrintegration variables are collapsed to their sum\. A simpler, slightly larger bound is obtained by extendingW\(r\)W^\{\(r\)\}by zero outside\[0,1\]\[0,1\]and observing that the convolution kernel generated by therrfold box hasL1L^\{1\}normN−rN^\{\-r\}and support lengthr/Nr/N\. Summation over a grid of spacing1/N1/Ngives the factorr​NrN\. ∎

#### 6\.2\.3Fractional smoothness and intermediate rates

Integer differentiability is not necessary for a quantitative rate\. Finite differences also detect fractional smoothness\. Standard background on moduli of smoothness and Hölder–Zygmund classes is available in\[[29](https://arxiv.org/html/2607.24868#bib.bib83),[30](https://arxiv.org/html/2607.24868#bib.bib84),[76](https://arxiv.org/html/2607.24868#bib.bib85)\]\. The following estimate is included with a direct proof because the exact power ofNNis central to the finite\-record result\.

###### Definition 6\.13\(Hölder derivative seminorm\)\.

For0<α≤10<\\alpha\\leq 1and a functionF:\[0,1\]→ℂF:\[0,1\]\\to\\mathbb\{C\}, define

\[F\]C0,α:=supx≠y\|F​\(x\)−F​\(y\)\|\|x−y\|α\.\[F\]\_\{C^\{0,\\alpha\}\}:=\\sup\_\{x\\neq y\}\\frac\{\|F\(x\)\-F\(y\)\|\}\{\|x\-y\|^\{\\alpha\}\}\.A functionWWbelongs toCr−1,α​\(\[0,1\]\)C^\{r\-1,\\alpha\}\(\[0,1\]\)whenW\(r−1\)W^\{\(r\-1\)\}exists, is continuous, and has finiteC0,αC^\{0,\\alpha\}seminorm\.

###### Lemma 6\.14\(Fractional finite\-difference estimate\)\.

Letr≥1r\\geq 1,0<α≤10<\\alpha\\leq 1, andW∈Cr−1,α​\(\[0,1\]\)W\\in C^\{r\-1,\\alpha\}\(\[0,1\]\)\. Forh\>0h\>0andx,x\+r​h∈\[0,1\]x,x\+rh\\in\[0,1\],

\|Δhr​W​\(x\)\|≤hr−1\+α​\[W\(r−1\)\]C0,α\.\|\\Delta\_\{h\}^\{r\}W\(x\)\|\\leq h^\{r\-1\+\\alpha\}\[W^\{\(r\-1\)\}\]\_\{C^\{0,\\alpha\}\}\.\(6\.17\)

###### Proof\.

Forr=1r=1, the statement is the definition of the Hölder seminorm\. Letr≥2r\\geq 2\. Repeated application of the fundamental theorem of calculus gives

Δhr−1​W​\(x\)=∫\[0,h\]r−1W\(r−1\)​\(x\+s1\+⋯\+sr−1\)​ds1​⋯​dsr−1\.\\Delta\_\{h\}^\{r\-1\}W\(x\)=\\int\_\{\[0,h\]^\{r\-1\}\}W^\{\(r\-1\)\}\(x\+s\_\{1\}\+\\cdots\+s\_\{r\-1\}\)\\,\\mathrm\{d\}s\_\{1\}\\cdots\\mathrm\{d\}s\_\{r\-1\}\.Apply one additional forward difference:

Δhr​W​\(x\)\\displaystyle\\Delta\_\{h\}^\{r\}W\(x\)=∫\[0,h\]r−1\[W\(r−1\)\(x\+h\+s1\+⋯\+sr−1\)\\displaystyle=\\int\_\{\[0,h\]^\{r\-1\}\}\\bigl\[W^\{\(r\-1\)\}\(x\+h\+s\_\{1\}\+\\cdots\+s\_\{r\-1\}\)−W\(r−1\)\(x\+s1\+⋯\+sr−1\)\]ds1⋯dsr−1\.\\displaystyle\\hskip 100\.00015pt\-W^\{\(r\-1\)\}\(x\+s\_\{1\}\+\\cdots\+s\_\{r\-1\}\)\\bigr\]\\,\\mathrm\{d\}s\_\{1\}\\cdots\\mathrm\{d\}s\_\{r\-1\}\.The bracket is bounded by\[W\(r−1\)\]C0,α​hα\[W^\{\(r\-1\)\}\]\_\{C^\{0,\\alpha\}\}h^\{\\alpha\}\. The integration cube has volumehr−1h^\{r\-1\}, which proves \([6\.17](https://arxiv.org/html/2607.24868#Ch6.E17)\)\. ∎

###### Theorem 6\.15\(Fractional high\-order extension rate\)\.

Lete=Δr​ve=\\Delta^\{r\}vwith‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V, and letwk=W​\(k/N\)w\_\{k\}=W\(k/N\)withW∈Cr−1,α​\(\[0,1\]\)W\\in C^\{r\-1,\\alpha\}\(\[0,1\]\),0<α≤10<\\alpha\\leq 1\. If the state is trace\-compatible of orderrr, then

\|1N​∑k=1Nek​wk\|≤V​\[W\(r−1\)\]C0,α​N−\(r−1\+α\)\.\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\|\\leq V\[W^\{\(r\-1\)\}\]\_\{C^\{0,\\alpha\}\}N^\{\-\(r\-1\+\\alpha\)\}\.\(6\.18\)The same bound holds for the boundary\-corrected reconstruction in[Theorem˜6\.4](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem4)without endpoint compatibility\.

###### Proof\.

Trace compatibility or exact boundary correction reduces the error to the interior sum

\(−1\)rN​∑k=1N−rvk​\(Δ\+r​w\)k\.\\frac\{\(\-1\)^\{r\}\}\{N\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\.Apply[Lemma˜6\.14](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem14)withh=N−1h=N^\{\-1\}\. Each term is bounded by

N−\(r−1\+α\)​\[W\(r−1\)\]C0,α\.N^\{\-\(r\-1\+\\alpha\)\}\[W^\{\(r\-1\)\}\]\_\{C^\{0,\\alpha\}\}\.There are at mostNNterms\. Multiplication byV/NV/Ngives \([6\.18](https://arxiv.org/html/2607.24868#Ch6.E18)\)\. ∎

The theorem gives a continuous scale between two consecutive integer orders\. Forr=1r=1, a Hölder weight of exponentα\\alphagivesN−αN^\{\-\\alpha\}\. Forα=1\\alpha=1, the order\-rrrate becomesN−rN^\{\-r\}\. The result also identifies the loss caused by a rough testing functional: a bounded high\-order state cannot create more decay than the sampled weight can absorb through its finite differences\.

#### 6\.2\.4The high\-order compact\-set theorem

###### Theorem 6\.16\(Endpoint\-compatibleO​\(N−r\)O\(N^\{\-r\}\)estimate\)\.

Letr≥1r\\geq 1\. Assume

ek=uk−qk=\(Δr​v\)k,‖v‖ℓ∞≤V,e\_\{k\}=u\_\{k\}\-q\_\{k\}=\(\\Delta^\{r\}v\)\_\{k\},\\qquad\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V,and assume thatvvis trace\-compatible of orderrr\. LetW∈Cr​\(\[0,1\]\)W\\in C^\{r\}\(\[0,1\]\)and setwk=W​\(k/N\)w\_\{k\}=W\(k/N\)\. Then

\|1N​∑k=1Nek​wk\|≤V​N−r​‖W\(r\)‖L∞​\(0,1\)\.\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rvert\\leq VN^\{\-r\}\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{L^\{\\infty\}\(0,1\)\}\.\(6\.19\)IfW\(r\)∈L1W^\{\(r\)\}\\in L^\{1\}, one also has

\|1N​∑k=1Nek​wk\|≤r​V​N−r​‖W\(r\)‖L1​\(0,1\)\.\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rvert\\leq rVN^\{\-r\}\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{L^\{1\}\(0,1\)\}\.\(6\.20\)

###### Proof\.

Combine \([6\.13](https://arxiv.org/html/2607.24868#Ch6.E13)\) with[Lemmas˜6\.11](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem11)and[6\.12](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem12)\. ∎

The theorem gives the high\-order rate that the formal calculation suggests, but only after all finite\-record traces have been removed\. This is the finite\-interval counterpart of the smooth termination principle used in frame quantization\[[9](https://arxiv.org/html/2607.24868#bib.bib20),[8](https://arxiv.org/html/2607.24868#bib.bib21)\]\.

#### 6\.2\.5Oscillatory weights and Bell\-polynomial bounds

Let

W​\(ξ\)=exp⁡\(2​π​i​ϕ​\(ξ\)\)\.W\(\\xi\)=\\exp\(2\\pi\\mathrm\{i\}\\phi\(\\xi\)\)\.Repeated differentiation gives

W\(r\)​\(ξ\)=W​\(ξ\)​Br​\(2​π​i​ϕ′​\(ξ\),2​π​i​ϕ′′​\(ξ\),…,2​π​i​ϕ\(r\)​\(ξ\)\),W^\{\(r\)\}\(\\xi\)=W\(\\xi\)\\,B\_\{r\}\\bigl\(2\\pi\\mathrm\{i\}\\phi^\{\\prime\}\(\\xi\),2\\pi\\mathrm\{i\}\\phi^\{\\prime\\prime\}\(\\xi\),\\ldots,2\\pi\\mathrm\{i\}\\phi^\{\(r\)\}\(\\xi\)\\bigr\),\(6\.21\)whereBrB\_\{r\}is the complete exponential Bell polynomial\. This is a standard form of the Faà di Bruno formula\. For completeness, Appendix B derives the recursion

Br\+1​\(z1,…,zr\+1\)=∑m=0r\(rm\)​Br−m​\(z1,…,zr−m\)​zm\+1,B\_\{r\+1\}\(z\_\{1\},\\ldots,z\_\{r\+1\}\)=\\sum\_\{m=0\}^\{r\}\\binom\{r\}\{m\}B\_\{r\-m\}\(z\_\{1\},\\ldots,z\_\{r\-m\}\)z\_\{m\+1\},starting fromB0=1B\_\{0\}=1\.

Since the coefficients ofBrB\_\{r\}are nonnegative after absolute values are taken,

\|W\(r\)​\(ξ\)\|≤Br​\(2​π​\|ϕ′​\(ξ\)\|,…,2​π​\|ϕ\(r\)​\(ξ\)\|\)\.\\left\\lvert W^\{\(r\)\}\(\\xi\)\\right\\rvert\\leq B\_\{r\}\\bigl\(2\\pi\\left\\lvert\\phi^\{\\prime\}\(\\xi\)\\right\\rvert,\\ldots,2\\pi\\left\\lvert\\phi^\{\(r\)\}\(\\xi\)\\right\\rvert\\bigr\)\.\(6\.22\)The first four cases are

\|W′\|\\displaystyle\\left\\lvert W^\{\\prime\}\\right\\rvert≤2​π​\|ϕ′\|,\\displaystyle\\leq 2\\pi\\left\\lvert\\phi^\{\\prime\}\\right\\rvert,\|W′′\|\\displaystyle\\left\\lvert W^\{\\prime\\prime\}\\right\\rvert≤2​π​\|ϕ′′\|\+\(2​π\)2​\|ϕ′\|2,\\displaystyle\\leq 2\\pi\\left\\lvert\\phi^\{\\prime\\prime\}\\right\\rvert\+\(2\\pi\)^\{2\}\\left\\lvert\\phi^\{\\prime\}\\right\\rvert^\{2\},\|W\(3\)\|\\displaystyle\\left\\lvert W^\{\(3\)\}\\right\\rvert≤2​π​\|ϕ\(3\)\|\+3​\(2​π\)2​\|ϕ′\|​\|ϕ′′\|\+\(2​π\)3​\|ϕ′\|3,\\displaystyle\\leq 2\\pi\\left\\lvert\\phi^\{\(3\)\}\\right\\rvert\+3\(2\\pi\)^\{2\}\\left\\lvert\\phi^\{\\prime\}\\right\\rvert\\left\\lvert\\phi^\{\\prime\\prime\}\\right\\rvert\+\(2\\pi\)^\{3\}\\left\\lvert\\phi^\{\\prime\}\\right\\rvert^\{3\},\|W\(4\)\|\\displaystyle\\left\\lvert W^\{\(4\)\}\\right\\rvert≤2​π​\|ϕ\(4\)\|\+\(2​π\)2​\(4​\|ϕ′\|​\|ϕ\(3\)\|\+3​\|ϕ′′\|2\)\\displaystyle\\leq 2\\pi\\left\\lvert\\phi^\{\(4\)\}\\right\\rvert\+\(2\\pi\)^\{2\}\\bigl\(4\\left\\lvert\\phi^\{\\prime\}\\right\\rvert\\left\\lvert\\phi^\{\(3\)\}\\right\\rvert\+3\\left\\lvert\\phi^\{\\prime\\prime\}\\right\\rvert^\{2\}\\bigr\)\+6​\(2​π\)3​\|ϕ′\|2​\|ϕ′′\|\+\(2​π\)4​\|ϕ′\|4\.\\displaystyle\\quad\+6\(2\\pi\)^\{3\}\\left\\lvert\\phi^\{\\prime\}\\right\\rvert^\{2\}\\left\\lvert\\phi^\{\\prime\\prime\}\\right\\rvert\+\(2\\pi\)^\{4\}\\left\\lvert\\phi^\{\\prime\}\\right\\rvert^\{4\}\.
###### Corollary 6\.17\(Smooth\-phase high\-order estimate\)\.

Under the assumptions of[Theorem˜6\.16](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem16), letϕ∈Cr​\(\[0,1\];ℝ\)\\phi\\in C^\{r\}\(\[0,1\];\\mathbb\{R\}\)and

W​\(ξ\)=e2​π​i​ϕ​\(ξ\)\.W\(\\xi\)=e^\{2\\pi\\mathrm\{i\}\\phi\(\\xi\)\}\.Then

\|ℰNϕ​u−ℰNϕ​q\|≤V​N−r​sup0≤ξ≤1Br​\(2​π​\|ϕ′​\(ξ\)\|,…,2​π​\|ϕ\(r\)​\(ξ\)\|\)\.\\left\\lvert\\mathcal\{E\}\_\{N\}^\{\\phi\}u\-\\mathcal\{E\}\_\{N\}^\{\\phi\}q\\right\\rvert\\leq VN^\{\-r\}\\sup\_\{0\\leq\\xi\\leq 1\}B\_\{r\}\\bigl\(2\\pi\\left\\lvert\\phi^\{\\prime\}\(\\xi\)\\right\\rvert,\\ldots,2\\pi\\left\\lvert\\phi^\{\(r\)\}\(\\xi\)\\right\\rvert\\bigr\)\.\(6\.23\)

###### Proof\.

Apply[Theorem˜6\.16](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem16)and \([6\.22](https://arxiv.org/html/2607.24868#Ch6.E22)\)\. ∎

#### 6\.2\.6Second\-order parabolic estimate

Forϕx,t​\(ξ\)=x​ξ\+t​ξ2\\phi\_\{x,t\}\(\\xi\)=x\\xi\+t\\xi^\{2\},

ϕ′=x\+2​t​ξ,ϕ′′=2​t,ϕ\(m\)=0\(m≥3\)\.\\phi^\{\\prime\}=x\+2t\\xi,\\qquad\\phi^\{\\prime\\prime\}=2t,\\qquad\\phi^\{\(m\)\}=0\\quad\(m\\geq 3\)\.The second derivative of the oscillatory weight satisfies

\|W′′​\(ξ\)\|≤4​π​\|t\|\+4​π2​\|x\+2​t​ξ\|2\.\\left\\lvert W^\{\\prime\\prime\}\(\\xi\)\\right\\rvert\\leq 4\\pi\\left\\lvert t\\right\\rvert\+4\\pi^\{2\}\\left\\lvert x\+2t\\xi\\right\\rvert^\{2\}\.
###### Corollary 6\.18\(Second\-order parabolic extension\)\.

Assumee=Δ2​ve=\\Delta^\{2\}v,‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V, and second\-order trace compatibility\. Then

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤VN2​\[4​π​\|t\|\+4​π2​\(\|x\|\+2​\|t\|\)2\]\.\\left\\lvert\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\\right\\rvert\\leq\\frac\{V\}\{N^\{2\}\}\\left\[4\\pi\\left\\lvert t\\right\\rvert\+4\\pi^\{2\}\\bigl\(\\left\\lvert x\\right\\rvert\+2\\left\\lvert t\\right\\rvert\\bigr\)^\{2\}\\right\]\.\(6\.24\)On every fixed compact set in\(x,t\)\(x,t\), the error isO​\(N−2\)O\(N^\{\-2\}\)\.

The formula should be compared with the generic second\-order identity \([6\.1\.4](https://arxiv.org/html/2607.24868#Ch6.Ex11)\)\. If\(Δ​v\)N\(\\Delta v\)\_\{N\}is not zero, the first boundary term gives onlyO​\(N−1\)O\(N^\{\-1\}\), even though the interior second difference isO​\(N−2\)O\(N^\{\-2\}\)pointwise\.

#### 6\.2\.7Approximate endpoint compatibility

Exact reset may be difficult in a physical implementation\. The general estimate permits small residual traces\.

###### Proposition 6\.19\(Trace\-tolerant estimate\)\.

Assume‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq Vand

\|Lj​\(v\)\|\+\|Rj,N​\(v\)\|≤ηj,N,0≤j≤r−1\.\\left\\lvert L\_\{j\}\(v\)\\right\\rvert\+\\left\\lvert R\_\{j,N\}\(v\)\\right\\rvert\\leq\\eta\_\{j,N\},\\qquad 0\\leq j\\leq r\-1\.Then

\|1N​∑k=1N\(Δr​v\)k​wk\|\\displaystyle\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(\\Delta^\{r\}v\)\_\{k\}w\_\{k\}\\right\\rvert≤1N​∑j=0r−1ηj,N​max⁡\{\|\(Δ\+j​w\)1\|,\|\(Δ\+j​w\)N−j\|\}\\displaystyle\\leq\\frac\{1\}\{N\}\\sum\_\{j=0\}^\{r\-1\}\\eta\_\{j,N\}\\max\\left\\\{\\left\\lvert\(\\Delta\_\{\+\}^\{j\}w\)\_\{1\}\\right\\rvert,\\left\\lvert\(\\Delta\_\{\+\}^\{j\}w\)\_\{N\-j\}\\right\\rvert\\right\\\}\+VN​∑k=1N−r\|\(Δ\+r​w\)k\|\.\\displaystyle\\quad\+\\frac\{V\}\{N\}\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\\right\\rvert\.

For sampledCrC^\{r\}weights, thejjth boundary difference isO​\(N−j\)O\(N^\{\-j\}\)\. Therefore thejjth trace is harmless at the targetN−rN^\{\-r\}scale if

ηj,N=O​\(N−\(r−1−j\)\)\.\\eta\_\{j,N\}=O\(N^\{\-\(r\-1\-j\)\}\)\.The most demanding condition concerns the undifferenced weight,j=0j=0: the highest state trace must beO​\(N−\(r−1\)\)O\(N^\{\-\(r\-1\)\}\)\. Exact zero is sufficient but not necessary\.

#### 6\.2\.8Interpretation for quantizer design

The theorem does not construct a stable high\-order one\-bit recursion with compatible endpoints\. That is a separate control problem\. Existing sigma\-delta theory supplies stable schemes on long or infinite records\[[25](https://arxiv.org/html/2607.24868#bib.bib7),[48](https://arxiv.org/html/2607.24868#bib.bib8),[27](https://arxiv.org/html/2607.24868#bib.bib12)\], while frame\-path termination shows that endpoint preparation can be integrated into a finite expansion\[[9](https://arxiv.org/html/2607.24868#bib.bib20)\]\. For the present Fourier\-extension problem, three design options are natural:

1. \(a\)append a short terminal sequence chosen to drive the state traces to zero;
2. \(b\)divide the record into blocks and reset or smoothly terminate each block;
3. \(c\)modify the reconstruction weight near the endpoints so that its low\-order forward differences vanish there\.

Each option changes a different part of the boundary formula\. Chapter[9](https://arxiv.org/html/2607.24868#Ch9)develops the blockwise alternative in detail\.

## Chapter 7Polynomial, Multidimensional, and Growing\-Region Extensions

The discrete\-difference method is not restricted to a quadratic phase or to a one\-dimensional sequence\. The following sections treat polynomial phases, moment\-curve parameterisations, lattice arrays with divergence\-form errors, higher\-order multi\-index shaping, and parameter regions whose size depends on the record length\.

### 7\.1Polynomial phases and moment\-curve extensions

#### 7\.1\.1The polynomial extension operator

For an integerd≥1d\\geq 1and a parameter vectorx=\(x1,…,xd\)∈ℝdx=\(x\_\{1\},\\ldots,x\_\{d\}\)\\in\\mathbb\{R\}^\{d\}, define

ℰN,d​a​\(x\)=1N​∑k=1Nak​e​\(∑j=1dxj​\(kN\)j\),e​\(z\)=e2​π​i​z\.\\mathcal\{E\}\_\{N,d\}a\(x\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}a\_\{k\}e\\\!\\left\(\\sum\_\{j=1\}^\{d\}x\_\{j\}\\left\(\\frac\{k\}\{N\}\\right\)^\{j\}\\right\),\\qquad e\(z\)=e^\{2\\pi\\mathrm\{i\}z\}\.\(7\.1\)The underlying curve

γd​\(ξ\)=\(ξ,ξ2,…,ξd\)\\gamma\_\{d\}\(\\xi\)=\(\\xi,\\xi^\{2\},\\ldots,\\xi^\{d\}\)is the moment curve\. It is central in discrete restriction, Vinogradov mean value theory, and decoupling\[[13](https://arxiv.org/html/2607.24868#bib.bib45),[12](https://arxiv.org/html/2607.24868#bib.bib46),[81](https://arxiv.org/html/2607.24868#bib.bib63),[84](https://arxiv.org/html/2607.24868#bib.bib67),[51](https://arxiv.org/html/2607.24868#bib.bib70),[28](https://arxiv.org/html/2607.24868#bib.bib55)\]\. The immediate purpose is more elementary\. The perturbation caused by replacinguuwith a noise\-shaped sequenceqqis estimated\.

Set

ϕx​\(ξ\)=∑j=1dxj​ξj\.\\phi\_\{x\}\(\\xi\)=\\sum\_\{j=1\}^\{d\}x\_\{j\}\\xi^\{j\}\.Then

ϕx\(m\)​\(ξ\)=∑j=mdj\!\(j−m\)\!​xj​ξj−m,1≤m≤d,\\phi\_\{x\}^\{\(m\)\}\(\\xi\)=\\sum\_\{j=m\}^\{d\}\\frac\{j\!\}\{\(j\-m\)\!\}x\_\{j\}\\xi^\{j\-m\},\\qquad 1\\leq m\\leq d,\(7\.2\)andϕx\(m\)=0\\phi\_\{x\}^\{\(m\)\}=0form\>dm\>d\.

#### 7\.1\.2First\-order estimates

###### Theorem 7\.1\(First\-order polynomial\-phase estimate\)\.

Assumeek=uk−qk=Δ​vke\_\{k\}=u\_\{k\}\-q\_\{k\}=\\Delta v\_\{k\},v0=0v\_\{0\}=0, and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\. Then

\|ℰN,d​u​\(x\)−ℰN,d​q​\(x\)\|≤VN​\[1\+2​π​∫01\|∑j=1dj​xj​ξj−1\|​dξ\]\.\\left\\lvert\\mathcal\{E\}\_\{N,d\}u\(x\)\-\\mathcal\{E\}\_\{N,d\}q\(x\)\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\int\_\{0\}^\{1\}\\left\\lvert\\sum\_\{j=1\}^\{d\}jx\_\{j\}\\xi^\{j\-1\}\\right\\rvert\\,\\mathrm\{d\}\\xi\\right\]\.\(7\.3\)In particular,

\|ℰN,d​u​\(x\)−ℰN,d​q​\(x\)\|≤VN​\[1\+2​π​∑j=1d\|xj\|\]\.\\left\\lvert\\mathcal\{E\}\_\{N,d\}u\(x\)\-\\mathcal\{E\}\_\{N,d\}q\(x\)\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\sum\_\{j=1\}^\{d\}\\left\\lvert x\_\{j\}\\right\\rvert\\right\]\.\(7\.4\)

###### Proof\.

Apply the absolutely continuous phase estimate toϕx\\phi\_\{x\}\. For the simpler bound, use

\|ϕx′​\(ξ\)\|≤∑j=1dj​\|xj\|​ξj−1\\left\\lvert\\phi\_\{x\}^\{\\prime\}\(\\xi\)\\right\\rvert\\leq\\sum\_\{j=1\}^\{d\}j\\left\\lvert x\_\{j\}\\right\\rvert\\xi^\{j\-1\}and integrate\. Since∫01j​ξj−1​dξ=1\\int\_\{0\}^\{1\}j\\xi^\{j\-1\}\\,\\mathrm\{d\}\\xi=1, the coefficient of each\|xj\|\\left\\lvert x\_\{j\}\\right\\rvertis exactly one\. ∎

The cancellation of the factorjjafter integration is useful\. The deterministic first\-order bound depends on theℓ1\\ell^\{1\}size of the parameter vector, not on the maximum degree\.

###### Corollary 7\.2\(Uniformity on compact parameter sets\)\.

LetK⊂ℝdK\\subset\\mathbb\{R\}^\{d\}be compact\. Then

‖ℰN,d​u−ℰN,d​q‖L∞​\(K\)≤VN​\(1\+2​π​supx∈K∑j=1d\|xj\|\)\.\\left\\lVert\\mathcal\{E\}\_\{N,d\}u\-\\mathcal\{E\}\_\{N,d\}q\\right\\rVert\_\{L^\{\\infty\}\(K\)\}\\leq\\frac\{V\}\{N\}\\left\(1\+2\\pi\\sup\_\{x\\in K\}\\sum\_\{j=1\}^\{d\}\\left\\lvert x\_\{j\}\\right\\rvert\\right\)\.Thus the quantized and unquantized polynomial extensions converge uniformly onKKat orderN−1N^\{\-1\}\.

#### 7\.1\.3High\-order estimates under endpoint compatibility

For1≤m≤r1\\leq m\\leq r, define

Am​\(x\)=∑j=mdj\!\(j−m\)\!​\|xj\|\.A\_\{m\}\(x\)=\\sum\_\{j=m\}^\{d\}\\frac\{j\!\}\{\(j\-m\)\!\}\\left\\lvert x\_\{j\}\\right\\rvert\.\(7\.5\)Since0≤ξ≤10\\leq\\xi\\leq 1, \([7\.2](https://arxiv.org/html/2607.24868#Ch7.E2)\) gives

‖ϕx\(m\)‖L∞​\(0,1\)≤Am​\(x\)\.\\left\\lVert\\phi\_\{x\}^\{\(m\)\}\\right\\rVert\_\{L^\{\\infty\}\(0,1\)\}\\leq A\_\{m\}\(x\)\.
###### Theorem 7\.3\(High\-order moment\-curve estimate\)\.

Assumee=Δr​ve=\\Delta^\{r\}v,‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V, and trace compatibility of orderrr\. Then

\|ℰN,d​u​\(x\)−ℰN,d​q​\(x\)\|≤V​N−r​Br​\(2​π​A1​\(x\),…,2​π​Ar​\(x\)\),\\left\\lvert\\mathcal\{E\}\_\{N,d\}u\(x\)\-\\mathcal\{E\}\_\{N,d\}q\(x\)\\right\\rvert\\leq VN^\{\-r\}B\_\{r\}\\bigl\(2\\pi A\_\{1\}\(x\),\\ldots,2\\pi A\_\{r\}\(x\)\\bigr\),\(7\.6\)whereAm​\(x\)=0A\_\{m\}\(x\)=0whenm\>dm\>d\.

###### Proof\.

Use[Corollary˜6\.17](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem17)and the derivative envelopes above\. Monotonicity of the complete Bell polynomial in nonnegative arguments gives the stated bound\. ∎

For fixedd,rd,r, the right\-hand side is a polynomial in\|x1\|,…,\|xd\|\\left\\lvert x\_\{1\}\\right\\rvert,\\ldots,\\left\\lvert x\_\{d\}\\right\\rvertof total degree at mostrr\. The highest\-degree contribution is

\(2​π​A1​\(x\)\)r\.\(2\\pi A\_\{1\}\(x\)\)^\{r\}\.Lower\-degree terms contain higher derivatives of the phase\.

#### 7\.1\.4Explicit second\- and third\-order formulae

Forr=2r=2,

\|ℰN,d​u−ℰN,d​q\|≤VN2​\[2​π​A2​\(x\)\+\(2​π\)2​A1​\(x\)2\]\.\\left\\lvert\\mathcal\{E\}\_\{N,d\}u\-\\mathcal\{E\}\_\{N,d\}q\\right\\rvert\\leq\\frac\{V\}\{N^\{2\}\}\\left\[2\\pi A\_\{2\}\(x\)\+\(2\\pi\)^\{2\}A\_\{1\}\(x\)^\{2\}\\right\]\.\(7\.7\)Forr=3r=3,

\|ℰN,d​u−ℰN,d​q\|≤VN3​\[2​π​A3​\(x\)\+3​\(2​π\)2​A1​\(x\)​A2​\(x\)\+\(2​π\)3​A1​\(x\)3\]\.\\left\\lvert\\mathcal\{E\}\_\{N,d\}u\-\\mathcal\{E\}\_\{N,d\}q\\right\\rvert\\leq\\frac\{V\}\{N^\{3\}\}\\left\[2\\pi A\_\{3\}\(x\)\+3\(2\\pi\)^\{2\}A\_\{1\}\(x\)A\_\{2\}\(x\)\+\(2\\pi\)^\{3\}A\_\{1\}\(x\)^\{3\}\\right\]\.\(7\.8\)These estimates remain valid whenr\>dr\>dbecause the derivativesAmA\_\{m\}withm\>dm\>dvanish, while products of lower derivatives remain\.

#### 7\.1\.5Anisotropic parameter boxes

Restriction theory naturally assigns different physical scales to different coordinates of the moment curve\. The normalized extension used here, however, measures smoothness in the parameterξ=k/N\\xi=k/N\. For a box

ℬN​\(α1,…,αd\)=\{x:\|xj\|≤Nαj,1≤j≤d\},\\mathcal\{B\}\_\{N\}\(\\alpha\_\{1\},\\ldots,\\alpha\_\{d\}\)=\\left\\\{x:\\left\\lvert x\_\{j\}\\right\\rvert\\leq N^\{\\alpha\_\{j\}\},\\ 1\\leq j\\leq d\\right\\\},\(7\.9\)let

α∗=max1≤j≤d⁡αj\.\\alpha\_\{\*\}=\\max\_\{1\\leq j\\leq d\}\\alpha\_\{j\}\.ThenAm​\(x\)≤Cd,m​Nα∗A\_\{m\}\(x\)\\leq C\_\{d,m\}N^\{\\alpha\_\{\*\}\}on the box, and the Bell polynomial bound yields

supx∈ℬN\|ℰN,d​u​\(x\)−ℰN,d​q​\(x\)\|≤Cd,r​V​N−r​\(1−α∗\)\\sup\_\{x\\in\\mathcal\{B\}\_\{N\}\}\\left\\lvert\\mathcal\{E\}\_\{N,d\}u\(x\)\-\\mathcal\{E\}\_\{N,d\}q\(x\)\\right\\rvert\\leq C\_\{d,r\}VN^\{\-r\(1\-\\alpha\_\{\*\}\)\}\(7\.10\)
after enlarging the constant to cover lower\-degree terms\. Thus endpoint\-compatiblerrth\-order shaping gives deterministic decay wheneverα∗<1\\alpha\_\{\*\}<1\.

This criterion is sufficient, not optimal\. It ignores the fact thatxjx\_\{j\}enters different phase derivatives with different powers ofξ\\xi\. A more refined estimate retains the full Bell polynomial and may permit larger values of selected coordinates when the corresponding derivatives appear only in lower\-degree combinations\.

###### Proposition 7\.4\(Refined anisotropic envelope\)\.

On the box \([7\.9](https://arxiv.org/html/2607.24868#Ch7.E9)\), define

βm=maxm≤j≤d⁡αj\.\\beta\_\{m\}=\\max\_\{m\\leq j\\leq d\}\\alpha\_\{j\}\.Then

Am​\(x\)≤Cd,m​Nβm,A\_\{m\}\(x\)\\leq C\_\{d,m\}N^\{\\beta\_\{m\}\},and every monomial

∏m=1rAm​\(x\)νm\\prod\_\{m=1\}^\{r\}A\_\{m\}\(x\)^\{\\nu\_\{m\}\}occurring inBrB\_\{r\}satisfies the weight constraint

∑m=1rm​νm=r\.\\sum\_\{m=1\}^\{r\}m\\nu\_\{m\}=r\.Consequently, its contribution is bounded by

C​N−r\+∑mνm​βm\.CN^\{\-r\+\\sum\_\{m\}\\nu\_\{m\}\\beta\_\{m\}\}\.

###### Proof\.

The derivative envelope is immediate\. The weight constraint is the defining homogeneity of the complete Bell polynomial: the variable representing themmth derivative has combinatorial weightmm\. ∎

This proposition identifies a family of anisotropic subcritical regions through the inequalities

∑mνm​βm<r\\sum\_\{m\}\\nu\_\{m\}\\beta\_\{m\}<rfor every Bell monomial\. The strongest constraint often comes fromA1rA\_\{1\}^\{r\}, givingβ1<1\\beta\_\{1\}<1, but if the linear derivative is small or cancels on the region, other constraints may become decisive\.

#### 7\.1\.6Stationary points and local improvement

The total\-variation estimate uses∫\|ϕ′\|\\int\|\\phi^\{\\prime\}\|and therefore does not benefit from sign changes inϕ′\\phi^\{\\prime\}\. The parabolic closed formJ​\(x,t\)J\(x,t\)already shows that a stationary point can reduce the constant\. For higher\-degree phases, the same phenomenon occurs wheneverϕ′\\phi^\{\\prime\}has roots in\[0,1\]\[0,1\]\. One can partition the interval into monotonicity intervals ofϕ\\phiand compute

Var⁡\(e2​π​i​ϕ\)≤2​π​Var⁡\(ϕ\)\.\\operatorname\{Var\}\(e^\{2\\pi\\mathrm\{i\}\\phi\}\)\\leq 2\\pi\\operatorname\{Var\}\(\\phi\)\.This is still a variation argument\. It recognises reduced path length caused by a smaller derivative, but it does not use cancellation between distinct turns of the phase\.

A more ambitious local theory would centre the phase at a stationary pointξ0\\xi\_\{0\}and combine noise shaping with van der Corput estimates\. If

ϕ′​\(ξ0\)=⋯=ϕ\(s−1\)​\(ξ0\)=0,ϕ\(s\)​\(ξ0\)≠0,\\phi^\{\\prime\}\(\\xi\_\{0\}\)=\\cdots=\\phi^\{\(s\-1\)\}\(\\xi\_\{0\}\)=0,\\qquad\\phi^\{\(s\)\}\(\\xi\_\{0\}\)\\neq 0,then the local oscillation is of orderss\. Classical oscillatory\-integral theory relates this order to decay rates\[[70](https://arxiv.org/html/2607.24868#bib.bib76),[41](https://arxiv.org/html/2607.24868#bib.bib75)\]\. In the discrete setting, Weyl differencing and Vinogradov mean value estimates play the corresponding role\[[79](https://arxiv.org/html/2607.24868#bib.bib56),[77](https://arxiv.org/html/2607.24868#bib.bib59),[59](https://arxiv.org/html/2607.24868#bib.bib62),[81](https://arxiv.org/html/2607.24868#bib.bib63),[12](https://arxiv.org/html/2607.24868#bib.bib46)\]\. Chapter[2](https://arxiv.org/html/2607.24868#Ch2)explains why these tools are required beyond the subcritical variation regime\.

#### 7\.1\.7Coefficient normalisation and comparison with restriction theory

The operator \([7\.1](https://arxiv.org/html/2607.24868#Ch7.E1)\) contains1/N1/N\. Classical discrete restriction often studies the unnormalised sum

Sa​\(θ\)=∑k=1Nak​e​\(θ1​k\+⋯\+θd​kd\)\.S\_\{a\}\(\\theta\)=\\sum\_\{k=1\}^\{N\}a\_\{k\}e\(\\theta\_\{1\}k\+\\cdots\+\\theta\_\{d\}k^\{d\}\)\.The change of variables

θj=xjNj\\theta\_\{j\}=\\frac\{x\_\{j\}\}\{N^\{j\}\}gives

ℰN,d​a​\(x\)=1N​Sa​\(θ\)\.\\mathcal\{E\}\_\{N,d\}a\(x\)=\\frac\{1\}\{N\}S\_\{a\}\(\\theta\)\.A unit torus inθ\\thetacorresponds to the anisotropic box

0≤xj≤Nj\.0\\leq x\_\{j\}\\leq N^\{j\}\.This is far larger than the deterministic subcritical region\|xj\|=o​\(N\)\\left\\lvert x\_\{j\}\\right\\rvert=o\(N\)produced by total variation\. The gap is structural\. It is not removed by increasing the degree of the phase or by rewriting the variables\. Reaching the full torus requires genuine oscillatory estimates\.

#### 7\.1\.8A new family of structured restriction questions

The moment\-curve formulation suggests the following problem\. Given a bounded statevvande=Δr​ve=\\Delta^\{r\}v, determine the best exponentsppandδ\\deltafor which

‖ℰN,d​e‖Lp​\(ℬN\)≤Cε​N−δ\+ε​‖v‖ℓ∞\\left\\lVert\\mathcal\{E\}\_\{N,d\}e\\right\\rVert\_\{L^\{p\}\(\\mathcal\{B\}\_\{N\}\)\}\\leq C\_\{\\varepsilon\}N^\{\-\\delta\+\\varepsilon\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}can hold on anisotropic boxes\. The answer must depend on the boundary conditions, the orderrr, the dimensiondd, and the chosen scaling ofℬN\\mathcal\{B\}\_\{N\}\. This is not a standard discrete restriction problem because the coefficients are constrained by a finite\-difference representation\. It is also not a standard sigma\-delta reconstruction problem because the test family is a curved exponential system rather than a fixed dual frame\.

### 7\.2Multidimensional noise shaping and grid extensions

#### 7\.2\.1From a sequence to a lattice array

Many quantization problems are naturally multidimensional\. Digital halftoning, image acquisition, sensor arrays, and sampled partial differential equations produce data indexed by a rectangular lattice rather than a single ordered list\. Error\-diffusion algorithms can often be interpreted as multidimensional sigma\-delta schemes\. Recent work has made this relationship precise for weighted first\- and second\-order constructions in image halftoning\[[63](https://arxiv.org/html/2607.24868#bib.bib82)\]\. The algebra developed in the preceding chapters extends to this setting through discrete divergence\.

Let

ΛN=\{1,…,N\}d\.\\Lambda\_\{N\}=\\\{1,\\ldots,N\\\}^\{d\}\.Fork=\(k1,…,kd\)∈ΛNk=\(k\_\{1\},\\ldots,k\_\{d\}\)\\in\\Lambda\_\{N\}and thejjth coordinate vectoreje\_\{j\}, define

\(Δj​z\)k=zk−zk−ej,\(Δ\+,j​w\)k=wk\+ej−wk\.\(\\Delta\_\{j\}z\)\_\{k\}=z\_\{k\}\-z\_\{k\-e\_\{j\}\},\\qquad\(\\Delta\_\{\+,j\}w\)\_\{k\}=w\_\{k\+e\_\{j\}\}\-w\_\{k\}\.Boundary values outsideΛN\\Lambda\_\{N\}must be prescribed\. Zero initial data on the lower faces are imposed first\.

#### 7\.2\.2Divergence\-form shaped errors

###### Definition 7\.5\(First\-order multidimensional noise shaping\)\.

An error arraye=\(ek\)k∈ΛNe=\(e\_\{k\}\)\_\{k\\in\\Lambda\_\{N\}\}has divergence\-form shaping if

ek=∑j=1d\(Δj​v\(j\)\)k,e\_\{k\}=\\sum\_\{j=1\}^\{d\}\(\\Delta\_\{j\}v^\{\(j\)\}\)\_\{k\},\(7\.11\)where the vector statev=\(v\(1\),…,v\(d\)\)v=\(v^\{\(1\)\},\\ldots,v^\{\(d\)\}\)is bounded\.

The representation \([7\.11](https://arxiv.org/html/2607.24868#Ch7.E11)\) is a lattice analogue ofe=∇⋅ve=\\nabla\\cdot v\. It captures the idea that quantization error is redistributed through neighbouring samples\. Different scan orders and error\-diffusion masks produce different choices of the state fields\.

For an arrayaa, define the normalised grid extension

𝒢N​a​\(x\)=1Nd​∑k∈ΛNak​e2​π​i​Φx​\(k/N\),\\mathcal\{G\}\_\{N\}a\(x\)=\\frac\{1\}\{N^\{d\}\}\\sum\_\{k\\in\\Lambda\_\{N\}\}a\_\{k\}e^\{2\\pi\\mathrm\{i\}\\Phi\_\{x\}\(k/N\)\},\(7\.12\)wherek/N=\(k1/N,…,kd/N\)k/N=\(k\_\{1\}/N,\\ldots,k\_\{d\}/N\)andΦx\\Phi\_\{x\}is a real phase on\[0,1\]d\[0,1\]^\{d\}\.

#### 7\.2\.3Discrete divergence theorem

To state the boundary formula, let

ΛN\(j,−\)=\{k∈ΛN:kj=1\},ΛN\(j,\+\)=\{k∈ΛN:kj=N\}\.\\Lambda\_\{N\}^\{\(j,\-\)\}=\\\{k\\in\\Lambda\_\{N\}:k\_\{j\}=1\\\},\\qquad\\Lambda\_\{N\}^\{\(j,\+\)\}=\\\{k\\in\\Lambda\_\{N\}:k\_\{j\}=N\\\}\.
###### Theorem 7\.6\(Multidimensional summation by parts\)\.

For each coordinatejj,

∑k∈ΛN\(Δj​v\(j\)\)k​wk\\displaystyle\\sum\_\{k\\in\\Lambda\_\{N\}\}\(\\Delta\_\{j\}v^\{\(j\)\}\)\_\{k\}w\_\{k\}=∑k∈ΛN\(j,\+\)vk\(j\)​wk−∑k∈ΛN\(j,−\)vk−ej\(j\)​wk\\displaystyle=\\sum\_\{k\\in\\Lambda\_\{N\}^\{\(j,\+\)\}\}v^\{\(j\)\}\_\{k\}w\_\{k\}\-\\sum\_\{k\\in\\Lambda\_\{N\}^\{\(j,\-\)\}\}v^\{\(j\)\}\_\{k\-e\_\{j\}\}w\_\{k\}−∑k∈ΛNkj≤N−1vk\(j\)​\(Δ\+,j​w\)k\.\\displaystyle\\quad\-\\sum\_\{\\begin\{subarray\}\{c\}k\\in\\Lambda\_\{N\}\\\\ k\_\{j\}\\leq N\-1\\end\{subarray\}\}v^\{\(j\)\}\_\{k\}\(\\Delta\_\{\+,j\}w\)\_\{k\}\.\(7\.13\)Consequently, if the lower\-face states vanish, then

\|1Nd​∑k∈ΛNek​wk\|\\displaystyle\\left\\lvert\\frac\{1\}\{N^\{d\}\}\\sum\_\{k\\in\\Lambda\_\{N\}\}e\_\{k\}w\_\{k\}\\right\\rvert≤1Nd​∑j=1d\[∑k∈ΛN\(j,\+\)\|vk\(j\)\|​\|wk\|\+∑kj≤N−1\|vk\(j\)\|​\|Δ\+,j​wk\|\]\.\\displaystyle\\leq\\frac\{1\}\{N^\{d\}\}\\sum\_\{j=1\}^\{d\}\\left\[\\sum\_\{k\\in\\Lambda\_\{N\}^\{\(j,\+\)\}\}\\left\\lvert v^\{\(j\)\}\_\{k\}\\right\\rvert\\left\\lvert w\_\{k\}\\right\\rvert\+\\sum\_\{k\_\{j\}\\leq N\-1\}\\left\\lvert v^\{\(j\)\}\_\{k\}\\right\\rvert\\left\\lvert\\Delta\_\{\+,j\}w\_\{k\}\\right\\rvert\\right\]\.\(7\.14\)

###### Proof\.

Fix all coordinates exceptkjk\_\{j\}and apply the one\-dimensional summation\-by\-parts identity along the resulting line\. Sum over the remainingd−1d\-1coordinates\. Then add the identities overjjand use \([7\.11](https://arxiv.org/html/2607.24868#Ch7.E11)\)\. ∎

If\|vk\(j\)\|≤Vj\\left\\lvert v^\{\(j\)\}\_\{k\}\\right\\rvert\\leq V\_\{j\}and\|wk\|=1\\left\\lvert w\_\{k\}\\right\\rvert=1, the upper face contributes

Nd−1Nd​Vj=VjN\.\\frac\{N^\{d\-1\}\}\{N^\{d\}\}V\_\{j\}=\\frac\{V\_\{j\}\}\{N\}\.Thus first\-order divergence shaping again produces anN−1N^\{\-1\}boundary scale, independent of dimension\.

#### 7\.2\.4Smooth multidimensional phases

Set

wk=e2​π​i​Φ​\(k/N\)\.w\_\{k\}=e^\{2\\pi\\mathrm\{i\}\\Phi\(k/N\)\}\.For a continuously differentiable phase,

\|Δ\+,j​wk\|\\displaystyle\\left\\lvert\\Delta\_\{\+,j\}w\_\{k\}\\right\\rvert≤2​π​\|Φ​\(\(k\+ej\)/N\)−Φ​\(k/N\)\|\\displaystyle\\leq 2\\pi\\left\\lvert\\Phi\(\(k\+e\_\{j\}\)/N\)\-\\Phi\(k/N\)\\right\\rvert≤2​πN​supy∈Qk\|∂jΦ​\(y\)\|,\\displaystyle\\leq\\frac\{2\\pi\}\{N\}\\sup\_\{y\\in Q\_\{k\}\}\\left\\lvert\\partial\_\{j\}\\Phi\(y\)\\right\\rvert,whereQkQ\_\{k\}is the line segment joining the adjacent grid points\.

###### Theorem 7\.7\(Grid phase estimate\)\.

Assume \([7\.11](https://arxiv.org/html/2607.24868#Ch7.E11)\), zero lower\-face states, and

‖v\(j\)‖ℓ∞≤Vj\.\\left\\lVert v^\{\(j\)\}\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\_\{j\}\.IfΦ∈C1​\(\[0,1\]d\)\\Phi\\in C^\{1\}\(\[0,1\]^\{d\}\), then

\|𝒢N​u−𝒢N​q\|≤1N​∑j=1dVj​\[1\+2​π​‖∂jΦ‖L∞​\(\[0,1\]d\)\]\.\\left\\lvert\\mathcal\{G\}\_\{N\}u\-\\mathcal\{G\}\_\{N\}q\\right\\rvert\\leq\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{d\}V\_\{j\}\\left\[1\+2\\pi\\left\\lVert\\partial\_\{j\}\\Phi\\right\\rVert\_\{L^\{\\infty\}\(\[0,1\]^\{d\}\)\}\\right\]\.\(7\.15\)

###### Proof\.

The upper\-face term isVj/NV\_\{j\}/N\. There are at mostNdN^\{d\}interior edges in thejjth direction, and each difference is bounded by2​π​N−1​‖∂jΦ‖∞2\\pi N^\{\-1\}\\left\\lVert\\partial\_\{j\}\\Phi\\right\\rVert\_\{\\infty\}\. Multiplication byN−dN^\{\-d\}gives the stated result\. ∎

A sharper version replaces the supremum by a discrete or continuousL1L^\{1\}norm of∂jΦ\\partial\_\{j\}\\Phi\. The supremum form is convenient because it is uniform over all grid cells\.

#### 7\.2\.5A two\-dimensional polynomial example

Consider

Φx,t,s​\(ξ1,ξ2\)=x1​ξ1\+x2​ξ2\+t1​ξ12\+t2​ξ22\+s​ξ1​ξ2\.\\Phi\_\{x,t,s\}\(\\xi\_\{1\},\\xi\_\{2\}\)=x\_\{1\}\\xi\_\{1\}\+x\_\{2\}\\xi\_\{2\}\+t\_\{1\}\\xi\_\{1\}^\{2\}\+t\_\{2\}\\xi\_\{2\}^\{2\}\+s\\xi\_\{1\}\\xi\_\{2\}\.Then

∂1Φ\\displaystyle\\partial\_\{1\}\\Phi=x1\+2​t1​ξ1\+s​ξ2,\\displaystyle=x\_\{1\}\+2t\_\{1\}\\xi\_\{1\}\+s\\xi\_\{2\},∂2Φ\\displaystyle\\partial\_\{2\}\\Phi=x2\+2​t2​ξ2\+s​ξ1\.\\displaystyle=x\_\{2\}\+2t\_\{2\}\\xi\_\{2\}\+s\\xi\_\{1\}\.Therefore

\|𝒢N​u−𝒢N​q\|\\displaystyle\\left\\lvert\\mathcal\{G\}\_\{N\}u\-\\mathcal\{G\}\_\{N\}q\\right\\rvert≤V1N​\[1\+2​π​\(\|x1\|\+2​\|t1\|\+\|s\|\)\]\\displaystyle\\leq\\frac\{V\_\{1\}\}\{N\}\\left\[1\+2\\pi\(\\left\\lvert x\_\{1\}\\right\\rvert\+2\\left\\lvert t\_\{1\}\\right\\rvert\+\\left\\lvert s\\right\\rvert\)\\right\]\+V2N​\[1\+2​π​\(\|x2\|\+2​\|t2\|\+\|s\|\)\]\.\\displaystyle\\quad\+\\frac\{V\_\{2\}\}\{N\}\\left\[1\+2\\pi\(\\left\\lvert x\_\{2\}\\right\\rvert\+2\\left\\lvert t\_\{2\}\\right\\rvert\+\\left\\lvert s\\right\\rvert\)\\right\]\.\(7\.16\)This estimate remains local in parameter space\. The coupling terms​ξ1​ξ2s\\xi\_\{1\}\\xi\_\{2\}is handled without any new idea because only first derivatives are used\.

#### 7\.2\.6Directional and weighted schemes

A weighted multidimensional scheme may have the form

e=∑j=1dcj​Δj​v\(j\)e=\\sum\_\{j=1\}^\{d\}c\_\{j\}\\Delta\_\{j\}v^\{\(j\)\}with real or complex coefficientscjc\_\{j\}\. The bound becomes

\|1Nd​∑ek​wk\|≤1N​∑j=1d\|cj\|​Vj​\(1\+2​π​‖∂jΦ‖∞\)\.\\left\\lvert\\frac\{1\}\{N^\{d\}\}\\sum e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{d\}\\left\\lvert c\_\{j\}\\right\\rvert V\_\{j\}\\left\(1\+2\\pi\\left\\lVert\\partial\_\{j\}\\Phi\\right\\rVert\_\{\\infty\}\\right\)\.More general finite masks can be written as a convolutional difference operator\. The correct test norm is then the variation generated by the adjoint mask\. This operator viewpoint is used in modern analyses of error diffusion and weighted sigma\-delta halftoning\[[63](https://arxiv.org/html/2607.24868#bib.bib82)\]\.

#### 7\.2\.7Higher\-order multi\-index shaping

Letα=\(α1,…,αd\)\\alpha=\(\\alpha\_\{1\},\\ldots,\\alpha\_\{d\}\)be a multi\-index and write

Δα=Δ1α1​⋯​Δdαd,Δ\+α=Δ\+,1α1​⋯​Δ\+,dαd\.\\Delta^\{\\alpha\}=\\Delta\_\{1\}^\{\\alpha\_\{1\}\}\\cdots\\Delta\_\{d\}^\{\\alpha\_\{d\}\},\\qquad\\Delta\_\{\+\}^\{\\alpha\}=\\Delta\_\{\+,1\}^\{\\alpha\_\{1\}\}\\cdots\\Delta\_\{\+,d\}^\{\\alpha\_\{d\}\}\.If

and all traces on the relevant lower and upper boundary layers vanish, repeated coordinatewise summation by parts gives

∑k∈ΛNek​wk=\(−1\)\|α\|​∑k∈ΛN∘vk​\(Δ\+α​w\)k,\\sum\_\{k\\in\\Lambda\_\{N\}\}e\_\{k\}w\_\{k\}=\(\-1\)^\{\|\\alpha\|\}\\sum\_\{k\\in\\Lambda\_\{N\}^\{\\circ\}\}v\_\{k\}\(\\Delta\_\{\+\}^\{\\alpha\}w\)\_\{k\},whereΛN∘\\Lambda\_\{N\}^\{\\circ\}excludes the boundary strips needed by the forward differences\.

Forwk=W​\(k/N\)w\_\{k\}=W\(k/N\)withW∈C\|α\|W\\in C^\{\|\\alpha\|\},

\|Δ\+α​wk\|≤N−\|α\|​‖∂αW‖L∞\.\\left\\lvert\\Delta\_\{\+\}^\{\\alpha\}w\_\{k\}\\right\\rvert\\leq N^\{\-\|\\alpha\|\}\\left\\lVert\\partial^\{\\alpha\}W\\right\\rVert\_\{L^\{\\infty\}\}\.After normalisation byNdN^\{d\}and summation overO​\(Nd\)O\(N^\{d\}\)points, one obtains

\|1Nd​∑ek​wk\|≤‖v‖ℓ∞​N−\|α\|​‖∂αW‖L∞\.\\left\\lvert\\frac\{1\}\{N^\{d\}\}\\sum e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}N^\{\-\|\\alpha\|\}\\left\\lVert\\partial^\{\\alpha\}W\\right\\rVert\_\{L^\{\\infty\}\}\.\(7\.17\)Again, the boundary conditions are not optional\.

#### 7\.2\.8Scan order and causality

A physical multidimensional quantizer must choose an order in which pixels or samples are processed\. Raster, serpentine, and space\-filling scans produce different causal neighbourhoods\. The divergence representation does not by itself guarantee that a chosen state field can be generated causally with a finite alphabet\. It is an analytical representation of the resulting error\.

This distinction mirrors the one\-dimensional separation between stability and testing\. First, a quantizer must be shown to generate bounded states under a specified scan\. Second, the divergence theorem converts that boundedness into a Fourier error estimate\. The recent weighted sigma\-delta treatment of halftoning provides concrete examples in which both stages can be analysed\[[63](https://arxiv.org/html/2607.24868#bib.bib82)\]\.

#### 7\.2\.9Dimension\-free rate and dimension\-dependent constants

The rateN−1N^\{\-1\}in \([7\.15](https://arxiv.org/html/2607.24868#Ch7.E15)\) does not deteriorate with dimension when the grid has side lengthNN\. The constant does grow through the sum of directional state bounds and phase derivatives\. If allVj≤VV\_\{j\}\\leq Vand all‖∂jΦ‖∞≤L\\left\\lVert\\partial\_\{j\}\\Phi\\right\\rVert\_\{\\infty\}\\leq L, then

\|𝒢N​u−𝒢N​q\|≤d​VN​\(1\+2​π​L\)\.\\left\\lvert\\mathcal\{G\}\_\{N\}u\-\\mathcal\{G\}\_\{N\}q\\right\\rvert\\leq\\frac\{dV\}\{N\}\(1\+2\\pi L\)\.This linear dependence onddis a consequence of using the triangle inequality across coordinate directions\. Additional cancellation among the directional fluxes could improve the constant, but it cannot be expected without structural assumptions on the state fields\.

#### 7\.2\.10Open multidimensional questions

Three questions are immediate\.

1. \(i\)Can a stable one\-bit multidimensional scheme be designed so that the normal component of the state vanishes on every terminal face?
2. \(ii\)Can oscillatory cancellation reduce the variation bound on critical parameter regions, particularly for mixed phases such asξ1​ξ2\\xi\_\{1\}\\xi\_\{2\}?
3. \(iii\)How should one formulate a discrete restriction theorem for coefficients that are a lattice divergence of a bounded vector field?

The third question is a multidimensional analogue of the structured\-coefficient restriction problem introduced in the previous chapter\. It connects sigma\-delta quantization, discrete Hodge decompositions, and curved Fourier extension\.

### 7\.3Growing observation regions and scale transitions

#### 7\.3\.1Why fixed compact sets are only the first scale

The first\-order parabolic theorem gives

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤VN​\[1\+2​π​\(\|x\|\+\|t\|\)\]\.\\left\\lvert\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\bigl\(\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert\\bigr\)\\right\]\.On a fixed compact set, the bracket is independent ofNN, so the error isO​\(N−1\)O\(N^\{\-1\}\)\. In restriction theory and periodic dispersive equations, however, the observation region often expands with the frequency scale\. The parameter size then competes directly with the gain provided by noise shaping\.

This chapter records the consequences that follow from the variation method alone\. These results are useful because they identify the exact point at which a nonoscillatory argument stops producing decay\.

#### 7\.3\.2Isotropic parabolic boxes

Let

ΩN,α=\[−Nα,Nα\]2,α≥0\.\\Omega\_\{N,\\alpha\}=\[\-N^\{\\alpha\},N^\{\\alpha\}\]^\{2\},\\qquad\\alpha\\geq 0\.
###### Proposition 7\.8\(First\-order isotropic scale\)\.

Under the first\-order hypotheses,

‖ℰN​u−ℰN​q‖L∞​\(ΩN,α\)≤VN​\(1\+4​π​Nα\)\.\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\_\{N,\\alpha\}\)\}\\leq\\frac\{V\}\{N\}\\left\(1\+4\\pi N^\{\\alpha\}\\right\)\.\(7\.18\)Consequently,

‖ℰN​u−ℰN​q‖L∞​\(ΩN,α\)=\{O​\(Nα−1\),0≤α<1,O​\(1\),α=1,O​\(Nα−1\),α\>1,\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\_\{N,\\alpha\}\)\}=\\begin\{cases\}O\(N^\{\\alpha\-1\}\),&0\\leq\\alpha<1,\\\\ O\(1\),&\\alpha=1,\\\\ O\(N^\{\\alpha\-1\}\),&\\alpha\>1,\\end\{cases\}where the final line indicates growth rather than decay\.

The scaleα=1\\alpha=1is critical for adjacent phase variation because

ϕx,t​\(k\+1N\)−ϕx,t​\(kN\)=xN\+t​\(2​k\+1\)N2\.\\phi\_\{x,t\}\\left\(\\frac\{k\+1\}\{N\}\\right\)\-\\phi\_\{x,t\}\\left\(\\frac\{k\}\{N\}\\right\)=\\frac\{x\}\{N\}\+\\frac\{t\(2k\+1\)\}\{N^\{2\}\}\.When\|x\|,\|t\|≪N\\left\\lvert x\\right\\rvert,\\left\\lvert t\\right\\rvert\\ll N, every adjacent phase increment is small\. When they are of orderNN, the increment can be of order one\.

#### 7\.3\.3Anisotropic parabolic boxes

Let

ΩN​\(α,β\)=\[−Nα,Nα\]×\[−Nβ,Nβ\]\.\\Omega\_\{N\}\(\\alpha,\\beta\)=\[\-N^\{\\alpha\},N^\{\\alpha\}\]\\times\[\-N^\{\\beta\},N^\{\\beta\}\]\.
###### Proposition 7\.9\(Anisotropic first\-order bound\)\.

For first\-order stable shaping,

‖ℰN​u−ℰN​q‖L∞​\(ΩN​\(α,β\)\)≤VN​\(1\+2​π​Nα\+2​π​Nβ\)\.\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\_\{N\}\(\\alpha,\\beta\)\)\}\\leq\\frac\{V\}\{N\}\\left\(1\+2\\pi N^\{\\alpha\}\+2\\pi N^\{\\beta\}\\right\)\.\(7\.19\)In particular, the deterministic error tends to zero when

max⁡\{α,β\}<1\.\\max\\\{\\alpha,\\beta\\\}<1\.

The condition is sufficient and sharp for this bound\. It is not a claim about the actual error on every larger region\. Oscillation may produce cancellation that the total variation does not detect\.

#### 7\.3\.4Exact sine variation

For the parabolic weight,

wk\+1−wk=wk​\[e​\(xN\+t​\(2​k\+1\)N2\)−1\]\.w\_\{k\+1\}\-w\_\{k\}=w\_\{k\}\\left\[e\\\!\\left\(\\frac\{x\}\{N\}\+\\frac\{t\(2k\+1\)\}\{N^\{2\}\}\\right\)\-1\\right\]\.Hence

\|wk\+1−wk\|=2​\|sin⁡π​\(xN\+t​\(2​k\+1\)N2\)\|\.\\left\\lvert w\_\{k\+1\}\-w\_\{k\}\\right\\rvert=2\\left\\lvert\\sin\\pi\\left\(\\frac\{x\}\{N\}\+\\frac\{t\(2k\+1\)\}\{N^\{2\}\}\\right\)\\right\\rvert\.\(7\.20\)The derivative estimate replaces the sine by the absolute value of its argument\. The exact formula is better when the increment is close to an integer, but it remains bounded below by a positive constant on many critical\-scale regions\. Summing absolute values can therefore still produce anO​\(N\)O\(N\)variation\.

A periodic resonance can also make the exact variation unexpectedly small\. For example, ifx/Nx/Nandt/Nt/Nare integers arranged so that every increment is integral, then the weight is constant on the sample grid even though the continuous phase varies rapidly\. Such arithmetic effects show that the parameter size alone does not determine the discrete variation\.

#### 7\.3\.5Higher\-order endpoint\-compatible scaling

Assumee=Δr​ve=\\Delta^\{r\}vwith trace compatibility\. For a parabolic phase, the Bell\-polynomial derivative bound has the form

‖W\(r\)‖∞≤Cr​\(1\+\|x\|\+\|t\|\)r,\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{\\infty\}\\leq C\_\{r\}\\left\(1\+\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert\\right\)^\{r\},\(7\.21\)whereCrC\_\{r\}depends only onrr\. Therefore

‖ℰN​u−ℰN​q‖L∞​\(ΩN,α\)≤Cr​V​N−r​\(1−α\)\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\_\{N,\\alpha\}\)\}\\leq C\_\{r\}VN^\{\-r\(1\-\\alpha\)\}\(7\.22\)for0≤α≤10\\leq\\alpha\\leq 1, after the constant is enlarged to cover bounded terms\.

###### Proposition 7\.10\(High\-order subcritical decay\)\.

Ifα<1\\alpha<1, endpoint\-compatible order\-rrshaping yields decay of order at least

N−r​\(1−α\)N^\{\-r\(1\-\\alpha\)\}onΩN,α\\Omega\_\{N,\\alpha\}\. Atα=1\\alpha=1, the derivative method gives onlyO​\(1\)O\(1\)for every fixedrr\.

High order improves the rate inside the subcritical regime but does not move the critical boundary\. This is an important structural conclusion\. Repeated finite differences create additional powers of the small adjacent phase increment\. Once that increment is no longer small, the additional order provides no deterministic gain\.

#### 7\.3\.6A boundary\-layer refinement

The parameter plane can be divided according to the size of

δk​\(x,t\)=xN\+t​\(2​k\+1\)N2\.\\delta\_\{k\}\(x,t\)=\\frac\{x\}\{N\}\+\\frac\{t\(2k\+1\)\}\{N^\{2\}\}\.For a threshold0<ρ<10<\\rho<1, define the good index set

Gρ​\(x,t\)=\{k:dist⁡\(δk​\(x,t\),ℤ\)≤ρ\}G\_\{\\rho\}\(x,t\)=\\\{k:\\operatorname\{dist\}\(\\delta\_\{k\}\(x,t\),\\mathbb\{Z\}\)\\leq\\rho\\\}and the complementary bad setBρ​\(x,t\)B\_\{\\rho\}\(x,t\)\. Then

TVN⁡\(w\)\\displaystyle\\operatorname\{TV\}\_\{N\}\(w\)≤2​π​ρ​\|Gρ​\(x,t\)\|\+2​\|Bρ​\(x,t\)\|\.\\displaystyle\\leq 2\\pi\\rho\\left\\lvert G\_\{\\rho\}\(x,t\)\\right\\rvert\+2\\left\\lvert B\_\{\\rho\}\(x,t\)\\right\\rvert\.This decomposition is exact up to the elementary sine inequality\. It converts the problem into counting how often an affine progression approaches integers\.

At subcritical scale, every index is good withρ≍Nα−1\\rho\\asymp N^\{\\alpha\-1\}\. At critical scale, the distribution of the increments modulo one matters\. This opens a number\-theoretic route: if the progression is equidistributed modulo one, then most sine values are not small, so total variation remains large\. If it is resonant, variation may collapse\. Either case confirms that critical behaviour cannot be described by a single smoothness parameter\.

#### 7\.3\.7Volume factors in localLpL^\{p\}estimates

OnΩN,α\\Omega\_\{N,\\alpha\}, the measure is4​N2​α4N^\{2\\alpha\}\. Combining theL∞L^\{\\infty\}estimate with the trivial embedding gives

‖ℰN​u−ℰN​q‖Lp​\(ΩN,α\)≤Cp,r​V​N2​α/p−r​\(1−α\)\\left\\lVert\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\right\\rVert\_\{L^\{p\}\(\\Omega\_\{N,\\alpha\}\)\}\\leq C\_\{p,r\}VN^\{2\\alpha/p\-r\(1\-\\alpha\)\}\(7\.23\)under endpoint\-compatible order\-rrshaping\. This tends to zero if

r​\(1−α\)\>2​αp\.r\(1\-\\alpha\)\>\\frac\{2\\alpha\}\{p\}\.\(7\.24\)For first order,

α<pp\+2\.\\alpha<\\frac\{p\}\{p\+2\}\.TheLpL^\{p\}condition is stricter than the pointwise condition because the region volume grows\.

This estimate follows from pointwise variation control and the measure of the parameter region\. OscillatoryLpL^\{p\}methods can improve the volume dependence or replace it with a scale\-sensitive extension norm\.

#### 7\.3\.8Physical and torus scalings

The normalized parabolic extension can be rewritten as

ℰN​a​\(x,t\)=1N​∑k=1Nak​e​\(xN​k\+tN2​k2\)\.\\mathcal\{E\}\_\{N\}a\(x,t\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}a\_\{k\}e\\\!\\left\(\\frac\{x\}\{N\}k\+\\frac\{t\}\{N^\{2\}\}k^\{2\}\\right\)\.Introduce torus variables

θ=xN,τ=tN2\.\\theta=\\frac\{x\}\{N\},\\qquad\\tau=\\frac\{t\}\{N^\{2\}\}\.A full period inθ\\thetacorresponds tox∈\[0,N\]x\\in\[0,N\], while a full period inτ\\taucorresponds tot∈\[0,N2\]t\\in\[0,N^\{2\}\]\. Thus the natural full periodic cell is anisotropic:

\[0,N\]×\[0,N2\]\.\[0,N\]\\times\[0,N^\{2\}\]\.The local variation theorem reaches onlyt=o​\(N\)t=o\(N\)without additional cancellation\. There is a wide gap between the variation scale and the full quadratic periodic scale\.

The gap reflects the difference between controlling neighbouring phase increments and controlling an oscillatory sum over a complete period\. Periodic Strichartz estimates, discrete restriction, and Vinogradov mean values operate on the latter scale\[[15](https://arxiv.org/html/2607.24868#bib.bib41),[56](https://arxiv.org/html/2607.24868#bib.bib43),[13](https://arxiv.org/html/2607.24868#bib.bib45),[12](https://arxiv.org/html/2607.24868#bib.bib46),[61](https://arxiv.org/html/2607.24868#bib.bib54)\]\.

#### 7\.3\.9A scale diagram

It is useful to distinguish four regimes\.

RegimeTypical sizeAvailable conclusionFixed local\|x\|\+\|t\|=O​\(1\)\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert=O\(1\)first order givesN−1N^\{\-1\}; compatible orderrrgivesN−rN^\{\-r\}Subcritical growing\|x\|\+\|t\|=O​\(Nα\)\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert=O\(N^\{\\alpha\}\),α<1\\alpha<1deterministic decay survives; rate weakens withα\\alphaCritical adjacent phase\|x\|\+\|t\|=O​\(N\)\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert=O\(N\)total variation gives no decay; arithmetic resonances matterFull quadratic periodx=O​\(N\)x=O\(N\),t=O​\(N2\)t=O\(N^\{2\}\)restriction, Strichartz, VMVT, or decoupling are requiredThe distinction between the critical adjacent\-phase scale and the full quadratic period is essential\. The time parameter reaches its complete torus scale only atN2N^\{2\}\.

#### 7\.3\.10Interpretation of the scale transition

The results of this chapter prove deterministic convergence on explicit subcritical regions\. They do not prove divergence at critical scale\. A large variation bound means only that the triangle\-inequality argument is inconclusive\. Actual cancellation may still occur for particular states, inputs, or parameter averages\.

Compact\-set approximation and global restriction behaviour occur on different scales\. Local sharpness is governed by neighbouring phase increments, whereas global estimates depend on curvature, periodicity, and arithmetic structure\. The next chapter quantifies this scale transition through exact orthogonality identities\.

## Chapter 8LocalLpL^\{p\}Analysis and Oscillatory Transfer

Pointwise variation estimates provide sharp deterministic control on fixed parameter sets, but they cease to yield decay when the observation region reaches the natural oscillatory scale\. The purpose of this chapter is to identify the exact point at which this transition occurs and to develop a rigorous collection of tools for passing from noise\-shaped difference structure to averaged Fourier bounds\. The analysis separates four mechanisms: exact orthogonality, cancellation of low\-order moments, local kernel estimates, and oscillatory transfer through state sums\. The resulting statements apply to arbitrary coefficient sequences where possible and retain the additional structure of errors of the form

whenever that structure produces a genuine improvement\.

The quadratic extension error is written as

FN​\(x,t\):=ℰN​e​\(x,t\)=1N​∑k=1Nek​e​\(x​kN\+t​k2N2\),F\_\{N\}\(x,t\):=\\mathcal\{E\}\_\{N\}e\(x,t\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}\\mathrm\{e\}\\\!\\left\(x\\frac\{k\}\{N\}\+t\\frac\{k^\{2\}\}\{N^\{2\}\}\\right\),\(8\.1\)wheree​\(s\)=e2​π​i​s\\mathrm\{e\}\(s\)=e^\{2\\pi\\mathrm\{i\}s\}\. Unless stated otherwise, the coefficient sequence is complex valued\. For sigma–delta data,ek=uk−qke\_\{k\}=u\_\{k\}\-q\_\{k\}\.

### 8\.1Exact orthogonality and baseline estimates

#### 8\.1\.1Orthogonality in the linear parameter

For fixedtt, the functions

x⟼e​\(x​kN\),k=1,…,N,x\\longmapsto\\mathrm\{e\}\\\!\\left\(x\\frac\{k\}\{N\}\\right\),\\qquad k=1,\\ldots,N,form an orthogonal family on every interval of lengthNN\. This elementary fact determines the exact globalLx2L^\{2\}\_\{x\}scale\.

###### Theorem 8\.1\(Exact linear\-parameter orthogonality\)\.

For everye=\(ek\)k=1Ne=\(e\_\{k\}\)\_\{k=1\}^\{N\}, everyt∈ℝt\\in\\mathbb\{R\}, and everya∈ℝa\\in\\mathbb\{R\},

∫aa\+N\|FN​\(x,t\)\|2​dx=1N​∑k=1N\|ek\|2\.\\int\_\{a\}^\{a\+N\}\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}x=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\\left\\lvert e\_\{k\}\\right\\rvert^\{2\}\.\(8\.2\)

###### Proof\.

Expanding the square gives

∫aa\+N\|FN​\(x,t\)\|2​dx\\displaystyle\\int\_\{a\}^\{a\+N\}\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}x=1N2​∑k,ℓ=1Nek​eℓ¯​e​\(t​k2−ℓ2N2\)​∫aa\+Ne​\(x​k−ℓN\)​dx\.\\displaystyle=\\frac\{1\}\{N^\{2\}\}\\sum\_\{k,\\ell=1\}^\{N\}e\_\{k\}\\overline\{e\_\{\\ell\}\}\\mathrm\{e\}\\\!\\left\(t\\frac\{k^\{2\}\-\\ell^\{2\}\}\{N^\{2\}\}\\right\)\\int\_\{a\}^\{a\+N\}\\mathrm\{e\}\\\!\\left\(x\\frac\{k\-\\ell\}\{N\}\\right\)\\,\\mathrm\{d\}x\.Ifk=ℓk=\\ell, the inner integral equalsNN\. Ifk≠ℓk\\neq\\ell, then

∫aa\+Ne​\(x​k−ℓN\)​dx=N2​π​i​\(k−ℓ\)​\[e​\(\(a\+N\)​k−ℓN\)−e​\(a​k−ℓN\)\]=0,\\int\_\{a\}^\{a\+N\}\\mathrm\{e\}\\\!\\left\(x\\frac\{k\-\\ell\}\{N\}\\right\)\\,\\mathrm\{d\}x=\\frac\{N\}\{2\\pi\\mathrm\{i\}\(k\-\\ell\)\}\\left\[\\mathrm\{e\}\\\!\\left\(\(a\+N\)\\frac\{k\-\\ell\}\{N\}\\right\)\-\\mathrm\{e\}\\\!\\left\(a\\frac\{k\-\\ell\}\{N\}\\right\)\\right\]=0,becausek−ℓ∈ℤk\-\\ell\\in\\mathbb\{Z\}\. Only the diagonal terms remain, which yields

1N2​∑k=1N\|ek\|2​N=1N​∑k=1N\|ek\|2\.\\frac\{1\}\{N^\{2\}\}\\sum\_\{k=1\}^\{N\}\\left\\lvert e\_\{k\}\\right\\rvert^\{2\}N=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\\left\\lvert e\_\{k\}\\right\\rvert^\{2\}\.∎

###### Corollary 8\.2\(Normalised root\-mean\-square scale\)\.

Under the assumptions of Theorem[8\.1](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem1),

\(1N​∫aa\+N\|FN​\(x,t\)\|2​dx\)1/2=‖e‖ℓ2N\.\\left\(\\frac\{1\}\{N\}\\int\_\{a\}^\{a\+N\}\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}x\\right\)^\{1/2\}=\\frac\{\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\}\{N\}\.\(8\.3\)In particular, if\|ek\|≤Ce\\left\\lvert e\_\{k\}\\right\\rvert\\leq C\_\{e\}, then

\(1N​∫aa\+N\|FN​\(x,t\)\|2​dx\)1/2≤Ce​N−1/2\.\\left\(\\frac\{1\}\{N\}\\int\_\{a\}^\{a\+N\}\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}x\\right\)^\{1/2\}\\leq C\_\{e\}N^\{\-1/2\}\.

The exponentN−1/2N^\{\-1/2\}in the normalised average is the generic orthogonality scale for bounded coefficients\. It does not by itself express a sigma–delta gain\. Any improvement that is genuinely caused by noise shaping must use more than theℓ2\\ell^\{2\}\-size ofee\.

#### 8\.1\.2Two\-parameter torus normalisation

Introduce

θ=xN,τ=tN2,\\theta=\\frac\{x\}\{N\},\\qquad\\tau=\\frac\{t\}\{N^\{2\}\},and define

Se\(θ,τ\):=∑k=1Neke\(θk\+τk2\)\.S\_\{e\}\(\\theta,\\tau\):=\\sum\_\{k=1\}^\{N\}e\_\{k\}\\mathrm\{e\}\(\\theta k\+\\tau k^\{2\}\)\.\(8\.4\)Then

FN​\(x,t\)=N−1​Se​\(x/N,t/N2\)\.F\_\{N\}\(x,t\)=N^\{\-1\}S\_\{e\}\(x/N,t/N^\{2\}\)\.The physical cell\[0,N\]×\[0,N2\]\[0,N\]\\times\[0,N^\{2\}\]corresponds to𝕋2\\mathbb\{T\}^\{2\}, and the Jacobian isN3N^\{3\}\. Therefore, for every1≤p<∞1\\leq p<\\infty,

‖FN‖Lp​\(\[0,N\]×\[0,N2\]\)=N3/p−1​‖Se‖Lp​\(𝕋2\)\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{p\}\(\[0,N\]\\times\[0,N^\{2\}\]\)\}=N^\{3/p\-1\}\\left\\lVert S\_\{e\}\\right\\rVert\_\{L^\{p\}\(\\mathbb\{T\}^\{2\}\)\}\.\(8\.5\)
###### Theorem 8\.3\(Exact full\-cellL2L^\{2\}identity\)\.

For every coefficient sequenceee,

∫0N∫0N2\|FN​\(x,t\)\|2​dt​dx=N​∑k=1N\|ek\|2\.\\int\_\{0\}^\{N\}\\int\_\{0\}^\{N^\{2\}\}\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}t\\,\\mathrm\{d\}x=N\\sum\_\{k=1\}^\{N\}\\left\\lvert e\_\{k\}\\right\\rvert^\{2\}\.\(8\.6\)Equivalently,

‖Se‖L2​\(𝕋2\)=‖e‖ℓ2\.\\left\\lVert S\_\{e\}\\right\\rVert\_\{L^\{2\}\(\\mathbb\{T\}^\{2\}\)\}=\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\.

###### Proof\.

The torus identity follows by expanding\|Se\|2\\left\\lvert S\_\{e\}\\right\\rvert^\{2\}and integrating first inθ\\theta:

∫𝕋e​\(θ​\(k−ℓ\)\)​dθ=𝟏\{k=ℓ\}\.\\int\_\{\\mathbb\{T\}\}\\mathrm\{e\}\(\\theta\(k\-\\ell\)\)\\,\\mathrm\{d\}\\theta=\\mathbf\{1\}\_\{\\\{k=\\ell\\\}\}\.The integral inτ\\tauis then equal to one on the remaining diagonal\. Formula \([8\.6](https://arxiv.org/html/2607.24868#Ch8.E6)\) follows from \([8\.5](https://arxiv.org/html/2607.24868#Ch8.E5)\) withp=2p=2\. ∎

#### 8\.1\.3The fourth moment and additive energy

The fourth moment already detects arithmetic interactions among the frequencies\. It is useful to record the exact identity because it distinguishes genericℓ2\\ell^\{2\}information from additional additive structure\.

###### Proposition 8\.4\(Exact fourth\-moment identity\)\.

ForSeS\_\{e\}defined by \([8\.4](https://arxiv.org/html/2607.24868#Ch8.E4)\),

‖Se‖L4​\(𝕋2\)4=∑k1\+k2=k3\+k4k12\+k22=k32\+k42ek1​ek2​ek3​ek4¯\.\\left\\lVert S\_\{e\}\\right\\rVert\_\{L^\{4\}\(\\mathbb\{T\}^\{2\}\)\}^\{4\}=\\sum\_\{\\begin\{subarray\}\{c\}k\_\{1\}\+k\_\{2\}=k\_\{3\}\+k\_\{4\}\\\\ k\_\{1\}^\{2\}\+k\_\{2\}^\{2\}=k\_\{3\}^\{2\}\+k\_\{4\}^\{2\}\\end\{subarray\}\}e\_\{k\_\{1\}\}e\_\{k\_\{2\}\}\\overline\{e\_\{k\_\{3\}\}e\_\{k\_\{4\}\}\}\.\(8\.7\)For indices in\{1,…,N\}\\\{1,\\ldots,N\\\}, the two Diophantine equations imply equality of the unordered pairs

\{k1,k2\}=\{k3,k4\}\.\\\{k\_\{1\},k\_\{2\}\\\}=\\\{k\_\{3\},k\_\{4\}\\\}\.Consequently,

‖Se‖L4​\(𝕋2\)4=2​\(∑k=1N\|ek\|2\)2−∑k=1N\|ek\|4\.\\left\\lVert S\_\{e\}\\right\\rVert\_\{L^\{4\}\(\\mathbb\{T\}^\{2\}\)\}^\{4\}=2\\left\(\\sum\_\{k=1\}^\{N\}\\left\\lvert e\_\{k\}\\right\\rvert^\{2\}\\right\)^\{2\}\-\\sum\_\{k=1\}^\{N\}\\left\\lvert e\_\{k\}\\right\\rvert^\{4\}\.\(8\.8\)

###### Proof\.

Expansion of the fourth power followed by integration over𝕋2\\mathbb\{T\}^\{2\}gives \([8\.7](https://arxiv.org/html/2607.24868#Ch8.E7)\)\. Suppose the two displayed Diophantine equations hold\. Set

s=k1\+k2=k3\+k4\.s=k\_\{1\}\+k\_\{2\}=k\_\{3\}\+k\_\{4\}\.Since

k12\+k22=s2−2​k1​k2andk32\+k42=s2−2​k3​k4,k\_\{1\}^\{2\}\+k\_\{2\}^\{2\}=s^\{2\}\-2k\_\{1\}k\_\{2\}\\quad\\text\{and\}\\quad k\_\{3\}^\{2\}\+k\_\{4\}^\{2\}=s^\{2\}\-2k\_\{3\}k\_\{4\},it follows thatk1​k2=k3​k4k\_\{1\}k\_\{2\}=k\_\{3\}k\_\{4\}\. Thus the pairs\(k1,k2\)\(k\_\{1\},k\_\{2\}\)and\(k3,k4\)\(k\_\{3\},k\_\{4\}\)are roots of the same quadratic polynomial

z2−s​z\+k1​k2,z^\{2\}\-sz\+k\_\{1\}k\_\{2\},so their unordered pairs coincide\. There are two ordered matches whenk1≠k2k\_\{1\}\\neq k\_\{2\}and one whenk1=k2k\_\{1\}=k\_\{2\}\. Summing the corresponding coefficient products yields \([8\.8](https://arxiv.org/html/2607.24868#Ch8.E8)\)\. ∎

###### Corollary 8\.5\(Sharp quadraticL4L^\{4\}bound\)\.

For everyee,

‖Se‖L4​\(𝕋2\)≤21/4​‖e‖ℓ2\.\\left\\lVert S\_\{e\}\\right\\rVert\_\{L^\{4\}\(\\mathbb\{T\}^\{2\}\)\}\\leq 2^\{1/4\}\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\.Hence

‖FN‖L4​\(\[0,N\]×\[0,N2\]\)≤21/4​N−1/4​‖e‖ℓ2\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{4\}\(\[0,N\]\\times\[0,N^\{2\}\]\)\}\\leq 2^\{1/4\}N^\{\-1/4\}\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\.\(8\.9\)

###### Proof\.

Equation \([8\.8](https://arxiv.org/html/2607.24868#Ch8.E8)\) gives

‖Se‖L44≤2​‖e‖ℓ24\.\\left\\lVert S\_\{e\}\\right\\rVert\_\{L^\{4\}\}^\{4\}\\leq 2\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}^\{4\}\.The physical estimate follows from \([8\.5](https://arxiv.org/html/2607.24868#Ch8.E5)\) withp=4p=4\. ∎

#### 8\.1\.4The sixth moment and quadratic discrete restriction

The critical moment for the parabola isp=6p=6\. The quadratic Vinogradov mean value theorem and theℓ2\\ell^\{2\}decoupling theorem imply that, for everyε\>0\\varepsilon\>0,

‖Se‖L6​\(𝕋2\)≤Cε​Nε​‖e‖ℓ2\.\\left\\lVert S\_\{e\}\\right\\rVert\_\{L^\{6\}\(\\mathbb\{T\}^\{2\}\)\}\\leq C\_\{\\varepsilon\}N^\{\\varepsilon\}\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\.\(8\.10\)This estimate belongs to the established theory of periodic Strichartz estimates, discrete restriction, Vinogradov mean values, and decoupling\[[15](https://arxiv.org/html/2607.24868#bib.bib41),[16](https://arxiv.org/html/2607.24868#bib.bib42),[56](https://arxiv.org/html/2607.24868#bib.bib43),[13](https://arxiv.org/html/2607.24868#bib.bib45),[12](https://arxiv.org/html/2607.24868#bib.bib46),[28](https://arxiv.org/html/2607.24868#bib.bib55),[81](https://arxiv.org/html/2607.24868#bib.bib63),[82](https://arxiv.org/html/2607.24868#bib.bib65)\]\.

###### Corollary 8\.6\(Critical sixth\-moment estimate on the physical cell\)\.

For everyε\>0\\varepsilon\>0,

‖FN‖L6​\(\[0,N\]×\[0,N2\]\)≤Cε​N−1/2\+ε​‖e‖ℓ2\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{6\}\(\[0,N\]\\times\[0,N^\{2\}\]\)\}\\leq C\_\{\\varepsilon\}N^\{\-1/2\+\\varepsilon\}\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\.\(8\.11\)If\|ek\|≤Ce\\left\\lvert e\_\{k\}\\right\\rvert\\leq C\_\{e\}, then

‖FN‖L6​\(\[0,N\]×\[0,N2\]\)≤Cε​Ce​Nε\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{6\}\(\[0,N\]\\times\[0,N^\{2\}\]\)\}\\leq C\_\{\\varepsilon\}C\_\{e\}N^\{\\varepsilon\}\.

###### Proof\.

Apply \([8\.5](https://arxiv.org/html/2607.24868#Ch8.E5)\) withp=6p=6:

N3/6−1=N−1/2\.N^\{3/6\-1\}=N^\{\-1/2\}\.Then use \([8\.10](https://arxiv.org/html/2607.24868#Ch8.E10)\)\. The final assertion follows from‖e‖ℓ2≤Ce​N1/2\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\\leq C\_\{e\}N^\{1/2\}\. ∎

The estimate shows that standard discrete restriction is scale sharp for arbitrary bounded coefficients, but it does not automatically provide decay for unit\-size sigma–delta errors on the full parabolic cell\. A stronger result must preserve the difference structure instead of replacing it immediately by‖e‖ℓ2\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\.

### 8\.2Interpolation and localLpL^\{p\}consequences

#### 8\.2\.1Interpolation on the full parabolic cell

The exactL2L^\{2\}identity, the sharpL4L^\{4\}estimate, and the criticalL6L^\{6\}estimate yield a complete family of baseline bounds\.

###### Proposition 8\.7\(Baseline bounds for2≤p≤62\\leq p\\leq 6\)\.

Let2≤p≤62\\leq p\\leq 6\. For everyε\>0\\varepsilon\>0,

‖Se‖Lp​\(𝕋2\)≤Cp,ε​Nε​‖e‖ℓ2\.\\left\\lVert S\_\{e\}\\right\\rVert\_\{L^\{p\}\(\\mathbb\{T\}^\{2\}\)\}\\leq C\_\{p,\\varepsilon\}N^\{\\varepsilon\}\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\.\(8\.12\)Consequently,

‖FN‖Lp​\(\[0,N\]×\[0,N2\]\)≤Cp,ε​N3/p−1\+ε​‖e‖ℓ2\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{p\}\(\[0,N\]\\times\[0,N^\{2\}\]\)\}\\leq C\_\{p,\\varepsilon\}N^\{3/p\-1\+\\varepsilon\}\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\.\(8\.13\)For2≤p≤42\\leq p\\leq 4, the factorNεN^\{\\varepsilon\}may be omitted\.

###### Proof\.

For2≤p≤42\\leq p\\leq 4, interpolate between Theorem[8\.3](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem3)and Corollary[8\.5](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem5)\. For4≤p≤64\\leq p\\leq 6, interpolate between Corollary[8\.5](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem5)and \([8\.10](https://arxiv.org/html/2607.24868#Ch8.E10)\)\. Formula \([8\.13](https://arxiv.org/html/2607.24868#Ch8.E13)\) follows from the scaling relation \([8\.5](https://arxiv.org/html/2607.24868#Ch8.E5)\)\. ∎

#### 8\.2\.2Fixed compact sets

On a fixed compact set, the deterministic pointwise theory is stronger than the baseline restriction estimates\. Suppose first\-order shaping satisfies

ek=vk−vk−1,‖v‖ℓ∞≤V,v0=0\.e\_\{k\}=v\_\{k\}\-v\_\{k\-1\},\\qquad\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V,\\qquad v\_\{0\}=0\.The parabolic variation bound from Chapter[5](https://arxiv.org/html/2607.24868#Ch5)gives

\|FN​\(x,t\)\|≤VN​\[1\+2​π​J​\(x,t\)\],\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert\\leq\\frac\{V\}\{N\}\\bigl\[1\+2\\pi J\(x,t\)\\bigr\],\(8\.14\)where

J​\(x,t\)=∫01\|x\+2​t​ξ\|​dξ\.J\(x,t\)=\\int\_\{0\}^\{1\}\\left\\lvert x\+2t\\xi\\right\\rvert\\,\\mathrm\{d\}\\xi\.
###### Corollary 8\.8\(Fixed\-setLpL^\{p\}convergence\)\.

LetK⊂ℝ2K\\subset\\mathbb\{R\}^\{2\}be measurable with finite measure, and set

MK:=sup\(x,t\)∈KJ​\(x,t\)<∞\.M\_\{K\}:=\\sup\_\{\(x,t\)\\in K\}J\(x,t\)<\\infty\.Then, for1≤p<∞1\\leq p<\\infty,

‖FN‖Lp​\(K\)≤\|K\|1/p​VN​\(1\+2​π​MK\),\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{p\}\(K\)\}\\leq\\left\\lvert K\\right\\rvert^\{1/p\}\\frac\{V\}\{N\}\(1\+2\\pi M\_\{K\}\),\(8\.15\)and

‖FN‖L∞​\(K\)≤VN​\(1\+2​π​MK\)\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{\\infty\}\(K\)\}\\leq\\frac\{V\}\{N\}\(1\+2\\pi M\_\{K\}\)\.

###### Proof\.

Apply \([8\.14](https://arxiv.org/html/2607.24868#Ch8.E14)\) and the elementary embedding

‖f‖Lp​\(K\)≤\|K\|1/p​‖f‖L∞​\(K\)\.\\left\\lVert f\\right\\rVert\_\{L^\{p\}\(K\)\}\\leq\\left\\lvert K\\right\\rvert^\{1/p\}\\left\\lVert f\\right\\rVert\_\{L^\{\\infty\}\(K\)\}\.∎

#### 8\.2\.3Growing rectangles and the deterministic threshold

Let

ΩRx,Rt:=\[−Rx,Rx\]×\[−Rt,Rt\]\.\\Omega\_\{R\_\{x\},R\_\{t\}\}:=\[\-R\_\{x\},R\_\{x\}\]\\times\[\-R\_\{t\},R\_\{t\}\]\.SinceJ​\(x,t\)≤\|x\|\+\|t\|J\(x,t\)\\leq\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert,

‖FN‖L∞​\(ΩRx,Rt\)≤VN​\[1\+2​π​\(Rx\+Rt\)\]\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\_\{R\_\{x\},R\_\{t\}\}\)\}\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\(R\_\{x\}\+R\_\{t\}\)\\right\]\.\(8\.16\)
###### Proposition 8\.9\(Anisotropic growing\-region estimate\)\.

For1≤p<∞1\\leq p<\\infty,

‖FN‖Lp​\(ΩRx,Rt\)≤\(4​Rx​Rt\)1/p​VN​\[1\+2​π​\(Rx\+Rt\)\]\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{p\}\(\\Omega\_\{R\_\{x\},R\_\{t\}\}\)\}\\leq\(4R\_\{x\}R\_\{t\}\)^\{1/p\}\\frac\{V\}\{N\}\\left\[1\+2\\pi\(R\_\{x\}\+R\_\{t\}\)\\right\]\.\(8\.17\)In particular, if

Rx=Nα,Rt=Nβ,R\_\{x\}=N^\{\\alpha\},\\qquad R\_\{t\}=N^\{\\beta\},then

‖FN‖Lp​\(ΩNα,Nβ\)≲pV​N−1\+\(α\+β\)/p\+max⁡\{α,β,0\}\.\\left\\lVert F\_\{N\}\\right\\rVert\_\{L^\{p\}\(\\Omega\_\{N^\{\\alpha\},N^\{\\beta\}\}\)\}\\lesssim\_\{p\}VN^\{\-1\+\(\\alpha\+\\beta\)/p\+\\max\\\{\\alpha,\\beta,0\\\}\}\.\(8\.18\)

###### Proof\.

The volume of the rectangle is4​Rx​Rt4R\_\{x\}R\_\{t\}\. Combining this fact with \([8\.16](https://arxiv.org/html/2607.24868#Ch8.E16)\) proves \([8\.17](https://arxiv.org/html/2607.24868#Ch8.E17)\)\. The power estimate follows by retaining the largest power among11,NαN^\{\\alpha\}, andNβN^\{\\beta\}\. ∎

Formula \([8\.18](https://arxiv.org/html/2607.24868#Ch8.E18)\) gives the exact threshold of the direct variation method\. Once the exponent on the right becomes nonnegative, further decay can only come from oscillatory cancellation, additional state regularity, stronger endpoint conditions, or an averaging norm that is adapted to the geometry\.

### 8\.3LocalL2L^\{2\}analysis by oscillatory kernels

#### 8\.3\.1The exact kernel identity

LetI=\[a,a\+R\]⊂ℝI=\[a,a\+R\]\\subset\\mathbb\{R\}, where0<R≤N0<R\\leq N, and define

KI​\(m\):=∫Ie​\(x​mN\)​dx,m∈ℤ\.K\_\{I\}\(m\):=\\int\_\{I\}\\mathrm\{e\}\\\!\\left\(x\\frac\{m\}\{N\}\\right\)\\,\\mathrm\{d\}x,\\qquad m\\in\\mathbb\{Z\}\.\(8\.19\)Then

and, form≠0m\\neq 0,

KI​\(m\)=e​\(mN​\(a\+R2\)\)​N​sin⁡\(π​m​R/N\)π​m\.K\_\{I\}\(m\)=\\mathrm\{e\}\\\!\\left\(\\frac\{m\}\{N\}\\left\(a\+\\frac\{R\}\{2\}\\right\)\\right\)\\frac\{N\\sin\(\\pi mR/N\)\}\{\\pi m\}\.\(8\.20\)Hence

\|KI​\(m\)\|≤min⁡\{R,Nπ​\|m\|\}\.\\left\\lvert K\_\{I\}\(m\)\\right\\rvert\\leq\\min\\left\\\{R,\\frac\{N\}\{\\pi\\left\\lvert m\\right\\rvert\}\\right\\\}\.\(8\.21\)
###### Proposition 8\.10\(Exact localLx2L^\{2\}\_\{x\}kernel formula\)\.

For every fixedt∈ℝt\\in\\mathbb\{R\},

∫I\|FN​\(x,t\)\|2​dx=1N2​∑k,ℓ=1Nek​eℓ¯​e​\(t​k2−ℓ2N2\)​KI​\(k−ℓ\)\.\\int\_\{I\}\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}x=\\frac\{1\}\{N^\{2\}\}\\sum\_\{k,\\ell=1\}^\{N\}e\_\{k\}\\overline\{e\_\{\\ell\}\}\\mathrm\{e\}\\\!\\left\(t\\frac\{k^\{2\}\-\\ell^\{2\}\}\{N^\{2\}\}\\right\)K\_\{I\}\(k\-\\ell\)\.\(8\.22\)

###### Proof\.

Expand\|FN​\(x,t\)\|2\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}, interchange the finite sum and the integral, and use the definition \([8\.19](https://arxiv.org/html/2607.24868#Ch8.E19)\)\. ∎

#### 8\.3\.2A deterministic localL2L^\{2\}bound

The kernel estimate produces a useful baseline inequality without any assumption on the difference structure\.

###### Theorem 8\.11\(Generic localL2L^\{2\}bound\)\.

For every intervalIIof length0<R≤N0<R\\leq N, everyt∈ℝt\\in\\mathbb\{R\}, and every coefficient sequenceee,

∫I\|FN​\(x,t\)\|2​dx≤R\+C​N​log⁡\(2​N\)N2​‖e‖ℓ22,\\int\_\{I\}\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}x\\leq\\frac\{R\+CN\\log\(2N\)\}\{N^\{2\}\}\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}^\{2\},\(8\.23\)whereC\>0C\>0is an absolute constant\.

###### Proof\.

LetA=\(Ak​ℓ\)A=\(A\_\{k\\ell\}\)be the matrix

Ak​ℓ=e​\(t​k2−ℓ2N2\)​KI​\(k−ℓ\)\.A\_\{k\\ell\}=\\mathrm\{e\}\\\!\\left\(t\\frac\{k^\{2\}\-\\ell^\{2\}\}\{N^\{2\}\}\\right\)K\_\{I\}\(k\-\\ell\)\.The phase factor has modulus one\. By \([8\.21](https://arxiv.org/html/2607.24868#Ch8.E21)\), every row satisfies

∑ℓ=1N\|Ak​ℓ\|\\displaystyle\\sum\_\{\\ell=1\}^\{N\}\\left\\lvert A\_\{k\\ell\}\\right\\rvert≤R\+2​∑m=1N−1min⁡\{R,Nπ​m\}\\displaystyle\\leq R\+2\\sum\_\{m=1\}^\{N\-1\}\\min\\left\\\{R,\\frac\{N\}\{\\pi m\}\\right\\\}≤R\+2​Nπ​∑m=1N−11m\\displaystyle\\leq R\+\\frac\{2N\}\{\\pi\}\\sum\_\{m=1\}^\{N\-1\}\\frac\{1\}\{m\}≤R\+C​N​log⁡\(2​N\)\.\\displaystyle\\leq R\+CN\\log\(2N\)\.The same bound holds for every column\. Schur’s test therefore gives

\|∑k,ℓek​eℓ¯​Ak​ℓ\|≤\(R\+C​N​log⁡\(2​N\)\)​‖e‖ℓ22\.\\left\\lvert\\sum\_\{k,\\ell\}e\_\{k\}\\overline\{e\_\{\\ell\}\}A\_\{k\\ell\}\\right\\rvert\\leq\\left\(R\+CN\\log\(2N\)\\right\)\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}^\{2\}\.Division byN2N^\{2\}proves the result\. ∎

The logarithmic loss is not sharp whenRRis comparable withNN, because exact orthogonality is then available\. Its role is to show explicitly that localising the linear parameter creates off\-diagonal interactions\. Removing the logarithm or extracting a difference\-structure gain requires a more refined analysis of the oscillatory matrix in \([8\.22](https://arxiv.org/html/2607.24868#Ch8.E22)\)\.

#### 8\.3\.3Double summation by parts for first\-order shaped errors

Assume

ek=Δ​vk=vk−vk−1,v0=0\.e\_\{k\}=\\Delta v\_\{k\}=v\_\{k\}\-v\_\{k\-1\},\\qquad v\_\{0\}=0\.For fixedtt, define

𝒦k​ℓ:=e​\(t​k2−ℓ2N2\)​KI​\(k−ℓ\)\.\\mathcal\{K\}\_\{k\\ell\}:=\\mathrm\{e\}\\\!\\left\(t\\frac\{k^\{2\}\-\\ell^\{2\}\}\{N^\{2\}\}\\right\)K\_\{I\}\(k\-\\ell\)\.\(8\.24\)Then \([8\.22](https://arxiv.org/html/2607.24868#Ch8.E22)\) is the pairing ofΔ​v\\Delta vwithΔ​v¯\\Delta\\overline\{v\}against𝒦\\mathcal\{K\}\. Repeated discrete summation by parts moves one difference from each state factor onto the two\-index kernel\. The following formula records the interior term and the boundary contributions separately\.

###### Proposition 8\.12\(Two\-index summation\-by\-parts identity\)\.

Letv0=0v\_\{0\}=0,ek=Δ​vke\_\{k\}=\\Delta v\_\{k\}, and let𝒦k​ℓ\\mathcal\{K\}\_\{k\\ell\}be any complex matrix indexed by1≤k,ℓ≤N1\\leq k,\\ell\\leq N\. Define

\(Δ\+,1​𝒦\)k​ℓ:=𝒦k\+1,ℓ−𝒦k​ℓ,\(Δ\+,2​𝒦\)k​ℓ:=𝒦k,ℓ\+1−𝒦k​ℓ\.\(\\Delta\_\{\+,1\}\\mathcal\{K\}\)\_\{k\\ell\}:=\\mathcal\{K\}\_\{k\+1,\\ell\}\-\\mathcal\{K\}\_\{k\\ell\},\\qquad\(\\Delta\_\{\+,2\}\\mathcal\{K\}\)\_\{k\\ell\}:=\\mathcal\{K\}\_\{k,\\ell\+1\}\-\\mathcal\{K\}\_\{k\\ell\}\.Then

∑k,ℓ=1Nek​eℓ¯​𝒦k​ℓ\\displaystyle\\sum\_\{k,\\ell=1\}^\{N\}e\_\{k\}\\overline\{e\_\{\\ell\}\}\\mathcal\{K\}\_\{k\\ell\}=vN​vN¯​𝒦N​N−vN​∑ℓ=1N−1vℓ¯​\(Δ\+,2​𝒦\)N,ℓ\\displaystyle=v\_\{N\}\\overline\{v\_\{N\}\}\\mathcal\{K\}\_\{NN\}\-v\_\{N\}\\sum\_\{\\ell=1\}^\{N\-1\}\\overline\{v\_\{\\ell\}\}\(\\Delta\_\{\+,2\}\\mathcal\{K\}\)\_\{N,\\ell\}−vN¯​∑k=1N−1vk​\(Δ\+,1​𝒦\)k,N\\displaystyle\\quad\-\\overline\{v\_\{N\}\}\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(\\Delta\_\{\+,1\}\\mathcal\{K\}\)\_\{k,N\}\+∑k,ℓ=1N−1vk​vℓ¯​\(Δ\+,1​Δ\+,2​𝒦\)k​ℓ\.\\displaystyle\\quad\+\\sum\_\{k,\\ell=1\}^\{N\-1\}v\_\{k\}\\overline\{v\_\{\\ell\}\}\(\\Delta\_\{\+,1\}\\Delta\_\{\+,2\}\\mathcal\{K\}\)\_\{k\\ell\}\.\(8\.25\)IfvN=0v\_\{N\}=0, only the final double sum remains\.

###### Proof\.

For each fixedℓ\\ell, one\-dimensional summation by parts inkkgives

∑k=1N\(vk−vk−1\)​𝒦k​ℓ=vN​𝒦N​ℓ−∑k=1N−1vk​\(Δ\+,1​𝒦\)k​ℓ\.\\sum\_\{k=1\}^\{N\}\(v\_\{k\}\-v\_\{k\-1\}\)\\mathcal\{K\}\_\{k\\ell\}=v\_\{N\}\\mathcal\{K\}\_\{N\\ell\}\-\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(\\Delta\_\{\+,1\}\\mathcal\{K\}\)\_\{k\\ell\}\.Multiply byeℓ¯\\overline\{e\_\{\\ell\}\}and sum inℓ\\ell\. Apply the same identity to the conjugate state difference in each of the two resulting terms\. The first produces

vN​\(vN¯​𝒦N​N−∑ℓ=1N−1vℓ¯​\(Δ\+,2​𝒦\)N,ℓ\),v\_\{N\}\\left\(\\overline\{v\_\{N\}\}\\mathcal\{K\}\_\{NN\}\-\\sum\_\{\\ell=1\}^\{N\-1\}\\overline\{v\_\{\\ell\}\}\(\\Delta\_\{\+,2\}\\mathcal\{K\}\)\_\{N,\\ell\}\\right\),while the second produces

−vN¯​∑k=1N−1vk​\(Δ\+,1​𝒦\)k,N\+∑k,ℓ=1N−1vk​vℓ¯​\(Δ\+,1​Δ\+,2​𝒦\)k​ℓ\.\-\\overline\{v\_\{N\}\}\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(\\Delta\_\{\+,1\}\\mathcal\{K\}\)\_\{k,N\}\+\\sum\_\{k,\\ell=1\}^\{N\-1\}v\_\{k\}\\overline\{v\_\{\\ell\}\}\(\\Delta\_\{\+,1\}\\Delta\_\{\+,2\}\\mathcal\{K\}\)\_\{k\\ell\}\.Combining the terms proves \([8\.12](https://arxiv.org/html/2607.24868#Ch8.Ex34)\)\. ∎

This identity is the correct starting point for a structured localL2L^\{2\}theorem\. A direct absolute\-value estimate of the final double sum generally loses the oscillatory gain\. The mixed differenceΔ\+,1​Δ\+,2​𝒦\\Delta\_\{\+,1\}\\Delta\_\{\+,2\}\\mathcal\{K\}must instead be analysed as a two\-parameter oscillatory kernel\.

### 8\.4Moment cancellation and endpoint\-compatible shaping

#### 8\.4\.1Exact annihilation of low\-degree polynomial tests

Higher\-order noise shaping acquires additional cancellation only when its finite\-record boundary traces are controlled\. Let

The finite summation\-by\-parts identity established in Chapter[6](https://arxiv.org/html/2607.24868#Ch6)consists of an interior term involvingΔ\+r​w\\Delta\_\{\+\}^\{r\}wand a collection of boundary traces\. If all traces of order belowrrvanish, the boundary terms disappear\.

###### Theorem 8\.13\(Discrete vanishing moments\)\.

Assumee=Δr​ve=\\Delta^\{r\}vand endpoint compatibility of orderrr, so that the pure interior identity

∑k=1Nek​wk=\(−1\)r​∑k=1N−rvk​\(Δ\+r​w\)k\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}=\(\-1\)^\{r\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\(8\.26\)holds\. Then

∑k=1Nek​P​\(k\)=0\\sum\_\{k=1\}^\{N\}e\_\{k\}P\(k\)=0\(8\.27\)for every polynomialPPof degree at mostr−1r\-1\.

###### Proof\.

Setwk=P​\(k\)w\_\{k\}=P\(k\)in \([8\.26](https://arxiv.org/html/2607.24868#Ch8.E26)\)\. Therrth forward difference of a polynomial of degree at mostr−1r\-1vanishes identically\. Hence the right\-hand side is zero\. ∎

#### 8\.4\.2Taylor subtraction for smooth weights

The vanishing moments permit a direct Taylor\-remainder argument\. This gives an alternative proof of high\-order decay and makes the role of low\-order polynomial modes transparent\.

###### Theorem 8\.15\(Taylor\-subtraction estimate\)\.

Assume the hypotheses of Theorem[8\.13](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem13)\. LetW∈Cr​\(\[0,1\]\)W\\in C^\{r\}\(\[0,1\]\)and setwk=W​\(k/N\)w\_\{k\}=W\(k/N\)\. Then

\|1N​∑k=1Nek​W​\(k/N\)\|≤‖v‖ℓ∞Nr​∫01\|W\(r\)​\(ξ\)\|​dξ\.\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}W\(k/N\)\\right\\rvert\\leq\\frac\{\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\}\{N^\{r\}\}\\int\_\{0\}^\{1\}\\left\\lvert W^\{\(r\)\}\(\\xi\)\\right\\rvert\\,\\mathrm\{d\}\\xi\.\(8\.28\)

###### Proof\.

The pure interior identity gives

1N​∑k=1Nek​wk=\(−1\)rN​∑k=1N−rvk​\(Δ\+r​w\)k\.\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}=\\frac\{\(\-1\)^\{r\}\}\{N\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\.The integral representation of the forward difference is

\(Δ\+r​w\)k=∫\[0,1/N\]rW\(r\)​\(kN\+s1\+⋯\+sr\)​ds1​⋯​dsr\.\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}=\\int\_\{\[0,1/N\]^\{r\}\}W^\{\(r\)\}\\\!\\left\(\\frac\{k\}\{N\}\+s\_\{1\}\+\\cdots\+s\_\{r\}\\right\)\\,\\mathrm\{d\}s\_\{1\}\\cdots\\,\\mathrm\{d\}s\_\{r\}\.\(8\.29\)Therefore

∑k=1N−r\|\(Δ\+r​w\)k\|\\displaystyle\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\\right\\rvert≤∫\[0,1/N\]r∑k=1N−r\|W\(r\)​\(k/N\+s1\+⋯\+sr\)\|​d​s1​⋯​d​sr\.\\displaystyle\\leq\\int\_\{\[0,1/N\]^\{r\}\}\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert W^\{\(r\)\}\(k/N\+s\_\{1\}\+\\cdots\+s\_\{r\}\)\\right\\rvert\\,\\mathrm\{d\}s\_\{1\}\\cdots\\,\\mathrm\{d\}s\_\{r\}\.For each fixed\(s1,…,sr\)\(s\_\{1\},\\ldots,s\_\{r\}\), the shifted sampling points have spacing1/N1/N\. Bounding the corresponding Riemann sum by the integral over\[0,1\]\[0,1\], with the standard endpoint enlargement already absorbed by the exact integral representation, gives

∑k=1N−r\|\(Δ\+r​w\)k\|≤N1−r​∫01\|W\(r\)​\(ξ\)\|​dξ\.\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\\right\\rvert\\leq N^\{1\-r\}\\int\_\{0\}^\{1\}\\left\\lvert W^\{\(r\)\}\(\\xi\)\\right\\rvert\\,\\mathrm\{d\}\\xi\.Multiplication byN−1​‖v‖ℓ∞N^\{\-1\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}proves \([8\.28](https://arxiv.org/html/2607.24868#Ch8.E28)\)\. ∎

#### 8\.4\.3Polynomial\-phase weights

Let

ϕ𝐚​\(ξ\)=a1​ξ\+a2​ξ2\+⋯\+ad​ξd,W𝐚​\(ξ\)=e​\(ϕ𝐚​\(ξ\)\)\.\\phi\_\{\\mathbf\{a\}\}\(\\xi\)=a\_\{1\}\\xi\+a\_\{2\}\\xi^\{2\}\+\\cdots\+a\_\{d\}\\xi^\{d\},\\qquad W\_\{\\mathbf\{a\}\}\(\\xi\)=\\mathrm\{e\}\(\\phi\_\{\\mathbf\{a\}\}\(\\xi\)\)\.Faà di Bruno’s formula expressesW𝐚\(r\)W\_\{\\mathbf\{a\}\}^\{\(r\)\}asW𝐚W\_\{\\mathbf\{a\}\}times a complete Bell polynomial in

2​π​i​ϕ𝐚′,…,2​π​i​ϕ𝐚\(r\)\.2\\pi\\mathrm\{i\}\\phi\_\{\\mathbf\{a\}\}^\{\\prime\},\\ldots,2\\pi\\mathrm\{i\}\\phi\_\{\\mathbf\{a\}\}^\{\(r\)\}\.Consequently,

\|W𝐚\(r\)​\(ξ\)\|≤Cr​∑m1\+2​m2\+⋯\+r​mr=r∏j=1r\|ϕ𝐚\(j\)​\(ξ\)\|mj,\\left\\lvert W\_\{\\mathbf\{a\}\}^\{\(r\)\}\(\\xi\)\\right\\rvert\\leq C\_\{r\}\\sum\_\{\\begin\{subarray\}\{c\}m\_\{1\}\+2m\_\{2\}\+\\cdots\+rm\_\{r\}=r\\end\{subarray\}\}\\prod\_\{j=1\}^\{r\}\\left\\lvert\\phi\_\{\\mathbf\{a\}\}^\{\(j\)\}\(\\xi\)\\right\\rvert^\{m\_\{j\}\},\(8\.30\)where the constant depends only onrr\. Combining this estimate with Theorem[8\.15](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem15)gives explicit high\-order compact\-set bounds for every polynomial phase\.

For the quadratic phaseϕx,t​\(ξ\)=x​ξ\+t​ξ2\\phi\_\{x,t\}\(\\xi\)=x\\xi\+t\\xi^\{2\}, the first two derivatives are

ϕx,t′​\(ξ\)=x\+2​t​ξ,ϕx,t′′​\(ξ\)=2​t,\\phi\_\{x,t\}^\{\\prime\}\(\\xi\)=x\+2t\\xi,\\qquad\\phi\_\{x,t\}^\{\\prime\\prime\}\(\\xi\)=2t,and all higher derivatives vanish\. ThusWx,t\(r\)W\_\{x,t\}^\{\(r\)\}is a polynomial inx\+2​t​ξx\+2t\\xiandtt, multiplied byWx,tW\_\{x,t\}\. On a fixed rectangle\|x\|≤Rx\\left\\lvert x\\right\\rvert\\leq R\_\{x\},\|t\|≤Rt\\left\\lvert t\\right\\rvert\\leq R\_\{t\},

supξ∈\[0,1\]\|Wx,t\(r\)​\(ξ\)\|≤Cr​\(1\+Rx\+Rt\)r\.\\sup\_\{\\xi\\in\[0,1\]\}\\left\\lvert W\_\{x,t\}^\{\(r\)\}\(\\xi\)\\right\\rvert\\leq C\_\{r\}\(1\+R\_\{x\}\+R\_\{t\}\)^\{r\}\.\(8\.31\)
###### Corollary 8\.17\(Endpoint\-compatible high\-order compact\-set bound\)\.

Assumee=Δr​ve=\\Delta^\{r\}v, endpoint compatibility of orderrr, and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\. Then, on every fixed rectangleΩRx,Rt\\Omega\_\{R\_\{x\},R\_\{t\}\},

‖ℰN​e‖L∞​\(ΩRx,Rt\)≤Cr,Rx,Rt​V​N−r\.\\left\\lVert\\mathcal\{E\}\_\{N\}e\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\_\{R\_\{x\},R\_\{t\}\}\)\}\\leq C\_\{r,R\_\{x\},R\_\{t\}\}VN^\{\-r\}\.\(8\.32\)For1≤p<∞1\\leq p<\\infty,

‖ℰN​e‖Lp​\(ΩRx,Rt\)≤\(4​Rx​Rt\)1/p​Cr,Rx,Rt​V​N−r\.\\left\\lVert\\mathcal\{E\}\_\{N\}e\\right\\rVert\_\{L^\{p\}\(\\Omega\_\{R\_\{x\},R\_\{t\}\}\)\}\\leq\(4R\_\{x\}R\_\{t\}\)^\{1/p\}C\_\{r,R\_\{x\},R\_\{t\}\}VN^\{\-r\}\.\(8\.33\)

###### Proof\.

Apply Theorem[8\.15](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem15)withW=Wx,tW=W\_\{x,t\}, then use \([8\.31](https://arxiv.org/html/2607.24868#Ch8.E31)\)\. TheLpL^\{p\}estimate follows from the volume of the rectangle\. ∎

### 8\.5Exact transfer formulas for noise\-shaped states

#### 8\.5\.1First\-order phase factorisation

Let

wk=e​\(ϕk\)w\_\{k\}=\\mathrm\{e\}\(\\phi\_\{k\}\)and define the phase increment

δk​ϕ:=ϕk\+1−ϕk\.\\delta\_\{k\}\\phi:=\\phi\_\{k\+1\}\-\\phi\_\{k\}\.Then

Δ\+​wk=wk​\[e​\(δk​ϕ\)−1\]\.\\Delta\_\{\+\}w\_\{k\}=w\_\{k\}\\left\[\\mathrm\{e\}\(\\delta\_\{k\}\\phi\)\-1\\right\]\.\(8\.34\)Set

mk:=e​\(δk​ϕ\)−1\.m\_\{k\}:=\\mathrm\{e\}\(\\delta\_\{k\}\\phi\)\-1\.\(8\.35\)Ifek=vk−vk−1e\_\{k\}=v\_\{k\}\-v\_\{k\-1\}, then finite summation by parts yields

ℰN​e=vN​wN−v0​w1N−1N​∑k=1N−1vk​mk​wk\.\\mathcal\{E\}\_\{N\}e=\\frac\{v\_\{N\}w\_\{N\}\-v\_\{0\}w\_\{1\}\}\{N\}\-\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}m\_\{k\}w\_\{k\}\.\(8\.36\)This identity is exact and contains the entire first\-order problem\. The total\-variation argument bounds the state sum by

‖v‖ℓ∞​∑\|mk\|\.\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\sum\\left\\lvert m\_\{k\}\\right\\rvert\.Oscillatory improvement requires a smaller estimate for the signed complex sum itself\.

For the parabolic phase

ϕk=x​kN\+t​k2N2,\\phi\_\{k\}=x\\frac\{k\}\{N\}\+t\\frac\{k^\{2\}\}\{N^\{2\}\},the increment is

δk​ϕ=xN\+t​\(2​k\+1\)N2\.\\delta\_\{k\}\\phi=\\frac\{x\}\{N\}\+\\frac\{t\(2k\+1\)\}\{N^\{2\}\}\.\(8\.37\)Thus the multiplier is a slowly varying chirp at fixed scale and becomes order one whenxxorttapproaches the critical scale\.

#### 8\.5\.2A deterministic transfer theorem

###### Theorem 8\.18\(First\-order oscillatory transfer\)\.

LetΩ\\Omegabe a parameter set and let𝒱N\\mathcal\{V\}\_\{N\}be a class of state sequences\. Suppose that

supλ∈Ω\|∑k=1N−1vk​mk​\(λ\)​wk​\(λ\)\|≤AN​‖v‖ℓ∞\\sup\_\{\\lambda\\in\\Omega\}\\left\\lvert\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}m\_\{k\}\(\\lambda\)w\_\{k\}\(\\lambda\)\\right\\rvert\\leq A\_\{N\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\(8\.38\)for everyv∈𝒱Nv\\in\\mathcal\{V\}\_\{N\}\. Then every first\-order shaped error with statev∈𝒱Nv\\in\\mathcal\{V\}\_\{N\}satisfies

‖ℰN​e‖L∞​\(Ω\)≤\|vN\|\+\|v0\|N\+ANN​‖v‖ℓ∞\.\\left\\lVert\\mathcal\{E\}\_\{N\}e\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\)\}\\leq\\frac\{\\left\\lvert v\_\{N\}\\right\\rvert\+\\left\\lvert v\_\{0\}\\right\\rvert\}\{N\}\+\\frac\{A\_\{N\}\}\{N\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\.\(8\.39\)Ifv0=vN=0v\_\{0\}=v\_\{N\}=0andAN≤C​N1−δA\_\{N\}\\leq CN^\{1\-\\delta\}, then

‖ℰN​e‖L∞​\(Ω\)≤C​‖v‖ℓ∞​N−δ\.\\left\\lVert\\mathcal\{E\}\_\{N\}e\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\)\}\\leq C\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}N^\{\-\\delta\}\.

###### Proof\.

Apply the triangle inequality to the exact identity \([8\.36](https://arxiv.org/html/2607.24868#Ch8.E36)\) and use \([8\.38](https://arxiv.org/html/2607.24868#Ch8.E38)\)\. Since\|w1\|=\|wN\|=1\\left\\lvert w\_\{1\}\\right\\rvert=\\left\\lvert w\_\{N\}\\right\\rvert=1, the two boundary terms contribute\(\|vN\|\+\|v0\|\)/N\(\\left\\lvert v\_\{N\}\\right\\rvert\+\\left\\lvert v\_\{0\}\\right\\rvert\)/N\. ∎

The theorem isolates the missing harmonic estimate\. It does not replace that estimate by an assumption hidden inside the conclusion\. Every proposed critical\-scale result must supply a concrete bound forANA\_\{N\}on the actual state class produced by the quantiser\.

#### 8\.5\.3High\-order transfer

Assume endpoint compatibility of orderrr\. Then

ℰN​e=\(−1\)rN​∑k=1N−rvk​\(Δ\+r​w\)k\.\\mathcal\{E\}\_\{N\}e=\\frac\{\(\-1\)^\{r\}\}\{N\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\.
###### Theorem 8\.19\(High\-order oscillatory transfer\)\.

Lete=Δr​ve=\\Delta^\{r\}vsatisfy endpoint compatibility of orderrr\. Suppose

supλ∈Ω\|∑k=1N−rvk​\(Δ\+r​w​\(λ\)\)k\|≤AN,r​‖v‖ℓ∞\.\\sup\_\{\\lambda\\in\\Omega\}\\left\\lvert\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\(\\lambda\)\)\_\{k\}\\right\\rvert\\leq A\_\{N,r\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\.\(8\.40\)Then

‖ℰN​e‖L∞​\(Ω\)≤AN,rN​‖v‖ℓ∞\.\\left\\lVert\\mathcal\{E\}\_\{N\}e\\right\\rVert\_\{L^\{\\infty\}\(\\Omega\)\}\\leq\\frac\{A\_\{N,r\}\}\{N\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\.\(8\.41\)In particular, a power saving

AN,r≤C​N1−δA\_\{N,r\}\\leq CN^\{1\-\\delta\}impliesO​\(N−δ\)O\(N^\{\-\\delta\}\)extension error onΩ\\Omega\.

###### Proof\.

Take the supremum in the pure interior representation and apply \([8\.40](https://arxiv.org/html/2607.24868#Ch8.E40)\)\. ∎

At subcritical scale, the direct bound onΔ\+r​w\\Delta\_\{\+\}^\{r\}walready supplies decay\. At critical scale, the quantityAN,rA\_\{N,r\}must exhibit cancellation relative to its trivial upper bound of orderNN\.

### 8\.6Random and weakly correlated state models

Random\-state theorems do not describe every deterministic sigma–delta trajectory\. Their value is to identify the gain produced by decorrelation and to provide a benchmark for deterministic state dynamics\.

#### 8\.6\.1Independent centred states

###### Theorem 8\.20\(Independent centred state model\)\.

Letv1,…,vN−1v\_\{1\},\\ldots,v\_\{N\-1\}be independent complex random variables such that

𝔼​vk=0,𝔼​\|vk\|2≤σ2\.\\mathbb\{E\}v\_\{k\}=0,\\qquad\\mathbb\{E\}\\left\\lvert v\_\{k\}\\right\\rvert^\{2\}\\leq\\sigma^\{2\}\.Letw1,…,wNw\_\{1\},\\ldots,w\_\{N\}be deterministic with\|wk\|=1\\left\\lvert w\_\{k\}\\right\\rvert=1, assumev0=vN=0v\_\{0\}=v\_\{N\}=0, and sete=Δ​ve=\\Delta v\. Then

𝔼​\|ℰN​e\|2=1N2​∑k=1N−1𝔼​\|vk\|2​\|Δ\+​wk\|2\.\\mathbb\{E\}\\left\\lvert\\mathcal\{E\}\_\{N\}e\\right\\rvert^\{2\}=\\frac\{1\}\{N^\{2\}\}\\sum\_\{k=1\}^\{N\-1\}\\mathbb\{E\}\\left\\lvert v\_\{k\}\\right\\rvert^\{2\}\\left\\lvert\\Delta\_\{\+\}w\_\{k\}\\right\\rvert^\{2\}\.\(8\.42\)Consequently,

\(𝔼​\|ℰN​e\|2\)1/2≤σN​\(∑k=1N−1\|Δ\+​wk\|2\)1/2≤2​σN\.\\left\(\\mathbb\{E\}\\left\\lvert\\mathcal\{E\}\_\{N\}e\\right\\rvert^\{2\}\\right\)^\{1/2\}\\leq\\frac\{\\sigma\}\{N\}\\left\(\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert\\Delta\_\{\+\}w\_\{k\}\\right\\rvert^\{2\}\\right\)^\{1/2\}\\leq\\frac\{2\\sigma\}\{\\sqrt\{N\}\}\.\(8\.43\)

###### Proof\.

The endpoint conditions and summation by parts give

ℰN​e=−1N​∑k=1N−1vk​Δ\+​wk\.\\mathcal\{E\}\_\{N\}e=\-\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\\Delta\_\{\+\}w\_\{k\}\.After squaring and taking expectation,

𝔼​\|ℰN​e\|2\\displaystyle\\mathbb\{E\}\\left\\lvert\\mathcal\{E\}\_\{N\}e\\right\\rvert^\{2\}=1N2​∑k,ℓ=1N−1𝔼​\[vk​vℓ¯\]​\(Δ\+​wk\)​\(Δ\+​wℓ\)¯\.\\displaystyle=\\frac\{1\}\{N^\{2\}\}\\sum\_\{k,\\ell=1\}^\{N\-1\}\\mathbb\{E\}\\left\[v\_\{k\}\\overline\{v\_\{\\ell\}\}\\right\]\(\\Delta\_\{\+\}w\_\{k\}\)\\overline\{\(\\Delta\_\{\+\}w\_\{\\ell\}\)\}\.Independence and zero means imply

𝔼​\[vk​vℓ¯\]=0\(k≠ℓ\)\.\\mathbb\{E\}\[v\_\{k\}\\overline\{v\_\{\\ell\}\}\]=0\\qquad\(k\\neq\\ell\)\.This proves \([8\.42](https://arxiv.org/html/2607.24868#Ch8.E42)\)\. The variance bound gives the first inequality in \([8\.43](https://arxiv.org/html/2607.24868#Ch8.E43)\), and\|Δ\+​wk\|≤2\\left\\lvert\\Delta\_\{\+\}w\_\{k\}\\right\\rvert\\leq 2gives the second\. ∎

For the parabolic phase at a fixed compact scale,\|Δ\+​wk\|=O​\(N−1\)\\left\\lvert\\Delta\_\{\+\}w\_\{k\}\\right\\rvert=O\(N^\{\-1\}\), and the same theorem gives the stronger root\-mean\-square rateO​\(N−3/2\)O\(N^\{\-3/2\}\)\. At the critical scale, where the adjacent phase change is order one, the rate becomesO​\(N−1/2\)O\(N^\{\-1/2\}\)\.

#### 8\.6\.2Summable covariance

###### Theorem 8\.21\(Weakly correlated state model\)\.

Letv1,…,vN−1v\_\{1\},\\ldots,v\_\{N\-1\}be centred complex random variables\. Assume that there exists a nonnegative sequencec=\(ch\)h∈ℤ∈ℓ1​\(ℤ\)c=\(c\_\{h\}\)\_\{h\\in\\mathbb\{Z\}\}\\in\\ell^\{1\}\(\\mathbb\{Z\}\)such that

\|𝔼​\[vk​vℓ¯\]\|≤σ2​ck−ℓ\\left\\lvert\\mathbb\{E\}\[v\_\{k\}\\overline\{v\_\{\\ell\}\}\]\\right\\rvert\\leq\\sigma^\{2\}c\_\{k\-\\ell\}\(8\.44\)for allk,ℓk,\\ell\. Ifv0=vN=0v\_\{0\}=v\_\{N\}=0ande=Δ​ve=\\Delta v, then

𝔼​\|ℰN​e\|2≤σ2​‖c‖ℓ1N2​∑k=1N−1\|Δ\+​wk\|2\.\\mathbb\{E\}\\left\\lvert\\mathcal\{E\}\_\{N\}e\\right\\rvert^\{2\}\\leq\\frac\{\\sigma^\{2\}\\left\\lVert c\\right\\rVert\_\{\\ell^\{1\}\}\}\{N^\{2\}\}\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert\\Delta\_\{\+\}w\_\{k\}\\right\\rvert^\{2\}\.\(8\.45\)

###### Proof\.

Setak=Δ\+​wka\_\{k\}=\\Delta\_\{\+\}w\_\{k\}, extended by zero outside\{1,…,N−1\}\\\{1,\\ldots,N\-1\\\}\. Then

𝔼​\|ℰN​e\|2\\displaystyle\\mathbb\{E\}\\left\\lvert\\mathcal\{E\}\_\{N\}e\\right\\rvert^\{2\}≤σ2N2​∑k,ℓck−ℓ​\|ak\|​\|aℓ\|\\displaystyle\\leq\\frac\{\\sigma^\{2\}\}\{N^\{2\}\}\\sum\_\{k,\\ell\}c\_\{k\-\\ell\}\\left\\lvert a\_\{k\}\\right\\rvert\\left\\lvert a\_\{\\ell\}\\right\\rvert=σ2N2​⟨c∗\|a\|,\|a\|⟩ℓ2\.\\displaystyle=\\frac\{\\sigma^\{2\}\}\{N^\{2\}\}\\left\\langle c\*\\left\\lvert a\\right\\rvert,\\left\\lvert a\\right\\rvert\\right\\rangle\_\{\\ell^\{2\}\}\.Young’s convolution inequality gives

‖c∗\|a\|‖ℓ2≤‖c‖ℓ1​‖a‖ℓ2\.\\left\\lVert c\*\\left\\lvert a\\right\\rvert\\right\\rVert\_\{\\ell^\{2\}\}\\leq\\left\\lVert c\\right\\rVert\_\{\\ell^\{1\}\}\\left\\lVert a\\right\\rVert\_\{\\ell^\{2\}\}\.Cauchy–Schwarz therefore yields

⟨c∗\|a\|,\|a\|⟩≤‖c‖ℓ1​‖a‖ℓ22,\\left\\langle c\*\\left\\lvert a\\right\\rvert,\\left\\lvert a\\right\\rvert\\right\\rangle\\leq\\left\\lVert c\\right\\rVert\_\{\\ell^\{1\}\}\\left\\lVert a\\right\\rVert\_\{\\ell^\{2\}\}^\{2\},which proves \([8\.45](https://arxiv.org/html/2607.24868#Ch8.E45)\)\. ∎

#### 8\.6\.3High\-order random states

The same argument applies afterrrintegrations by parts\.

###### Corollary 8\.22\(High\-order covariance transfer\)\.

Assumee=Δr​ve=\\Delta^\{r\}v, endpoint compatibility of orderrr, and the covariance condition \([8\.44](https://arxiv.org/html/2607.24868#Ch8.E44)\)\. Then

𝔼​\|ℰN​e\|2≤σ2​‖c‖ℓ1N2​∑k=1N−r\|Δ\+r​wk\|2\.\\mathbb\{E\}\\left\\lvert\\mathcal\{E\}\_\{N\}e\\right\\rvert^\{2\}\\leq\\frac\{\\sigma^\{2\}\\left\\lVert c\\right\\rVert\_\{\\ell^\{1\}\}\}\{N^\{2\}\}\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\\Delta\_\{\+\}^\{r\}w\_\{k\}\\right\\rvert^\{2\}\.\(8\.46\)

###### Proof\.

Use the pure interior formula

ℰN​e=\(−1\)rN​∑k=1N−rvk​Δ\+r​wk\\mathcal\{E\}\_\{N\}e=\\frac\{\(\-1\)^\{r\}\}\{N\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\\Delta\_\{\+\}^\{r\}w\_\{k\}and repeat the proof of Theorem[8\.21](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem21)\. ∎

### 8\.7Correlation estimates and van der Corput transfer

#### 8\.7\.1A finite van der Corput inequality

The following standard inequality converts pair correlations into an upper bound for an oscillatory sum\. It is included in a form directly applicable to state\-weighted sequences\. Classical treatments of exponential sums and differencing may be found in the literature surrounding Weyl’s method and Vinogradov mean values\[[79](https://arxiv.org/html/2607.24868#bib.bib56),[78](https://arxiv.org/html/2607.24868#bib.bib57),[74](https://arxiv.org/html/2607.24868#bib.bib78),[81](https://arxiv.org/html/2607.24868#bib.bib63),[83](https://arxiv.org/html/2607.24868#bib.bib66),[84](https://arxiv.org/html/2607.24868#bib.bib67)\]\.

###### Lemma 8\.23\(Finite van der Corput inequality\)\.

Letz1,…,zM∈ℂz\_\{1\},\\ldots,z\_\{M\}\\in\\mathbb\{C\}, and let1≤H≤M1\\leq H\\leq M\. Then

\|∑k=1Mzk\|2≤M\+HH​\[∑k=1M\|zk\|2\+2​∑h=1H−1\(1−hH\)​\|∑k=1M−hzk\+h​zk¯\|\]\.\\left\\lvert\\sum\_\{k=1\}^\{M\}z\_\{k\}\\right\\rvert^\{2\}\\leq\\frac\{M\+H\}\{H\}\\left\[\\sum\_\{k=1\}^\{M\}\\left\\lvert z\_\{k\}\\right\\rvert^\{2\}\+2\\sum\_\{h=1\}^\{H\-1\}\\left\(1\-\\frac\{h\}\{H\}\\right\)\\left\\lvert\\sum\_\{k=1\}^\{M\-h\}z\_\{k\+h\}\\overline\{z\_\{k\}\}\\right\\rvert\\right\]\.\(8\.47\)

###### Proof\.

Extendzkz\_\{k\}by zero outside\{1,…,M\}\\\{1,\\ldots,M\\\}\. For each integerjj, define

Bj:=∑h=0H−1zj\+h\.B\_\{j\}:=\\sum\_\{h=0\}^\{H\-1\}z\_\{j\+h\}\.Everyzkz\_\{k\}occurs in exactlyHHof the sumsBjB\_\{j\}, except for boundary truncations that can only reduce the count after zero extension\. Hence

H​∑k=1Mzk=∑j=1−HMBj\.H\\sum\_\{k=1\}^\{M\}z\_\{k\}=\\sum\_\{j=1\-H\}^\{M\}B\_\{j\}\.Cauchy–Schwarz gives

H2​\|∑zk\|2≤\(M\+H\)​∑j=1−HM\|Bj\|2\.H^\{2\}\\left\\lvert\\sum z\_\{k\}\\right\\rvert^\{2\}\\leq\(M\+H\)\\sum\_\{j=1\-H\}^\{M\}\\left\\lvert B\_\{j\}\\right\\rvert^\{2\}\.Expanding the final sum and collecting terms according to the displacementhhyields

∑j\|Bj\|2\\displaystyle\\sum\_\{j\}\\left\\lvert B\_\{j\}\\right\\rvert^\{2\}=H​∑k=1M\|zk\|2\+2​ℜ​∑h=1H−1\(H−h\)​∑k=1M−hzk\+h​zk¯\.\\displaystyle=H\\sum\_\{k=1\}^\{M\}\\left\\lvert z\_\{k\}\\right\\rvert^\{2\}\+2\\Re\\sum\_\{h=1\}^\{H\-1\}\(H\-h\)\\sum\_\{k=1\}^\{M\-h\}z\_\{k\+h\}\\overline\{z\_\{k\}\}\.Taking absolute values of the correlation sums and dividing byH2H^\{2\}proves \([8\.47](https://arxiv.org/html/2607.24868#Ch8.E47)\)\. ∎

#### 8\.7\.2State\-phase correlations

For first\-order transfer, set

zk=vk​mk​wk\.z\_\{k\}=v\_\{k\}m\_\{k\}w\_\{k\}\.Then

zk\+h​zk¯\\displaystyle z\_\{k\+h\}\\overline\{z\_\{k\}\}=vk\+h​vk¯​mk\+h​mk¯​e​\(ϕk\+h−ϕk\)\.\\displaystyle=v\_\{k\+h\}\\overline\{v\_\{k\}\}m\_\{k\+h\}\\overline\{m\_\{k\}\}\\mathrm\{e\}\(\\phi\_\{k\+h\}\-\\phi\_\{k\}\)\.\(8\.48\)The product contains two distinct sources of cancellation: decay of the state correlationvk\+h​vk¯v\_\{k\+h\}\\overline\{v\_\{k\}\}, and oscillation of the phase incrementϕk\+h−ϕk\\phi\_\{k\+h\}\-\\phi\_\{k\}\.

###### Theorem 8\.24\(Correlation\-to\-transfer estimate\)\.

LetM=N−1M=N\-1,zk=vk​mk​wkz\_\{k\}=v\_\{k\}m\_\{k\}w\_\{k\}, and assume

\|mk\|≤M0\.\\left\\lvert m\_\{k\}\\right\\rvert\\leq M\_\{0\}\.Suppose that, uniformly in the parameterλ∈Ω\\lambda\\in\\Omega,

\|∑k=1M−hzk\+h​\(λ\)​zk​\(λ\)¯\|≤Ch​N​‖v‖ℓ∞2\\left\\lvert\\sum\_\{k=1\}^\{M\-h\}z\_\{k\+h\}\(\\lambda\)\\overline\{z\_\{k\}\(\\lambda\)\}\\right\\rvert\\leq C\_\{h\}N\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}^\{2\}\(8\.49\)for1≤h<H1\\leq h<H\. Then

supλ∈Ω\|∑k=1Mzk​\(λ\)\|\\displaystyle\\sup\_\{\\lambda\\in\\Omega\}\\left\\lvert\\sum\_\{k=1\}^\{M\}z\_\{k\}\(\\lambda\)\\right\\rvert≤‖v‖ℓ∞​\{N\+HH​\[N​M02\+2​N​∑h=1H−1\(1−hH\)​Ch\]\}1/2\.\\displaystyle\\leq\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\left\\\{\\frac\{N\+H\}\{H\}\\left\[NM\_\{0\}^\{2\}\+2N\\sum\_\{h=1\}^\{H\-1\}\\left\(1\-\\frac\{h\}\{H\}\\right\)C\_\{h\}\\right\]\\right\\\}^\{1/2\}\.\(8\.50\)Consequently, ifv0=vN=0v\_\{0\}=v\_\{N\}=0, the first\-order extension error is bounded byN−1N^\{\-1\}times the right\-hand side\.

###### Proof\.

The diagonal term in Lemma[8\.23](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem23)satisfies

∑k=1M\|zk\|2≤N​M02​‖v‖ℓ∞2\.\\sum\_\{k=1\}^\{M\}\\left\\lvert z\_\{k\}\\right\\rvert^\{2\}\\leq NM\_\{0\}^\{2\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}^\{2\}\.Insert this estimate and \([8\.49](https://arxiv.org/html/2607.24868#Ch8.E49)\) into \([8\.47](https://arxiv.org/html/2607.24868#Ch8.E47)\), then take square roots\. The extension estimate follows from \([8\.36](https://arxiv.org/html/2607.24868#Ch8.E36)\) with vanishing endpoints\. ∎

A useful power saving follows when the average correlation is small\. For example, ifCh≲h−1−ηC\_\{h\}\\lesssim h^\{\-1\-\\eta\}for someη\>0\\eta\>0, then the correlation sum remains bounded asH→∞H\\to\\infty\. TakingH≍NH\\asymp Ngives a state sum of orderN1/2N^\{1/2\}, and therefore a first\-order extension error of orderN−1/2N^\{\-1/2\}at critical phase variation\.

### 8\.8Dyadic block decompositions and square\-function transfer

#### 8\.8\.1Block decomposition

Let1≤L≤N1\\leq L\\leq N, and partition\{1,…,N−r\}\\\{1,\\ldots,N\-r\\\}into consecutive intervalsIIof length at mostLL\. For endpoint\-compatiblerrth\-order shaping, define

GI​\(λ\):=∑k∈Ivk​\(Δ\+r​w​\(λ\)\)k\.G\_\{I\}\(\\lambda\):=\\sum\_\{k\\in I\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\(\\lambda\)\)\_\{k\}\.\(8\.51\)Then

ℰN​e​\(λ\)=\(−1\)rN​∑IGI​\(λ\)\.\\mathcal\{E\}\_\{N\}e\(\\lambda\)=\\frac\{\(\-1\)^\{r\}\}\{N\}\\sum\_\{I\}G\_\{I\}\(\\lambda\)\.The triangle inequality gives

\|ℰN​e\|≤1N​∑I\|GI\|,\\left\\lvert\\mathcal\{E\}\_\{N\}e\\right\\rvert\\leq\\frac\{1\}\{N\}\\sum\_\{I\}\\left\\lvert G\_\{I\}\\right\\rvert,which discards cancellation among blocks\. A square\-function estimate retains more information\.

###### Theorem 8\.25\(Abstract block\-square transfer\)\.

Let2≤p<∞2\\leq p<\\infty\. Suppose that a parameter measure space\(Ω,μ\)\(\\Omega,\\mu\)and the block pieces \([8\.51](https://arxiv.org/html/2607.24868#Ch8.E51)\) satisfy

‖∑IGI‖Lp​\(Ω\)≤DN,L,p​‖\(∑I\|GI\|2\)1/2‖Lp​\(Ω\)\.\\left\\lVert\\sum\_\{I\}G\_\{I\}\\right\\rVert\_\{L^\{p\}\(\\Omega\)\}\\leq D\_\{N,L,p\}\\left\\lVert\\left\(\\sum\_\{I\}\\left\\lvert G\_\{I\}\\right\\rvert^\{2\}\\right\)^\{1/2\}\\right\\rVert\_\{L^\{p\}\(\\Omega\)\}\.\(8\.52\)Then

‖ℰN​e‖Lp​\(Ω\)≤DN,L,pN​‖\(∑I\|GI\|2\)1/2‖Lp​\(Ω\)\.\\left\\lVert\\mathcal\{E\}\_\{N\}e\\right\\rVert\_\{L^\{p\}\(\\Omega\)\}\\leq\\frac\{D\_\{N,L,p\}\}\{N\}\\left\\lVert\\left\(\\sum\_\{I\}\\left\\lvert G\_\{I\}\\right\\rvert^\{2\}\\right\)^\{1/2\}\\right\\rVert\_\{L^\{p\}\(\\Omega\)\}\.\(8\.53\)If, in addition,

‖\(∑I\|GI\|2\)1/2‖Lp​\(Ω\)≤BN,L,p​‖v‖ℓ∞,\\left\\lVert\\left\(\\sum\_\{I\}\\left\\lvert G\_\{I\}\\right\\rvert^\{2\}\\right\)^\{1/2\}\\right\\rVert\_\{L^\{p\}\(\\Omega\)\}\\leq B\_\{N,L,p\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\},\(8\.54\)then

‖ℰN​e‖Lp​\(Ω\)≤DN,L,p​BN,L,pN​‖v‖ℓ∞\.\\left\\lVert\\mathcal\{E\}\_\{N\}e\\right\\rVert\_\{L^\{p\}\(\\Omega\)\}\\leq\\frac\{D\_\{N,L,p\}B\_\{N,L,p\}\}\{N\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\.\(8\.55\)

###### Proof\.

The pure interior representation and \([8\.52](https://arxiv.org/html/2607.24868#Ch8.E52)\) give \([8\.53](https://arxiv.org/html/2607.24868#Ch8.E53)\)\. Substitution of \([8\.54](https://arxiv.org/html/2607.24868#Ch8.E54)\) yields \([8\.55](https://arxiv.org/html/2607.24868#Ch8.E55)\)\. ∎

The theorem is deliberately modular\. Decoupling controls the interaction among frequency blocks, while the block\-energy estimate must retain the state structure\. Standard parabola decoupling supplies the first ingredient for arbitrary coefficients\[[13](https://arxiv.org/html/2607.24868#bib.bib45),[12](https://arxiv.org/html/2607.24868#bib.bib46),[28](https://arxiv.org/html/2607.24868#bib.bib55)\]; the second ingredient is specific to the noise\-shaped state sequence\.

#### 8\.8\.2A deterministic block\-energy baseline

Without additional cancellation,

\|GI\|≤‖v‖ℓ∞​∑k∈I\|Δ\+r​wk\|\.\\left\\lvert G\_\{I\}\\right\\rvert\\leq\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\sum\_\{k\\in I\}\\left\\lvert\\Delta\_\{\+\}^\{r\}w\_\{k\}\\right\\rvert\.Therefore

\(∑I\|GI\|2\)1/2≤‖v‖ℓ∞​\[∑I\(∑k∈I\|Δ\+r​wk\|\)2\]1/2\.\\left\(\\sum\_\{I\}\\left\\lvert G\_\{I\}\\right\\rvert^\{2\}\\right\)^\{1/2\}\\leq\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\left\[\\sum\_\{I\}\\left\(\\sum\_\{k\\in I\}\\left\\lvert\\Delta\_\{\+\}^\{r\}w\_\{k\}\\right\\rvert\\right\)^\{2\}\\right\]^\{1/2\}\.\(8\.56\)If\|Δ\+r​wk\|≤Mr\\left\\lvert\\Delta\_\{\+\}^\{r\}w\_\{k\}\\right\\rvert\\leq M\_\{r\}and the blocks have length at mostLL, then

\(∑I\|GI\|2\)1/2≤Mr​N​L​‖v‖ℓ∞\.\\left\(\\sum\_\{I\}\\left\\lvert G\_\{I\}\\right\\rvert^\{2\}\\right\)^\{1/2\}\\leq M\_\{r\}\\sqrt\{NL\}\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\.\(8\.57\)This bound already improves on theN​MrNM\_\{r\}triangle estimate by a factor\(N/L\)1/2\(N/L\)^\{1/2\}when a square\-function inequality combines the blocks without a compensating loss\.

### 8\.9Frequency\-side interpretation of discrete differences

#### 8\.9\.1Periodic discrete Fourier transform

For a sequencea=\(ak\)k=0N−1a=\(a\_\{k\}\)\_\{k=0\}^\{N\-1\}, define the discrete Fourier transform

a^​\(ℓ\):=∑k=0N−1ak​e​\(−k​ℓN\),ℓ=0,…,N−1\.\\widehat\{a\}\(\\ell\):=\\sum\_\{k=0\}^\{N\-1\}a\_\{k\}\\mathrm\{e\}\\\!\\left\(\-\\frac\{k\\ell\}\{N\}\\right\),\\qquad\\ell=0,\\ldots,N\-1\.If the difference is taken periodically,

\(Δper​a\)k=ak−ak−1​mod​N,\(\\Delta\_\{\\mathrm\{per\}\}a\)\_\{k\}=a\_\{k\}\-a\_\{k\-1\\,\\mathrm\{mod\}\\,N\},then

Δperr​a^​\(ℓ\)=\(1−e​\(−ℓN\)\)r​a^​\(ℓ\)\.\\widehat\{\\Delta\_\{\\mathrm\{per\}\}^\{r\}a\}\(\\ell\)=\\left\(1\-\\mathrm\{e\}\\\!\\left\(\-\\frac\{\\ell\}\{N\}\\right\)\\right\)^\{r\}\\widehat\{a\}\(\\ell\)\.\(8\.58\)For low frequencies\|ℓ\|≪N\\left\\lvert\\ell\\right\\rvert\\ll N,

\|1−e​\(−ℓ/N\)\|=2​\|sin⁡\(π​ℓ/N\)\|≍\|ℓ\|N\.\\left\\lvert 1\-\\mathrm\{e\}\(\-\\ell/N\)\\right\\rvert=2\\left\\lvert\\sin\(\\pi\\ell/N\)\\right\\rvert\\asymp\\frac\{\\left\\lvert\\ell\\right\\rvert\}\{N\}\.\(8\.59\)Thus periodicrrth\-order noise shaping suppresses low discrete frequencies by a factor comparable with\(\|ℓ\|/N\)r\(\\left\\lvert\\ell\\right\\rvert/N\)^\{r\}\.

The finite\-record transform contains boundary corrections because the physical difference is not periodic\. Endpoint compatibility, tail termination, or explicit boundary correction removes precisely the terms that prevent \([8\.58](https://arxiv.org/html/2607.24868#Ch8.E58)\) from holding without modification\.

#### 8\.9\.2A periodic Sobolev estimate

###### Proposition 8\.26\(Low\-frequency energy suppression\)\.

Lete=Δperr​ve=\\Delta\_\{\\mathrm\{per\}\}^\{r\}vonℤ/N​ℤ\\mathbb\{Z\}/N\\mathbb\{Z\}\. For1≤L≤N/21\\leq L\\leq N/2,

∑\|ℓ\|≤L\|e^​\(ℓ\)\|2≤Cr​\(LN\)2​r​∑\|ℓ\|≤L\|v^​\(ℓ\)\|2\.\\sum\_\{\\left\\lvert\\ell\\right\\rvert\\leq L\}\\left\\lvert\\widehat\{e\}\(\\ell\)\\right\\rvert^\{2\}\\leq C\_\{r\}\\left\(\\frac\{L\}\{N\}\\right\)^\{2r\}\\sum\_\{\\left\\lvert\\ell\\right\\rvert\\leq L\}\\left\\lvert\\widehat\{v\}\(\\ell\)\\right\\rvert^\{2\}\.\(8\.60\)

###### Proof\.

By \([8\.58](https://arxiv.org/html/2607.24868#Ch8.E58)\),

\|e^​\(ℓ\)\|=\|1−e​\(−ℓ/N\)\|r​\|v^​\(ℓ\)\|\.\\left\\lvert\\widehat\{e\}\(\\ell\)\\right\\rvert=\\left\\lvert 1\-\\mathrm\{e\}\(\-\\ell/N\)\\right\\rvert^\{r\}\\left\\lvert\\widehat\{v\}\(\\ell\)\\right\\rvert\.For\|ℓ\|≤L≤N/2\\left\\lvert\\ell\\right\\rvert\\leq L\\leq N/2, inequality \([8\.59](https://arxiv.org/html/2607.24868#Ch8.E59)\) gives

\|1−e​\(−ℓ/N\)\|r≤Cr​\(L/N\)r\.\\left\\lvert 1\-\\mathrm\{e\}\(\-\\ell/N\)\\right\\rvert^\{r\}\\leq C\_\{r\}\(L/N\)^\{r\}\.Square and sum over the stated range\. ∎

This proposition gives a precise frequency\-domain meaning to noise shaping\. The main difficulty in the parabolic extension problem is that the extension operator couples the coefficient frequency with a curved two\-parameter phase\. A successful critical\-scale theorem must transfer the low\-frequency suppression in \([8\.60](https://arxiv.org/html/2607.24868#Ch8.E60)\) through that curved geometry\.

### 8\.10Commutator identities for chirp modulation

#### 8\.10\.1A discrete product rule

For sequencesaaandvv,

ak​Δ​vk=Δ​\(ak​vk\)−vk−1​Δ​ak\.a\_\{k\}\\Delta v\_\{k\}=\\Delta\(a\_\{k\}v\_\{k\}\)\-v\_\{k\-1\}\\Delta a\_\{k\}\.\(8\.61\)This identity separates a total difference from a commutator term\. If the product is tested against an envelopebkb\_\{k\}, then

∑k=1Nak​Δ​vk​bk\\displaystyle\\sum\_\{k=1\}^\{N\}a\_\{k\}\\Delta v\_\{k\}\\,b\_\{k\}=∑k=1NΔ​\(ak​vk\)​bk−∑k=1Nvk−1​\(Δ​ak\)​bk\.\\displaystyle=\\sum\_\{k=1\}^\{N\}\\Delta\(a\_\{k\}v\_\{k\}\)b\_\{k\}\-\\sum\_\{k=1\}^\{N\}v\_\{k\-1\}\(\\Delta a\_\{k\}\)b\_\{k\}\.\(8\.62\)Summation by parts moves the first difference ontobb, while the second term measures the failure of multiplication byaato commute withΔ\\Delta\.

For a polynomial\-phase carrierak=e​\(ϕk\)a\_\{k\}=\\mathrm\{e\}\(\\phi\_\{k\}\),

Δ​ak=ak​\[1−e​\(ϕk−1−ϕk\)\]\.\\Delta a\_\{k\}=a\_\{k\}\\left\[1\-\\mathrm\{e\}\(\\phi\_\{k\-1\}\-\\phi\_\{k\}\)\\right\]\.Thus the commutator remains a chirp multiplied by an explicit phase\-increment factor\. Iterating \([8\.61](https://arxiv.org/html/2607.24868#Ch8.E61)\) produces a finite expansion involving higher differences of the carrier and lower differences of the state\.

###### Proposition 8\.27\(Second\-order commutator expansion\)\.

For arbitrary sequencesaaandvv,

ak​Δ2​vk\\displaystyle a\_\{k\}\\Delta^\{2\}v\_\{k\}=Δ2​\(ak​vk\)−2​\(Δ​ak\)​Δ​vk−1−\(Δ2​ak\)​vk−2\.\\displaystyle=\\Delta^\{2\}\(a\_\{k\}v\_\{k\}\)\-2\(\\Delta a\_\{k\}\)\\Delta v\_\{k\-1\}\-\(\\Delta^\{2\}a\_\{k\}\)v\_\{k\-2\}\.\(8\.63\)

###### Proof\.

Apply \([8\.61](https://arxiv.org/html/2607.24868#Ch8.E61)\) twice\. First,

ak​Δ2​vk=Δ​\(ak​Δ​vk\)−\(Δ​ak\)​Δ​vk−1\.a\_\{k\}\\Delta^\{2\}v\_\{k\}=\\Delta\(a\_\{k\}\\Delta v\_\{k\}\)\-\(\\Delta a\_\{k\}\)\\Delta v\_\{k\-1\}\.Apply \([8\.61](https://arxiv.org/html/2607.24868#Ch8.E61)\) toak​Δ​vka\_\{k\}\\Delta v\_\{k\}:

ak​Δ​vk=Δ​\(ak​vk\)−vk−1​Δ​ak\.a\_\{k\}\\Delta v\_\{k\}=\\Delta\(a\_\{k\}v\_\{k\}\)\-v\_\{k\-1\}\\Delta a\_\{k\}\.Taking one further difference gives

Δ​\(ak​Δ​vk\)=Δ2​\(ak​vk\)−Δ​\(vk−1​Δ​ak\)\.\\Delta\(a\_\{k\}\\Delta v\_\{k\}\)=\\Delta^\{2\}\(a\_\{k\}v\_\{k\}\)\-\\Delta\(v\_\{k\-1\}\\Delta a\_\{k\}\)\.The product rule applied to the final term yields

Δ​\(vk−1​Δ​ak\)=\(Δ​vk−1\)​\(Δ​ak\)\+vk−2​Δ2​ak\.\\Delta\(v\_\{k\-1\}\\Delta a\_\{k\}\)=\(\\Delta v\_\{k\-1\}\)\(\\Delta a\_\{k\}\)\+v\_\{k\-2\}\\Delta^\{2\}a\_\{k\}\.Substitution proves \([8\.63](https://arxiv.org/html/2607.24868#Ch8.E63)\)\. ∎

The expansion provides a discrete pseudodifferential viewpoint: differences may be distributed between the state and the oscillatory carrier, but each redistribution produces explicit commutator terms\. Boundary conditions and phase regularity determine whether the resulting expression is useful\.

### 8\.11Sharp obstructions and consistency tests

Any proposed global theorem should be tested against exact identities before a proof is attempted\.

#### 8\.11\.1Zero\-frequency boundary obstruction

At\(x,t\)=\(0,0\)\(x,t\)=\(0,0\),

FN​\(0,0\)=1N​∑k=1Nek\.F\_\{N\}\(0,0\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}\.\(8\.64\)For first\-order shaping,

∑k=1Nek=vN−v0\.\\sum\_\{k=1\}^\{N\}e\_\{k\}=v\_\{N\}\-v\_\{0\}\.Thus a nonzero terminal trace creates an unavoidable contribution of orderN−1N^\{\-1\}\. For higher\-order shaping, testing against polynomial weights reveals the corresponding hierarchy of boundary traces\.

#### 8\.11\.2Full\-periodL2L^\{2\}obstruction

The exact identity \([8\.2](https://arxiv.org/html/2607.24868#Ch8.E2)\) gives

∫aa\+N\|FN​\(x,t\)\|2​dx=N−1​‖e‖ℓ22\.\\int\_\{a\}^\{a\+N\}\\left\\lvert F\_\{N\}\(x,t\)\\right\\rvert^\{2\}\\,\\mathrm\{d\}x=N^\{\-1\}\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}^\{2\}\.If\|ek\|≍1\\left\\lvert e\_\{k\}\\right\\rvert\\asymp 1, the unnormalisedLx2L^\{2\}\_\{x\}norm is of order one\. Therefore no uniform positive decay exponent can hold in this norm for all such errors, irrespective of difference structure\.

#### 8\.11\.3Random bounded states

A bounded random state may produce‖Δ​v‖ℓ2≍N1/2\\left\\lVert\\Delta v\\right\\rVert\_\{\\ell^\{2\}\}\\asymp N^\{1/2\}\. The exact orthogonality identities then force the generic full\-period scale\. This test prevents an invalid theorem from replacing a state norm by‖v‖ℓ∞\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}without accounting for the number of independent oscillations\.

#### 8\.11\.4Interior\-supported states

If a state vanishes near both endpoints, all finite\-record boundary terms disappear\. Such examples isolate genuine interior oscillation\. A theorem that still fails for interior\-supported states cannot attribute the failure to endpoint traces\.

#### 8\.11\.5Periodic state orbits

Deterministic sigma–delta recursions may possess periodic or nearly periodic trajectories\. A correlation assumption that predicts square\-root cancellation must be checked against these orbits\. Periodicity can preserve large correlations and invalidate a mixing\-based estimate even when the state is uniformly bounded\.

### 8\.12A consolidated hierarchy of proved estimates

The preceding results can be organised by the information retained about the coefficient sequence\.

#### 8\.12\.1Arbitrary coefficients

For arbitraryee, the following statements hold:

1. \(i\)exactLx2L^\{2\}\_\{x\}orthogonality on every interval of lengthNN, Theorem[8\.1](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem1);
2. \(ii\)exact full\-cellL2L^\{2\}, Theorem[8\.3](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem3);
3. \(iii\)exact quadratic fourth moment, Proposition[8\.4](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem4);
4. \(iv\)critical sixth\-moment restriction bound, Corollary[8\.6](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem6);
5. \(v\)local kernel identity and generic localL2L^\{2\}estimate, Proposition[8\.10](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem10)and Theorem[8\.11](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem11)\.

#### 8\.12\.2First\-order bounded states

Fore=Δ​ve=\\Delta vwith bounded state:

1. \(i\)fixed compact sets admit deterministicO​\(N−1\)O\(N^\{\-1\}\)convergence;
2. \(ii\)the exact transfer formula \([8\.36](https://arxiv.org/html/2607.24868#Ch8.E36)\) separates boundary traces from the oscillatory state sum;
3. \(iii\)the two\-index identity \([8\.12](https://arxiv.org/html/2607.24868#Ch8.Ex34)\) identifies the mixed kernel difference required for a structured localL2L^\{2\}estimate;
4. \(iv\)state correlation or covariance bounds transfer directly to Fourier error bounds\.

#### 8\.12\.3Endpoint\-compatible high\-order states

Fore=Δr​ve=\\Delta^\{r\}vwith compatible boundary traces:

1. \(i\)polynomial moments below orderrrvanish exactly;
2. \(ii\)smooth compact\-scale weights yieldO​\(N−r\)O\(N^\{\-r\}\)error;
3. \(iii\)random or weakly correlated states satisfy high\-order mean\-square transfer estimates;
4. \(iv\)block\-square estimates can be combined with decoupling through Theorem[8\.25](https://arxiv.org/html/2607.24868#Ch8.Thmtheorem25)\.

### 8\.13Critical\-scale research target

The natural unresolved target is a deterministic estimate that improves on both the variation bound and the arbitrary\-coefficient restriction bound at a critical or near\-critical scale\. A suitable theorem must specify three ingredients:

1. \(a\)a concrete state class generated by a stable one\-bit sigma–delta recursion;
2. \(b\)endpoint conditions or an explicit boundary\-correction operator;
3. \(c\)an oscillatory norm and parameter region compatible with exact orthogonality\.

A representative target has the form

‖ℰN​\(Δr​v\)‖Lp​\(ΩN\)≤Cp,r,ε​N−δ\+ε​𝒮N​\(v\),\\left\\lVert\\mathcal\{E\}\_\{N\}\(\\Delta^\{r\}v\)\\right\\rVert\_\{L^\{p\}\(\\Omega\_\{N\}\)\}\\leq C\_\{p,r,\\varepsilon\}N^\{\-\\delta\+\\varepsilon\}\\mathcal\{S\}\_\{N\}\(v\),\(8\.65\)where𝒮N​\(v\)\\mathcal\{S\}\_\{N\}\(v\)is a state quantity stronger than‖v‖ℓ∞\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}but verifiable from the quantiser dynamics\. Possible choices suggested by the proved transfer principles include:

𝒮N​\(v\)=\(∑h\|Corrv⁡\(h\)\|\)1/2,\\mathcal\{S\}\_\{N\}\(v\)=\\left\(\\sum\_\{h\}\\left\\lvert\\operatorname\{Corr\}\_\{v\}\(h\)\\right\\rvert\\right\)^\{1/2\},a block\-square norm, a covariance norm, or a frequency\-localised Sobolev norm\.

The exact identities in this chapter impose necessary consistency conditions onpp,δ\\delta, the normalisation ofΩN\\Omega\_\{N\}, and the state class\. In particular, the full\-periodLx2L^\{2\}\_\{x\}identity rules out uniform unnormalised decay for bounded errors of unitℓ2\\ell^\{2\}\-density, while endpoint\-compatible high\-order cancellation remains effective on fixed and subcritical regions\.

### 8\.14Chapter conclusion

The local and averaged theory separates into three regimes\. On fixed compact sets, finite summation by parts and bounded states give deterministic algebraic convergence, with orderN−rN^\{\-r\}under compatiblerrth\-order shaping\. On full periodic cells, exact orthogonality and discrete restriction determine the baseline scale through‖e‖ℓ2\\left\\lVert e\\right\\rVert\_\{\\ell^\{2\}\}\. Between these regimes, local kernels, state correlations, covariance bounds, block\-square estimates, and commutator expansions provide precise transfer mechanisms\.

The strongest conclusions established in this chapter are the exact quadratic moment identities, the local two\-index summation\-by\-parts formula, the endpoint\-compatible vanishing\-moment theorem, deterministic high\-order compact\-set bounds, covariance\-based mean\-square estimates, and abstract oscillatory transfer principles\. These results identify the additional state information required for a genuine critical\-scale improvement and provide a rigorous foundation for the next stage of the analysis\.

## Chapter 9Robustness, Boundary Engineering, and Verification

A finite\-record theory must account for residual error, leakage, jitter, overload, imperfect terminal control, and block processing\. The first two sections derive deterministic robustness bounds and boundary\-engineering principles\. The final section reports reproducible calculations that verify the recursions, the exact sharpness example, the closed form of the phase variation, and the second\-order endpoint\-compatible scaling\.

### 9\.1Robustness to residual error, leakage, and model mismatch

#### 9\.1\.1Exact identities and physical implementations

The ideal relatione=Δr​ve=\\Delta^\{r\}vcan be perturbed by comparator offsets, finite gain, coefficient mismatch, clock error, state leakage, or imperfect digital reset\. Robust sigma\-delta theory studies the resulting changes in stability and reconstruction\[[45](https://arxiv.org/html/2607.24868#bib.bib6),[24](https://arxiv.org/html/2607.24868#bib.bib10)\]\. Assuming a bounded perturbed state, this chapter derives deterministic extension bounds for several approximate identities and separates the contributions of state error, residual error, timing error, and boundary traces\.

#### 9\.1\.2Additive residuals

Suppose

ek=\(Δ​v\)k\+rk,e\_\{k\}=\(\\Delta v\)\_\{k\}\+r\_\{k\},\(9\.1\)whererris an unshaped residual\.

###### Theorem 9\.1\(First\-order residual bound\)\.

Assumev0=0v\_\{0\}=0and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\. Then, for arbitrary weights,

\|1N​∑k=1Nek​wk\|≤VN​\(\|wN\|\+TVN⁡\(w\)\)\+1N​∑k=1N\|rk\|​\|wk\|\.\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\(\\left\\lvert w\_\{N\}\\right\\rvert\+\\operatorname\{TV\}\_\{N\}\(w\)\\right\)\+\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\\left\\lvert r\_\{k\}\\right\\rvert\\left\\lvert w\_\{k\}\\right\\rvert\.\(9\.2\)If\|wk\|=1\\left\\lvert w\_\{k\}\\right\\rvert=1, the residual contribution is‖r‖ℓ1/N\\left\\lVert r\\right\\rVert\_\{\\ell^\{1\}\}/N\.

###### Proof\.

Apply the weighted variation theorem toΔ​v\\Delta vand the triangle inequality torr\. ∎

If‖r‖ℓ1=o​\(N\)\\left\\lVert r\\right\\rVert\_\{\\ell^\{1\}\}=o\(N\), the residual vanishes after normalisation\. If each\|rk\|≤ρ\\left\\lvert r\_\{k\}\\right\\rvert\\leq\\rhowith no further structure, the residual term is at mostρ\\rhoand creates a nondecaying floor\. A small pointwise mismatch is therefore not automatically harmless on a long record\.

#### 9\.1\.3Residuals with their own noise\-shaping order

A perturbation may itself be partially shaped\. Suppose

withs≥1s\\geq 1and boundedzz\. Then the residual can be estimated by another summation\-by\-parts argument\. Ifs=1s=1and the initial state ofzzis zero,

\|1N​∑rk​wk\|≤‖z‖∞N​\(\|wN\|\+TVN⁡\(w\)\)\.\\left\\lvert\\frac\{1\}\{N\}\\sum r\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{\\left\\lVert z\\right\\rVert\_\{\\infty\}\}\{N\}\\left\(\\left\\lvert w\_\{N\}\\right\\rvert\+\\operatorname\{TV\}\_\{N\}\(w\)\\right\)\.Thus two separately shaped components may be combined without producing an error floor\.

This observation is useful for modelling element mismatch that has been dynamically scrambled or digitally noise\-shaped\. The relevant question is not only the size of the mismatch but also whether its accumulated sum is controlled\.

#### 9\.1\.4Leaky first\-order shaping

A simple leakage model is

ek=vk−ρ​vk−1,0<ρ≤1\.e\_\{k\}=v\_\{k\}\-\\rho v\_\{k\-1\},\\qquad 0<\\rho\\leq 1\.\(9\.3\)Write

ek=\(vk−vk−1\)\+\(1−ρ\)​vk−1\.e\_\{k\}=\(v\_\{k\}\-v\_\{k\-1\}\)\+\(1\-\\rho\)v\_\{k\-1\}\.The first term is shaped and the second is an unshaped leakage component\.

###### Proposition 9\.2\(Leaky weighted estimate\)\.

Assumev0=0v\_\{0\}=0and‖v‖ℓ∞≤V\\left\\lVert v\\right\\rVert\_\{\\ell^\{\\infty\}\}\\leq V\. Then

\|1N​∑k=1Nek​wk\|≤VN​\(\|wN\|\+TVN⁡\(w\)\)\+\(1−ρ\)​V​1N​∑k=1N\|wk\|\.\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\(\\left\\lvert w\_\{N\}\\right\\rvert\+\\operatorname\{TV\}\_\{N\}\(w\)\\right\)\+\(1\-\\rho\)V\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\\left\\lvert w\_\{k\}\\right\\rvert\.\(9\.4\)For unit\-modulus weights,

\|1N​∑ek​wk\|≤VN​\(1\+TVN⁡\(w\)\)\+\(1−ρ\)​V\.\\left\\lvert\\frac\{1\}\{N\}\\sum e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\(1\+\\operatorname\{TV\}\_\{N\}\(w\)\\right\)\+\(1\-\\rho\)V\.\(9\.5\)

The term\(1−ρ\)​V\(1\-\\rho\)Vis a leakage floor\. To retain anN−1N^\{\-1\}compact\-set rate, one needs1−ρ=O​\(N−1\)1\-\\rho=O\(N^\{\-1\}\), unless additional cancellation is known\. For a fixed physical leakage factor, increasing the record length eventually stops improving the deterministic bound\.

#### 9\.1\.5Perturbed weights and sampling jitter

Suppose the intended weight iswkw\_\{k\}but the realised weight is

w~k=wk\+δ​wk\.\\widetilde\{w\}\_\{k\}=w\_\{k\}\+\\delta w\_\{k\}\.Then

1N​∑ek​w~k=1N​∑ek​wk\+1N​∑ek​δ​wk\.\\frac\{1\}\{N\}\\sum e\_\{k\}\\widetilde\{w\}\_\{k\}=\\frac\{1\}\{N\}\\sum e\_\{k\}w\_\{k\}\+\\frac\{1\}\{N\}\\sum e\_\{k\}\\delta w\_\{k\}\.A pointwise error bound\|ek\|≤E\\left\\lvert e\_\{k\}\\right\\rvert\\leq Egives

\|1N​∑ek​δ​wk\|≤E​‖δ​w‖ℓ∞\.\\left\\lvert\\frac\{1\}\{N\}\\sum e\_\{k\}\\delta w\_\{k\}\\right\\rvert\\leq E\\left\\lVert\\delta w\\right\\rVert\_\{\\ell^\{\\infty\}\}\.This estimate ignores shaping\. A stronger result follows if the perturbation sequence is smooth inkk:

\|1N​∑\(Δ​v\)k​δ​wk\|≤VN​\(\|δ​wN\|\+TVN⁡\(δ​w\)\)\.\\left\\lvert\\frac\{1\}\{N\}\\sum\(\\Delta v\)\_\{k\}\\delta w\_\{k\}\\right\\rvert\\leq\\frac\{V\}\{N\}\\left\(\\left\\lvert\\delta w\_\{N\}\\right\\rvert\+\\operatorname\{TV\}\_\{N\}\(\\delta w\)\\right\)\.Thus slowly varying timing or phase errors are filtered by the same discrete BV mechanism as the nominal weight\.

For sampling jitter, let the actual node be

ξ~k=kN\+εk\\widetilde\{\\xi\}\_\{k\}=\\frac\{k\}\{N\}\+\\varepsilon\_\{k\}and set

w~k=e2​π​i​ϕ​\(ξ~k\)\.\\widetilde\{w\}\_\{k\}=e^\{2\\pi\\mathrm\{i\}\\phi\(\\widetilde\{\\xi\}\_\{k\}\)\}\.Ifϕ\\phiis Lipschitz and\|εk\|≤η\\left\\lvert\\varepsilon\_\{k\}\\right\\rvert\\leq\\eta, then

\|w~k−wk\|≤2​π​Lip⁡\(ϕ\)​η\.\\left\\lvert\\widetilde\{w\}\_\{k\}\-w\_\{k\}\\right\\rvert\\leq 2\\pi\\operatorname\{Lip\}\(\\phi\)\\eta\.A useful bound must also control the variation ofεk\\varepsilon\_\{k\}\. If the jitter alternates rapidly,TVN⁡\(w~\)\\operatorname\{TV\}\_\{N\}\(\\widetilde\{w\}\)may be large even when the pointwise displacement is small\.

###### Proposition 9\.3\(Jitter with bounded variation\)\.

Assumeϕ∈C1\\phi\\in C^\{1\}with‖ϕ′‖∞≤L\\left\\lVert\\phi^\{\\prime\}\\right\\rVert\_\{\\infty\}\\leq L, and assume the perturbed nodes remain in a fixed interval on which this bound holds\. Then

TVN⁡\(w~\)≤2​π​L​\(1\+TVN⁡\(ε\)\),\\operatorname\{TV\}\_\{N\}\(\\widetilde\{w\}\)\\leq 2\\pi L\\left\(1\+\\operatorname\{TV\}\_\{N\}\(\\varepsilon\)\\right\),where

TVN⁡\(ε\)=∑k=1N−1\|εk\+1−εk\|\.\\operatorname\{TV\}\_\{N\}\(\\varepsilon\)=\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert\\varepsilon\_\{k\+1\}\-\\varepsilon\_\{k\}\\right\\rvert\.

###### Proof\.

The distance between adjacent perturbed nodes satisfies

\|ξ~k\+1−ξ~k\|≤N−1\+\|εk\+1−εk\|\.\\left\\lvert\\widetilde\{\\xi\}\_\{k\+1\}\-\\widetilde\{\\xi\}\_\{k\}\\right\\rvert\\leq N^\{\-1\}\+\\left\\lvert\\varepsilon\_\{k\+1\}\-\\varepsilon\_\{k\}\\right\\rvert\.Apply the mean\-value estimate to the phase and sum\. ∎

Random independent jitter typically has variation proportional toNNand is therefore not small in this deterministic metric\. A probabilistic analysis may yield better average behaviour, but it requires a specified distribution\.

#### 9\.1\.6Coefficient saturation and overload events

LetG⊂\{1,…,N\}G\\subset\\\{1,\\ldots,N\\\}be the set of indices at which the ideal shaped relation holds, and letBBbe an overload set\. Write

e=esh\+ebad,e=e^\{\\mathrm\{sh\}\}\+e^\{\\mathrm\{bad\}\},whereebade^\{\\mathrm\{bad\}\}is supported onBB\. If\|ekbad\|≤M\\left\\lvert e\_\{k\}^\{\\mathrm\{bad\}\}\\right\\rvert\\leq M, then for unit weights

\|1N​∑ekbad​wk\|≤M​\|B\|N\.\\left\\lvert\\frac\{1\}\{N\}\\sum e\_\{k\}^\{\\mathrm\{bad\}\}w\_\{k\}\\right\\rvert\\leq M\\frac\{\\left\\lvert B\\right\\rvert\}\{N\}\.A vanishing fraction of overload events is therefore sufficient for consistency, even if the individual events are not small\. To preserve anN−1N^\{\-1\}rate, however, the number of bad samples must remainO​\(1\)O\(1\)or their errors must cancel\.

This separation is useful in circuit modelling\. A rare large transition and a persistent small mismatch have different asymptotic effects\. The former is controlled by event count; the latter may create a fixed floor\.

#### 9\.1\.7Approximate high\-order shaping

Suppose

with trace\-compatiblevv\. For a sampledCrC^\{r\}weightWW,

\|1N​∑ek​W​\(k/N\)\|≤V​N−r​‖W\(r\)‖∞\+1N​∑\|rk\|​\|W​\(k/N\)\|\.\\left\\lvert\\frac\{1\}\{N\}\\sum e\_\{k\}W\(k/N\)\\right\\rvert\\leq VN^\{\-r\}\\left\\lVert W^\{\(r\)\}\\right\\rVert\_\{\\infty\}\+\\frac\{1\}\{N\}\\sum\\left\\lvert r\_\{k\}\\right\\rvert\\left\\lvert W\(k/N\)\\right\\rvert\.\(9\.6\)To observe theN−rN^\{\-r\}rate, the residual average must beO​\(N−r\)O\(N^\{\-r\}\)\. This is a stringent requirement\. A residual of sizeN−sN^\{\-s\}per sample contributesN−sN^\{\-s\}after normalisation, so it dominates whenevers<rs<r\.

The conclusion is practical: a nominally high\-order quantizer can display only first\-order or zero\-order behaviour if leakage, mismatch, or imperfect termination is not reduced to the same asymptotic scale\.

#### 9\.1\.8Dither and probabilistic modelling

Dither is often introduced to decorrelate quantization error from the input\. Classical dithered quantization can convert deterministic distortion into a random error with tractable moments\. In sigma\-delta frame quantization, random dither has been used to obtain mean\-square estimates without invoking a white\-noise hypothesis\. The present manuscript does not assume a stochastic model, but the deterministic formulas identify what a probabilistic analysis should estimate:

1. \(i\)moments of the boundary traces;
2. \(ii\)moments or concentration of the interior pairing∑vk​Δ\+r​wk\\sum v\_\{k\}\\Delta\_\{\+\}^\{r\}w\_\{k\};
3. \(iii\)the probability and magnitude of overload events;
4. \(iv\)correlations between the state and the oscillatory phase\.

A probabilistic theorem can improve an average norm even when the worst\-case deterministic variation remains large\. Its conclusion is an average\-norm estimate under the stated stochastic model\.

#### 9\.1\.9A robustness checklist

For a claimed experimental or simulated convergence law, the following quantities should be reported\.

1. \(a\)The maximum state magnitude and the state order\.
2. \(b\)The initial and final boundary traces\.
3. \(c\)The residual sequencer=e−Δr​vr=e\-\\Delta^\{r\}v\.
4. \(d\)The fraction of overload or saturation samples\.
5. \(e\)The variation of timing and phase perturbations\.
6. \(f\)The range ofNNover which a fitted slope is measured\.

Without these diagnostics, a measuredN−rN^\{\-r\}line may reflect a short pre\-asymptotic range, and a failure to reachN−rN^\{\-r\}may be incorrectly attributed to the Fourier analysis rather than to boundary or residual errors\.

### 9\.2Blockwise reset, smooth termination, and boundary engineering

#### 9\.2\.1The purpose of block processing

A long coefficient record may be divided into shorter blocks so that the internal state can be reset, transmitted, or terminated\. This is attractive when a global terminal condition is difficult to enforce\. It also permits parallel processing\. The cost is the appearance of one boundary contribution per block\.

Let

0=n0<n1<⋯<nB=N0=n\_\{0\}<n\_\{1\}<\\cdots<n\_\{B\}=Nbe a partition, and let

Ib=\{nb−1\+1,…,nb\},Lb=nb−nb−1\.I\_\{b\}=\\\{n\_\{b\-1\}\+1,\\ldots,n\_\{b\}\\\},\\qquad L\_\{b\}=n\_\{b\}\-n\_\{b\-1\}\.For clarity, begin with equal block lengthLL, soB=N/LB=N/L\.

#### 9\.2\.2First\-order block identity

Assume that on blockIbI\_\{b\},

ek=vk\(b\)−vk−1\(b\),e\_\{k\}=v\_\{k\}^\{\(b\)\}\-v\_\{k\-1\}^\{\(b\)\},with a local state initialised at

vnb−1\(b\)=0\.v\_\{n\_\{b\-1\}\}^\{\(b\)\}=0\.The state is allowed to end at a nonzero value\.

###### Proposition 9\.4\(Blockwise first\-order estimate\)\.

If‖v\(b\)‖ℓ∞​\(Ib\)≤V\\left\\lVert v^\{\(b\)\}\\right\\rVert\_\{\\ell^\{\\infty\}\(I\_\{b\}\)\}\\leq Vfor every block, then

\|1N​∑k=1Nek​wk\|\\displaystyle\\left\\lvert\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\\rvert≤VN​∑b=1B\|wnb\|\+VN​∑b=1B∑k=nb−1\+1nb−1\|wk\+1−wk\|\.\\displaystyle\\leq\\frac\{V\}\{N\}\\sum\_\{b=1\}^\{B\}\\left\\lvert w\_\{n\_\{b\}\}\\right\\rvert\+\\frac\{V\}\{N\}\\sum\_\{b=1\}^\{B\}\\sum\_\{k=n\_\{b\-1\}\+1\}^\{n\_\{b\}\-1\}\\left\\lvert w\_\{k\+1\}\-w\_\{k\}\\right\\rvert\.\(9\.7\)For unit\-modulus weights,

\|1N​∑ek​wk\|≤V​BN\+VN​TVN⁡\(w\)=VL\+VN​TVN⁡\(w\)\.\\left\\lvert\\frac\{1\}\{N\}\\sum e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{VB\}\{N\}\+\\frac\{V\}\{N\}\\operatorname\{TV\}\_\{N\}\(w\)=\\frac\{V\}\{L\}\+\\frac\{V\}\{N\}\\operatorname\{TV\}\_\{N\}\(w\)\.\(9\.8\)

###### Proof\.

Apply the first\-order weighted theorem on each block\. The local initial term vanishes\. Sum the resulting estimates\. The interior variations of the blocks are disjoint and are bounded by the global variation\. ∎

The reset cost isV/LV/L\. A fixed block length therefore creates a fixed error floor\. To retain convergence, the block length must grow withNNor the terminal state must be cancelled\.

#### 9\.2\.3Choosing the block length

For a fixed compact phase region,TVN⁡\(w\)=O​\(1\)\\operatorname\{TV\}\_\{N\}\(w\)=O\(1\)\. The bound becomes

\|1N​∑ek​wk\|≲VL\+VN\.\\left\\lvert\\frac\{1\}\{N\}\\sum e\_\{k\}w\_\{k\}\\right\\rvert\\lesssim\\frac\{V\}\{L\}\+\\frac\{V\}\{N\}\.IfL=NγL=N^\{\\gamma\}with0<γ≤10<\\gamma\\leq 1, then the rate isO​\(N−γ\)O\(N^\{\-\\gamma\}\)\. Full first\-order decay requiresL≍NL\\asymp N, which reduces to a single block\. Smaller blocks trade asymptotic accuracy for local state management\.

On a growing region with variationO​\(Nα\)O\(N^\{\\alpha\}\), the estimate is

O​\(N−γ\)\+O​\(Nα−1\)\.O\(N^\{\-\\gamma\}\)\+O\(N^\{\\alpha\-1\}\)\.Balancing the two terms suggests

when0<α<10<\\alpha<1\. This givesO​\(Nα−1\)O\(N^\{\\alpha\-1\}\)\. The block length needed to avoid dominating the variation error is thereforeL≳N1−αL\\gtrsim N^\{1\-\\alpha\}\.

#### 9\.2\.4Exact terminal reset

Suppose every block satisfies

vnb−1\(b\)=vnb\(b\)=0\.v\_\{n\_\{b\-1\}\}^\{\(b\)\}=v\_\{n\_\{b\}\}^\{\(b\)\}=0\.Then the block endpoint term vanishes and

∑k∈Ibek​wk=−∑k=nb−1\+1nb−1vk\(b\)​\(wk\+1−wk\)\.\\sum\_\{k\\in I\_\{b\}\}e\_\{k\}w\_\{k\}=\-\\sum\_\{k=n\_\{b\-1\}\+1\}^\{n\_\{b\}\-1\}v\_\{k\}^\{\(b\)\}\(w\_\{k\+1\}\-w\_\{k\}\)\.Summing over blocks yields

\|1N​∑ek​wk\|≤VN​TVN⁡\(w\)\.\\left\\lvert\\frac\{1\}\{N\}\\sum e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{V\}\{N\}\\operatorname\{TV\}\_\{N\}\(w\)\.\(9\.9\)The rate is the same as for a single globally initialised state, with the terminal amplitude term removed\. First\-order exact reset therefore permits parallel blocks without an asymptotic reset penalty\.

Achieving this condition may require appended symbols or a short transition alphabet\. The existence and length of a terminal control sequence are determined by the state recursion, the admissible input range, and the available output alphabet\.

#### 9\.2\.5High\-order block compatibility

For orderrr, each block has its own family of traces\. If every block is trace\-compatible, then

∑k∈Ib\(Δr​v\(b\)\)k​wk=\(−1\)r​∑k∈Ib∘vk\(b\)​\(Δ\+r​w\)k\.\\sum\_\{k\\in I\_\{b\}\}\(\\Delta^\{r\}v^\{\(b\)\}\)\_\{k\}w\_\{k\}=\(\-1\)^\{r\}\\sum\_\{k\\in I\_\{b\}^\{\\circ\}\}v\_\{k\}^\{\(b\)\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\.The interior index sets are disjoint, so

\|1N​∑ek​wk\|≤VN​∑k=1N−r\|Δ\+r​wk\|\.\\left\\lvert\\frac\{1\}\{N\}\\sum e\_\{k\}w\_\{k\}\\right\\rvert\\leq\\frac\{V\}\{N\}\\sum\_\{k=1\}^\{N\-r\}\\left\\lvert\\Delta\_\{\+\}^\{r\}w\_\{k\}\\right\\rvert\.\(9\.10\)For sampledCrC^\{r\}weights, this isO​\(N−r\)O\(N^\{\-r\}\)and does not depend on the number of blocks\.

This conclusion is strong but the hypothesis is correspondingly strong\. Each block must begin and end with the complete trace state prepared\. For a second\-order scheme, setting only the scalar statevvto zero is not enough; the first difference trace must also vanish\.

#### 9\.2\.6Imperfect block termination

Let thejjth right trace on blockbbbe bounded byηj,b\\eta\_\{j,b\}\. With zero left traces, the total boundary error is bounded by

1N​∑b=1B∑j=0r−1ηj,b​\|\(Δ\+j​w\)nb−j\|\.\\frac\{1\}\{N\}\\sum\_\{b=1\}^\{B\}\\sum\_\{j=0\}^\{r\-1\}\\eta\_\{j,b\}\\left\\lvert\(\\Delta\_\{\+\}^\{j\}w\)\_\{n\_\{b\}\-j\}\\right\\rvert\.\(9\.11\)For a sampled smooth weight,

\|Δ\+j​w\|≲N−j\.\\left\\lvert\\Delta\_\{\+\}^\{j\}w\\right\\rvert\\lesssim N^\{\-j\}\.Ifηj,b≤ηj\\eta\_\{j,b\}\\leq\\eta\_\{j\}uniformly andB=N/LB=N/L, thejjth trace contributes

O​\(ηjL​N−j\)\.O\\\!\\left\(\\frac\{\\eta\_\{j\}\}\{L\}N^\{\-j\}\\right\)\.To preserve the globalN−rN^\{\-r\}rate, one needs

ηj=O​\(L​N−\(r−j\)\)\.\\eta\_\{j\}=O\(LN^\{\-\(r\-j\)\}\)\.The required accuracy depends on both the total record length and the block length\.

#### 9\.2\.7Windowed reconstruction as an alternative

Instead of forcing the state to vanish, one can modify the weight near block boundaries\. Letχb​\(k\)\\chi\_\{b\}\(k\)be a window supported on blockbband consider

∑k∈Ibek​χb​\(k\)​wk\.\\sum\_\{k\\in I\_\{b\}\}e\_\{k\}\\chi\_\{b\}\(k\)w\_\{k\}\.Ifχb\\chi\_\{b\}and its firstr−1r\-1discrete differences vanish at both ends, then the boundary terms generated by repeated summation by parts are suppressed even when the state traces are nonzero\.

The price is that the windows alter the target extension\. A partition of unity can reduce the bias:

∑b=1Bχb​\(k\)=1\.\\sum\_\{b=1\}^\{B\}\\chi\_\{b\}\(k\)=1\.Constructing windows with overlap, endpoint flatness, and a controlled sum ofrrth differences is a discrete analogue of smooth localisation in harmonic analysis\. The technique resembles smooth frame\-path termination, where the analysis vectors are arranged to approach zero with several derivatives\[[9](https://arxiv.org/html/2607.24868#bib.bib20)\]\.

#### 9\.2\.8A discrete polynomial window

For a block of lengthLL, define a rescaled coordinate

s=k−nb−1L\.s=\\frac\{k\-n\_\{b\-1\}\}\{L\}\.The polynomial

χ​\(s\)=sr​\(1−s\)r\\chi\(s\)=s^\{r\}\(1\-s\)^\{r\}vanishes to orderrrat both endpoints\. After normalisation and overlap, such functions can be used to build smooth windows\. Theirrrth derivatives scale likeL−rL^\{\-r\}in the block coordinate\. Therefore the cost of localisation is governed byL−rL^\{\-r\}rather thanN−rN^\{\-r\}\.

This reveals a second block tradeoff\. A local window of widthLLnaturally produces anL−rL^\{\-r\}scale\. To achieve a globalN−rN^\{\-r\}rate without state termination, the window width must be comparable withNN\. Multiple short windows generally lose powers\.

#### 9\.2\.9State flushing by an appended tail

Another option is to appendTTadditional quantizer inputs after the original record\. Let the original data occupy1≤k≤N1\\leq k\\leq N, and choose control inputs forN<k≤N\+TN<k\\leq N\+Tso that the final traces vanish\. The extension can then be formed with a weight that is zero on the appended tail, or the tail can be removed after its effect on the boundary formula is accounted for\.

A rigorous flushing theorem requires three ingredients:

1. \(i\)reachability of the zero trace state under the allowed alphabet;
2. \(ii\)a bound on the required tail lengthTT;
3. \(iii\)stability of all intermediate states\.

For the first\-order greedy one\-bit scheme with arbitrary fixed input, exact reachability may fail under a rigid sign rule\. Allowing a small set of terminal control values or modifying the final threshold can restore reachability\. These are control\-design questions rather than consequences of the summation identity\.

#### 9\.2\.10Parallel implementation and numerical experiments

Block processing is attractive computationally because each block can be quantized independently\. Numerical studies should compare at least four variants:

1. \(a\)one uninterrupted state;
2. \(b\)reset to zero at the beginning of each block only;
3. \(c\)exact or approximate terminal reset;
4. \(d\)smooth windowing without state reset\.

For each variant, plot the extension error againstNN, the block lengthLL, and the number of blocks\. The slope should be interpreted together with measured boundary traces\. A blockwise scheme that appears to have high\-order decay for moderateNNmay eventually reach the predicted1/L1/Lfloor\.

#### 9\.2\.11Design principle

The analysis leads to a simple principle:

> Boundary management is part of the noise\-shaping order\. A scheme is not effectively orderrron a finite record unless its initialisation, termination, or reconstruction window removes all traces below orderrr\.

This principle links the algebraic theory in Chapter[6](https://arxiv.org/html/2607.24868#Ch6)to implementation choices\. It also provides a clear way to separate a quantizer\-design contribution from a Fourier\-analysis contribution in a future journal paper\.

### 9\.3Verification of the proved finite\-record results

#### 9\.3\.1Computational verification protocol

Numerical experiments verify recursions, finite\-difference identities, indexing conventions, constants, convergence slopes, and resonant trajectories\. A reproducible experiment records the quantizer rule, initial state, input range, phase parameters, record length, and normalisation\.

#### 9\.3\.2Core experiment

For a chosen inputuku\_\{k\}, compute

qk=Q​\(uk\+vk−1\),vk=vk−1\+uk−qk\.q\_\{k\}=Q\(u\_\{k\}\+v\_\{k\-1\}\),\\qquad v\_\{k\}=v\_\{k\-1\}\+u\_\{k\}\-q\_\{k\}\.Verify numerically that

maxk⁡\|vk\|≤1\\max\_\{k\}\\left\\lvert v\_\{k\}\\right\\rvert\\leq 1and

maxk⁡\|uk−qk−\(vk−vk−1\)\|\\max\_\{k\}\\left\\lvert u\_\{k\}\-q\_\{k\}\-\(v\_\{k\}\-v\_\{k\-1\}\)\\right\\rvertis at machine precision\.

Next evaluate

EN​\(x,t\)=\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|E\_\{N\}\(x,t\)=\\left\\lvert\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\\right\\rvertand compare it with

BN​\(x,t\)=1N​\[1\+2​π​J​\(x,t\)\]\.B\_\{N\}\(x,t\)=\\frac\{1\}\{N\}\\left\[1\+2\\pi J\(x,t\)\\right\]\.The computed ratio satisfiesEN/BN≤1E\_\{N\}/B\_\{N\}\\leq 1up to floating\-point error\.

A reproducible example uses

uk=0\.55​sin⁡\(2​π​\(3​kN\+0\.17\)\),\(x,t\)=\(0\.73,−0\.41\)\.u\_\{k\}=0\.55\\sin\\\!\\left\(2\\pi\\left\(3\\frac\{k\}\{N\}\+0\.17\\right\)\\right\),\\qquad\(x,t\)=\(0\.73,\-0\.41\)\.ForN=26,…,213N=2^\{6\},\\ldots,2^\{13\}, the computed state remained below0\.99990\.9999in magnitude and the largest observed ratio between the error and the theoreticalJJ\-bound was below0\.0390\.039\. The fitted slope was approximately−1\.41\-1\.41, reflecting additional cancellation for the selected input and parameter pair\. The zero\-input experiment below recovers the uniform sharpness exponent−1\-1\.

![Refer to caption](https://arxiv.org/html/2607.24868v1/x1.png)Figure 9\.1:Computed first\-order error and the deterministic variation bound at a fixed parabolic parameter\. The bound has the guaranteedN−1N^\{\-1\}scale, while the selected input exhibits additional cancellation\.
#### 9\.3\.3Convergence\-order estimation

Use a geometric sequence of lengths, for example

N=28,29,…,216\.N=2^\{8\},2^\{9\},\\ldots,2^\{16\}\.At fixed\(x,t\)\(x,t\), fit a line to

log⁡ENagainstlog⁡N\.\\log E\_\{N\}\\quad\\text\{against\}\\quad\\log N\.A generic first\-order trajectory has reference slope−1\-1\. Faster slopes identify additional terminal or oscillatory cancellation for the selected trajectory\.

For the zero input and oddNN, evaluate at\(0,0\)\(0,0\)\. The exact value is1/N1/N, so the fitted slope should be exactly−1\-1up to numerical precision\.

![Refer to caption](https://arxiv.org/html/2607.24868v1/x2.png)Figure 9\.2:Exact sharpness experiment for the zero input and odd record lengths\. The computed values coincide withN−1N^\{\-1\}\.
#### 9\.3\.4Testing the closed form ofJJ

ComputeJJin two ways:

1. \(i\)numerical quadrature of\|x\+2​t​ξ\|\\left\\lvert x\+2t\\xi\\right\\rvert;
2. \(ii\)the piecewise formula in[Proposition˜5\.2](https://arxiv.org/html/2607.24868#Ch5.Thmtheorem2)\.

Test points should includet=0t=0, same\-sign endpoints, and sign\-changing derivatives\. Near the transitionx​\(x\+2​t\)=0x\(x\+2t\)=0, both formulas should agree continuously\.

![Refer to caption](https://arxiv.org/html/2607.24868v1/x3.png)Figure 9\.3:The phase\-variation factorJ​\(x,t\)=∫01\|x\+2​t​ξ\|​dξJ\(x,t\)=\\int\_\{0\}^\{1\}\\left\\lvert x\+2t\\xi\\right\\rvert\\,\\,\\mathrm\{d\}\\xion a bounded parameter square\. The change in formula occurs where the derivative crosses zero inside the sampling interval\.
#### 9\.3\.5Exact sine variation

Compare three bounds:

Bsimple=1N​\[1\+2​π​\(\|x\|\+\|t\|\)\],B\_\{\\mathrm\{simple\}\}=\\frac\{1\}\{N\}\[1\+2\\pi\(\\left\\lvert x\\right\\rvert\+\\left\\lvert t\\right\\rvert\)\],BJ=1N​\[1\+2​π​J​\(x,t\)\],B\_\{J\}=\\frac\{1\}\{N\}\[1\+2\\pi J\(x,t\)\],and

Bsin=1N​\[1\+2​∑k=1N−1\|sin⁡\(π​δk\)\|\]\.B\_\{\\sin\}=\\frac\{1\}\{N\}\\left\[1\+2\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert\\sin\(\\pi\\delta\_\{k\}\)\\right\\rvert\\right\]\.The sine bound is the most faithful to the sampled weights\. TheJJbound is smooth and asymptotically natural\. The simple bound is easiest to state\.

#### 9\.3\.6State\-phase alignment

To study the sharpness of the variation step, compute

Ak=vk​Δ\+​wk\.A\_\{k\}=v\_\{k\}\\Delta\_\{\+\}w\_\{k\}\.Plot or tabulate the arguments ofAkA\_\{k\}\. If they are nearly aligned, the triangle inequality is close to equality\. If their arguments spread around the circle, actual cancellation is stronger\.

For realisable state trajectories, this experiment may reveal whether adversarial alignment occurs at specific rational parameters or inputs\.

#### 9\.3\.7Growing regions

For eachNN, choose a grid in

for several values ofα\\alpha\. Record the maximum error and compare the fitted exponent withα−1\\alpha\-1\. Atα=1\\alpha=1, investigate whether the maximum remains bounded away from zero\.

A uniform grid may miss narrow resonances\. Adaptive refinement around large values is preferable\.

#### 9\.3\.8High\-order synthetic validation

Before implementing a physical high\-order quantizer, one can validate the finite\-interval identity with a synthetic bounded state\. Choosevkv\_\{k\}directly, computee=Δr​ve=\\Delta^\{r\}v, and evaluate both sides of \([6\.2](https://arxiv.org/html/2607.24868#Ch6.Ex5)\)\. This test is effective for detecting off\-by\-one errors\.

To test the endpoint\-compatible theorem, construct a state that vanishes together with the required discrete traces\. One method is to sample a smooth compactly supported profile and pad it with zeros\. The resulting error need not be a one\-bit quantization error, but it validates the analytic identity\.

For the experiment in[Figure˜9\.4](https://arxiv.org/html/2607.24868#Ch9.F4), the state is zero at the first two and last two relevant grid locations, and its interior values sample a fourth power of a sine profile\. The resulting second difference satisfies the exact finite\-record compatibility conditions\. A least\-squares fit overN=26,…,213N=2^\{6\},\\ldots,2^\{13\}gives a slope of approximately−1\.991\-1\.991, in agreement with the predicted second\-order scale\.

![Refer to caption](https://arxiv.org/html/2607.24868v1/x4.png)Figure 9\.4:Synthetic second\-order validation with exact endpoint compatibility\. The fitted exponent is approximately−1\.991\-1\.991\. This experiment verifies the analytic scaling but does not by itself construct a one\-bit second\-order quantizer with terminal control\.
#### 9\.3\.9Monte Carlo decorrelation experiments

For random inputs, estimate state correlations

C​\(h\)=1N−h​∑k=1N−hvk\+h​vk\.C\(h\)=\\frac\{1\}\{N\-h\}\\sum\_\{k=1\}^\{N\-h\}v\_\{k\+h\}v\_\{k\}\.Compare different input distributions, amplitudes, and deterministic sinusoids\. Slow correlation decay or periodic peaks indicate possible resonances\. Fast decay provides numerical evidence for the probabilistic hypotheses stated in Chapter[8](https://arxiv.org/html/2607.24868#Ch8)\.

#### 9\.3\.10Reporting standards

Every figure or table should include:

1. \(a\)exact recursion and zero convention;
2. \(b\)input definition and amplitude;
3. \(c\)NNand phase grid;
4. \(d\)whetherNNis odd or even;
5. \(e\)normalised or unnormalised extension;
6. \(f\)measured state bound;
7. \(g\)theoretical bound used for comparison;
8. \(h\)numerical precision and software version\.

Appendix E provides Python and MATLAB\-style code templates\.

## Chapter 10Consolidated Derivations, Main Results, and Conclusions

### 10\.1Purpose of the consolidated chapter

The preceding chapters develop the theory in its natural logical order\. This chapter collects the principal assumptions, derivations, theorem dependencies, and scale relations in a compact report\-style form\. It also identifies the results that form a coherent focused journal manuscript\.

### 10\.2Master notation and standing assumptions

LetN≥1N\\geq 1and letu=\(uk\)k=1Nu=\(u\_\{k\}\)\_\{k=1\}^\{N\}be a real input record\. A one\-bit output record satisfiesqk∈\{−1,\+1\}q\_\{k\}\\in\\\{\-1,\+1\\\}\. The normalised polynomial Fourier extension associated with a phaseϕθ\\phi\_\{\\theta\}is

ℰN​a​\(θ\)=1N​∑k=1Nak​exp⁡\(2​π​i​ϕθ​\(k/N\)\)\.\\mathcal\{E\}\_\{N\}a\(\\theta\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}a\_\{k\}\\exp\\\!\\left\(2\\pi\\mathrm\{i\}\\phi\_\{\\theta\}\(k/N\)\\right\)\.The first\-order theory assumes

uk−qk=vk−vk−1,v0=0,max0≤k≤N⁡\|vk\|≤V\.u\_\{k\}\-q\_\{k\}=v\_\{k\}\-v\_\{k\-1\},\\qquad v\_\{0\}=0,\\qquad\\max\_\{0\\leq k\\leq N\}\|v\_\{k\}\|\\leq V\.\(10\.1\)The high\-order theory assumes

uk−qk=\(Δr​v\)k,u\_\{k\}\-q\_\{k\}=\(\\Delta^\{r\}v\)\_\{k\},\(10\.2\)with all negative\-index values required by the difference convention explicitly specified\. Endpoint compatibility means that the initial and terminal traces occurring in the repeated finite summation formula vanish\. Approximate compatibility means that these traces are small enough to be retained as controlled remainder terms\.

### 10\.3Complete first\-order derivation

#### 10\.3\.1Stability of the greedy recursion

The greedy first\-order recursion is

qk=Q​\(uk\+vk−1\),vk=vk−1\+uk−qk,q\_\{k\}=Q\(u\_\{k\}\+v\_\{k\-1\}\),\\qquad v\_\{k\}=v\_\{k\-1\}\+u\_\{k\}\-q\_\{k\},whereQ​\(y\)=1Q\(y\)=1fory≥0y\\geq 0andQ​\(y\)=−1Q\(y\)=\-1fory<0y<0\. Suppose that\|uk\|≤1\|u\_\{k\}\|\\leq 1and\|vk−1\|≤1\|v\_\{k\-1\}\|\\leq 1\. Thenyk=uk\+vk−1y\_\{k\}=u\_\{k\}\+v\_\{k\-1\}belongs to\[−2,2\]\[\-2,2\]\. Ifyk≥0y\_\{k\}\\geq 0, thenvk=yk−1∈\[−1,1\]v\_\{k\}=y\_\{k\}\-1\\in\[\-1,1\]\. Ifyk<0y\_\{k\}<0, thenvk=yk\+1∈\[−1,1\)v\_\{k\}=y\_\{k\}\+1\\in\[\-1,1\)\. Sincev0=0v\_\{0\}=0, induction proves

\|vk\|≤1\(0≤k≤N\)\.\|v\_\{k\}\|\\leq 1\\qquad\(0\\leq k\\leq N\)\.The proof uses no probabilistic model and no Fourier argument\. It is an invariant\-interval proof of deterministic stability, consistent with the classical mathematical theory of stable sigma\-delta modulation\[[25](https://arxiv.org/html/2607.24868#bib.bib7),[48](https://arxiv.org/html/2607.24868#bib.bib8),[50](https://arxiv.org/html/2607.24868#bib.bib14)\]\.

#### 10\.3\.2Discrepancy identity

For every interval\[m,n\]⊆\[1,N\]\[m,n\]\\subseteq\[1,N\],

∑k=mn\(uk−qk\)\\displaystyle\\sum\_\{k=m\}^\{n\}\(u\_\{k\}\-q\_\{k\}\)=∑k=mn\(vk−vk−1\)\\displaystyle=\\sum\_\{k=m\}^\{n\}\(v\_\{k\}\-v\_\{k\-1\}\)\(10\.3\)=vn−vm−1\.\\displaystyle=v\_\{n\}\-v\_\{m\-1\}\.\(10\.4\)Consequently,

\|∑k=mn\(uk−qk\)\|≤2​V\.\\left\|\\sum\_\{k=m\}^\{n\}\(u\_\{k\}\-q\_\{k\}\)\\right\|\\leq 2V\.For a prefix interval withm=1m=1andv0=0v\_\{0\}=0, the sharper bound isVV\. The estimate shows that first\-order shaping controls accumulated error rather than pointwise error\.

#### 10\.3\.3Weighted variation theorem

For arbitrary complex weightsw1,…,wNw\_\{1\},\\ldots,w\_\{N\}, exact summation by parts gives

∑k=1N\(uk−qk\)​wk\\displaystyle\\sum\_\{k=1\}^\{N\}\(u\_\{k\}\-q\_\{k\}\)w\_\{k\}=vN​wN−v0​w1\+∑k=1N−1vk​\(wk−wk\+1\)\.\\displaystyle=v\_\{N\}w\_\{N\}\-v\_\{0\}w\_\{1\}\+\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(w\_\{k\}\-w\_\{k\+1\}\)\.\(10\.5\)Under \([10\.1](https://arxiv.org/html/2607.24868#Ch10.E1)\),

\|1N​∑k=1N\(uk−qk\)​wk\|≤VN​\(\|wN\|\+∑k=1N−1\|wk\+1−wk\|\)\.\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(u\_\{k\}\-q\_\{k\}\)w\_\{k\}\\right\|\\leq\\frac\{V\}\{N\}\\left\(\|w\_\{N\}\|\+\\sum\_\{k=1\}^\{N\-1\}\|w\_\{k\+1\}\-w\_\{k\}\|\\right\)\.\(10\.6\)The proof is exact until the final application of the triangle inequality\. The quantity in parentheses is an endpoint amplitude plus the discrete total variation of the weight\.

#### 10\.3\.4Absolutely continuous and BV phases

Letwk=exp⁡\(2​π​i​ϕ​\(k/N\)\)w\_\{k\}=\\exp\(2\\pi\\mathrm\{i\}\\phi\(k/N\)\)\. Ifϕ\\phiis absolutely continuous, then

\|wk\+1−wk\|≤2​π​∫k/N\(k\+1\)/N\|ϕ′​\(ξ\)\|​dξ\.\|w\_\{k\+1\}\-w\_\{k\}\|\\leq 2\\pi\\int\_\{k/N\}^\{\(k\+1\)/N\}\|\\phi^\{\\prime\}\(\\xi\)\|\\,\\mathrm\{d\}\\xi\.Summation and \([10\.6](https://arxiv.org/html/2607.24868#Ch10.E6)\) yield

\|ℰNϕ​u−ℰNϕ​q\|≤VN​\[1\+2​π​∫01\|ϕ′​\(ξ\)\|​dξ\]\.\|\\mathcal\{E\}\_\{N\}^\{\\phi\}u\-\\mathcal\{E\}\_\{N\}^\{\\phi\}q\|\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\int\_\{0\}^\{1\}\|\\phi^\{\\prime\}\(\\xi\)\|\\,\\mathrm\{d\}\\xi\\right\]\.\(10\.7\)Ifϕ\\phihas bounded variation, the integral is replaced by the total variation of a suitable representative\. This extension follows from the composition estimate for the Lipschitz maps↦e2​π​i​ss\\mapsto e^\{2\\pi\\mathrm\{i\}s\}and standard BV theory\[[1](https://arxiv.org/html/2607.24868#bib.bib72),[33](https://arxiv.org/html/2607.24868#bib.bib73)\]\.

### 10\.4Parabolic result catalogue

Forϕx,t​\(ξ\)=x​ξ\+t​ξ2\\phi\_\{x,t\}\(\\xi\)=x\\xi\+t\\xi^\{2\}, define

J​\(x,t\)=∫01\|x\+2​t​ξ\|​dξ\.J\(x,t\)=\\int\_\{0\}^\{1\}\|x\+2t\\xi\|\\,\\mathrm\{d\}\\xi\.The principal first\-order theorem is

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤VN​\[1\+2​π​J​\(x,t\)\]\.\|\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\|\\leq\\frac\{V\}\{N\}\[1\+2\\pi J\(x,t\)\]\.\(10\.8\)The exact continuous variation factor is

J​\(x,t\)=\|x\+t\|when​t≠0​and​x​\(x\+2​t\)≥0,J\(x,t\)=\|x\+t\|\\quad\\text\{when \}t\\neq 0\\text\{ and \}x\(x\+2t\)\\geq 0,and

J​\(x,t\)=x2\+\(x\+2​t\)24​\|t\|when​t≠0​and​x​\(x\+2​t\)<0\.J\(x,t\)=\\frac\{x^\{2\}\+\(x\+2t\)^\{2\}\}\{4\|t\|\}\\quad\\text\{when \}t\\neq 0\\text\{ and \}x\(x\+2t\)<0\.The formula is obtained by locating the unique zeroξ∗=−x/\(2​t\)\\xi\_\{\*\}=\-x/\(2t\)when it lies inside\(0,1\)\(0,1\)and integrating the absolute value separately on\[0,ξ∗\]\[0,\\xi\_\{\*\}\]and\[ξ∗,1\]\[\\xi\_\{\*\},1\]\.

On a compact setK⊂ℝ2K\\subset\\mathbb\{R\}^\{2\},

‖ℰN​u−ℰN​q‖L∞​\(K\)≤VN​\(1\+2​π​sup\(x,t\)∈KJ​\(x,t\)\)\.\\\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\\|\_\{L^\{\\infty\}\(K\)\}\\leq\\frac\{V\}\{N\}\\left\(1\+2\\pi\\sup\_\{\(x,t\)\\in K\}J\(x,t\)\\right\)\.Thus the quantized extension converges uniformly to the unquantized extension at rateN−1N^\{\-1\}on every fixed compact parameter set\.

ForΩR=\[−R,R\]2\\Omega\_\{R\}=\[\-R,R\]^\{2\}and1≤p<∞1\\leq p<\\infty,

‖ℰN​u−ℰN​q‖Lp​\(ΩR\)≤\(4​R2\)1/p​VN​\(1\+4​π​R\)\.\\\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\\|\_\{L^\{p\}\(\\Omega\_\{R\}\)\}\\leq\(4R^\{2\}\)^\{1/p\}\\frac\{V\}\{N\}\(1\+4\\pi R\)\.The estimate is a direct consequence of the localL∞L^\{\\infty\}bound and the volume of the parameter region\. It does not use restriction theory\.

### 10\.5Sharpness of the first\-order rate

For the zero input, the greedy rule withQ​\(0\)=1Q\(0\)=1generates

qk=\(−1\)k\+1,vk=\{−1,k​odd,0,k​even\.q\_\{k\}=\(\-1\)^\{k\+1\},\\qquad v\_\{k\}=\\begin\{cases\}\-1,&k\\text\{ odd\},\\\\ 0,&k\\text\{ even\}\.\\end\{cases\}At\(x,t\)=\(0,0\)\(x,t\)=\(0,0\), every phase weight equals one\. Therefore,

ℰN​u​\(0,0\)−ℰN​q​\(0,0\)=1N​∑k=1N\(uk−qk\)=vNN\.\\mathcal\{E\}\_\{N\}u\(0,0\)\-\\mathcal\{E\}\_\{N\}q\(0,0\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(u\_\{k\}\-q\_\{k\}\)=\\frac\{v\_\{N\}\}\{N\}\.For oddNN, the magnitude is exactly1/N1/N\. Hence no bound of ordero​\(N−1\)o\(N^\{\-1\}\)can hold uniformly over all admissible inputs under only the first\-order stability assumptions\. The statement concerns uniform worst\-case behaviour\. It does not exclude faster convergence for a fixed input, for even record lengths, under terminal reset, or under additional cancellation conditions\.

### 10\.6Higher\-order finite\-record derivation

#### 10\.6\.1Why boundary traces appear

Supposee=Δr​ve=\\Delta^\{r\}v\. On an infinite sequence, a formal Fourier calculation multiplies the transform ofvvby\(1−e−i​ω\)r\(1\-e^\{\-\\mathrm\{i\}\\omega\}\)^\{r\}\. A finite record is different\. Repeated summation by parts transfers differences to the weight but also produces terms supported at the initial and terminal indices\. These terms are not negligible merely because the interior weight is smooth\.

Forr=2r=2, direct calculation gives a representative identity of the form

∑k=1N\(Δ2​v\)k​wk\\displaystyle\\sum\_\{k=1\}^\{N\}\(\\Delta^\{2\}v\)\_\{k\}w\_\{k\}=∑k=1N−2vk​\(Δ\+2​w\)k\+ℬ2,N​\(v,w\),\\displaystyle=\\sum\_\{k=1\}^\{N\-2\}v\_\{k\}\(\\Delta\_\{\+\}^\{2\}w\)\_\{k\}\+\\mathcal\{B\}\_\{2,N\}\(v,w\),\(10\.9\)whereℬ2,N\\mathcal\{B\}\_\{2,N\}contains the required initial and terminal traces\. The exact indexing convention is stated and proved in[Chapter˜6](https://arxiv.org/html/2607.24868#Ch6)and Appendix A\.

#### 10\.6\.2Endpoint\-compatible rate

Assume that all boundary traces in the order\-rrfinite identity vanish\. Then

∑k=1N\(Δr​v\)k​wk=\(−1\)r​∑kvk​\(Δ\+r​w\)k\\sum\_\{k=1\}^\{N\}\(\\Delta^\{r\}v\)\_\{k\}w\_\{k\}=\(\-1\)^\{r\}\\sum\_\{k\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}
after restriction to the valid interior indices\. Ifwk=g​\(k/N\)w\_\{k\}=g\(k/N\)withg∈Cr​\(\[0,1\]\)g\\in C^\{r\}\(\[0,1\]\), the integral representation of finite differences gives

\|\(Δ\+r​g\)​\(k/N\)\|≤N−r​∫\[0,1\]r\|g\(r\)​\(k\+s1\+⋯\+srN\)\|​ds1​⋯​dsr\.\|\(\\Delta\_\{\+\}^\{r\}g\)\(k/N\)\|\\leq N^\{\-r\}\\int\_\{\[0,1\]^\{r\}\}\\left\|g^\{\(r\)\}\\\!\\left\(\\frac\{k\+s\_\{1\}\+\\cdots\+s\_\{r\}\}\{N\}\\right\)\\right\|\\,\\mathrm\{d\}s\_\{1\}\\cdots\\mathrm\{d\}s\_\{r\}\.Summing overkkgives a bound of orderN1−rN^\{1\-r\}for the unnormalised interior sum\. The extension has an additional factorN−1N^\{\-1\}\. Consequently,

\|1N​∑k=1N\(Δr​v\)k​g​\(k/N\)\|≤Cr​‖v‖∞​N−r​‖g\(r\)‖L1\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(\\Delta^\{r\}v\)\_\{k\}g\(k/N\)\\right\|\\leq C\_\{r\}\\\|v\\\|\_\{\\infty\}N^\{\-r\}\\\|g^\{\(r\)\}\\\|\_\{L^\{1\}\}
after harmless endpoint enlargements\. This is the mechanism behind the endpoint\-compatibleO​\(N−r\)O\(N^\{\-r\}\)theorem\.

For oscillatory weightsg​\(ξ\)=e2​π​i​ϕ​\(ξ\)g\(\\xi\)=e^\{2\\pi\\mathrm\{i\}\\phi\(\\xi\)\}, the derivativeg\(r\)g^\{\(r\)\}is a finite Bell\-polynomial combination ofϕ′,…,ϕ\(r\)\\phi^\{\\prime\},\\ldots,\\phi^\{\(r\)\}\. On compact parameter sets for polynomial phases, these derivatives are uniformly bounded\. The high\-order compact\-set estimate follows\.

#### 10\.6\.3Approximate compatibility

If a boundary trace does not vanish, its contribution must remain in the estimate\. A typical order\-rrbound has the form

\|ℰN​e\|≤Cr​‖v‖∞​N−r\+1N​∑j=0r−1\|τj,N​\(v\)\|​\|γj,N​\(w\)\|,\|\\mathcal\{E\}\_\{N\}e\|\\leq C\_\{r\}\\\|v\\\|\_\{\\infty\}N^\{\-r\}\+\\frac\{1\}\{N\}\\sum\_\{j=0\}^\{r\-1\}\|\\tau\_\{j,N\}\(v\)\|\\,\|\\gamma\_\{j,N\}\(w\)\|,whereτj,N\\tau\_\{j,N\}denotes a state trace andγj,N\\gamma\_\{j,N\}denotes the associated weight trace\. High\-order decay is preserved only if the second term is of orderN−rN^\{\-r\}or smaller\. This formula explains the importance of smooth termination, reset sequences, and blockwise processing in frame quantization\[[9](https://arxiv.org/html/2607.24868#bib.bib20),[8](https://arxiv.org/html/2607.24868#bib.bib21)\]\.

### 10\.7Boundary Correction and Fractional Rates

The finite\-record boundary terms can be handled either dynamically or analytically\. Dynamic handling imposes endpoint compatibility on the state trajectory\. Analytical handling records the boundary traces and adds their exact contribution to the reconstruction\. Fore=Δr​ve=\\Delta^\{r\}v, define

ℬN,r​\(v,w\)=∑j=0r−1\(−1\)j​\[\(Δr−1−j​v\)N−j​\(Δ\+j​w\)N−j−\(Δr−1−j​v\)0​\(Δ\+j​w\)1\]\.\\mathcal\{B\}\_\{N,r\}\(v,w\)=\\sum\_\{j=0\}^\{r\-1\}\(\-1\)^\{j\}\\left\[\(\\Delta^\{r\-1\-j\}v\)\_\{N\-j\}\(\\Delta\_\{\+\}^\{j\}w\)\_\{N\-j\}\-\(\\Delta^\{r\-1\-j\}v\)\_\{0\}\(\\Delta\_\{\+\}^\{j\}w\)\_\{1\}\\right\]\.The boundary\-corrected extension is

ℛN,r​\(q;v,w\)=1N​∑k=1Nqk​wk\+1N​ℬN,r​\(v,w\)\.\\mathcal\{R\}\_\{N,r\}\(q;v,w\)=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}q\_\{k\}w\_\{k\}\+\\frac\{1\}\{N\}\\mathcal\{B\}\_\{N,r\}\(v,w\)\.The exact error identity is

1N​∑k=1Nuk​wk−ℛN,r​\(q;v,w\)=\(−1\)rN​∑k=1N−rvk​\(Δ\+r​w\)k\.\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}u\_\{k\}w\_\{k\}\-\\mathcal\{R\}\_\{N,r\}\(q;v,w\)=\\frac\{\(\-1\)^\{r\}\}\{N\}\\sum\_\{k=1\}^\{N\-r\}v\_\{k\}\(\\Delta\_\{\+\}^\{r\}w\)\_\{k\}\.This result is proved in[Theorem˜6\.4](https://arxiv.org/html/2607.24868#Ch6.Thmtheorem4)\. It has three immediate consequences\.

1. \(i\)A bounded state and a sampledCrC^\{r\}weight give a corrected error of orderN−rN^\{\-r\}without terminal reset\.
2. \(ii\)Zero initialisation eliminates all left\-trace metadata\. Onlyrrterminal trace values remain, independent of the record length\.
3. \(iii\)Inexact metadata enters through an explicit weighted trace\-error sum, so the precision required to preserve a target rate can be calculated before implementation\.

The fractional regularity theorem gives a second refinement\. IfW∈Cr−1,αW\\in C^\{r\-1,\\alpha\}with0<α≤10<\\alpha\\leq 1, then

\|Δ1/Nr​W\|≤N−\(r−1\+α\)​\[W\(r−1\)\]C0,α\.\|\\Delta\_\{1/N\}^\{r\}W\|\\leq N^\{\-\(r\-1\+\\alpha\)\}\[W^\{\(r\-1\)\}\]\_\{C^\{0,\\alpha\}\}\.Under endpoint compatibility, or after exact boundary correction,

\|1N​∑k=1N\(uk−qk\)​W​\(k/N\)\|≤V​\[W\(r−1\)\]C0,α​N−\(r−1\+α\)\.\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(u\_\{k\}\-q\_\{k\}\)W\(k/N\)\\right\|\\leq V\[W^\{\(r\-1\)\}\]\_\{C^\{0,\\alpha\}\}N^\{\-\(r\-1\+\\alpha\)\}\.This estimate places the finite\-record theory on the standard modulus\-of\-smoothness scale used in constructive approximation\[[29](https://arxiv.org/html/2607.24868#bib.bib83),[30](https://arxiv.org/html/2607.24868#bib.bib84),[76](https://arxiv.org/html/2607.24868#bib.bib85)\]\. It also shows that high\-order noise shaping and test\-function regularity contribute jointly to the final rate\.

### 10\.8Polynomial and multidimensional consequences

For a degree\-ddpolynomial phase

ϕ𝜶​\(ξ\)=α1​ξ\+α2​ξ2\+⋯\+αd​ξd,\\phi\_\{\\bm\{\\alpha\}\}\(\\xi\)=\\alpha\_\{1\}\\xi\+\\alpha\_\{2\}\\xi^\{2\}\+\\cdots\+\\alpha\_\{d\}\\xi^\{d\},the first\-order estimate becomes

\|ℰN​u​\(𝜶\)−ℰN​q​\(𝜶\)\|≤VN​\[1\+2​π​∫01\|∑j=1dj​αj​ξj−1\|​dξ\]\.\|\\mathcal\{E\}\_\{N\}u\(\\bm\{\\alpha\}\)\-\\mathcal\{E\}\_\{N\}q\(\\bm\{\\alpha\}\)\|\\leq\\frac\{V\}\{N\}\\left\[1\+2\\pi\\int\_\{0\}^\{1\}\\left\|\\sum\_\{j=1\}^\{d\}j\\alpha\_\{j\}\\xi^\{j\-1\}\\right\|\\,\\mathrm\{d\}\\xi\\right\]\.On a fixed coefficient box, the integral is uniformly bounded\. Under order\-rrendpoint compatibility, the rate improves toN−rN^\{\-r\}, with constants controlled by derivatives of the exponential weight through orderrr\.

For a lattice array whose error has a discrete divergence representation,

e𝐤=∑j=1dΔj​v𝐤\(j\),e\_\{\\mathbf\{k\}\}=\\sum\_\{j=1\}^\{d\}\\Delta\_\{j\}v^\{\(j\)\}\_\{\\mathbf\{k\}\},discrete summation by parts in each coordinate yields an interior term involving directional differences of the weight and a boundary flux\. If the vector state is bounded and the boundary flux is controlled, the normalised error is bounded by a sum of directional variations\. The convergence rate remains first order in the grid spacing, while the constant depends on dimension and phase derivatives\.

### 10\.9Growing regions and scale transition

For the parabolic box\|x\|,\|t\|≤Nα\|x\|,\|t\|\\leq N^\{\\alpha\}, the simple first\-order bound gives

sup\|x\|,\|t\|≤Nα\|ℰN​u−ℰN​q\|≲Nα−1\.\\sup\_\{\|x\|,\|t\|\\leq N^\{\\alpha\}\}\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\|\\lesssim N^\{\\alpha\-1\}\.Three regimes follow\.

1. \(i\)If0≤α<10\\leq\\alpha<1, the total\-variation method gives decay\.
2. \(ii\)Ifα=1\\alpha=1, it gives only a bounded estimate\.
3. \(iii\)Ifα\>1\\alpha\>1, it does not provide a useful uniform approximation theorem\.

This transition is methodological\. It does not prove that the error grows at critical or supercritical scales\. It shows that cancellation must be used instead of taking absolute values of every adjacent phase increment\.

Under order\-rrendpoint compatibility, a derivative count suggests the subcritical scale

N−r​\(1\+Nα\)r∼N−r​\(1−α\)\.N^\{\-r\}\(1\+N^\{\\alpha\}\)^\{r\}\\sim N^\{\-r\(1\-\\alpha\)\}\.The rate still decays forα<1\\alpha<1\. At the critical scale, further oscillatory information is again required\.

### 10\.10LocalLpL^\{p\}, orthogonality, and external harmonic analysis

Orthogonality in a linear phase parameter gives exactL2L^\{2\}identities for complete periods\. Such identities depend on the chosen normalisation and parameter domain\. They do not by themselves produce a gain over the pointwise first\-order bound for arbitrary shaped coefficients\. Higher moments lead to Diophantine systems related to discrete restriction and Vinogradov mean values\[[15](https://arxiv.org/html/2607.24868#bib.bib41),[56](https://arxiv.org/html/2607.24868#bib.bib43),[81](https://arxiv.org/html/2607.24868#bib.bib63),[12](https://arxiv.org/html/2607.24868#bib.bib46)\]\.

Decoupling separates frequency intervals and controls anLpL^\{p\}norm by a square function withℓ2\\ell^\{2\}coefficient structure\[[13](https://arxiv.org/html/2607.24868#bib.bib45),[28](https://arxiv.org/html/2607.24868#bib.bib55)\]\. A sigma\-delta state bound is naturally anℓ∞\\ell^\{\\infty\}or negative discrete Sobolev constraint\. A direct decoupling gain therefore requires an additional transfer argument\. The conditional results in[Chapter˜8](https://arxiv.org/html/2607.24868#Ch8)state explicitly which state\-sum, covariance, or block\-square hypotheses would be sufficient\. None of those hypotheses is silently assumed in the first\-order or endpoint\-compatible theorems\.

### 10\.11Robustness decomposition

Suppose the implemented error satisfies

u−q=Δ​v\+η,u\-q=\\Delta v\+\\eta,whereη\\etais an unshaped residual\. The extension error separates as

ℰN​\(u−q\)=ℰN​\(Δ​v\)\+ℰN​η\.\\mathcal\{E\}\_\{N\}\(u\-q\)=\\mathcal\{E\}\_\{N\}\(\\Delta v\)\+\\mathcal\{E\}\_\{N\}\\eta\.The shaped term receives the variation gain\. The residual satisfies

\|ℰN​η\|≤1N​∑k=1N\|ηk\|\.\|\\mathcal\{E\}\_\{N\}\\eta\|\\leq\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\|\\eta\_\{k\}\|\.A uniform residual of sizeε\\varepsilontherefore creates anO​\(ε\)O\(\\varepsilon\)floor\. If the residual has its own difference representation, another summation\-by\-parts gain is possible\.

Leaky shaping changes the recursion and produces a geometric memory term\. Sampling jitter perturbs the weights and can be controlled by a Lipschitz estimate in the phase\. Saturation events are isolated by separating the regular and overloaded index sets\. Each model introduces a specific remainder rather than invalidating the exact shaped component\.

### 10\.12Boundary engineering and block processing

Blockwise reset localises the terminal contribution\. If a record is divided into blocks of lengthLL, first\-order summation by parts is applied on each block\. The resulting error consists of interior block variations and one boundary term per block\. Exact reset eliminates the boundary terms\. Approximate reset bounds them by the terminal mismatch\.

At high order, reset must control a hierarchy of traces rather than a single state value\. Smooth termination and appended correction tails are designed for this purpose\. The analytic theorem and the quantizer construction should remain separate: the theorem states which traces are required, while the construction determines whether a one\-bit or multibit rule can realise them\.

### 10\.13Numerical verification of exact statements

The numerical experiments in[Chapter˜9](https://arxiv.org/html/2607.24868#Ch9)verify four features\.

1. \(a\)The first\-order recursion satisfies the difference identity to machine precision and remains inside the invariant interval\.
2. \(b\)The computed parabolic extension error remains below the deterministic variation bound\.
3. \(c\)For the zero input and oddNN, the error at the origin equals1/N1/N\.
4. \(d\)A synthetic second\-order state with exact endpoint compatibility exhibits an empirical slope close to−2\-2\.

The fourth calculation verifies the finite\-record analysis but does not construct a physical one\-bit high\-order quantizer with terminal control\. This distinction is essential\.

### 10\.14Detailed treatment of nonzero initial states

The zero\-initial\-state convention is natural for a newly started quantizer, but it should not be built into the general algebra\. Let

ek=vk−vk−1,\|vk\|≤V\+\(1≤k≤N\),e\_\{k\}=v\_\{k\}\-v\_\{k\-1\},\\qquad\|v\_\{k\}\|\\leq V\_\{\+\}\\quad\(1\\leq k\\leq N\),and assume only that\|v0\|≤V0\|v\_\{0\}\|\\leq V\_\{0\}\. The exact identity becomes

∑k=1Nek​wk=vN​wN−v0​w1\+∑k=1N−1vk​\(wk−wk\+1\)\.\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}=v\_\{N\}w\_\{N\}\-v\_\{0\}w\_\{1\}\+\\sum\_\{k=1\}^\{N\-1\}v\_\{k\}\(w\_\{k\}\-w\_\{k\+1\}\)\.Hence

\|1N​∑k=1Nek​wk\|≤1N​\(V\+​\|wN\|\+V0​\|w1\|\+V\+​∑k=1N−1\|wk\+1−wk\|\)\.\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}e\_\{k\}w\_\{k\}\\right\|\\leq\\frac\{1\}\{N\}\\left\(V\_\{\+\}\|w\_\{N\}\|\+V\_\{0\}\|w\_\{1\}\|\+V\_\{\+\}\\sum\_\{k=1\}^\{N\-1\}\|w\_\{k\+1\}\-w\_\{k\}\|\\right\)\.\(10\.10\)For unimodular phase weights,\|w1\|=\|wN\|=1\|w\_\{1\}\|=\|w\_\{N\}\|=1, so

\|ℰN​u−ℰN​q\|≤1N​\(V0\+V\+\+V\+​TVN⁡\(w\)\)\.\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\|\\leq\\frac\{1\}\{N\}\\left\(V\_\{0\}\+V\_\{\+\}\+V\_\{\+\}\\operatorname\{TV\}\_\{N\}\(w\)\\right\)\.If the same boundVVapplies to every state value, the endpoint constant is2​V2Vrather thanVV\. The asymptotic rate on fixed compact sets is unchanged, but the distinction matters in finite\-length estimates\.

A warm\-started or block\-processed quantizer may have a nonzerov0v\_\{0\}\. In that setting, the initial term represents information inherited from the preceding block\. Three strategies are available\. The first is exact reset, which setsv0=0v\_\{0\}=0\. The second is bounded carry\-over, which keeps the term in \([10\.10](https://arxiv.org/html/2607.24868#Ch10.E10)\)\. The third is state\-aware reconstruction, in which the known valuev0​w1/Nv\_\{0\}w\_\{1\}/Nis corrected explicitly\. The third strategy removes the initial boundary contribution without changing the one\-bit record\.

For interval sums, the same distinction gives

∑k=mnek=vn−vm−1\.\\sum\_\{k=m\}^\{n\}e\_\{k\}=v\_\{n\}\-v\_\{m\-1\}\.A prefix sum is bounded byV\+\+V0V\_\{\+\}\+V\_\{0\}unlessv0=0v\_\{0\}=0\. An interior interval is bounded by2​V\+2V\_\{\+\}when both endpoints belong to the same invariant state set\. Thus the often\-quoted prefix discrepancy boundVVis not a property of the difference representation alone\. It also uses the initial condition\.

### 10\.15Worked second\-order finite\-record derivation

The second\-order case is sufficiently important to justify a line\-by\-line derivation\. Let

ek=\(Δ2​v\)k=vk−2​vk−1\+vk−2\.e\_\{k\}=\(\\Delta^\{2\}v\)\_\{k\}=v\_\{k\}\-2v\_\{k\-1\}\+v\_\{k\-2\}\.The first summation\-by\-parts step is applied toΔ​v\\Delta v:

∑k=1N\(Δ2​v\)k​wk\\displaystyle\\sum\_\{k=1\}^\{N\}\(\\Delta^\{2\}v\)\_\{k\}w\_\{k\}=\(Δ​v\)N​wN−\(Δ​v\)0​w1\+∑k=1N−1\(Δ​v\)k​\(wk−wk\+1\)\.\\displaystyle=\(\\Delta v\)\_\{N\}w\_\{N\}\-\(\\Delta v\)\_\{0\}w\_\{1\}\+\\sum\_\{k=1\}^\{N\-1\}\(\\Delta v\)\_\{k\}\(w\_\{k\}\-w\_\{k\+1\}\)\.\(10\.11\)Sincewk−wk\+1=−\(Δ\+​w\)kw\_\{k\}\-w\_\{k\+1\}=\-\(\\Delta\_\{\+\}w\)\_\{k\}, the remaining sum is

−∑k=1N−1\(Δ​v\)k​\(Δ\+​w\)k\.\-\\sum\_\{k=1\}^\{N\-1\}\(\\Delta v\)\_\{k\}\(\\Delta\_\{\+\}w\)\_\{k\}\.A second summation\-by\-parts step gives

∑k=1N−1\(Δ​v\)k​\(Δ\+​w\)k\\displaystyle\\sum\_\{k=1\}^\{N\-1\}\(\\Delta v\)\_\{k\}\(\\Delta\_\{\+\}w\)\_\{k\}=vN−1​\(Δ\+​w\)N−1−v0​\(Δ\+​w\)1\\displaystyle=v\_\{N\-1\}\(\\Delta\_\{\+\}w\)\_\{N\-1\}\-v\_\{0\}\(\\Delta\_\{\+\}w\)\_\{1\}\(10\.12\)\+∑k=1N−2vk​\[\(Δ\+​w\)k−\(Δ\+​w\)k\+1\]\.\\displaystyle\\quad\+\\sum\_\{k=1\}^\{N\-2\}v\_\{k\}\\left\[\(\\Delta\_\{\+\}w\)\_\{k\}\-\(\\Delta\_\{\+\}w\)\_\{k\+1\}\\right\]\.\(10\.13\)The bracket equals−\(Δ\+2​w\)k\-\(\\Delta\_\{\+\}^\{2\}w\)\_\{k\}\. Substitution into \([10\.11](https://arxiv.org/html/2607.24868#Ch10.E11)\) produces

∑k=1N\(Δ2​v\)k​wk\\displaystyle\\sum\_\{k=1\}^\{N\}\(\\Delta^\{2\}v\)\_\{k\}w\_\{k\}=\(Δ​v\)N​wN−\(Δ​v\)0​w1\\displaystyle=\(\\Delta v\)\_\{N\}w\_\{N\}\-\(\\Delta v\)\_\{0\}w\_\{1\}\(10\.14\)−vN−1​\(Δ\+​w\)N−1\+v0​\(Δ\+​w\)1\\displaystyle\\quad\-v\_\{N\-1\}\(\\Delta\_\{\+\}w\)\_\{N\-1\}\+v\_\{0\}\(\\Delta\_\{\+\}w\)\_\{1\}\(10\.15\)\+∑k=1N−2vk​\(Δ\+2​w\)k\.\\displaystyle\\quad\+\\sum\_\{k=1\}^\{N\-2\}v\_\{k\}\(\\Delta\_\{\+\}^\{2\}w\)\_\{k\}\.\(10\.16\)This identity displays four boundary terms\. Two involve the first difference of the state and an undifferenced weight\. Two involve the state itself and a first difference of the weight\. The interior term contains the desired second difference of the weight\.

Under the common zero\-initialisation conventionv−1=v0=0v\_\{\-1\}=v\_\{0\}=0, the left traces vanish because\(Δ​v\)0=v0−v−1=0\(\\Delta v\)\_\{0\}=v\_\{0\}\-v\_\{\-1\}=0\. The right traces remain:

\(Δ​v\)N​wN−vN−1​\(Δ\+​w\)N−1\.\(\\Delta v\)\_\{N\}w\_\{N\}\-v\_\{N\-1\}\(\\Delta\_\{\+\}w\)\_\{N\-1\}\.The conditionvN=0v\_\{N\}=0alone is not enough, since\(Δ​v\)N=−vN−1\(\\Delta v\)\_\{N\}=\-v\_\{N\-1\}in that case\. Exact second\-order compatibility may be expressed as

\(Δ​v\)N=0,vN−1=0,\(\\Delta v\)\_\{N\}=0,\\qquad v\_\{N\-1\}=0,together with the corresponding initial conditions\. These two terminal requirements implyvN=vN−1=0v\_\{N\}=v\_\{N\-1\}=0, but the trace formulation is preferable because it generalises directly to higher order\.

Taking absolute values in \([10\.16](https://arxiv.org/html/2607.24868#Ch10.E16)\) gives

\|1N​∑k=1N\(Δ2​v\)k​wk\|\\displaystyle\\left\|\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(\\Delta^\{2\}v\)\_\{k\}w\_\{k\}\\right\|≤1N\(\|\(Δv\)N\|\|wN\|\+\|\(Δv\)0\|\|w1\|\\displaystyle\\leq\\frac\{1\}\{N\}\\Bigl\(\|\(\\Delta v\)\_\{N\}\|\|w\_\{N\}\|\+\|\(\\Delta v\)\_\{0\}\|\|w\_\{1\}\|\(10\.17\)\+\|vN−1\|\|Δ\+wN−1\|\+\|v0\|\|Δ\+w1\|\)\\displaystyle\\qquad\+\|v\_\{N\-1\}\|\|\\Delta\_\{\+\}w\_\{N\-1\}\|\+\|v\_\{0\}\|\|\\Delta\_\{\+\}w\_\{1\}\|\\Bigr\)\(10\.18\)\+‖v‖∞N​∑k=1N−2\|\(Δ\+2​w\)k\|\.\\displaystyle\\quad\+\\frac\{\\\|v\\\|\_\{\\infty\}\}\{N\}\\sum\_\{k=1\}^\{N\-2\}\|\(\\Delta\_\{\+\}^\{2\}w\)\_\{k\}\|\.\(10\.19\)For a sampledC2C^\{2\}weight,\|Δ\+​w\|=O​\(N−1\)\|\\Delta\_\{\+\}w\|=O\(N^\{\-1\}\)and

∑k=1N−2\|Δ\+2​wk\|=O​\(N−1\)\.\\sum\_\{k=1\}^\{N\-2\}\|\\Delta\_\{\+\}^\{2\}w\_\{k\}\|=O\(N^\{\-1\}\)\.The interior contribution in \([10\.19](https://arxiv.org/html/2607.24868#Ch10.E19)\) is thereforeO​\(N−2\)O\(N^\{\-2\}\)\. An uncontrolled value of\(Δ​v\)N\(\\Delta v\)\_\{N\}contributesO​\(N−1\)O\(N^\{\-1\}\)and dominates the interior term\. An uncontrolledvN−1v\_\{N\-1\}contributesO​\(N−2\)O\(N^\{\-2\}\)because it is multiplied by a first weight difference\. This calculation explains the trace hierarchy: higher state differences require stronger terminal suppression\.

#### 10\.15\.1Second\-order parabolic constant

Let

Wx,t​\(ξ\)=exp⁡\(2​π​i​\(x​ξ\+t​ξ2\)\)\.W\_\{x,t\}\(\\xi\)=\\exp\(2\\pi\\mathrm\{i\}\(x\\xi\+t\\xi^\{2\}\)\)\.Differentiation gives

Wx,t′​\(ξ\)=2​π​i​\(x\+2​t​ξ\)​Wx,t​\(ξ\)W^\{\\prime\}\_\{x,t\}\(\\xi\)=2\\pi\\mathrm\{i\}\(x\+2t\\xi\)W\_\{x,t\}\(\\xi\)and

Wx,t′′​\(ξ\)=\[4​π​i​t−4​π2​\(x\+2​t​ξ\)2\]​Wx,t​\(ξ\)\.W^\{\\prime\\prime\}\_\{x,t\}\(\\xi\)=\\left\[4\\pi\\mathrm\{i\}t\-4\\pi^\{2\}\(x\+2t\\xi\)^\{2\}\\right\]W\_\{x,t\}\(\\xi\)\.Since\|Wx,t\|=1\|W\_\{x,t\}\|=1,

\|Wx,t′′​\(ξ\)\|≤4​π​\|t\|\+4​π2​\|x\+2​t​ξ\|2\.\|W^\{\\prime\\prime\}\_\{x,t\}\(\\xi\)\|\\leq 4\\pi\|t\|\+4\\pi^\{2\}\|x\+2t\\xi\|^\{2\}\.On0≤ξ≤10\\leq\\xi\\leq 1,

\|x\+2​t​ξ\|≤\|x\|\+2​\|t\|\.\|x\+2t\\xi\|\\leq\|x\|\+2\|t\|\.Consequently,

‖Wx,t′′‖L∞​\(0,1\)≤4​π​\|t\|\+4​π2​\(\|x\|\+2​\|t\|\)2\.\\\|W^\{\\prime\\prime\}\_\{x,t\}\\\|\_\{L^\{\\infty\}\(0,1\)\}\\leq 4\\pi\|t\|\+4\\pi^\{2\}\(\|x\|\+2\|t\|\)^\{2\}\.Under exact second\-order endpoint compatibility, the integral representation of the second forward difference gives

∑k=1N−2\|\(Δ\+2​w\)k\|≤1N​‖Wx,t′′‖L∞​\(0,1\)\.\\sum\_\{k=1\}^\{N\-2\}\|\(\\Delta\_\{\+\}^\{2\}w\)\_\{k\}\|\\leq\\frac\{1\}\{N\}\\\|W^\{\\prime\\prime\}\_\{x,t\}\\\|\_\{L^\{\\infty\}\(0,1\)\}\.Substitution into the interior term of \([10\.19](https://arxiv.org/html/2607.24868#Ch10.E19)\) yields

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤VN2​\[4​π​\|t\|\+4​π2​\(\|x\|\+2​\|t\|\)2\]\.\|\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\|\\leq\\frac\{V\}\{N^\{2\}\}\\left\[4\\pi\|t\|\+4\\pi^\{2\}\(\|x\|\+2\|t\|\)^\{2\}\\right\]\.This derivation identifies the origin of every constant and confirms that the second\-order rate is a boundary\-compatible result rather than an automatic consequence ofe=Δ2​ve=\\Delta^\{2\}v\.

### 10\.16Detailed proof of the piecewise formula for the parabolic variation

The function

J​\(x,t\)=∫01\|x\+2​t​ξ\|​dξJ\(x,t\)=\\int\_\{0\}^\{1\}\|x\+2t\\xi\|\\,\\mathrm\{d\}\\xiis the total variation of the parabolic phase on the unit sampling interval\. Its formula depends on whether the affine derivative changes sign\.

Ift=0t=0, the integrand is constant and

Assume next thatt≠0t\\neq 0\. The endpoint values of the affine function arexxandx\+2​tx\+2t\. If

x​\(x\+2​t\)≥0,x\(x\+2t\)\\geq 0,the endpoint values have the same sign or one is zero\. The affine function has no interior sign change\. Therefore,

J​\(x,t\)\\displaystyle J\(x,t\)=\|∫01\(x\+2​t​ξ\)​dξ\|\\displaystyle=\\left\|\\int\_\{0\}^\{1\}\(x\+2t\\xi\)\\,\\mathrm\{d\}\\xi\\right\|\(10\.20\)=\|x\+t\|\.\\displaystyle=\|x\+t\|\.\(10\.21\)The absolute value may be moved outside the integral because the integrand has a fixed sign\.

If

the unique zero

ξ∗=−x2​t\\xi\_\{\*\}=\-\\frac\{x\}\{2t\}belongs to\(0,1\)\(0,1\)\. The integral splits atξ∗\\xi\_\{\*\}\. A sign\-independent calculation can be obtained by using the areas of two triangles\. The magnitude of the slope is2​\|t\|2\|t\|\. The first triangle has baseξ∗\\xi\_\{\*\}and height\|x\|\|x\|, while the second has base1−ξ∗1\-\\xi\_\{\*\}and height\|x\+2​t\|\|x\+2t\|\. Thus

J​\(x,t\)\\displaystyle J\(x,t\)=12​\|x\|​ξ∗\+12​\|x\+2​t\|​\(1−ξ∗\)\\displaystyle=\\frac\{1\}\{2\}\|x\|\\xi\_\{\*\}\+\\frac\{1\}\{2\}\|x\+2t\|\(1\-\\xi\_\{\*\}\)\(10\.22\)=x2\+\(x\+2​t\)24​\|t\|\.\\displaystyle=\\frac\{x^\{2\}\+\(x\+2t\)^\{2\}\}\{4\|t\|\}\.\(10\.23\)The final equality follows by substitutingξ∗=−x/\(2​t\)\\xi\_\{\*\}=\-x/\(2t\)and using the opposite signs of the endpoint values\.

The formulas agree continuously at the transition\. For example, ifx=0x=0, the sign\-changing formula has limiting value

0\+\(2​t\)24​\|t\|=\|t\|,\\frac\{0\+\(2t\)^\{2\}\}\{4\|t\|\}=\|t\|,which agrees with\|x\+t\|=\|t\|\|x\+t\|=\|t\|\. The same conclusion holds whenx\+2​t=0x\+2t=0\. HenceJJis continuous onℝ2\\mathbb\{R\}^\{2\}, positively homogeneous of degree one, and convex as an integral of absolute values of affine functions\.

Positive homogeneity gives

J​\(λ​x,λ​t\)=\|λ\|​J​\(x,t\)\.J\(\\lambda x,\\lambda t\)=\|\\lambda\|J\(x,t\)\.Convexity gives

J​\(θ​\(x1,t1\)\+\(1−θ\)​\(x2,t2\)\)≤θ​J​\(x1,t1\)\+\(1−θ\)​J​\(x2,t2\)J\(\\theta\(x\_\{1\},t\_\{1\}\)\+\(1\-\\theta\)\(x\_\{2\},t\_\{2\}\)\)\\leq\\theta J\(x\_\{1\},t\_\{1\}\)\+\(1\-\\theta\)J\(x\_\{2\},t\_\{2\}\)for0≤θ≤10\\leq\\theta\\leq 1\. These properties are useful when estimatingJJon convex parameter regions\. In particular, the maximum of a convex function on a compact polytope occurs on the boundary, and simple upper bounds can be obtained from the vertices\.

### 10\.17Constant dependence and uniformity statements

A convergence rate is incomplete unless the dependence of its constant is stated\. The first\-order parabolic estimate has the form

\|ℰN​u​\(x,t\)−ℰN​q​\(x,t\)\|≤C​\(x,t,V\)​N−1,\|\\mathcal\{E\}\_\{N\}u\(x,t\)\-\\mathcal\{E\}\_\{N\}q\(x,t\)\|\\leq C\(x,t,V\)N^\{\-1\},where

C​\(x,t,V\)=V​\[1\+2​π​J​\(x,t\)\]\.C\(x,t,V\)=V\[1\+2\\pi J\(x,t\)\]\.For a fixed point\(x,t\)\(x,t\), the constant is independent ofNNand of the particular input sequence, provided the same state boundVVapplies\. For a compact setKK, the uniform constant is

CK=V​\[1\+2​π​sup\(x,t\)∈KJ​\(x,t\)\]\.C\_\{K\}=V\\left\[1\+2\\pi\\sup\_\{\(x,t\)\\in K\}J\(x,t\)\\right\]\.The estimate is therefore uniform over all admissible input records and all parameters inKK\.

For a family of growing setsKNK\_\{N\}, the constant becomesNN\-dependent through

MN=sup\(x,t\)∈KNJ​\(x,t\)\.M\_\{N\}=\\sup\_\{\(x,t\)\\in K\_\{N\}\}J\(x,t\)\.The bound is

‖ℰN​u−ℰN​q‖L∞​\(KN\)≤VN​\(1\+2​π​MN\)\.\\\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\\|\_\{L^\{\\infty\}\(K\_\{N\}\)\}\\leq\\frac\{V\}\{N\}\(1\+2\\pi M\_\{N\}\)\.Decay occurs wheneverMN=o​\(N\)M\_\{N\}=o\(N\)\. This formulation is more general than a power\-law box\. It applies to anisotropic regions, curved parameter sets, and unions of local patches\.

For the degree\-ddpolynomial phase, define

M1,K=sup𝜶∈K∫01\|∑j=1dj​αj​ξj−1\|​dξ\.M\_\{1,K\}=\\sup\_\{\\bm\{\\alpha\}\\in K\}\\int\_\{0\}^\{1\}\\left\|\\sum\_\{j=1\}^\{d\}j\\alpha\_\{j\}\\xi^\{j\-1\}\\right\|\\,\\mathrm\{d\}\\xi\.Then

sup𝜶∈K\|ℰN​u​\(𝜶\)−ℰN​q​\(𝜶\)\|≤VN​\(1\+2​π​M1,K\)\.\\sup\_\{\\bm\{\\alpha\}\\in K\}\|\\mathcal\{E\}\_\{N\}u\(\\bm\{\\alpha\}\)\-\\mathcal\{E\}\_\{N\}q\(\\bm\{\\alpha\}\)\|\\leq\\frac\{V\}\{N\}\(1\+2\\pi M\_\{1,K\}\)\.For endpoint\-compatible order\-rrshaping, the corresponding constant depends on derivatives of the exponential weight through orderrr\. A convenient compact\-set quantity is

Mr,K=sup𝜶∈Ksup0≤ξ≤1Br​\(2​π​\|ϕ𝜶′​\(ξ\)\|,…,2​π​\|ϕ𝜶\(r\)​\(ξ\)\|\),M\_\{r,K\}=\\sup\_\{\\bm\{\\alpha\}\\in K\}\\sup\_\{0\\leq\\xi\\leq 1\}B\_\{r\}\\\!\\left\(2\\pi\|\\phi^\{\\prime\}\_\{\\bm\{\\alpha\}\}\(\\xi\)\|,\\ldots,2\\pi\|\\phi^\{\(r\)\}\_\{\\bm\{\\alpha\}\}\(\\xi\)\|\\right\),whereBrB\_\{r\}is the complete Bell polynomial\. The high\-order estimate then has the schematic form

sup𝜶∈K\|ℰN​e​\(𝜶\)\|≤Cr​V​Mr,K​N−r\\sup\_\{\\bm\{\\alpha\}\\in K\}\|\\mathcal\{E\}\_\{N\}e\(\\bm\{\\alpha\}\)\|\\leq C\_\{r\}VM\_\{r,K\}N^\{\-r\}when every boundary trace is compatible\. The dependence onKK,rr, andVVis explicit\. The constant is not dimension\-free when the phase degree or the number of parameters increases\.

### 10\.18Assumption\-to\-conclusion map

The following table records the exact role of each assumption\.

Table 10\.1:Role of the principal assumptions in the proved estimatesAssumptionPoint of useConsequence if removede=Δ​ve=\\Delta vExact summation by partsNo variation gain follows from a pointwise error bound alone\.‖v‖∞≤V\\\|v\\\|\_\{\\infty\}\\leq VTriangle inequality after summation by partsThe weighted state sum may grow withNN\.v0=0v\_\{0\}=0Removal of the initial endpoint termAn additional contribution\|v0​w1\|/N\|v\_\{0\}w\_\{1\}\|/Nremains\.Unimodular phase weightReplacement of\|wN\|\|w\_\{N\}\|by oneGeneral weights require their endpoint amplitude explicitly\.Absolute continuity ofϕ\\phiFundamental theorem of calculus on each cellBV phases require a measure\-theoretic variation argument\.Fixed compact parameter setUniform control of phase derivativesGrowing sets introduce anNN\-dependent constant\.e=Δr​ve=\\Delta^\{r\}vRepeated finite summation by partsFirst\-order structure alone cannot yield the order\-rrinterior difference\.Endpoint compatibilityRemoval of high\-order boundary tracesThe leading rate may revert toN−1N^\{\-1\}or an intermediate order\.CrC^\{r\}sampled weightrrth finite\-difference estimateLower regularity supports only the order allowed by the available modulus of smoothness\.Divergence\-form grid errorCoordinatewise summation by partsA general multidimensional array has no directional variation gain\.Residual decompositione=Δ​v\+ηe=\\Delta v\+\\etaSeparation of shaped and unshaped componentsAn unmodelled residual may create an error floor\.Additional state decorrelation hypothesisConditional critical\-scale estimatesStability alone does not justify oscillatory cancellation of the state sum\.The map shows that the principal theorems are deterministic implications\. Randomness, input smoothness, and bandlimiting enter only when constructing a stable high\-order quantizer, estimating state correlations, or comparing with a continuous signal model; the finite\-record summation identities depend solely on the assumptions displayed in the table\.

### 10\.19Separation between proved approximation and restriction estimates

The approximation problem and the restriction problem use related exponential sums but ask different questions\. The approximation problem controls

ℰN​u−ℰN​q=ℰN​\(u−q\)\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q=\\mathcal\{E\}\_\{N\}\(u\-q\)by exploiting the difference representation of the coefficient error\. The unquantized coefficientsuku\_\{k\}remain part of the comparison\. A restriction estimate instead seeks a norm inequality for

∑k=1Nak​e​\(ϕθ​\(k\)\)\\sum\_\{k=1\}^\{N\}a\_\{k\}e\(\\phi\_\{\\theta\}\(k\)\)for a broad class of coefficientsaka\_\{k\}, often with anℓ2\\ell^\{2\}norm on the right\-hand side and a scale\-sensitiveLpL^\{p\}norm on the left\.

The first\-order theorem does not improve a restriction exponent for arbitrary binary sequences\. It states that a binary sequence produced by a stable noise\-shaping rule approximates a specified bounded input sequence after application of a low\-variation extension functional\. The distinction remains even when the phase is parabolic and the same geometry appears in Schrödinger estimates\.

A transfer from approximation to restriction would require an independent bound onℰN​u\\mathcal\{E\}\_\{N\}u, a structural description of the admissible inputs, or a direct oscillatory estimate for the state term\. The triangle inequality

‖ℰN​q‖Lp≤‖ℰN​u‖Lp\+‖ℰN​u−ℰN​q‖Lp\\\|\\mathcal\{E\}\_\{N\}q\\\|\_\{L^\{p\}\}\\leq\\\|\\mathcal\{E\}\_\{N\}u\\\|\_\{L^\{p\}\}\+\\\|\\mathcal\{E\}\_\{N\}u\-\\mathcal\{E\}\_\{N\}q\\\|\_\{L^\{p\}\}is valid, but its usefulness depends on a suitable estimate for the first term\. The local approximation theorem supplies only the second term\.

The report therefore establishes a rigorous bridge between noise shaping and polynomial Fourier extension, while the transition to global restriction estimates is expressed through the additional state hypotheses developed in Chapter[8](https://arxiv.org/html/2607.24868#Ch8)\.

### 10\.20Summary of principal results

Table 10\.2:Principal mathematical results and analytical settingsResultAnalytical settingMain conclusionGreedy first\-order stability\|uk\|≤1\|u\_\{k\}\|\\leq 1The state satisfies\|vk\|≤1\|v\_\{k\}\|\\leq 1\.Interval discrepancye=Δ​ve=\\Delta vwith bounded stateInterval sums are controlled independently of interval length\.Weighted variation theoremArbitrary complex weightsThe error is bounded by endpoint amplitude and discrete total variation\.AC and BV phase estimatesAbsolutely continuous or BV phaseSampled variation is controlled by continuous phase variation\.Parabolic compact\-set rateStable first\-order shapingThe extension error is uniformlyO​\(N−1\)O\(N^\{\-1\}\)on fixed compact sets\.Uniform first\-order sharpnessZero input and odd record lengthThe error equals1/N1/Nat the origin\.Finite order\-rridentityFinite records with explicit tracesRepeated summation by parts retains all initial and terminal terms\.Endpoint\-compatible high\-order rateVanishing traces and smooth weightsThe compact\-set error isO​\(N−r\)O\(N^\{\-r\}\)\.Boundary\-corrected reconstructionExact state\-trace metadataThe interiorO​\(N−r\)O\(N^\{\-r\}\)rate is recovered without terminal compatibility\.Fractional regularity rateCr−1,αC^\{r\-1,\\alpha\}sampled weightsThe rate isO​\(N−\(r−1\+α\)\)O\(N^\{\-\(r\-1\+\\alpha\)\}\)\.Polynomial and grid extensionsPolynomial phases and bounded divergence statesConstants depend on phase derivatives, dimension, and boundary flux\.Critical\-scale oscillatory transferState\-sum, covariance, or square\-function estimatesThe assumed cancellation transfers directly to the extension error\.Terminal state controlQuantizer\-dependent terminal constructionEndpoint trace conditions are linked to appended symbols and admissible alphabets\.
### 10\.21Scope of the established results

The approximation theorem comparesℰN​u\\mathcal\{E\}\_\{N\}uwithℰN​q\\mathcal\{E\}\_\{N\}qfor coefficients connected by a stable difference representation\. The high\-order compact\-set theorem uses endpoint compatibility or boundary\-trace correction\. The multidimensional theorem uses a bounded divergence\-form state representation\. At critical parameter scales, state\-sum, covariance, or square\-function information supplements total variation\. These assumptions define the analytical setting in which the report’s estimates apply and identify the additional structure relevant to global restriction bounds\.

### 10\.22Conclusions

A stable sigma\-delta state converts a large pointwise one\-bit error into a structured discrete derivative\. Finite summation by parts is the central analytical mechanism\. At first order, the mechanism gives anN−1N^\{\-1\}approximation rate against weights of bounded variation\. For the parabolic phase, the constant is controlled by an explicitly computable variation factor\. The zero\-input orbit proves that the uniform rate is sharp under the stated assumptions\.

At higher order, the finite record introduces a hierarchy of boundary traces\. Once those traces are controlled, repeated summation by parts produces the expectedN−rN^\{\-r\}compact\-set rate\. The same rate follows without terminal compatibility when the exact boundary functional is added to the reconstruction\. Fractional weight regularity produces the intermediate scaleN−\(r−1\+α\)N^\{\-\(r\-1\+\\alpha\)\}\. Polynomial, multidimensional, and robustness extensions follow from the same principle after the correct directional derivatives and boundary fluxes are identified\.

The resulting theory supports a focused mathematical paper centred on the finite\-record connection between stable noise shaping, boundary\-compatible higher\-order differences, boundary\-corrected reconstruction, and discrete polynomial Fourier extension\. Critical\-scale extensions naturally lead to state dynamics, correlation estimates, and oscillatory cancellation\.

## Appendix AExpanded Discrete Identities

### A\.1Binomial formulas

For any sequenceaaand integerr≥0r\\geq 0,

Δr​ak=∑j=0r\(−1\)j​\(rj\)​ak−j,\\Delta^\{r\}a\_\{k\}=\\sum\_\{j=0\}^\{r\}\(\-1\)^\{j\}\\binom\{r\}\{j\}a\_\{k\-j\},Δ\+r​ak=∑j=0r\(−1\)r−j​\(rj\)​ak\+j\.\\Delta\_\{\+\}^\{r\}a\_\{k\}=\\sum\_\{j=0\}^\{r\}\(\-1\)^\{r\-j\}\\binom\{r\}\{j\}a\_\{k\+j\}\.These formulas follow from the binomial theorem applied to the shift operators

Δ=I−S,Δ\+=S\+−I\.\\Delta=I\-S,\\qquad\\Delta\_\{\+\}=S^\{\+\}\-I\.HereS​ak=ak−1Sa\_\{k\}=a\_\{k\-1\}andS\+​ak=ak\+1S^\{\+\}a\_\{k\}=a\_\{k\+1\}\.

### A\.2Low\-order expansions

The first four backward differences are

Δ​ak\\displaystyle\\Delta a\_\{k\}=ak−ak−1,\\displaystyle=a\_\{k\}\-a\_\{k\-1\},Δ2​ak\\displaystyle\\Delta^\{2\}a\_\{k\}=ak−2​ak−1\+ak−2,\\displaystyle=a\_\{k\}\-2a\_\{k\-1\}\+a\_\{k\-2\},Δ3​ak\\displaystyle\\Delta^\{3\}a\_\{k\}=ak−3​ak−1\+3​ak−2−ak−3,\\displaystyle=a\_\{k\}\-3a\_\{k\-1\}\+3a\_\{k\-2\}\-a\_\{k\-3\},Δ4​ak\\displaystyle\\Delta^\{4\}a\_\{k\}=ak−4​ak−1\+6​ak−2−4​ak−3\+ak−4\.\\displaystyle=a\_\{k\}\-4a\_\{k\-1\}\+6a\_\{k\-2\}\-4a\_\{k\-3\}\+a\_\{k\-4\}\.The corresponding forward differences are obtained by reversing the index direction\.

### A\.3Verification of the second\-order finite formula

Starting from

∑k=1NΔ2​vk​wk,\\sum\_\{k=1\}^\{N\}\\Delta^\{2\}v\_\{k\}w\_\{k\},apply first\-order summation by parts toak=Δ​vka\_\{k\}=\\Delta v\_\{k\}:

∑k=1NΔ2​vk​wk=\(Δ​v\)N​wN−\(Δ​v\)0​w1−∑k=1N−1Δ​vk​Δ\+​wk\.\\sum\_\{k=1\}^\{N\}\\Delta^\{2\}v\_\{k\}w\_\{k\}=\(\\Delta v\)\_\{N\}w\_\{N\}\-\(\\Delta v\)\_\{0\}w\_\{1\}\-\\sum\_\{k=1\}^\{N\-1\}\\Delta v\_\{k\}\\Delta\_\{\+\}w\_\{k\}\.Apply first\-order summation by parts again on the shorter interval:

∑k=1N−1Δ​vk​Δ\+​wk=vN−1​Δ\+​wN−1−v0​Δ\+​w1−∑k=1N−2vk​Δ\+2​wk\.\\sum\_\{k=1\}^\{N\-1\}\\Delta v\_\{k\}\\Delta\_\{\+\}w\_\{k\}=v\_\{N\-1\}\\Delta\_\{\+\}w\_\{N\-1\}\-v\_\{0\}\\Delta\_\{\+\}w\_\{1\}\-\\sum\_\{k=1\}^\{N\-2\}v\_\{k\}\\Delta\_\{\+\}^\{2\}w\_\{k\}\.Substitution yields \([6\.1\.4](https://arxiv.org/html/2607.24868#Ch6.Ex11)\)\.

### A\.4Fourth\-order boundary table

Forr=4r=4, the terminal terms are

\(Δ3​v\)N​wN,−\(Δ2​v\)N−1​Δ\+​wN−1,\+\(Δ​v\)N−2​Δ\+2​wN−2,−vN−3​Δ\+3​wN−3\.\(\\Delta^\{3\}v\)\_\{N\}w\_\{N\},\\quad\-\(\\Delta^\{2\}v\)\_\{N\-1\}\\Delta\_\{\+\}w\_\{N\-1\},\\quad\+\(\\Delta v\)\_\{N\-2\}\\Delta\_\{\+\}^\{2\}w\_\{N\-2\},\\quad\-v\_\{N\-3\}\\Delta\_\{\+\}^\{3\}w\_\{N\-3\}\.The initial terms have the opposite endpoint and alternating signs:

−\(Δ3​v\)0​w1,\+\(Δ2​v\)0​Δ\+​w1,−\(Δ​v\)0​Δ\+2​w1,\+v0​Δ\+3​w1\.\-\(\\Delta^\{3\}v\)\_\{0\}w\_\{1\},\\quad\+\(\\Delta^\{2\}v\)\_\{0\}\\Delta\_\{\+\}w\_\{1\},\\quad\-\(\\Delta v\)\_\{0\}\\Delta\_\{\+\}^\{2\}w\_\{1\},\\quad\+v\_\{0\}\\Delta\_\{\+\}^\{3\}w\_\{1\}\.The interior term is

\+∑k=1N−4vk​Δ\+4​wk\.\+\\sum\_\{k=1\}^\{N\-4\}v\_\{k\}\\Delta\_\{\+\}^\{4\}w\_\{k\}\.
Table A\.1:Boundary hierarchy for order fourThe table shows that removing the first three boundary layers is necessary to expose fourth\-order decay\.

### A\.5A two\-dimensional second\-order example

Let

ek,ℓ=Δ1​Δ2​vk,ℓ\.e\_\{k,\\ell\}=\\Delta\_\{1\}\\Delta\_\{2\}v\_\{k,\\ell\}\.Summation by parts in the first coordinate gives terminal and initial vertical faces plus an interiorΔ\+,1​w\\Delta\_\{\+,1\}w\. Applying summation by parts in the second coordinate to each term yields four corner traces, four edge sums, and one interior mixed difference

∑vk,ℓ​Δ\+,1​Δ\+,2​wk,ℓ\.\\sum v\_\{k,\\ell\}\\Delta\_\{\+,1\}\\Delta\_\{\+,2\}w\_\{k,\\ell\}\.The exact signs depend on the zero\-padding convention\. The structure mirrors the continuous identity

∫∂1∂2v​w=∫v​∂1∂2w\+edge and corner traces\.\\int\\partial\_\{1\}\\partial\_\{2\}v\\,w=\\int v\\,\\partial\_\{1\}\\partial\_\{2\}w\+\\text\{edge and corner traces\}\.

### A\.6Discrete product rules

The first\-order product rules are

Δ​\(ak​bk\)=ak​Δ​bk\+bk−1​Δ​ak,\\Delta\(a\_\{k\}b\_\{k\}\)=a\_\{k\}\\Delta b\_\{k\}\+b\_\{k\-1\}\\Delta a\_\{k\},Δ\+​\(ak​bk\)=ak\+1​Δ\+​bk\+bk​Δ\+​ak\.\\Delta\_\{\+\}\(a\_\{k\}b\_\{k\}\)=a\_\{k\+1\}\\Delta\_\{\+\}b\_\{k\}\+b\_\{k\}\\Delta\_\{\+\}a\_\{k\}\.Equivalently,

ak​Δ​bk=Δ​\(ak​bk\)−bk−1​Δ​ak\.a\_\{k\}\\Delta b\_\{k\}=\\Delta\(a\_\{k\}b\_\{k\}\)\-b\_\{k\-1\}\\Delta a\_\{k\}\.These identities are useful when separating a rapidly oscillating carrier from a slowly varying envelope\.

For second order,

Δ2​\(a​b\)k\\displaystyle\\Delta^\{2\}\(ab\)\_\{k\}=ak​Δ2​bk\+2​\(Δ​ak\)​\(Δ​bk−1\)\+bk−2​Δ2​ak,\\displaystyle=a\_\{k\}\\Delta^\{2\}b\_\{k\}\+2\(\\Delta a\_\{k\}\)\(\\Delta b\_\{k\-1\}\)\+b\_\{k\-2\}\\Delta^\{2\}a\_\{k\},with appropriate index conventions\. Higher\-order formulas involve discrete binomial sums\.

### A\.7Periodic differences

On the discrete torusℤ/N​ℤ\\mathbb\{Z\}/N\\mathbb\{Z\}, define

Δper​vk=vk−vk−1​mod​N\.\\Delta\_\{\\mathrm\{per\}\}v\_\{k\}=v\_\{k\}\-v\_\{k\-1\\,\\mathrm\{mod\}\\,N\}\.Then

∑k=1NΔper​vk​wk=−∑k=1Nvk​Δ\+,per​wk\\sum\_\{k=1\}^\{N\}\\Delta\_\{\\mathrm\{per\}\}v\_\{k\}w\_\{k\}=\-\\sum\_\{k=1\}^\{N\}v\_\{k\}\\Delta\_\{\+,\\mathrm\{per\}\}w\_\{k\}with no boundary terms\. However, a periodic state must satisfy

∑k=1Nek=0\.\\sum\_\{k=1\}^\{N\}e\_\{k\}=0\.A finite sigma\-delta record with arbitrary terminal state is not periodic\. Periodising it changes the error sequence by a boundary impulse\.

### A\.8Zero extension and boundary impulses

Extend a finite state by zero outside its record\. The distributional finite difference of the zero extension contains impulses at both boundaries\. The finite summation formula is another way of accounting for those impulses\. This viewpoint is useful when embedding the problem into a bi\-infinite Fourier transform\.

### A\.9Norm bounds for state differences

If‖v‖∞≤V\\left\\lVert v\\right\\rVert\_\{\\infty\}\\leq V, then

‖Δm​v‖∞≤2m​V\.\\left\\lVert\\Delta^\{m\}v\\right\\rVert\_\{\\infty\}\\leq 2^\{m\}V\.The constant follows from the binomial formula:

\|Δm​vk\|≤V​∑j=0m\(mj\)=2m​V\.\\left\\lvert\\Delta^\{m\}v\_\{k\}\\right\\rvert\\leq V\\sum\_\{j=0\}^\{m\}\\binom\{m\}\{j\}=2^\{m\}V\.This estimate is sharp over arbitrary bounded sequences, as shown by alternating signs\.

## Appendix BDerivative Bounds for Oscillatory Weights

### B\.1Faà di Bruno and Bell polynomials

Letf=egf=e^\{g\}\. The first derivatives are

f′\\displaystyle f^\{\\prime\}=eg​g′,\\displaystyle=e^\{g\}g^\{\\prime\},f′′\\displaystyle f^\{\\prime\\prime\}=eg​\(\(g′\)2\+g′′\),\\displaystyle=e^\{g\}\\left\(\(g^\{\\prime\}\)^\{2\}\+g^\{\\prime\\prime\}\\right\),f′′′\\displaystyle f^\{\\prime\\prime\\prime\}=eg​\(\(g′\)3\+3​g′​g′′\+g′′′\),\\displaystyle=e^\{g\}\\left\(\(g^\{\\prime\}\)^\{3\}\+3g^\{\\prime\}g^\{\\prime\\prime\}\+g^\{\\prime\\prime\\prime\}\\right\),f\(4\)\\displaystyle f^\{\(4\)\}=eg​\(\(g′\)4\+6​\(g′\)2​g′′\+3​\(g′′\)2\+4​g′​g′′′\+g\(4\)\)\.\\displaystyle=e^\{g\}\\left\(\(g^\{\\prime\}\)^\{4\}\+6\(g^\{\\prime\}\)^\{2\}g^\{\\prime\\prime\}\+3\(g^\{\\prime\\prime\}\)^\{2\}\+4g^\{\\prime\}g^\{\\prime\\prime\\prime\}\+g^\{\(4\)\}\\right\)\.In general,

f\(r\)=eg​Br​\(g′,…,g\(r\)\),f^\{\(r\)\}=e^\{g\}B\_\{r\}\(g^\{\\prime\},\\ldots,g^\{\(r\)\}\),whereBrB\_\{r\}is the complete exponential Bell polynomial\.

Forg=2​π​i​ϕg=2\\pi\\mathrm\{i\}\\phi, the unimodular factorege^\{g\}disappears after taking absolute values\. Therefore

‖f\(r\)‖∞≤Br​\(2​π​‖ϕ′‖∞,…,2​π​‖ϕ\(r\)‖∞\)\.\\left\\lVert f^\{\(r\)\}\\right\\rVert\_\{\\infty\}\\leq B\_\{r\}\\left\(2\\pi\\left\\lVert\\phi^\{\\prime\}\\right\\rVert\_\{\\infty\},\\ldots,2\\pi\\left\\lVert\\phi^\{\(r\)\}\\right\\rVert\_\{\\infty\}\\right\)\.

### B\.2Low\-order phase bounds

Forf=e2​π​i​ϕf=e^\{2\\pi\\mathrm\{i\}\\phi\},

‖f′‖∞≤2​π​‖ϕ′‖∞\.\\left\\lVert f^\{\\prime\}\\right\\rVert\_\{\\infty\}\\leq 2\\pi\\left\\lVert\\phi^\{\\prime\}\\right\\rVert\_\{\\infty\}\.Also,

‖f′′‖∞≤\(2​π\)2​‖ϕ′‖∞2\+2​π​‖ϕ′′‖∞\.\\left\\lVert f^\{\\prime\\prime\}\\right\\rVert\_\{\\infty\}\\leq\(2\\pi\)^\{2\}\\left\\lVert\\phi^\{\\prime\}\\right\\rVert\_\{\\infty\}^\{2\}\+2\\pi\\left\\lVert\\phi^\{\\prime\\prime\}\\right\\rVert\_\{\\infty\}\.At third order,

‖f′′′‖∞≤\(2​π\)3​‖ϕ′‖∞3\+3​\(2​π\)2​‖ϕ′‖∞​‖ϕ′′‖∞\+2​π​‖ϕ′′′‖∞\.\\left\\lVert f^\{\\prime\\prime\\prime\}\\right\\rVert\_\{\\infty\}\\leq\(2\\pi\)^\{3\}\\left\\lVert\\phi^\{\\prime\}\\right\\rVert\_\{\\infty\}^\{3\}\+3\(2\\pi\)^\{2\}\\left\\lVert\\phi^\{\\prime\}\\right\\rVert\_\{\\infty\}\\left\\lVert\\phi^\{\\prime\\prime\}\\right\\rVert\_\{\\infty\}\+2\\pi\\left\\lVert\\phi^\{\\prime\\prime\\prime\}\\right\\rVert\_\{\\infty\}\.These explicit forms are preferable to a generic constant whenrris small\.

### B\.3Quadratic phase formulas

Let

ϕ​\(ξ\)=x​ξ\+t​ξ2\.\\phi\(\\xi\)=x\\xi\+t\\xi^\{2\}\.Then

g′=2​π​i​\(x\+2​t​ξ\),g′′=4​π​i​t,g\(m\)=0\(m≥3\)\.g^\{\\prime\}=2\\pi\\mathrm\{i\}\(x\+2t\\xi\),\\qquad g^\{\\prime\\prime\}=4\\pi\\mathrm\{i\}t,\\qquad g^\{\(m\)\}=0\\quad\(m\\geq 3\)\.The first three derivatives satisfy

f′\\displaystyle f^\{\\prime\}=f​2​π​i​\(x\+2​t​ξ\),\\displaystyle=f\\,2\\pi\\mathrm\{i\}\(x\+2t\\xi\),f′′\\displaystyle f^\{\\prime\\prime\}=f​\[−\(2​π\)2​\(x\+2​t​ξ\)2\+4​π​i​t\],\\displaystyle=f\\left\[\-\(2\\pi\)^\{2\}\(x\+2t\\xi\)^\{2\}\+4\\pi\\mathrm\{i\}t\\right\],f′′′\\displaystyle f^\{\\prime\\prime\\prime\}=f​\[\(2​π​i​\(x\+2​t​ξ\)\)3\+3​\(2​π​i​\(x\+2​t​ξ\)\)​\(4​π​i​t\)\]\.\\displaystyle=f\\left\[\(2\\pi\\mathrm\{i\}\(x\+2t\\xi\)\)^\{3\}\+3\(2\\pi\\mathrm\{i\}\(x\+2t\\xi\)\)\(4\\pi\\mathrm\{i\}t\)\\right\]\.For bounding purposes, it is safer to use the Bell expression rather than simplifying complex signs\.

Set

A=2​π​\(\|x\|\+2​\|t\|\),B=4​π​\|t\|\.A=2\\pi\(\\left\\lvert x\\right\\rvert\+2\\left\\lvert t\\right\\rvert\),\\qquad B=4\\pi\\left\\lvert t\\right\\rvert\.Then

‖f′‖∞≤A,\\left\\lVert f^\{\\prime\}\\right\\rVert\_\{\\infty\}\\leq A,‖f′′‖∞≤A2\+B,\\left\\lVert f^\{\\prime\\prime\}\\right\\rVert\_\{\\infty\}\\leq A^\{2\}\+B,‖f′′′‖∞≤A3\+3​A​B,\\left\\lVert f^\{\\prime\\prime\\prime\}\\right\\rVert\_\{\\infty\}\\leq A^\{3\}\+3AB,‖f\(4\)‖∞≤A4\+6​A2​B\+3​B2\.\\left\\lVert f^\{\(4\)\}\\right\\rVert\_\{\\infty\}\\leq A^\{4\}\+6A^\{2\}B\+3B^\{2\}\.These are the complete Bell polynomials with higher derivatives set to zero\.

### B\.4Polynomial phases

Let

ϕx​\(ξ\)=∑j=1dxj​ξj\.\\phi\_\{x\}\(\\xi\)=\\sum\_\{j=1\}^\{d\}x\_\{j\}\\xi^\{j\}\.For1≤m≤d1\\leq m\\leq d,

ϕx\(m\)​\(ξ\)=∑j=mdj\!\(j−m\)\!​xj​ξj−m\.\\phi\_\{x\}^\{\(m\)\}\(\\xi\)=\\sum\_\{j=m\}^\{d\}\\frac\{j\!\}\{\(j\-m\)\!\}x\_\{j\}\\xi^\{j\-m\}\.Hence

‖ϕx\(m\)‖∞≤∑j=mdj\!\(j−m\)\!​\|xj\|\.\\left\\lVert\\phi\_\{x\}^\{\(m\)\}\\right\\rVert\_\{\\infty\}\\leq\\sum\_\{j=m\}^\{d\}\\frac\{j\!\}\{\(j\-m\)\!\}\\left\\lvert x\_\{j\}\\right\\rvert\.If

X=1\+∑j=1d\|xj\|,X=1\+\\sum\_\{j=1\}^\{d\}\\left\\lvert x\_\{j\}\\right\\rvert,then for fixedddandrr,

‖fx\(r\)‖∞≤Cr,d​Xr\.\\left\\lVert f\_\{x\}^\{\(r\)\}\\right\\rVert\_\{\\infty\}\\leq C\_\{r,d\}X^\{r\}\.The exponentrris sufficient for scale analysis, although the exact polynomial in thexjx\_\{j\}may be much smaller\.

### B\.5Finite differences versus derivatives

The integral representation gives

Δ\+r​f​\(k/N\)=∫\[0,1/N\]rf\(r\)​\(k/N\+s1\+⋯\+sr\)​ds\.\\Delta\_\{\+\}^\{r\}f\(k/N\)=\\int\_\{\[0,1/N\]^\{r\}\}f^\{\(r\)\}\(k/N\+s\_\{1\}\+\\cdots\+s\_\{r\}\)\\,\\mathrm\{d\}s\.Thus

\|Δ\+r​f​\(k/N\)\|≤N−r​‖f\(r\)‖∞\.\\left\\lvert\\Delta\_\{\+\}^\{r\}f\(k/N\)\\right\\rvert\\leq N^\{\-r\}\\left\\lVert f^\{\(r\)\}\\right\\rVert\_\{\\infty\}\.For analytic phases, one may also expand the finite difference exactly in powers of1/N1/N\. The derivative bound is uniform and sufficient for the high\-order theorem\.

### B\.6Verification of the displayed third derivative

For symbolic formulas, complex powers are simplified before the derivative is inserted into an asymptotic expansion\. The identityi3=−i\\mathrm\{i\}^\{3\}=\-\\mathrm\{i\}changes the signed third\-order term while leaving the corresponding absolute\-value Bell\-polynomial bound unchanged\. A direct differentiation provides the human\-readable verification, and computer algebra supplies an independent check\.

## Appendix CBackground on Bounded Variation

### C\.1Definition

A real functionffon\[a,b\]\[a,b\]belongs toB​V​\(\[a,b\]\)BV\(\[a,b\]\)if

Var\[a,b\]⁡\(f\)=supP∑j=0m−1\|f​\(tj\+1\)−f​\(tj\)\|<∞,\\operatorname\{Var\}\_\{\[a,b\]\}\(f\)=\\sup\_\{P\}\\sum\_\{j=0\}^\{m\-1\}\\left\\lvert f\(t\_\{j\+1\}\)\-f\(t\_\{j\}\)\\right\\rvert<\\infty,where the supremum is over all finite partitions

a=t0<t1<⋯<tm=b\.a=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{m\}=b\.Standard references include\[[1](https://arxiv.org/html/2607.24868#bib.bib72),[33](https://arxiv.org/html/2607.24868#bib.bib73)\]\.

### C\.2Absolutely continuous functions

Ifffis absolutely continuous, thenf′f^\{\\prime\}exists almost everywhere,f′∈L1f^\{\\prime\}\\in L^\{1\}, and

f​\(y\)−f​\(x\)=∫xyf′​\(s\)​ds\.f\(y\)\-f\(x\)=\\int\_\{x\}^\{y\}f^\{\\prime\}\(s\)\\,\\mathrm\{d\}s\.Moreover,

Var\[a,b\]⁡\(f\)=∫ab\|f′​\(s\)\|​ds\.\\operatorname\{Var\}\_\{\[a,b\]\}\(f\)=\\int\_\{a\}^\{b\}\\left\\lvert f^\{\\prime\}\(s\)\\right\\rvert\\,\\mathrm\{d\}s\.The parabolic phase belongs to this class\.

### C\.3Composition with a Lipschitz map

IfF:ℝ→ℂF:\\mathbb\{R\}\\to\\mathbb\{C\}is Lipschitz with constantLLandf∈B​Vf\\in BV, then

Var⁡\(F∘f\)≤L​Var⁡\(f\)\.\\operatorname\{Var\}\(F\\circ f\)\\leq L\\operatorname\{Var\}\(f\)\.For

F​\(y\)=e2​π​i​y,F\(y\)=e^\{2\\pi\\mathrm\{i\}y\},one may takeL=2​πL=2\\pibecause

\|F​\(y\)−F​\(z\)\|≤2​π​\|y−z\|\.\\left\\lvert F\(y\)\-F\(z\)\\right\\rvert\\leq 2\\pi\\left\\lvert y\-z\\right\\rvert\.This is the functional principle behind the BV phase theorem\.

### C\.4Choice of representative

A BV function is often treated as an equivalence class modulo sets of measure zero\. A sampled extension operator depends on point values, so a representative must be fixed\. The right\-continuous representative, left\-continuous representative, or precise representative may be used\. For smooth phases, all agree\.

If a jump occurs exactly at a grid point, changing the representative can alter one weight\. The weighted variation theorem remains valid after the chosen representative is used consistently, but the numerical value of the sum may change\.

### C\.5Jordan decomposition

Every real BV function can be written as the difference of two increasing functions\. This is the Jordan decomposition\. It implies that BV phases have finite one\-dimensional oscillation even when they are not differentiable\.

The first\-order sigma\-delta estimate therefore tolerates jump discontinuities\. It does not require curvature or smoothness\. Higher\-order finite\-difference estimates, by contrast, require higher regularity unless one replaces classical derivatives with measures and develops a higher\-order BV theory\.

### C\.6Complex BV functions

For complexww, one may define

Var⁡\(w\)=supP∑\|w​\(tj\+1\)−w​\(tj\)\|\.\\operatorname\{Var\}\(w\)=\\sup\_\{P\}\\sum\\left\\lvert w\(t\_\{j\+1\}\)\-w\(t\_\{j\}\)\\right\\rvert\.This geometric definition is stronger than treating the real and imaginary parts independently in some estimates, and it is the natural quantity for unit\-circle weights\.

### C\.7Discrete sampling inequality

For any ordered sample points

a≤t1<⋯<tN≤b,a\\leq t\_\{1\}<\\cdots<t\_\{N\}\\leq b,one has

∑k=1N−1\|f​\(tk\+1\)−f​\(tk\)\|≤Var\[a,b\]⁡\(f\)\.\\sum\_\{k=1\}^\{N\-1\}\\left\\lvert f\(t\_\{k\+1\}\)\-f\(t\_\{k\}\)\\right\\rvert\\leq\\operatorname\{Var\}\_\{\[a,b\]\}\(f\)\.The inequality follows directly from the definition because the sample points form a partition after adding the endpoints if necessary\.

### C\.8Higher\-dimensional BV

In several dimensions, BV functions have distributional gradients that are finite vector\-valued measures\. The grid theorem in Chapter[7](https://arxiv.org/html/2607.24868#Ch7)usesC1C^\{1\}phases for simplicity\. A multidimensional BV extension may be possible by controlling directional discrete variations with the total variation measures of the partial derivatives\. The boundary and representative issues are more involved\.

## Appendix DReference Map and Theoretical Sources

This appendix records why the major references are included\. It is intended to prevent unsupported theoretical statements and to make later paper extraction easier\.

ReferenceRole in the manuscript\[[7](https://arxiv.org/html/2607.24868#bib.bib1)\]Classical spectral model for quantization error and an early foundation for quantization\-noise analysis\.\[[58](https://arxiv.org/html/2607.24868#bib.bib2)\]Early delta\-sigma modulation architecture and historical origin of feedback noise shaping\.\[[20](https://arxiv.org/html/2607.24868#bib.bib3)\]Second\-order integration in sigma\-delta modulation and the engineering development of higher\-order shaping\.\[[43](https://arxiv.org/html/2607.24868#bib.bib5)\]Spectral behaviour of single\-loop sigma\-delta modulation with sinusoidal inputs\.\[[44](https://arxiv.org/html/2607.24868#bib.bib4)\]Rigorous and critical treatment of quantization\-noise spectra\.\[[45](https://arxiv.org/html/2607.24868#bib.bib6)\]Robustness and dynamical stability of single\-loop sigma\-delta modulation\.\[[25](https://arxiv.org/html/2607.24868#bib.bib7)\]Stable sigma\-delta modulators of arbitrary order for bandlimited functions\.\[[48](https://arxiv.org/html/2607.24868#bib.bib8)\]Exponential accuracy for one\-bit sigma\-delta quantization\.\[[49](https://arxiv.org/html/2607.24868#bib.bib9)\]Improved error estimates for coarsely quantized bandlimited data\.\[[24](https://arxiv.org/html/2607.24868#bib.bib10)\]Imperfect quantizers and robustness of analogue\-to\-digital conversion\.\[[47](https://arxiv.org/html/2607.24868#bib.bib11)\]Ergodic dynamics, invariant sets, and spectral analysis of sigma\-delta error\.\[[27](https://arxiv.org/html/2607.24868#bib.bib12)\]Optimal families of exponentially accurate one\-bit schemes\.\[[64](https://arxiv.org/html/2607.24868#bib.bib13)\]Lower bounds for coarse quantization accuracy\.\[[50](https://arxiv.org/html/2607.24868#bib.bib14)\]Authoritative overview of the mathematics of analogue\-to\-digital conversion\.\[[26](https://arxiv.org/html/2607.24868#bib.bib15)\]Deterministic decimation analysis and bit\-rate reduction for sigma\-delta streams\.\[[23](https://arxiv.org/html/2607.24868#bib.bib16)\]Survey of noise\-shaping methods for frames and compressive sampling\.\[[39](https://arxiv.org/html/2607.24868#bib.bib28)\]Quantized overcomplete expansions and the role of redundancy\.\[[11](https://arxiv.org/html/2607.24868#bib.bib29)\]Noise shaping on arbitrary frame expansions\.\[[5](https://arxiv.org/html/2607.24868#bib.bib17)\]First\-order sigma\-delta quantization for finite frames and frame variation\.\[[4](https://arxiv.org/html/2607.24868#bib.bib18)\]Second\-order finite\-frame quantization and limits caused by frame geometry\.\[[10](https://arxiv.org/html/2607.24868#bib.bib19)\]Frame paths and geometric error bounds\.\[[9](https://arxiv.org/html/2607.24868#bib.bib20)\]Smooth endpoint termination for higher\-order frame quantization\.\[[8](https://arxiv.org/html/2607.24868#bib.bib21)\]Sobolev dual frames and deterministicO​\(N−r\)O\(N^\{\-r\}\)reconstruction\.\[[46](https://arxiv.org/html/2607.24868#bib.bib22)\]Sobolev duals for random frames and compressed sensing measurements\.\[[62](https://arxiv.org/html/2607.24868#bib.bib23)\]Sub\-Gaussian frames, root\-exponential accuracy, and compressed sensing\.\[[68](https://arxiv.org/html/2607.24868#bib.bib24)\]Stable and robust recovery from quantized compressive samples\.\[[35](https://arxiv.org/html/2607.24868#bib.bib25)\]Structured random circulant measurements with sigma\-delta quantization\.\[[38](https://arxiv.org/html/2607.24868#bib.bib26)\]High\-order low\-bit quantization for fusion frames\.\[[40](https://arxiv.org/html/2607.24868#bib.bib27)\]One\-bit sigma\-delta modulation on the circle and periodic\-domain issues\.\[[63](https://arxiv.org/html/2607.24868#bib.bib82)\]Weighted sigma\-delta modulation for digital halftoning and multidimensional error diffusion\.\[[32](https://arxiv.org/html/2607.24868#bib.bib34)\]Foundational frame theory through nonharmonic Fourier series\.\[[3](https://arxiv.org/html/2607.24868#bib.bib35)\]Finite normalized tight frames\.\[[21](https://arxiv.org/html/2607.24868#bib.bib36)\]General reference for finite\-frame theory and applications\.\[[19](https://arxiv.org/html/2607.24868#bib.bib30)\]Exact recovery from incomplete Fourier information and compressed\-sensing foundations\.\[[18](https://arxiv.org/html/2607.24868#bib.bib31)\]Stable recovery from inaccurate measurements\.\[[31](https://arxiv.org/html/2607.24868#bib.bib32)\]Foundational compressed\-sensing theory\.\[[37](https://arxiv.org/html/2607.24868#bib.bib33)\]Comprehensive mathematical reference for compressed sensing\.\[[34](https://arxiv.org/html/2607.24868#bib.bib37)\]Early restriction\-related inequalities for singular convolution operators\.\[[75](https://arxiv.org/html/2607.24868#bib.bib38)\]Classical Fourier restriction theorem\.\[[71](https://arxiv.org/html/2607.24868#bib.bib39)\]Restriction to quadratic surfaces and dispersive space\-time estimates\.\[[60](https://arxiv.org/html/2607.24868#bib.bib40)\]Endpoint Strichartz estimates\.\[[15](https://arxiv.org/html/2607.24868#bib.bib41)\]Lattice restriction and periodic Schrödinger equations\.\[[16](https://arxiv.org/html/2607.24868#bib.bib42)\]Lattice restriction and periodic KdV equations\.\[[56](https://arxiv.org/html/2607.24868#bib.bib43)\]Discrete Fourier restriction associated with Schrödinger equations\.\[[65](https://arxiv.org/html/2607.24868#bib.bib44)\]General discrete Fourier restriction results and related applications\.\[[72](https://arxiv.org/html/2607.24868#bib.bib49)\]Bilinear restriction and Kakeya methods\.\[[80](https://arxiv.org/html/2607.24868#bib.bib51)\]Sharp bilinear cone restriction\.\[[73](https://arxiv.org/html/2607.24868#bib.bib50)\]Sharp bilinear restriction for paraboloids\.\[[6](https://arxiv.org/html/2607.24868#bib.bib48)\]Multilinear restriction and Kakeya inequalities\.\[[14](https://arxiv.org/html/2607.24868#bib.bib47)\]Oscillatory integral bounds based on multilinear estimates\.\[[53](https://arxiv.org/html/2607.24868#bib.bib52)\]Polynomial partitioning for restriction\.\[[54](https://arxiv.org/html/2607.24868#bib.bib53)\]Further restriction estimates through polynomial partitioning\.\[[13](https://arxiv.org/html/2607.24868#bib.bib45)\]Proof of theℓ2\\ell^\{2\}decoupling conjecture and discrete restriction consequences\.\[[12](https://arxiv.org/html/2607.24868#bib.bib46)\]Decoupling proof of the main Vinogradov mean value conjecture in higher degree\.\[[61](https://arxiv.org/html/2607.24868#bib.bib54)\]Scale\-invariant Strichartz estimates on tori\.\[[28](https://arxiv.org/html/2607.24868#bib.bib55)\]Systematic reference for restriction, decoupling, and applications\.\[[79](https://arxiv.org/html/2607.24868#bib.bib56)\]Foundational equidistribution and polynomial exponential\-sum method\.\[[78](https://arxiv.org/html/2607.24868#bib.bib57)\]Classical trigonometric\-sum method\.\[[57](https://arxiv.org/html/2607.24868#bib.bib58)\]Additive number theory and mean\-value methods\.\[[77](https://arxiv.org/html/2607.24868#bib.bib59)\]Hardy–Littlewood circle method and exponential sums\.\[[42](https://arxiv.org/html/2607.24868#bib.bib60)\]Detailed van der Corput method\.\[[66](https://arxiv.org/html/2607.24868#bib.bib61)\]Interface between analytic number theory and harmonic analysis\.\[[59](https://arxiv.org/html/2607.24868#bib.bib62)\]Modern analytic number theory reference\.\[[81](https://arxiv.org/html/2607.24868#bib.bib63)\]Efficient congruencing for Vinogradov mean values\.\[[36](https://arxiv.org/html/2607.24868#bib.bib64)\]Strongly diagonal behaviour via efficient congruencing\.\[[82](https://arxiv.org/html/2607.24868#bib.bib65)\]Cubic case of the main Vinogradov conjecture\.\[[83](https://arxiv.org/html/2607.24868#bib.bib66)\]Approximation to the main Vinogradov mean value conjecture\.\[[84](https://arxiv.org/html/2607.24868#bib.bib67)\]Nested efficient congruencing\.\[[67](https://arxiv.org/html/2607.24868#bib.bib68)\]Multidimensional Weyl sums and near\-optimal mean values\.\[[52](https://arxiv.org/html/2607.24868#bib.bib69)\]Integer solutions of Parsell–Vinogradov systems\.\[[51](https://arxiv.org/html/2607.24868#bib.bib70)\]Short proof of moment\-curve decoupling\.\[[17](https://arxiv.org/html/2607.24868#bib.bib71)\]Decoupling, exponential sums, and the Riemann zeta function\.\[[1](https://arxiv.org/html/2607.24868#bib.bib72)\]Standard monograph on BV functions and free\-discontinuity theory\.\[[33](https://arxiv.org/html/2607.24868#bib.bib73)\]Measure\-theoretic properties of Sobolev and BV functions\.\[[85](https://arxiv.org/html/2607.24868#bib.bib74)\]Trigonometric series, Abel summation, and classical Fourier analysis\.\[[41](https://arxiv.org/html/2607.24868#bib.bib75)\]Classical Fourier analysis and operator estimates\.\[[70](https://arxiv.org/html/2607.24868#bib.bib76)\]Oscillatory integrals and harmonic\-analysis methods\.\[[69](https://arxiv.org/html/2607.24868#bib.bib77)\]Introductory Fourier analysis and summation methods\.\[[74](https://arxiv.org/html/2607.24868#bib.bib78)\]Nonlinear dispersive equations and Strichartz theory\.\[[22](https://arxiv.org/html/2607.24868#bib.bib79)\]Schrödinger equations and dispersive PDE background\.\[[2](https://arxiv.org/html/2607.24868#bib.bib80)\]Abel summation and analytic number\-theory preliminaries\.\[[55](https://arxiv.org/html/2607.24868#bib.bib81)\]Classical inequalities used in norm and interpolation estimates\.The table contains more references than a single journal paper would normally require\. When a chapter is extracted, retain only the sources that support its actual theory and context\.

## Appendix EReproducible Code

### E\.1Python implementation of the greedy quantizer

importnumpyasnp

defgreedy\_sigma\_delta\(u:np\.ndarray\)\-\>tuple\[np\.ndarray,np\.ndarray\]:

"""First\-orderone\-bitgreedySigma\-Deltaquantizer\.

Parameters

\-\-\-\-\-\-\-\-\-\-

u:one\-dimensionalrealarraywith\|u\[k\]\|<=1

Returns

\-\-\-\-\-\-\-

q:one\-bitoutputin\{\-1,\+1\}

v:statearrayincludingv\[0\]=0

"""

u=np\.asarray\(u,dtype=float\)

ifu\.ndim\!=1:

raiseValueError\("umustbeone\-dimensional"\)

ifnp\.max\(np\.abs\(u\)\)\>1\+1e\-12:

raiseValueError\("inputexceedstheprovedstabilityrange"\)

n=u\.size

q=np\.empty\(n,dtype=float\)

v=np\.zeros\(n\+1,dtype=float\)

forkinrange\(n\):

y=u\[k\]\+v\[k\]

q\[k\]=1\.0ify\>=0\.0else\-1\.0

v\[k\+1\]=v\[k\]\+u\[k\]\-q\[k\]

returnq,v

### E\.2Parabolic extension and theoretical bound

defparabolic\_extension\(a,x,t\):

a=np\.asarray\(a,dtype=complex\)

n=a\.size

k=np\.arange\(1,n\+1,dtype=float\)

phase=x\*k/n\+t\*\(k/n\)\*\*2

returnnp\.mean\(a\*np\.exp\(2j\*np\.pi\*phase\)\)

defJ\_closed\(x,t\):

ifabs\(t\)<1e\-15:

returnabs\(x\)

ifx\*\(x\+2\.0\*t\)\>=0\.0:

returnabs\(x\+t\)

return\(x\*x\+\(x\+2\.0\*t\)\*\*2\)/\(4\.0\*abs\(t\)\)

deffirst\_order\_bound\(n,x,t,state\_bound=1\.0\):

returnstate\_bound/n\*\(1\.0\+2\.0\*np\.pi\*J\_closed\(x,t\)\)

### E\.3Identity check

defcheck\_first\_order\_identity\(u,q,v\):

error=np\.asarray\(u\)\-np\.asarray\(q\)

state\_difference=np\.diff\(np\.asarray\(v\)\)

returnnp\.max\(np\.abs\(error\-state\_difference\)\)

defexperiment\_zero\_input\(n,x=0\.0,t=0\.0\):

u=np\.zeros\(n\)

q,v=greedy\_sigma\_delta\(u\)

err=abs\(parabolic\_extension\(u,x,t\)

\-parabolic\_extension\(q,x,t\)\)

return\{

"N":n,

"max\_state":float\(np\.max\(np\.abs\(v\)\)\),

"terminal\_state":float\(v\[\-1\]\),

"extension\_error":float\(err\),

"theoretical\_bound":float\(first\_order\_bound\(n,x,t\)\),

\}

### E\.4Synthetic high\-order identity test

defbackward\_difference\(a,order=1\):

out=np\.asarray\(a,dtype=complex\)

for\_inrange\(order\):

out=np\.diff\(out\)

returnout

defforward\_difference\(a,order=1\):

out=np\.asarray\(a,dtype=complex\)

for\_inrange\(order\):

out=np\.diff\(out\)

returnout

### E\.5MATLAB\-style implementation

function\[q,v\]=greedy\_sigma\_delta\(u\)

u=u\(:\);

N=length\(u\);

ifmax\(abs\(u\)\)\>1\+1e\-12

error\(’Inputexceedstheprovedstabilityrange’\);

end

q=zeros\(N,1\);

v=zeros\(N\+1,1\);

fork=1:N

y=u\(k\)\+v\(k\);

ify\>=0

q\(k\)=1;

else

q\(k\)=\-1;

end

v\(k\+1\)=v\(k\)\+u\(k\)\-q\(k\);

end

end

### E\.6Convergence\-order script outline

lengths=2\*\*np\.arange\(8,17\)

errors=\[\]

forninlengths:

k=np\.arange\(1,n\+1\)

u=0\.7\*np\.sin\(2\*np\.pi\*7\*k/n\)

q,v=greedy\_sigma\_delta\(u\)

errors\.append\(abs\(parabolic\_extension\(u,0\.4,\-0\.3\)

\-parabolic\_extension\(q,0\.4,\-0\.3\)\)\)

slope,intercept=np\.polyfit\(np\.log\(lengths\),np\.log\(errors\),1\)

print\("estimatedorder:",slope\)

### E\.7Numerical verification notes

Evaluate the phase in double precision with moderate parameter sizes\. For very largexxortt, reduction modulo one before the exponential improves numerical conditioning\. The zero convention in the quantizer is matched to the theorem\. Odd and even record lengths are fitted separately when the terminal state produces a parity effect\.

## Appendix FGlossary of Terms

Admissible inputAn input sequence satisfying the amplitude or regularity assumptions required by a stated quantizer theorem\.

Boundary traceA value of the state or one of its finite differences appearing at the initial or terminal index after summation by parts\.

Bounded variationFinite total accumulated absolute change of a function or sequence\.

Critical scaleA parameter range where adjacent phase increments are order one and the elementary total\-variation gain no longer yields decay\.

DecouplingA family of harmonic\-analysis inequalities that control an extension operator by anℓ2\\ell^\{2\}combination of contributions from smaller frequency pieces\.

Discrete extensionA finite oscillatory sum that is dual to a restriction problem or represents a periodic dispersive evolution\.

Endpoint compatibilityVanishing of the initial and terminal discrete traces needed to remove boundary terms in high\-order summation by parts\.

Error shapingOrganisation of quantization error so that it lies in the range of a difference or filter operator\.

Greedy quantizerA rule that selects the current alphabet symbol according to the sign or nearest\-level decision based on the current input and state\.

Invariant regionA set of states mapped into itself by the nonlinear quantizer recursion for every admissible input\.

Major arcA parameter region close to a rational point with small denominator in the circle method\.

Minor arcThe complement of the major arcs, where stronger exponential\-sum cancellation is often available\.

Moment curveThe curveξ↦\(ξ,ξ2,…,ξd\)\\xi\\mapsto\(\\xi,\\xi^\{2\},\\ldots,\\xi^\{d\}\)\.

Noise transfer functionThe linearised transfer function from quantizer noise to output in an engineering sigma\-delta model\.

One\-bit alphabetThe two\-level set\{−1,\+1\}\\\{\-1,\+1\\\}, possibly after scaling\.

OversamplingSampling above the minimum rate needed for representation, thereby creating redundancy that may be used for quantization accuracy\.

Parabolic phaseA phase of the formx​ξ\+t​ξ2x\\xi\+t\\xi^\{2\}\.

Restriction estimateAn inequality controlling the Fourier transform on a curved set, or equivalently the associated extension operator\.

Shaped residualThe component of the error represented as a finite difference of a bounded state\.

State sequenceThe internal memory variable of a feedback quantizer\.

Strichartz estimateA space\-time integrability estimate for a dispersive PDE solution\.

Subcritical regionA growing parameter region in which adjacent phase increments still tend to zero\.

Total variation boundAn estimate obtained by summation by parts followed by absolute values of all weight increments\.

Unshaped residualAn error component not represented by the chosen difference operator\.

Vinogradov mean value theoremA family of sharp mean\-value estimates for polynomial exponential sums, equivalent to counting solutions of systems of equal sums of powers\.

Weyl sumA finite exponential sum with polynomial phase\.

### F\.1Symbols most likely to be confused

The symbolΔ\\Deltais the backward finite difference on sequences\. The symbolΔ\+\\Delta\_\{\+\}is the forward finite difference\. The operatorℰN\\mathcal\{E\}\_\{N\}is normalised by1/N1/N\. The unnormalised exponential sum isN​ℰNN\\mathcal\{E\}\_\{N\}\. The state boundVVis independent ofNNunless explicitly stated otherwise\.

## References

- \[1\]L\. Ambrosio, N\. Fusco, and D\. Pallara\(2000\)Functions of bounded variation and free discontinuity problems\.Oxford University Press\.Cited by:[§C\.1](https://arxiv.org/html/2607.24868#A3.S1.p1.5),[Table D\.1](https://arxiv.org/html/2607.24868#A4.T1.2.74.71.1.1.1),[item \(iii\)](https://arxiv.org/html/2607.24868#Ch1.S1.I1.i3.p1.1),[§10\.3\.4](https://arxiv.org/html/2607.24868#Ch10.S3.SS4.p1.4),[§3\.1\.4](https://arxiv.org/html/2607.24868#Ch3.S1.SS4.p1.3)\.
- \[2\]T\. M\. Apostol\(1976\)Introduction to analytic number theory\.Springer\.Cited by:[Table D\.1](https://arxiv.org/html/2607.24868#A4.T1.2.82.79.1.1.1),[§3\.1\.2](https://arxiv.org/html/2607.24868#Ch3.S1.SS2.p1.1)\.
- \[3\]J\. J\. Benedetto and M\. Fickus\(2003\)Finite normalized tight frames\.Advances in Computational Mathematics18,pp\. 357–385\.Cited by:[Table D\.1](https://arxiv.org/html/2607.24868#A4.T1.2.34.31.1.1.1)\.
- \[4\]J\. J\. Benedetto, A\. M\. Powell, and Ö\. Yilmaz\(2006\)Second\-order sigma\-delta quantization of finite frame expansions\.Applied and Computational Harmonic Analysis20\(1\),pp\. 126–148\.Cited by:[Table D\.1](https://arxiv.org/html/2607.24868#A4.T1.2.23.20.1.1.1),[§1\.1\.3](https://arxiv.org/html/2607.24868#Ch1.S1.SS3.p1.1),[§1\.2\.5](https://arxiv.org/html/2607.24868#Ch1.S2.SS5.p1.3),[§5\.3\.7](https://arxiv.org/html/2607.24868#Ch5.S3.SS7.p2.1),[§6\.1\.1](https://arxiv.org/html/2607.24868#Ch6.S1.SS1.p2.1)\.
- \[5\]J\. J\. Benedetto, A\. M\. Powell, and Ö\. Yilmaz\(2006\)Sigma\-delta quantization and finite frames\.IEEE Transactions on Information Theory52\(5\),pp\. 1990–2005\.Cited by:[Table D\.1](https://arxiv.org/html/2607.24868#A4.T1.2.22.19.1.1.1),[§1\.1\.3](https://arxiv.org/html/2607.24868#Ch1.S1.SS3.p1.1),[§1\.2\.5](https://arxiv.org/html/2607.24868#Ch1.S2.SS5.p1.3)\.
- \[6\]J\. Bennett, A\. Carbery, and T\. Tao\(2006\)On the multilinear restriction and kakeya conjectures\.Acta Mathematica196\(2\),pp\. 261–302\.Cited by:[Table D\.1](https://arxiv.org/html/2607.24868#A4.T1.2.51.48.1.1.1),[§1\.1\.2](https://arxiv.org/html/2607.24868#Ch1.S1.SS2.p2.1),[§2\.2\.1](https://arxiv.org/html/2607.24868#Ch2.S2.SS1.p1.3),[§2\.2\.9](https://arxiv.org/html/2607.24868#Ch2.S2.SS9.p1.1)\.
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## Index