SHiPPO: Recurrent Memory with Transported Polynomial Projections

# SHiPPO: Recurrent Memory with Transported Polynomial Projections
Source: [https://arxiv.org/html/2607.03055](https://arxiv.org/html/2607.03055)
Tomoya Mizuguchi Kyoto University, Japan &Bum Jun Kim The University of Tokyo, Japan

###### Abstract

HiPPO gives recurrent states memory semantics as coefficients of online polynomial projections, but in fixed channel coordinates\. Modern selective SSMs, by contrast, rely on token\-dependent control and channel interaction\. We introduce SHiPPO \(Sylvester HiPPO\), a transported projection\-memory prior that lifts HiPPO coefficient memories into a moving channel frame\. For any fixed or realized right\-transport path, SHiPPO transports the approximation family and channel metric together; conditional on that path, the state is ordinary HiPPO in a tied moving frame and follows Sylvester coefficient dynamics, preserving the left online\-memory operator while adding right\-action transport\. For selective\-SSM execution, we derive a restricted group\-local realization with controller\-compatible right actions, exponential\-adjusted updates, exact block\-affine scan, and recurrent decoding\. We also give a simultaneous\-reducibility criterion identifying when right transports collapse to static mixing plus independent scalar or blockwise banks\. Controlled diagnostics show that larger current\-token write rank improves ordinary prediction error but cannot recover order\-sensitive changes to already\-written memory; transported\-memory variants recover this signal, which disappears when the transport pathway is removed\. A finite\-field associative\-recall diagnostic with interleaved bindings, operations, and queries provides complementary autoregressive evidence while leaving the preferred right\-action realization open\. Taken together, these results support SHiPPO as a mechanistically grounded transported\-memory prior, with evidence focused on memory mechanisms rather than broad sequence\-modeling dominance\.

## 1Introduction

State space models \(SSMs\) have re\-emerged as efficient sequence backbones, combining recurrent inference with long\-context scalability\. We frame recurrent sequence modeling as*inductive\-bias selection*: choosing an ODE, recurrence family, parameterization, discretization, or initialization selects a structured family of causal dynamics and favors particular spectra, timescales, and optimization paths\[[41](https://arxiv.org/html/2607.03055#bib.bib21),[17](https://arxiv.org/html/2607.03055#bib.bib22),[63](https://arxiv.org/html/2607.03055#bib.bib25)\]\. For recurrent memory, a stronger model\-based bias is possible: the hidden state can be designed to represent an approximation of the revealed history rather than an unconstrained latent vector\[[67](https://arxiv.org/html/2607.03055#bib.bib26),[52](https://arxiv.org/html/2607.03055#bib.bib28)\]\. The Legendre Memory Unit and HiPPO are canonical examples: LMU represents a sliding history window in a Legendre basis, while HiPPO derives online compression dynamics from projection onto polynomial bases\[[65](https://arxiv.org/html/2607.03055#bib.bib41),[19](https://arxiv.org/html/2607.03055#bib.bib1)\]\. Throughout, we use “prior” in this inductive\- or modeling\-bias sense, not necessarily as a Bayesian prior over parameters\[[14](https://arxiv.org/html/2607.03055#bib.bib27)\]\.

In HiPPO, the recurrent ODE is not merely an architectural template; it is the coefficient dynamics induced by an online approximation problem, so the state stores coefficients of a history approximation\[[19](https://arxiv.org/html/2607.03055#bib.bib1)\]\. This objective\-derived view has shaped structured state\-space models, including S4, generalized\-basis interpretations of HiPPO, and diagonal simplifications such as DSS and S4D\[[21](https://arxiv.org/html/2607.03055#bib.bib2),[23](https://arxiv.org/html/2607.03055#bib.bib3),[24](https://arxiv.org/html/2607.03055#bib.bib4),[22](https://arxiv.org/html/2607.03055#bib.bib5)\]\. Recent work further studies how memory priors interact with initialization, timescales, diagonalization, robust parameterization, and spectral or frequency bias\[[36](https://arxiv.org/html/2607.03055#bib.bib12),[34](https://arxiv.org/html/2607.03055#bib.bib18),[76](https://arxiv.org/html/2607.03055#bib.bib10),[75](https://arxiv.org/html/2607.03055#bib.bib13),[74](https://arxiv.org/html/2607.03055#bib.bib14),[58](https://arxiv.org/html/2607.03055#bib.bib17)\]\. Our goal is to extend the underlying principle: an online approximation objective should justify the recurrent memory dynamics, while parameterization and initialization determine how a trainable model explores the resulting family\.

Modern SSM and linear\-recurrence blocks have moved beyond the original one\-sided independent\-memory setting\. S5 replaces many independent SISO SSMs by a MIMO state\-space layer\[[57](https://arxiv.org/html/2607.03055#bib.bib6)\]\. H3 introduces multiplicative interactions between SSM outputs and input projections for language modeling\[[15](https://arxiv.org/html/2607.03055#bib.bib7)\]\. Mamba makes SSM parameters input\-dependent and pairs the resulting selective recurrence with a hardware\-aware scan\[[20](https://arxiv.org/html/2607.03055#bib.bib8)\]\. SSD/Mamba\-2 analyzes selective recurrences through semiseparable structure and efficient algorithms\[[11](https://arxiv.org/html/2607.03055#bib.bib9)\]\. Recent gated linear\-attention and linear\-recurrence models combine gates, delta\-style updates, or state expansion for adaptive memory control\[[72](https://arxiv.org/html/2607.03055#bib.bib48),[73](https://arxiv.org/html/2607.03055#bib.bib47),[45](https://arxiv.org/html/2607.03055#bib.bib51),[71](https://arxiv.org/html/2607.03055#bib.bib16)\], and recent variants such as Mamba\-3 further broaden the expressive recurrence design space\[[33](https://arxiv.org/html/2607.03055#bib.bib20)\]\. These developments make clear that channel interaction and token\-dependent control are not themselves the missing novelty\. The open question for memory\-prior design is different: can channel interaction be made part of the online\-approximation memory semantics itself, rather than added only as an architectural mixer, projection, gate, or controller?

We introduce*SHiPPO*\(*Sylvester HiPPO*\), a transported online approximation framework for channel\-interacting recurrent memory\. At the operator level, SHiPPO is defined by a projection problem, not by postulating a recurrence in isolation\. Given an online\-memory basis and an admissible right\-transport path, SHiPPO jointly transports the channel metric and the approximation family\. The resulting coefficient matrix satisfies the Sylvester dynamics

C˙​\(t\)=AL​\(t\)​C​\(t\)\+BL​\(t\)​f​\(t\)⊤\+C​\(t\)​AR​\(t\)\.\\dot\{C\}\(t\)=A\_\{L\}\(t\)C\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\+C\(t\)A\_\{R\}\(t\)\.HereAL,BLA\_\{L\},B\_\{L\}are supplied by the chosen online approximation problem, whileARA\_\{R\}describes how channel coordinates are transported along the history\. The inner projection problem minimizes only over the coefficient stateCC; the transport path may be fixed, parameterized, or generated by a causal controller in an outer trainable model\. Thus, whenARA\_\{R\}is input\-dependent, the semantics is pathwise: conditional on the realized transport path, the state is the coefficient of the corresponding transported approximation problem, even though the full input–output map may be nonlinear\. For a fixed realized path, SHiPPO is ordinary HiPPO in a tied moving channel frame, with the history encoder, coefficient decoder, and Sylvester gauge term determined by the same transport path\.

This gives a lifting principle for deep SSM layers whose dynamics, structure, or initialization are motivated by online\-memory priors\. The lift does not prescribe a universal right generator or a particular right\-transport initialization\. Rather, it replaces a one\-sided online\-memory coefficient equation by a transported coefficient equation, separating the memory basis supplied by the left operator from the channel\-frame evolution supplied by the right transport\. For S4/S4D\-style models, this corresponds to adding a right\-action transport term to an existing coefficient equation\. For Mamba\-style selective SSMs, the constraint is stronger: the layer must preserve tokenwise parameter generation, a lightweight exact scan, and recurrent decoding\. A fully channelwise lift of a diagonal selective recurrence does not, in general, preserve a small scan algebra under nontrivial right transport\. We therefore study a restricted scan\-compatible SHiPPO\-derived realization: channels are partitioned into small transport groups, the left dynamics remain diagonal in the state dimension and tied within each group, and the right transport is controller\-compatible and group\-local\. These restrictions are not part of the abstract SHiPPO definition; they are the computational price paid to obtain exponential\-adjusted updates, exact group\-local block\-affine scan, and recurrent decoding for the chosen right action\. We also identify a simultaneous\-reducibility collapse criterion showing when right transports reduce, up to static channel mixing, to independent scalar or blockwise transported banks\. This motivates non\-reducible transport parameterizations in the scan\-compatible realization\.

This distinction also has empirical content\. High\-rank current\-step source/write updates can increase what is written into memory at the current token, but they do not by themselves implement a later right action on memory that has already been written\. We therefore use paired noncommutative transport diagnostics to separate source/write rank from future transport of stored memory, rather than treating broad associative recall as the main evidence\. We then use Transport\-MQAR as a complementary autoregressive finite\-field diagnostic in which bindings, operations, and queries are interleaved\.

![Refer to caption](https://arxiv.org/html/2607.03055v1/x1.png)Figure 1:Overview of the SHiPPO lift\. Ordinary HiPPO gives one\-sided online projection memory in fixed channel coordinates\. SHiPPO transports the channel frame while preserving the left online\-memory operator, yielding the right\-action termC​ARCA\_\{R\}on stored memory\. The selective cell is a scan\-compatible restriction with controller\-compatible group\-local right transport and exact block\-affine scan\.#### Contributions\.

\(i\) We formulate SHiPPO as a transported online approximation problem for channel\-interacting memory and prove that coupling the transported approximation family with the transported channel metric yields Sylvester coefficient dynamics\. \(ii\) We derive a pathwise lifting principle for online\-memory coefficient equations: a one\-sided projection memory is lifted by keeping the left online\-memory operator fixed and adding a right\-action transport path as an external modeling or learning choice\. \(iii\) We instantiate this prior in a restricted scan\-compatible selective SSM cell with group\-tied diagonal\-left dynamics and controller\-compatible group\-local right transport, deriving exponential\-adjusted discrete updates and proving exact group\-local block\-affine scan closure for the resulting recurrence\. \(iv\) We identify a simultaneous\-reducibility collapse criterion under which right transports are equivalent, up to static channel mixing, to independent scalar or blockwise transported banks; this motivates non\-reducible split\-flow transport designs\. \(v\) We provide paired noncommutative diagnostics separating high\-rank source/write updates from future right transport of already\-written memory, showing that source rank and transported memory are distinct mechanisms rather than interchangeable forms of channel interaction\. \(vi\) We evaluate scan\-compatible SHiPPO realizations on Transport\-MQAR and a controller\-suffix intervention as complementary autoregressive diagnostics of learned right\-action pathways\.

## 2SHiPPO \(Sylvester HiPPO\): Transported Online Projection Memories

SHiPPO \(*Sylvester HiPPO*\) builds on the HiPPO framework of online polynomial projection operators\[[19](https://arxiv.org/html/2607.03055#bib.bib1),[23](https://arxiv.org/html/2607.03055#bib.bib3)\]by adding a chosen right\-transport path to the channel frame\. We define SHiPPO as an online approximation problem for that realized transport path, rather than by postulating a recurrence in isolation\. Throughout this section, fix an admissible pathAR∈L1​\(\[0,T\];ℝd×d\)A\_\{R\}\\in L^\{1\}\(\[0,T\];\\mathbb\{R\}^\{d\\times d\}\); all identities are conditional on this realized path, and we suppress this dependence in the notation\. IfARA\_\{R\}is generated by a causal controller in a trainable model, the same projection semantics and coefficient dynamics hold*pathwise*for each realized trajectory\. The inner optimization is always over the coefficient matrixCC, not over the transport path\.

Letf:\[0,∞\)→ℝdf:\[0,\\infty\)\\to\\mathbb\{R\}^\{d\}be add\-channel signal\. For eacht\>0t\>0, letμt\\mu\_\{t\}be a measure on\[0,t\]\[0,t\], and letΦt:\[0,t\]→ℝN\\Phi\_\{t\}:\[0,t\]\\to\\mathbb\{R\}^\{N\}be a basis with invertible Gram matrix

G​\(t\):=∫0tΦt​\(τ\)​Φt​\(τ\)⊤​𝑑μt​\(τ\)\.G\(t\):=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)\\Phi\_\{t\}\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\.Whend​μt​\(τ\)=wt​\(τ\)​d​τd\\mu\_\{t\}\(\\tau\)=w\_\{t\}\(\\tau\)d\\tau, define

ψ​\(t,τ\):=wt​\(τ\)​Φt​\(τ\),∂tψ​\(t,τ\)=AL​\(t\)​ψ​\(t,τ\)\(τ<t\),BL​\(t\):=ψ​\(t,t\),\\psi\(t,\\tau\):=w\_\{t\}\(\\tau\)\\Phi\_\{t\}\(\\tau\),\\qquad\\partial\_\{t\}\\psi\(t,\\tau\)=A\_\{L\}\(t\)\\psi\(t,\\tau\)\\quad\(\\tau<t\),\\qquad B\_\{L\}\(t\):=\\psi\(t,t\),where the middle identity is the usual HiPPO closure condition\. Ordinary vector\-valued HiPPO approximatesf\|\[0,t\]f\|\_\{\[0,t\]\}byC⊤​Φt​\(τ\)C^\{\\top\}\\Phi\_\{t\}\(\\tau\)and has coefficient matrixCH​\(t\)C\_\{H\}\(t\)satisfying

G​\(t\)​CH​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​𝑑μt​\(τ\)\.G\(t\)C\_\{H\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\.In the orthonormalized caseG​\(t\)=ING\(t\)=I\_\{N\}, this gives the closed one\-sided coefficient dynamics

C˙H​\(t\)=AL​\(t\)​CH​\(t\)\+BL​\(t\)​f​\(t\)⊤\.\\dot\{C\}\_\{H\}\(t\)=A\_\{L\}\(t\)C\_\{H\}\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\.Appendix[A\.2](https://arxiv.org/html/2607.03055#A1.SS2)recalls the variational derivation and the non\-orthonormal form\.

We now add right transport\. LetP​\(t,τ\)∈G​L​\(d\)P\(t,\\tau\)\\in GL\(d\)be the state\-transition family for the right generator,

∂tP​\(t,τ\)=P​\(t,τ\)​AR​\(t\),P​\(τ,τ\)=Id,\\partial\_\{t\}P\(t,\\tau\)=P\(t,\\tau\)A\_\{R\}\(t\),\\qquad P\(\\tau,\\tau\)=I\_\{d\},which exists and is invertible under the integrability assumption onARA\_\{R\}\[[10](https://arxiv.org/html/2607.03055#bib.bib29)\]\. Set

MP​\(t,τ\):=P​\(t,τ\)​P​\(t,τ\)⊤,‖u‖MP​\(t,τ\)2:=u⊤​MP​\(t,τ\)​u\.M\_\{P\}\(t,\\tau\):=P\(t,\\tau\)P\(t,\\tau\)^\{\\top\},\\qquad\\\|u\\\|\_\{M\_\{P\}\(t,\\tau\)\}^\{2\}:=u^\{\\top\}M\_\{P\}\(t,\\tau\)u\.
###### Definition 2\.1\(SHiPPO online approximation problem\)\.

For eacht\>0t\>0, define the transported approximation family

𝒢tSH=\{τ↦P​\(t,τ\)−⊤​C⊤​Φt​\(τ\):C∈ℝN×d\}\.\\mathcal\{G\}\_\{t\}^\{\\mathrm\{SH\}\}=\\\{\\,\\tau\\mapsto P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\):C\\in\\mathbb\{R\}^\{N\\times d\}\\,\\\}\.The SHiPPO coefficient matrix is

CS​\(t\)=arg⁡minC∈ℝN×d​∫0t‖f​\(τ\)−P​\(t,τ\)−⊤​C⊤​Φt​\(τ\)‖MP​\(t,τ\)2​𝑑μt​\(τ\)\.C\_\{S\}\(t\)=\\arg\\min\_\{C\\in\\mathbb\{R\}^\{N\\times d\}\}\\int\_\{0\}^\{t\}\\left\\\|f\(\\tau\)\-P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\\right\\\|\_\{M\_\{P\}\(t,\\tau\)\}^\{2\}\\,d\\mu\_\{t\}\(\\tau\)\.We writeCS​\(t\)=shippot⁡\(f\)C\_\{S\}\(t\)=\\operatorname\{shippo\}\_\{t\}\(f\)\.

Definition[2\.1](https://arxiv.org/html/2607.03055#S2.Thmassumption1)transports the approximation family and the channel metric together\. This coupling is essential: transporting only the channel metric while keeping the ordinary HiPPO family generally destroys finite HiPPO\-style coefficient closure when the metric depends onτ\\tau\. Appendix[A\.6](https://arxiv.org/html/2607.03055#A1.SS6)gives the stationary calculation and theτ\\tau\-independent special case\.

The coupled construction is conjugate to ordinary HiPPO\. For fixedtt, define

\(𝒯t​u\)​\(τ\):=P​\(t,τ\)⊤​u​\(τ\)\.\(\\mathcal\{T\}\_\{t\}u\)\(\\tau\):=P\(t,\\tau\)^\{\\top\}u\(\\tau\)\.Then𝒯t\\mathcal\{T\}\_\{t\}is an isometry from the SHiPPO metric to the Euclidean HiPPO metric and maps𝒢tSH\\mathcal\{G\}\_\{t\}^\{\\mathrm\{SH\}\}bijectively onto the ordinary HiPPO family:

∫0tu​\(τ\)⊤​MP​\(t,τ\)​v​\(τ\)​𝑑μt​\(τ\)=∫0t\(𝒯t​u\)​\(τ\)⊤​\(𝒯t​v\)​\(τ\)​𝑑μt​\(τ\),shippot⁡\(f\)=hippot⁡\(𝒯t​f\)\.\\int\_\{0\}^\{t\}u\(\\tau\)^\{\\top\}M\_\{P\}\(t,\\tau\)v\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\)=\\int\_\{0\}^\{t\}\(\\mathcal\{T\}\_\{t\}u\)\(\\tau\)^\{\\top\}\(\\mathcal\{T\}\_\{t\}v\)\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\),\\qquad\\operatorname\{shippo\}\_\{t\}\(f\)=\\operatorname\{hippo\}\_\{t\}\(\\mathcal\{T\}\_\{t\}f\)\.The final equality is an identity of coefficient matrices; Appendix[A\.4](https://arxiv.org/html/2607.03055#A1.SS4)gives the full argument\.

###### Theorem 2\.2\(Normal equation and Sylvester coefficient dynamics\)\.

The SHiPPO coefficient matrix satisfies

G​\(t\)​CS​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑μt​\(τ\)\.G\(t\)C\_\{S\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.If, in addition,G​\(t\)=ING\(t\)=I\_\{N\},d​μt​\(τ\)=wt​\(τ\)​d​τd\\mu\_\{t\}\(\\tau\)=w\_\{t\}\(\\tau\)d\\tau, the HiPPO closure condition above holds, and the usual Leibniz\-rule regularity assumptions are satisfied, then, at differentiability times,

CS​\(t\)=∫0tψ​\(t,τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑τC\_\{S\}\(t\)=\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\tauand

C˙S​\(t\)=AL​\(t\)​CS​\(t\)\+BL​\(t\)​f​\(t\)⊤\+CS​\(t\)​AR​\(t\)\.\\dot\{C\}\_\{S\}\(t\)=A\_\{L\}\(t\)C\_\{S\}\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\+C\_\{S\}\(t\)A\_\{R\}\(t\)\.

###### Proof sketch\.

By the conjugacy above,CS​\(t\)C\_\{S\}\(t\)is the ordinary HiPPO coefficient matrix of the transformed signal\(𝒯t​f\)​\(τ\)=P​\(t,τ\)⊤​f​\(τ\)\(\\mathcal\{T\}\_\{t\}f\)\(\\tau\)=P\(t,\\tau\)^\{\\top\}f\(\\tau\), which gives the normal equation\. In the orthonormalized case, differentiate the integral representation using Leibniz’ rule,∂tψ=AL​ψ\\partial\_\{t\}\\psi=A\_\{L\}\\psi,∂tP=P​AR\\partial\_\{t\}P=PA\_\{R\}, andP​\(t,t\)=IdP\(t,t\)=I\_\{d\}\. The boundary term givesBL​f⊤B\_\{L\}f^\{\\top\}, while the two interior derivatives giveAL​CSA\_\{L\}C\_\{S\}andCS​ARC\_\{S\}A\_\{R\}\. Full regularity and first\-variation details are in Appendix[A\.5](https://arxiv.org/html/2607.03055#A1.SS5)\. ∎

In the orthonormalized case, the coefficient ODE has the standard differential Sylvester form\[[4](https://arxiv.org/html/2607.03055#bib.bib31),[56](https://arxiv.org/html/2607.03055#bib.bib30)\]\. Theorem[2\.2](https://arxiv.org/html/2607.03055#S2.Thmassumption2)therefore shows that SHiPPO is not an arbitrary modification of a state equation: the left operatorAL,BLA\_\{L\},B\_\{L\}is inherited from the underlying online approximation problem, while the right action enters only through the chosen transport path\. This is the sense in which SHiPPO lifts one\-sided online\-memory dynamics to channel\-interacting memory\.

###### Corollary 2\.3\(Pathwise transported lift of closed online\-memory dynamics\)\.

Any closed one\-sided online\-memory coefficient equation arising from the projection setup above,

C˙​\(t\)=AL​\(t\)​C​\(t\)\+BL​\(t\)​f​\(t\)⊤,\\dot\{C\}\(t\)=A\_\{L\}\(t\)C\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\},admits, for any admissible right\-transport pathARA\_\{R\}, the pathwise transported dynamics

C˙S​\(t\)=AL​\(t\)​CS​\(t\)\+BL​\(t\)​f​\(t\)⊤\+CS​\(t\)​AR​\(t\)\.\\dot\{C\}\_\{S\}\(t\)=A\_\{L\}\(t\)C\_\{S\}\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\+C\_\{S\}\(t\)A\_\{R\}\(t\)\.The left operator is inherited from the underlying online approximation problem, whereas the right transport is an external modeling choice\. Architectural restrictions onARA\_\{R\}are introduced only in the scan\-compatible realization of Section[3](https://arxiv.org/html/2607.03055#S3), not in the abstract SHiPPO definition\.

SHiPPO reduces exactly to ordinary HiPPO whenAR≡0A\_\{R\}\\equiv 0, equivalentlyP​\(t,τ\)≡IdP\(t,\\tau\)\\equiv I\_\{d\}\. For a nontrivial realized transport path, SHiPPO is ordinary HiPPO in a tied moving channel frame, not a static channel mixer or an arbitrary encoder–decoder factorization\. Appendix[B](https://arxiv.org/html/2607.03055#A2)gives the exact moving\-frame factorization and non\-reduction statements\. WhenARA\_\{R\}is input\-dependent, the same interpretation remains valid pathwise, but it does not yield a fixed input\-independent encoder–HiPPO–decoder reduction\.

## 3A Scan\-Compatible SHiPPO Lift for Mamba\-Style Selective SSMs

Section[2](https://arxiv.org/html/2607.03055#S2)defines SHiPPO as an operator\-level transported projection memory\. A selective SSM layer cannot use this full operator without additional computational restrictions\. We therefore study one scan\-compatible SHiPPO\-derived realization: channels are partitioned into small transport groups, the left dynamics are diagonal in the state dimension and tied across channels within each group, and the right transport is group\-local and controller\-compatible\. These restrictions are not part of the abstract SHiPPO definition; they are the price of exact scan and recurrent decoding\.

LetDDdenote the inner channel width of the selective branch, and partition the channels intoGGgroups of widthPP, soD=G​PD=GP\. We write formulas for one group and omit the group index\. Thus the group\-local input isxt∈ℝPx\_\{t\}\\in\\mathbb\{R\}^\{P\}, the memory state isHt∈ℝN×PH\_\{t\}\\in\\mathbb\{R\}^\{N\\times P\}, and the right transport acts byRt∈G​L​\(P\)R\_\{t\}\\in GL\(P\)\. The ungrouped case is recovered by takingP=DP=D\. A fully channelwise lift of a diagonal selective recurrence generally requires row\-dependent right transition products and does not close in the small block\-affine scan algebra used below; Appendix[C\.1](https://arxiv.org/html/2607.03055#A3.SS1)gives the two\-step obstruction\. The realization in this section should therefore be read as a restricted Mamba\-style cell derived from the SHiPPO prior, not as the full operator\-level SHiPPO construction and not as the original per\-channel Mamba parameterization\.

### 3\.1From the SHiPPO prior to a scan\-friendly selective recurrence

Our restricted continuous\-time selective lift is

H˙​\(t\)=Diag⁡\(a​\(t\)\)​H​\(t\)\+b​\(t\)​x​\(t\)⊤\+H​\(t\)​AR​\(t\),\\dot\{H\}\(t\)=\\operatorname\{Diag\}\(a\(t\)\)\\,H\(t\)\+b\(t\)x\(t\)^\{\\top\}\+H\(t\)A\_\{R\}\(t\),\(1\)wherea​\(t\)∈ℝNa\(t\)\\in\\mathbb\{R\}^\{N\},b​\(t\)∈ℝNb\(t\)\\in\\mathbb\{R\}^\{N\},x​\(t\)∈ℝPx\(t\)\\in\\mathbb\{R\}^\{P\}, andAR​\(t\)∈ℝP×PA\_\{R\}\(t\)\\in\\mathbb\{R\}^\{P\\times P\}\. The left dynamics remain diagonal in the state dimension, as in diagonal selective SSMs, but are tied across thePPchannels in the transport group\. The factorized sourceb​\(t\)​x​\(t\)⊤b\(t\)x\(t\)^\{\\top\}keeps the HiPPO\-like interpretation that the left operator injects the current input into a temporal memory basis\. More general additive sourcesU​\(t\)∈ℝN×PU\(t\)\\in\\mathbb\{R\}^\{N\\times P\}are compatible with the same scan algebra; the factorized form is the lightweight selective\-cell instance used below\.

Equation \([1](https://arxiv.org/html/2607.03055#S3.E1)\) is Mamba\-style only in this restricted sense\. It preserves diagonal state\-timescale dynamics on the left, while the SHiPPO right action supplies channel interaction inside the memory state itself\. The right\-transport family is a modeling restrictive bias; preferential choices such as analytic initialization are orthogonal to the scan\-compatible realization\.

### 3\.2Controller\-compatible transport and block\-affine scan closure

The right transport may be token\-dependent, but an exact finite scan summary requires it to be fixed with respect to the main memory variable being scanned\. We encode this through a causal controller pathξ1:T\\xi\_\{1:T\}, computed either input\-only,ξt=ϕ​\(xt\)\\xi\_\{t\}=\\phi\(x\_\{t\}\), or by an auxiliary causal module independent of the main memory recurrence\. Conditional on this path, the main recurrence is a sequence of affine two\-sided maps\.

###### Definition 3\.1\(Controller\-compatible right transport\)\.

A right transportRt∈G​L​\(P\)R\_\{t\}\\in GL\(P\)is*controller\-compatible*if, conditional on a precomputed causal controller pathξ1:T\\xi\_\{1:T\},

Rt=R​\(ξt\)R\_\{t\}=R\(\\xi\_\{t\}\)andRtR\_\{t\}is independent of the main memory stateHt−1H\_\{t\-1\}\. A transport of the formRt=R​\(ξt,Ht−1\)R\_\{t\}=R\(\\xi\_\{t\},H\_\{t\-1\}\)is state\-coupled and is not controller\-compatible unless the dependence onHt−1H\_\{t\-1\}is degenerate\.

###### Proposition 3\.2\(Exact block\-affine scan closure\)\.

Consider the recurrence

Ht=Lt​Ht−1​Rt\+Ut,H\_\{t\}=L\_\{t\}H\_\{t\-1\}R\_\{t\}\+U\_\{t\},whereLt∈ℝN×NL\_\{t\}\\in\\mathbb\{R\}^\{N\\times N\},Rt∈G​L​\(P\)R\_\{t\}\\in GL\(P\), andUt∈ℝN×PU\_\{t\}\\in\\mathbb\{R\}^\{N\\times P\}are fixed conditional on a precomputed controller path\. Then each step is affine inHt−1H\_\{t\-1\}, and summaries compose as

\(L2,R2,U2\)⋆\(L1,R1,U1\)=\(L2​L1,R1​R2,L2​U1​R2\+U2\)\.\(L\_\{2\},R\_\{2\},U\_\{2\}\)\\star\(L\_\{1\},R\_\{1\},U\_\{1\}\)=\(L\_\{2\}L\_\{1\},\\;R\_\{1\}R\_\{2\},\\;L\_\{2\}U\_\{1\}R\_\{2\}\+U\_\{2\}\)\.Thus the recurrence admits an exact block\-affine prefix scan\. IfRtR\_\{t\}depends directly onHt−1H\_\{t\-1\}, the step map is generically nonlinear inHt−1H\_\{t\-1\}, so this finite affine summary algebra is not closed without augmenting the state or imposing special degeneracies\.

This is an associative prefix\-scan computation\[[5](https://arxiv.org/html/2607.03055#bib.bib72)\]; related parallel scans have been used to parallelize linear recurrent networks over sequence length\[[38](https://arxiv.org/html/2607.03055#bib.bib73)\]\. Appendix[C](https://arxiv.org/html/2607.03055#A3)gives the proof and a concrete state\-coupled failure mode\. The scan above is not the original elementwise selective\-scan algebra of Mamba\. The original one\-channel scan is recovered in the trivial limitP=1P=1\. If the right transport is diagonal or identity, the recurrence decouples across channels within each transport group, although the group\-tied left restriction remains\. In the nontrivial transported case, exact scan is retained as a group\-local block\-affine scan\.

### 3\.3Degenerate versus nontrivial transport families

Controller compatibility preserves computation, but it does not guarantee expressive transport\. A common source of misleading adaptivity is token\-dependent coefficients over a generator family that is simultaneously reducible in a fixed channel basis\.

###### Proposition 3\.3\(Collapse under fixed simultaneous block reduction\)\.

Suppose all right\-generator basis matrices preserve the same nontrivial block decomposition: for every generatorGmG\_\{m\}there is a fixedQ∈G​L​\(P\)Q\\in GL\(P\)such that

Gm=Q​Λm​Q−1,G\_\{m\}=Q\\Lambda\_\{m\}Q^\{\-1\},where allΛm\\Lambda\_\{m\}are diagonal or block\-diagonal with the same fixed nontrivial block partition\. If

AR,t=∑mρt,m​Gm,A\_\{R,t\}=\\sum\_\{m\}\\rho\_\{t,m\}G\_\{m\},then every dense exponentialRt=exp⁡\(Δt​AR,t\)R\_\{t\}=\\exp\(\\Delta\_\{t\}A\_\{R,t\}\)and every fixed\-order split product formed from these generators is diagonal or block\-diagonal in the same fixed basis\. Consequently, the change of variablesH~t=Ht​Q\\widetilde\{H\}\_\{t\}=H\_\{t\}Qdecomposes

Ht=Lt​Ht−1​Rt\+UtH\_\{t\}=L\_\{t\}H\_\{t\-1\}R\_\{t\}\+U\_\{t\}into independent scalar or blockwise transported banks, up to a static change of channel coordinates\. IfUt=bt​xt⊤U\_\{t\}=b\_\{t\}x\_\{t\}^\{\\top\}, thenUt​Q=bt​\(Q⊤​xt\)⊤U\_\{t\}Q=b\_\{t\}\(Q^\{\\top\}x\_\{t\}\)^\{\\top\}, so the same fixed basis change can be absorbed into the group input coordinates\.

This proposition is a sufficient collapse criterion, not a complete classification of all degeneracies\. It rules out the case where token\-dependent coefficients only move within a simultaneously reducible generator family\. Noncommutativity alone is insufficient: noncommuting generators may still preserve a common block decomposition\. The scan\-compatible realization should therefore use right\-action families whose realized transports admit no fixed nontrivial common block decomposition, if the goal is genuinely channel\-interacting transported memory\.

### 3\.4Controlled split\-flow transport and exponential\-adjusted discrete cell

For implementation, each token first fixes a group\-local right actionRtR\_\{t\}, either by a dense frozen\-ODE exponential or by a fixed\-order split product of structured factors\. We use the structured generator family

AR,t=−Diag⁡\(dt\)\+∑mθt,m​Ωm\+∑nηt,n​Nn\+∑ℓζt,ℓ​pℓ​qℓ⊤,dt∈ℝ\+P,A\_\{R,t\}=\-\\operatorname\{Diag\}\(d\_\{t\}\)\+\\sum\_\{m\}\\theta\_\{t,m\}\\Omega\_\{m\}\+\\sum\_\{n\}\\eta\_\{t,n\}N\_\{n\}\+\\sum\_\{\\ell\}\\zeta\_\{t,\\ell\}p\_\{\\ell\}q\_\{\\ell\}^\{\\top\},\\qquad d\_\{t\}\\in\\mathbb\{R\}\_\{\+\}^\{P\},\(2\)whereΩm⊤=−Ωm\\Omega\_\{m\}^\{\\top\}=\-\\Omega\_\{m\},Nn2=0N\_\{n\}^\{2\}=0, and optional low\-rank terms add additional channel interaction\. The dense backend setsRt=exp⁡\(Δt​AR,t\)R\_\{t\}=\\exp\(\\Delta\_\{t\}A\_\{R,t\}\); a split backend instead implements a fixed\-order product of cheap factor exponentials\. Appendix[C](https://arxiv.org/html/2607.03055#A3)gives the factor actions, split\-product error, and dense\-backend local quadrature analysis\. The scan algebra below is exact for whichever discrete right actionRtR\_\{t\}is implemented\.

LetΔt\>0\\Delta\_\{t\}\>0,λt∈\[0,1\]\\lambda\_\{t\}\\in\[0,1\], andUt=bt​xt⊤U\_\{t\}=b\_\{t\}x\_\{t\}^\{\\top\}\. We use

Lt\\displaystyle L\_\{t\}:=exp⁡\(Δt​Diag⁡\(at\)\),Rt:=\{exp⁡\(Δt​AR,t\),dense backend,Rtsplit,split backend,\\displaystyle:=\\exp\\\!\\bigl\(\\Delta\_\{t\}\\operatorname\{Diag\}\(a\_\{t\}\)\\bigr\),\\qquad R\_\{t\}:=\\begin\{cases\}\\exp\(\\Delta\_\{t\}A\_\{R,t\}\),&\\text\{dense backend\},\\\\ R\_\{t\}^\{\\mathrm\{split\}\},&\\text\{split backend\},\\end\{cases\}U^t\\displaystyle\\widehat\{U\}\_\{t\}:=\(1−λt\)​Δt​Lt​Ut−1​Rt\+λt​Δt​Ut,Ht:=Lt​Ht−1​Rt\+U^t\.\\displaystyle:=\(1\-\\lambda\_\{t\}\)\\Delta\_\{t\}L\_\{t\}U\_\{t\-1\}R\_\{t\}\+\\lambda\_\{t\}\\Delta\_\{t\}U\_\{t\},\\qquad H\_\{t\}:=L\_\{t\}H\_\{t\-1\}R\_\{t\}\+\\widehat\{U\}\_\{t\}\.\(3\)For the first step, implementations may set the boundary sourceU0=0U\_\{0\}=0\.

###### Theorem 3\.4\(Exponential\-adjusted SHiPPO cell\)\.

For the dense backendRt=exp⁡\(Δt​AR,t\)R\_\{t\}=\\exp\(\\Delta\_\{t\}A\_\{R,t\}\), the update \([3](https://arxiv.org/html/2607.03055#S3.E3)\) is a two\-point exponential source discretization of \([1](https://arxiv.org/html/2607.03055#S3.E1)\) under step\-frozenata\_\{t\}andAR,tA\_\{R,t\}\. For a split backend, the same formula defines the exact discrete recurrence for the implemented split right action\. IfAR,t≡0A\_\{R,t\}\\equiv 0, thenRt=IPR\_\{t\}=I\_\{P\}and the cell reduces to the corresponding one\-sided diagonal\-left selective update; withP=1P=1andλt=1\\lambda\_\{t\}=1, this recovers the corresponding one\-channel exponential\-Euler selective update\.

Thus nontrivial right transport does not preserve the original elementwise Mamba scan\. Instead, the restricted SHiPPO realization yields an exact group\-local block\-affine scan, with computation controlled by the group widthPPand with the right action interpreted as learned transport of memory coordinates rather than as a generic channel mixer\.

## 4Experiments: Separating Source Writes from Right Transport

We evaluate SHiPPO as a transported\-memory prior\. The experiments ask a mechanistic question: can high\-rank current\-step source/write updates substitute for future right transport of memory that has already been written? We first use a paired noncommutative diagnostic that blocks payload\-dependent source injection at operation tokens\. We then use Transport\-MQAR as a complementary autoregressive recall diagnostic\.

The no\-right source/write baseline removes the right action,

Ht=Lt​Ht−1\+U^t\(r\),U^t\(r\)=1r​Bt​Xt⊤,H\_\{t\}=L\_\{t\}H\_\{t\-1\}\+\\widehat\{U\}\_\{t\}^\{\(r\)\},\\qquad\\widehat\{U\}\_\{t\}^\{\(r\)\}=\\frac\{1\}\{\\sqrt\{r\}\}B\_\{t\}X\_\{t\}^\{\\top\},\(4\)where increasingrrincreases the rank of the source written at the current step\. SHiPPO\-style variants instead use

Ht=Lt​Ht−1​Rt\+U^t,H\_\{t\}=L\_\{t\}H\_\{t\-1\}R\_\{t\}\+\\widehat\{U\}\_\{t\},\(5\)so that a future operation can act on the channel coordinates of memory that was written earlier\. We report means and sample standard deviations across random seeds\. Appendix[D](https://arxiv.org/html/2607.03055#A4)gives the paired\-diagnostic task definitions, model families, metrics, training protocol, numerical summaries, and the group\-local audit; Appendix[E](https://arxiv.org/html/2607.03055#A5)gives the Transport\-MQAR generator, controls, length sweeps, and controller counterfactuals\.

### 4\.1Paired noncommutative transport diagnostic

The primary diagnostic writes a payload vectorvvinto memory and then applies two operation tokens\. The two paired examples share the samevvand operation parameters but reverse the order of the operations,a​babversusb​aba\. The target paired difference is

Δtrue=v⊤​\(Ra​Rb−Rb​Ra\)\.\\Delta\_\{\\mathrm\{true\}\}=v^\{\\top\}\(R\_\{a\}R\_\{b\}\-R\_\{b\}R\_\{a\}\)\.\(6\)We measure the normalized mean\-squared error of this paired difference,

Pair​Δ​NMSE=∥Δpred−Δtrue∥22max⁡\{∥Δtrue∥22,ϵden\},ϵden=10−8\.\\mathrm\{Pair\}\\,\\Delta\\mathrm\{NMSE\}=\\frac\{\\lVert\\Delta\_\{\\mathrm\{pred\}\}\-\\Delta\_\{\\mathrm\{true\}\}\\rVert\_\{2\}^\{2\}\}\{\\max\\\{\\lVert\\Delta\_\{\\mathrm\{true\}\}\\rVert\_\{2\}^\{2\},\\epsilon\_\{\\mathrm\{den\}\}\\\}\},\\qquad\\epsilon\_\{\\mathrm\{den\}\}=10^\{\-8\}\.\(7\)Source injection is allowed at WRITE tokens, but operation tokens do not inject payload\-dependent source\. Thus a no\-right model can increase the rank of what it writes, but it cannot make a later operation right\-multiply previously written channel coordinates\. In contrast, a SHiPPO\-style model can applyRtR\_\{t\}to already\-written memory\.

### 4\.2Right\-transport parameterizations and intervention

The oracle\-RRvariant tests whether the true right action can solve the paired\-transport diagnostic\. A stronger test is whether learned and selective right\-transport parameterizations can do so as well\. We therefore compare oracle transport, learned Lie transports initialized from true, zero, or random generators, and a selective right\-transport controller\. We also perform an evaluation\-time intervention replacingRtR\_\{t\}by identity while leaving all other learned weights fixed; this tests whether the paired\-difference behavior is mediated by the right action\.

![Refer to caption](https://arxiv.org/html/2607.03055v1/x2.png)Figure 2:Main paired\-transport diagnostic\. \(a\) Paired examples share the same payload and operation coefficients but reverse order; operation tokens inject no payload\-dependent source\. \(b,c\) Increasing no\-right source rank improves ordinary NMSE but leaves PairΔ\\DeltaNMSE near one\. \(d\) Right\-transport variants recover the paired difference, while the evaluation\-timeRt→IR\_\{t\}\\\!\\to Iintervention returns PairΔ\\DeltaNMSE to one\. Points show means over seeds; error bars show sample standard deviations\.The key comparison is the intervention in Figure[2](https://arxiv.org/html/2607.03055#S4.F2)\(d\): the same trained models lose the paired\-difference signal whenRtR\_\{t\}is replaced by identity\. This makes the diagnostic more than a capacity comparison, because the recovered signal is tied to the presence of a future right action on stored memory\. Full numerical summaries are in Appendix[D\.5](https://arxiv.org/html/2607.03055#A4.SS5)\.

### 4\.3Transport\-MQAR as a complementary autoregressive diagnostic

The paired diagnostic isolates the mechanism in a minimal controlled setting\. We next ask whether scan\-compatible SHiPPO realizations also help in an autoregressive finite\-field recall task where bindings, operations, and queries are interleaved\. Transport\-MQAR inserts invertible operation tokens between key–value bindings and queries, so the target must be recovered in the current transported frame\. Appendix[E](https://arxiv.org/html/2607.03055#A5)gives the generator, metrics, model geometry, training protocol, full length sweep, and the controller\-suffix counterfactual; the main text reports only the length\-4096 summary needed for the mechanistic comparison\. Transport\-MQAR is a finite\-field, right\-transport modification of multi\-query associative recall \(MQAR\), a diagnostic introduced to study recall in efficient language models\[[2](https://arxiv.org/html/2607.03055#bib.bib76)\]\. The comparison includes standard GRU\[[8](https://arxiv.org/html/2607.03055#bib.bib77)\]and Transformer\[[64](https://arxiv.org/html/2607.03055#bib.bib79)\]baselines\.

Table 1:Transport\-MQAR summary at length 4096\. Full length sweeps, standard deviations, model geometry, and training protocol are in Appendix[E](https://arxiv.org/html/2607.03055#A5)\.ModelCoord\.ExactGRU0\.0950\.016Transformer0\.0420\.002Free enc/dec0\.0920\.009No\-right0\.1020\.016DirectGen\-SingleExp0\.1030\.036StructGen\-Split0\.1100\.033StructGen\-Split \+ suffix zeroing0\.1040\.018Table[1](https://arxiv.org/html/2607.03055#S4.T1)summarizes the length\-4096 Transport\-MQAR results\. StructGen\-Split modestly improves coordinate accuracy over no\-right and static\-basis controls, while DirectGen\-SingleExp obtains the strongest exact accuracy\. The suffix\-zeroing counterfactual reduces exact accuracy from 0\.033 to 0\.018, suggesting that the learned controller coordinates are used, but not identifying the learned transport geometry\.

### 4\.4Scope of the evidence

The experiments support a mechanistic claim in controlled diagnostics: high\-rank current\-step source/write updates and future right transport are different operations on matrix memory\. They do not establish broad sequence\-modeling superiority\. The formal separation is clearest in the full\-transport paired diagnostic; Transport\-MQAR provides complementary autoregressive evidence but does not identify the learned transport geometry\. The group\-local restriction is audited in Appendix[D\.6](https://arxiv.org/html/2607.03055#A4.SS6)\.

## 5Discussion

SHiPPO should be viewed as a transported online\-memory prior\. Once a right\-transport path is realized, the state has ordinary HiPPO coefficients in a tied moving channel frame, and the right\-action term is the gauge term induced by transporting the approximation family and metric together\. This distinguishes SHiPPO from untied matrix\-state recurrences, high\-rank source/write updates, or encoder–decoder factorizations that are not tied by a common transport path\.

The scan\-compatible cell of Section[3](https://arxiv.org/html/2607.03055#S3)is one restricted realization, chosen to retain exact block\-affine scan and recurrent decoding\. The collapse criterion gives the complementary limitation: simultaneously reducible right transports collapse to static mixing plus independent scalar or blockwise banks\.

The experiments support this distinction in controlled diagnostics rather than claiming broad sequence\-modeling dominance\. Paired diagnostics separate current\-step writes from future right transport, while Transport\-MQAR provides complementary autoregressive evidence and leaves the preferred right\-action realization open\.

## Impact Statement

This paper presents foundational work on recurrent memory priors for sequence models\. The empirical evaluation is primarily synthetic and diagnostic: we do not release a large pretrained generative model, and we do not claim deployment\-ready language\-modeling performance\. Potential positive impacts include better tools for understanding memorization, structured generalization, and memory mechanisms in efficient sequence models\. Potential negative impacts are indirect and similar to those of improved sequence\-modeling methods more broadly, including possible use in more capable generative systems\. We therefore emphasize diagnostic scope, transparent limitations, and reproducibility\.

## Acknowledgements

#### Funding\.

This work was supported by JSPS KAKENHI Grant Number JP26K21295\.

#### Computing resources\.

This research used resources of the Argonne Leadership Computing Facility under ALCF Allocation IDs 15652 and 15654, and resources of the Oak Ridge Leadership Computing Facility under OLCF Project ID CSC704 through Director’s Discretionary allocation awards\.

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## Appendix ADerivations for Section[2](https://arxiv.org/html/2607.03055#S2)

This appendix supports the operator\-level claims of Section[2](https://arxiv.org/html/2607.03055#S2)\. We first recall the ordinary vector\-valued HiPPO variational equations, then derive the SHiPPO normal equation directly from the transported online approximation objective\. We next record the transport\-isometry conjugacy, prove Theorem[2\.2](https://arxiv.org/html/2607.03055#S2.Thmassumption2)and Corollary[2\.3](https://arxiv.org/html/2607.03055#S2.Thmassumption3), and finally explain why transporting only the channel metric generally fails to preserve a finite HiPPO\-style coefficient closure\.

### A\.1Notation, standing assumptions, and pathwise semantics

Fix a timet\>0t\>0\. Throughout, signalsu​\(τ\),v​\(τ\),f​\(τ\)∈ℝdu\(\\tau\),v\(\\tau\),f\(\\tau\)\\in\\mathbb\{R\}^\{d\}are treated as column vectors, and the basisΦt​\(τ\)∈ℝN\\Phi\_\{t\}\(\\tau\)\\in\\mathbb\{R\}^\{N\}is also a column vector\. Coefficient matrices areC∈ℝN×dC\\in\\mathbb\{R\}^\{N\\times d\}, so thatC⊤​Φt​\(τ\)∈ℝdC^\{\\top\}\\Phi\_\{t\}\(\\tau\)\\in\\mathbb\{R\}^\{d\}\.

We use the Frobenius inner product

⟨A,B⟩F:=tr​\(A⊤​B\)\(A,B∈ℝN×d\),\\langle A,B\\rangle\_\{F\}:=\\mathrm\{tr\}\(A^\{\\top\}B\)\\qquad\(A,B\\in\\mathbb\{R\}^\{N\\times d\}\),so that first variations can be written as

δ​J​\(C;Δ\)=⟨∇CJ​\(C\),Δ⟩F\.\\delta J\(C;\\Delta\)=\\langle\\nabla\_\{C\}J\(C\),\\Delta\\rangle\_\{F\}\.
#### Measures and closure\.

For eachtt, letμt\\mu\_\{t\}be a measure on\[0,t\]\[0,t\], and define the Gram matrix

G​\(t\):=∫0tΦt​\(τ\)​Φt​\(τ\)⊤​𝑑μt​\(τ\)\.G\(t\):=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)\\Phi\_\{t\}\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\.We assumeG​\(t\)G\(t\)is invertible for allttof interest\. Whend​μt​\(τ\)=wt​\(τ\)​d​τd\\mu\_\{t\}\(\\tau\)=w\_\{t\}\(\\tau\)d\\tau, define

ψ​\(t,τ\):=wt​\(τ\)​Φt​\(τ\),\\psi\(t,\\tau\):=w\_\{t\}\(\\tau\)\\Phi\_\{t\}\(\\tau\),and assume the standard HiPPO closure condition

∂tψ​\(t,τ\)=AL​\(t\)​ψ​\(t,τ\)\(τ<t\),BL​\(t\):=ψ​\(t,t\)\.\\partial\_\{t\}\\psi\(t,\\tau\)=A\_\{L\}\(t\)\\psi\(t,\\tau\)\\quad\(\\tau<t\),\\qquad B\_\{L\}\(t\):=\\psi\(t,t\)\.We also assume sufficient regularity to justify the first variations and Leibniz\-rule differentiations below, for instance the usual dominated convergence hypotheses\. Under only weak regularity, the displayed coefficient dynamics are interpreted at times where the relevant derivatives exist, or almost everywhere\.

#### Right\-transport paths and pathwise semantics\.

For the SHiPPO construction, letAR:\[0,T\]→ℝd×dA\_\{R\}:\[0,T\]\\to\\mathbb\{R\}^\{d\\times d\}be an admissible integrable right\-generator path, and letP​\(t,τ\)P\(t,\\tau\)solve

∂tP​\(t,τ\)=P​\(t,τ\)​AR​\(t\),P​\(τ,τ\)=Id\.\\partial\_\{t\}P\(t,\\tau\)=P\(t,\\tau\)A\_\{R\}\(t\),\\qquad P\(\\tau,\\tau\)=I\_\{d\}\.Standard linear ODE theory givesP​\(t,τ\)∈G​L​\(d\)P\(t,\\tau\)\\in GL\(d\)\. The reduction and invertibility details are collected in Appendix[B](https://arxiv.org/html/2607.03055#A2)\. We use the induced metric

MP​\(t,τ\):=P​\(t,τ\)​P​\(t,τ\)⊤\.M\_\{P\}\(t,\\tau\):=P\(t,\\tau\)P\(t,\\tau\)^\{\\top\}\.
SHiPPO is defined relative to such a chosen right\-generator path\. For each chosen path, the transportP​\(t,τ\)P\(t,\\tau\), the metricMP​\(t,τ\)M\_\{P\}\(t,\\tau\), and the transported approximation family are determined, and the inner online projection problem minimizes only over the coefficient matrixCC\. IfARA\_\{R\}is generated by a causal controller, the variational identities, normal equations, and coefficient dynamics below apply pathwise for the realized transport path\. Learning or parameterizingARA\_\{R\}is an outer modeling problem, not part of the inner projection optimization\.

### A\.2Ordinary vector\-valued HiPPO

For ordinary HiPPO at timett, the approximation family is

𝒢tH=\{τ↦C⊤​Φt​\(τ\):C∈ℝN×d\},\\mathcal\{G\}\_\{t\}^\{\\mathrm\{H\}\}=\\\{\\,\\tau\\mapsto C^\{\\top\}\\Phi\_\{t\}\(\\tau\):C\\in\\mathbb\{R\}^\{N\\times d\}\\,\\\},and the Euclidean\-channel objective is

JtH​\(C\):=∫0t‖f​\(τ\)−C⊤​Φt​\(τ\)‖22​𝑑μt​\(τ\)\.J\_\{t\}^\{\\mathrm\{H\}\}\(C\):=\\int\_\{0\}^\{t\}\\bigl\\\|f\(\\tau\)\-C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\\bigr\\\|\_\{2\}^\{2\}\\,d\\mu\_\{t\}\(\\tau\)\.
###### Proposition A\.1\(Ordinary HiPPO normal equation\)\.

Any minimizerCH​\(t\)C\_\{H\}\(t\)ofJtHJ\_\{t\}^\{\\mathrm\{H\}\}satisfies

G​\(t\)​CH​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​𝑑μt​\(τ\)\.G\(t\)C\_\{H\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\.IfG​\(t\)G\(t\)is invertible, then the minimizer is unique\.

###### Proof\.

Fixtt, and abbreviateΦ​\(τ\)=Φt​\(τ\)\\Phi\(\\tau\)=\\Phi\_\{t\}\(\\tau\)\. For a perturbationC↦C\+ε​ΔC\\mapsto C\+\\varepsilon\\Delta, with arbitraryΔ∈ℝN×d\\Delta\\in\\mathbb\{R\}^\{N\\times d\}, define the residual

e​\(τ;C\):=f​\(τ\)−C⊤​Φ​\(τ\)\.e\(\\tau;C\):=f\(\\tau\)\-C^\{\\top\}\\Phi\(\\tau\)\.Then

e​\(τ;C\+ε​Δ\)=e​\(τ;C\)−ε​Δ⊤​Φ​\(τ\)\.e\(\\tau;C\+\\varepsilon\\Delta\)=e\(\\tau;C\)\-\\varepsilon\\,\\Delta^\{\\top\}\\Phi\(\\tau\)\.Using

JtH​\(C\)=∫0te​\(τ;C\)⊤​e​\(τ;C\)​𝑑μt​\(τ\),J\_\{t\}^\{\\mathrm\{H\}\}\(C\)=\\int\_\{0\}^\{t\}e\(\\tau;C\)^\{\\top\}e\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\),we obtain

δ​JtH​\(C;Δ\)\\displaystyle\\delta J\_\{t\}^\{\\mathrm\{H\}\}\(C;\\Delta\)=2​∫0t\(δ​e​\(τ;C\)\)⊤​e​\(τ;C\)​𝑑μt​\(τ\)\\displaystyle=2\\int\_\{0\}^\{t\}\\bigl\(\\delta e\(\\tau;C\)\\bigr\)^\{\\top\}e\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\)=−2​∫0t\(Δ⊤​Φ​\(τ\)\)⊤​e​\(τ;C\)​𝑑μt​\(τ\)\.\\displaystyle=\-2\\int\_\{0\}^\{t\}\\bigl\(\\Delta^\{\\top\}\\Phi\(\\tau\)\\bigr\)^\{\\top\}e\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\)\.Since

\(Δ⊤​Φ\)⊤​e=tr​\(Δ⊤​Φ​e⊤\),\\bigl\(\\Delta^\{\\top\}\\Phi\\bigr\)^\{\\top\}e=\\mathrm\{tr\}\\\!\\left\(\\Delta^\{\\top\}\\Phi e^\{\\top\}\\right\),we can write

δ​JtH​\(C;Δ\)=−2​tr​\[Δ⊤​∫0tΦ​\(τ\)​e​\(τ;C\)⊤​𝑑μt​\(τ\)\]\.\\delta J\_\{t\}^\{\\mathrm\{H\}\}\(C;\\Delta\)=\-2\\,\\mathrm\{tr\}\\\!\\left\[\\Delta^\{\\top\}\\int\_\{0\}^\{t\}\\Phi\(\\tau\)e\(\\tau;C\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\\right\]\.Stationarity for allΔ\\Deltais equivalent to

∫0tΦ​\(τ\)​e​\(τ;C\)⊤​𝑑μt​\(τ\)=0\.\\int\_\{0\}^\{t\}\\Phi\(\\tau\)e\(\\tau;C\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)=0\.Expanding

e​\(τ;C\)⊤=f​\(τ\)⊤−Φ​\(τ\)⊤​Ce\(\\tau;C\)^\{\\top\}=f\(\\tau\)^\{\\top\}\-\\Phi\(\\tau\)^\{\\top\}Cgives

∫0tΦ​\(τ\)​f​\(τ\)⊤​𝑑μt​\(τ\)−\(∫0tΦ​\(τ\)​Φ​\(τ\)⊤​𝑑μt​\(τ\)\)​C=0\.\\int\_\{0\}^\{t\}\\Phi\(\\tau\)f\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\-\\left\(\\int\_\{0\}^\{t\}\\Phi\(\\tau\)\\Phi\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\\right\)C=0\.Thus

G​\(t\)​C=∫0tΦt​\(τ\)​f​\(τ\)⊤​𝑑μt​\(τ\)\.G\(t\)C=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\.Uniqueness follows becauseJtHJ\_\{t\}^\{\\mathrm\{H\}\}is a strictly convex quadratic inCCwheneverG​\(t\)G\(t\)is invertible\. ∎

###### Proposition A\.2\(Ordinary HiPPO coefficient dynamics under closure\)\.

Assumed​μt​\(τ\)=wt​\(τ\)​d​τd\\mu\_\{t\}\(\\tau\)=w\_\{t\}\(\\tau\)d\\tauand the closure condition above\. IfG​\(t\)=ING\(t\)=I\_\{N\}, then

CH​\(t\)=∫0tψ​\(t,τ\)​f​\(τ\)⊤​𝑑τ,C\_\{H\}\(t\)=\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}\\,d\\tau,and

C˙H​\(t\)=AL​\(t\)​CH​\(t\)\+BL​\(t\)​f​\(t\)⊤\.\\dot\{C\}\_\{H\}\(t\)=A\_\{L\}\(t\)C\_\{H\}\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\.

###### Proof\.

WithG​\(t\)=ING\(t\)=I\_\{N\}, Proposition[A\.1](https://arxiv.org/html/2607.03055#A1.Thmassumption1)gives

CH​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​𝑑μt​\(τ\)=∫0tψ​\(t,τ\)​f​\(τ\)⊤​𝑑τ\.C\_\{H\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)=\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}\\,d\\tau\.Differentiate using Leibniz’ rule:

C˙H​\(t\)=ψ​\(t,t\)​f​\(t\)⊤\+∫0t∂tψ​\(t,τ\)​f​\(τ\)⊤​d​τ\.\\dot\{C\}\_\{H\}\(t\)=\\psi\(t,t\)f\(t\)^\{\\top\}\+\\int\_\{0\}^\{t\}\\partial\_\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}\\,d\\tau\.Substituting

∂tψ​\(t,τ\)=AL​\(t\)​ψ​\(t,τ\)\(τ<t\),BL​\(t\)=ψ​\(t,t\),\\partial\_\{t\}\\psi\(t,\\tau\)=A\_\{L\}\(t\)\\psi\(t,\\tau\)\\quad\(\\tau<t\),\\qquad B\_\{L\}\(t\)=\\psi\(t,t\),yields

C˙H​\(t\)=BL​\(t\)​f​\(t\)⊤\+AL​\(t\)​∫0tψ​\(t,τ\)​f​\(τ\)⊤​𝑑τ=AL​\(t\)​CH​\(t\)\+BL​\(t\)​f​\(t\)⊤\.\\dot\{C\}\_\{H\}\(t\)=B\_\{L\}\(t\)f\(t\)^\{\\top\}\+A\_\{L\}\(t\)\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}\\,d\\tau=A\_\{L\}\(t\)C\_\{H\}\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\.∎

#### Non\-orthonormal bases\.

IfG​\(t\)≠ING\(t\)\\neq I\_\{N\}, then

CH​\(t\)=G​\(t\)−1​b​\(t\),b​\(t\):=∫0tΦt​\(τ\)​f​\(τ\)⊤​𝑑μt​\(τ\)\.C\_\{H\}\(t\)=G\(t\)^\{\-1\}b\(t\),\\qquad b\(t\):=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\.Differentiating gives

C˙H​\(t\)=G​\(t\)−1​b˙​\(t\)−G​\(t\)−1​G˙​\(t\)​CH​\(t\)\.\\dot\{C\}\_\{H\}\(t\)=G\(t\)^\{\-1\}\\dot\{b\}\(t\)\-G\(t\)^\{\-1\}\\dot\{G\}\(t\)C\_\{H\}\(t\)\.The main text focuses on the orthonormalized caseG​\(t\)=ING\(t\)=I\_\{N\}, where the coefficient dynamics take the clean HiPPO form\.

### A\.3Direct stationarity of the SHiPPO online approximation problem

For fixedtt, the SHiPPO approximation family is

𝒢tSH=\{τ↦P​\(t,τ\)−⊤​C⊤​Φt​\(τ\):C∈ℝN×d\}\.\\mathcal\{G\}\_\{t\}^\{\\mathrm\{SH\}\}=\\\{\\,\\tau\\mapsto P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\):C\\in\\mathbb\{R\}^\{N\\times d\}\\,\\\}\.Equivalently, the SHiPPO coefficient matrix is the minimizer of

JtSH​\(C\):=∫0teS​\(τ;C\)⊤​MP​\(t,τ\)​eS​\(τ;C\)​𝑑μt​\(τ\),J\_\{t\}^\{\\mathrm\{SH\}\}\(C\):=\\int\_\{0\}^\{t\}e\_\{S\}\(\\tau;C\)^\{\\top\}M\_\{P\}\(t,\\tau\)e\_\{S\}\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\),where

eS​\(τ;C\):=f​\(τ\)−P​\(t,τ\)−⊤​C⊤​Φt​\(τ\)\.e\_\{S\}\(\\tau;C\):=f\(\\tau\)\-P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\.
###### Proposition A\.3\(SHiPPO normal equation from stationarity\)\.

Any minimizerCS​\(t\)C\_\{S\}\(t\)ofJtSHJ\_\{t\}^\{\\mathrm\{SH\}\}satisfies

G​\(t\)​CS​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑μt​\(τ\)\.G\(t\)C\_\{S\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.IfG​\(t\)G\(t\)is invertible, then the minimizer is unique\.

###### Proof\.

Fixtt, abbreviate

P​\(τ\):=P​\(t,τ\),MP​\(τ\):=MP​\(t,τ\),Φ​\(τ\):=Φt​\(τ\)\.P\(\\tau\):=P\(t,\\tau\),\\qquad M\_\{P\}\(\\tau\):=M\_\{P\}\(t,\\tau\),\\qquad\\Phi\(\\tau\):=\\Phi\_\{t\}\(\\tau\)\.For a perturbationC↦C\+ε​ΔC\\mapsto C\+\\varepsilon\\Delta,

δ​eS​\(τ;C\)=−P​\(τ\)−⊤​Δ⊤​Φ​\(τ\)\.\\delta e\_\{S\}\(\\tau;C\)=\-P\(\\tau\)^\{\-\\top\}\\Delta^\{\\top\}\\Phi\(\\tau\)\.Therefore

δ​JtSH​\(C;Δ\)\\displaystyle\\delta J\_\{t\}^\{\\mathrm\{SH\}\}\(C;\\Delta\)=2​∫0t\(δ​eS​\(τ;C\)\)⊤​MP​\(τ\)​eS​\(τ;C\)​𝑑μt​\(τ\)\\displaystyle=2\\int\_\{0\}^\{t\}\\bigl\(\\delta e\_\{S\}\(\\tau;C\)\\bigr\)^\{\\top\}M\_\{P\}\(\\tau\)e\_\{S\}\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\)=−2​∫0t\(Δ⊤​Φ​\(τ\)\)⊤​P​\(τ\)−1​MP​\(τ\)​eS​\(τ;C\)​𝑑μt​\(τ\)\.\\displaystyle=\-2\\int\_\{0\}^\{t\}\\bigl\(\\Delta^\{\\top\}\\Phi\(\\tau\)\\bigr\)^\{\\top\}P\(\\tau\)^\{\-1\}M\_\{P\}\(\\tau\)e\_\{S\}\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\)\.Since

P​\(τ\)−1​MP​\(τ\)=P​\(τ\)−1​P​\(τ\)​P​\(τ\)⊤=P​\(τ\)⊤,P\(\\tau\)^\{\-1\}M\_\{P\}\(\\tau\)=P\(\\tau\)^\{\-1\}P\(\\tau\)P\(\\tau\)^\{\\top\}=P\(\\tau\)^\{\\top\},we get

δ​JtSH​\(C;Δ\)=−2​∫0t\(Δ⊤​Φ​\(τ\)\)⊤​P​\(τ\)⊤​eS​\(τ;C\)​𝑑μt​\(τ\)\.\\delta J\_\{t\}^\{\\mathrm\{SH\}\}\(C;\\Delta\)=\-2\\int\_\{0\}^\{t\}\\bigl\(\\Delta^\{\\top\}\\Phi\(\\tau\)\\bigr\)^\{\\top\}P\(\\tau\)^\{\\top\}e\_\{S\}\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\)\.Using the trace identity

\(Δ⊤​Φ\)⊤​P⊤​eS=tr​\(Δ⊤​Φ​eS⊤​P\),\\bigl\(\\Delta^\{\\top\}\\Phi\\bigr\)^\{\\top\}P^\{\\top\}e\_\{S\}=\\mathrm\{tr\}\\\!\\left\(\\Delta^\{\\top\}\\Phi e\_\{S\}^\{\\top\}P\\right\),this becomes

δ​JtSH​\(C;Δ\)=−2​tr​\[Δ⊤​∫0tΦ​\(τ\)​eS​\(τ;C\)⊤​P​\(τ\)​𝑑μt​\(τ\)\]\.\\delta J\_\{t\}^\{\\mathrm\{SH\}\}\(C;\\Delta\)=\-2\\,\\mathrm\{tr\}\\\!\\left\[\\Delta^\{\\top\}\\int\_\{0\}^\{t\}\\Phi\(\\tau\)e\_\{S\}\(\\tau;C\)^\{\\top\}P\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\\right\]\.Stationarity for allΔ\\Deltais equivalent to

∫0tΦ​\(τ\)​eS​\(τ;C\)⊤​P​\(τ\)​𝑑μt​\(τ\)=0\.\\int\_\{0\}^\{t\}\\Phi\(\\tau\)e\_\{S\}\(\\tau;C\)^\{\\top\}P\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\)=0\.Now

eS​\(τ;C\)⊤​P​\(τ\)=f​\(τ\)⊤​P​\(τ\)−Φ​\(τ\)⊤​C,e\_\{S\}\(\\tau;C\)^\{\\top\}P\(\\tau\)=f\(\\tau\)^\{\\top\}P\(\\tau\)\-\\Phi\(\\tau\)^\{\\top\}C,because

\(P​\(τ\)−⊤​C⊤​Φ​\(τ\)\)⊤​P​\(τ\)=Φ​\(τ\)⊤​C​P​\(τ\)−1​P​\(τ\)=Φ​\(τ\)⊤​C\.\\bigl\(P\(\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\(\\tau\)\\bigr\)^\{\\top\}P\(\\tau\)=\\Phi\(\\tau\)^\{\\top\}CP\(\\tau\)^\{\-1\}P\(\\tau\)=\\Phi\(\\tau\)^\{\\top\}C\.Thus stationarity gives

∫0tΦ​\(τ\)​f​\(τ\)⊤​P​\(τ\)​𝑑μt​\(τ\)−\(∫0tΦ​\(τ\)​Φ​\(τ\)⊤​𝑑μt​\(τ\)\)​C=0,\\int\_\{0\}^\{t\}\\Phi\(\\tau\)f\(\\tau\)^\{\\top\}P\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\-\\left\(\\int\_\{0\}^\{t\}\\Phi\(\\tau\)\\Phi\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\\right\)C=0,or equivalently

G​\(t\)​C=∫0tΦt​\(τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑μt​\(τ\)\.G\(t\)C=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.This proves the displayed normal equation forC=CS​\(t\)C=C\_\{S\}\(t\)\.

Uniqueness follows from strict convexity\. Indeed, the mapC↦P​\(t,τ\)−⊤​C⊤​Φt​\(τ\)C\\mapsto P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\)is injective moduloμt\\mu\_\{t\}\-null sets whenG​\(t\)G\(t\)is invertible andP​\(t,τ\)∈G​L​\(d\)P\(t,\\tau\)\\in GL\(d\), and the metricMP​\(t,τ\)M\_\{P\}\(t,\\tau\)is positive definite for eachτ\\tau\. ∎

### A\.4Transport isometry and SHiPPO–HiPPO conjugacy

The direct stationarity calculation above gives the SHiPPO normal equation\. We now record the equivalent conjugacy view, which explains why the coupled transport of the metric and approximation family preserves the HiPPO coefficient structure\.

For fixedtt, define the transport operator

\(𝒯t​u\)​\(τ\):=P​\(t,τ\)⊤​u​\(τ\),\(𝒯t−1​u\)​\(τ\):=P​\(t,τ\)−⊤​u​\(τ\)\.\(\\mathcal\{T\}\_\{t\}u\)\(\\tau\):=P\(t,\\tau\)^\{\\top\}u\(\\tau\),\\qquad\(\\mathcal\{T\}\_\{t\}^\{\-1\}u\)\(\\tau\):=P\(t,\\tau\)^\{\-\\top\}u\(\\tau\)\.Also define the metric inner product

⟨u,v⟩t,MP:=∫0tu​\(τ\)⊤​MP​\(t,τ\)​v​\(τ\)​𝑑μt​\(τ\),\\langle u,v\\rangle\_\{t,M\_\{P\}\}:=\\int\_\{0\}^\{t\}u\(\\tau\)^\{\\top\}M\_\{P\}\(t,\\tau\)v\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\),and the Euclidean\-channel inner product

⟨u,v⟩t,I:=∫0tu​\(τ\)⊤​v​\(τ\)​𝑑μt​\(τ\)\.\\langle u,v\\rangle\_\{t,I\}:=\\int\_\{0\}^\{t\}u\(\\tau\)^\{\\top\}v\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.
###### Proposition A\.4\(Transport isometry and SHiPPO–HiPPO conjugacy\)\.

For anyu,v:\[0,t\]→ℝdu,v:\[0,t\]\\to\\mathbb\{R\}^\{d\},

⟨u,v⟩t,MP=⟨𝒯t​u,𝒯t​v⟩t,I\.\\langle u,v\\rangle\_\{t,M\_\{P\}\}=\\langle\\mathcal\{T\}\_\{t\}u,\\mathcal\{T\}\_\{t\}v\\rangle\_\{t,I\}\.Moreover,

𝒯t​\(P​\(t,⋅\)−⊤​C⊤​Φt​\(⋅\)\)=C⊤​Φt​\(⋅\),\\mathcal\{T\}\_\{t\}\\Bigl\(P\(t,\\cdot\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\cdot\)\\Bigr\)=C^\{\\top\}\\Phi\_\{t\}\(\\cdot\),so𝒯t\\mathcal\{T\}\_\{t\}maps𝒢tSH\\mathcal\{G\}\_\{t\}^\{\\mathrm\{SH\}\}bijectively onto𝒢tH\\mathcal\{G\}\_\{t\}^\{\\mathrm\{H\}\}\. Consequently,

shippot⁡\(f\)=hippot⁡\(𝒯t​f\)\.\\operatorname\{shippo\}\_\{t\}\(f\)=\\operatorname\{hippo\}\_\{t\}\(\\mathcal\{T\}\_\{t\}f\)\.

###### Proof\.

For the isometry, compute pointwise:

u​\(τ\)⊤​MP​\(t,τ\)​v​\(τ\)=u​\(τ\)⊤​P​\(t,τ\)​P​\(t,τ\)⊤​v​\(τ\)=\(𝒯t​u\)​\(τ\)⊤​\(𝒯t​v\)​\(τ\)\.u\(\\tau\)^\{\\top\}M\_\{P\}\(t,\\tau\)v\(\\tau\)=u\(\\tau\)^\{\\top\}P\(t,\\tau\)P\(t,\\tau\)^\{\\top\}v\(\\tau\)=\(\\mathcal\{T\}\_\{t\}u\)\(\\tau\)^\{\\top\}\(\\mathcal\{T\}\_\{t\}v\)\(\\tau\)\.Integrating over\[0,t\]\[0,t\]gives

⟨u,v⟩t,MP=⟨𝒯t​u,𝒯t​v⟩t,I\.\\langle u,v\\rangle\_\{t,M\_\{P\}\}=\\langle\\mathcal\{T\}\_\{t\}u,\\mathcal\{T\}\_\{t\}v\\rangle\_\{t,I\}\.
For the mapping of approximation families,

𝒯t​\(P​\(t,τ\)−⊤​C⊤​Φt​\(τ\)\)=P​\(t,τ\)⊤​P​\(t,τ\)−⊤​C⊤​Φt​\(τ\)=C⊤​Φt​\(τ\)\.\\mathcal\{T\}\_\{t\}\\left\(P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\\right\)=P\(t,\\tau\)^\{\\top\}P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\)=C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\.Thus

𝒯t​\(𝒢tSH\)=𝒢tH\.\\mathcal\{T\}\_\{t\}\(\\mathcal\{G\}\_\{t\}^\{\\mathrm\{SH\}\}\)=\\mathcal\{G\}\_\{t\}^\{\\mathrm\{H\}\}\.SinceP​\(t,τ\)∈G​L​\(d\)P\(t,\\tau\)\\in GL\(d\), the inverse map is𝒯t−1\\mathcal\{T\}\_\{t\}^\{\-1\}, so this correspondence is bijective\.

Finally, for anyg∈𝒢tSHg\\in\\mathcal\{G\}\_\{t\}^\{\\mathrm\{SH\}\},

‖f−g‖t,MP2=‖𝒯t​f−𝒯t​g‖t,I2\.\\\|f\-g\\\|\_\{t,M\_\{P\}\}^\{2\}=\\\|\\mathcal\{T\}\_\{t\}f\-\\mathcal\{T\}\_\{t\}g\\\|\_\{t,I\}^\{2\}\.Minimizing overg∈𝒢tSHg\\in\\mathcal\{G\}\_\{t\}^\{\\mathrm\{SH\}\}is therefore equivalent to minimizing over

g~:=𝒯t​g∈𝒢tH\.\\tilde\{g\}:=\\mathcal\{T\}\_\{t\}g\\in\\mathcal\{G\}\_\{t\}^\{\\mathrm\{H\}\}\.This is precisely the ordinary HiPPO projection of𝒯t​f\\mathcal\{T\}\_\{t\}f\. The coefficient matrix is unchanged under the map between the two approximation families, giving

shippot⁡\(f\)=hippot⁡\(𝒯t​f\)\.\\operatorname\{shippo\}\_\{t\}\(f\)=\\operatorname\{hippo\}\_\{t\}\(\\mathcal\{T\}\_\{t\}f\)\.∎

#### Relation to the normal equation\.

Proposition[A\.4](https://arxiv.org/html/2607.03055#A1.Thmassumption4)gives another way to obtain Proposition[A\.3](https://arxiv.org/html/2607.03055#A1.Thmassumption3): apply the ordinary HiPPO normal equation to the transformed signal

\(𝒯t​f\)​\(τ\)=P​\(t,τ\)⊤​f​\(τ\)\.\(\\mathcal\{T\}\_\{t\}f\)\(\\tau\)=P\(t,\\tau\)^\{\\top\}f\(\\tau\)\.More importantly, the conjugacy shows that SHiPPO is ordinary HiPPO in a transported channel frame\. This is the operator\-level reason that Definition[2\.1](https://arxiv.org/html/2607.03055#S2.Thmassumption1)transports the metric and the approximation family together, rather than modifying the channel metric alone\.

### A\.5Proof of Theorem[2\.2](https://arxiv.org/html/2607.03055#S2.Thmassumption2)and Corollary[2\.3](https://arxiv.org/html/2607.03055#S2.Thmassumption3)

###### Proof\.

By Proposition[A\.3](https://arxiv.org/html/2607.03055#A1.Thmassumption3), the SHiPPO minimizer satisfies

G​\(t\)​CS​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑μt​\(τ\)\.G\(t\)C\_\{S\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.This is the first displayed equation in Theorem[2\.2](https://arxiv.org/html/2607.03055#S2.Thmassumption2)\.

Now assumeG​\(t\)=ING\(t\)=I\_\{N\}andd​μt​\(τ\)=wt​\(τ\)​d​τd\\mu\_\{t\}\(\\tau\)=w\_\{t\}\(\\tau\)d\\tau, so thatψ​\(t,τ\)=wt​\(τ\)​Φt​\(τ\)\\psi\(t,\\tau\)=w\_\{t\}\(\\tau\)\\Phi\_\{t\}\(\\tau\)\. Then the normal equation becomes

CS​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑μt​\(τ\)=∫0tψ​\(t,τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑τ\.C\_\{S\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)=\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\tau\.Differentiating this representation using Leibniz’ rule gives

C˙S​\(t\)\\displaystyle\\dot\{C\}\_\{S\}\(t\)=ψ​\(t,t\)​f​\(t\)⊤​P​\(t,t\)\\displaystyle=\\psi\(t,t\)f\(t\)^\{\\top\}P\(t,t\)\+∫0t∂tψ​\(t,τ\)​f​\(τ\)⊤​P​\(t,τ\)​d​τ\\displaystyle\\quad\+\\int\_\{0\}^\{t\}\\partial\_\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\tau\+∫0tψ​\(t,τ\)​f​\(τ\)⊤​∂tP​\(t,τ\)​d​τ\.\\displaystyle\\quad\+\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}\\partial\_\{t\}P\(t,\\tau\)\\,d\\tau\.Using

P​\(t,t\)=Id,ψ​\(t,t\)=BL​\(t\),P\(t,t\)=I\_\{d\},\\qquad\\psi\(t,t\)=B\_\{L\}\(t\),the boundary term is

BL​\(t\)​f​\(t\)⊤\.B\_\{L\}\(t\)f\(t\)^\{\\top\}\.Using the closure condition

∂tψ​\(t,τ\)=AL​\(t\)​ψ​\(t,τ\),\\partial\_\{t\}\\psi\(t,\\tau\)=A\_\{L\}\(t\)\\psi\(t,\\tau\),the second term is

AL​\(t\)​∫0tψ​\(t,τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑τ=AL​\(t\)​CS​\(t\)\.A\_\{L\}\(t\)\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\tau=A\_\{L\}\(t\)C\_\{S\}\(t\)\.Using the transport equation

∂tP​\(t,τ\)=P​\(t,τ\)​AR​\(t\),\\partial\_\{t\}P\(t,\\tau\)=P\(t,\\tau\)A\_\{R\}\(t\),the third term is

∫0tψ​\(t,τ\)​f​\(τ\)⊤​P​\(t,τ\)​AR​\(t\)​𝑑τ=CS​\(t\)​AR​\(t\),\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)A\_\{R\}\(t\)\\,d\\tau=C\_\{S\}\(t\)A\_\{R\}\(t\),sinceAR​\(t\)A\_\{R\}\(t\)does not depend onτ\\tauinside the integral\. Therefore

C˙S​\(t\)=AL​\(t\)​CS​\(t\)\+BL​\(t\)​f​\(t\)⊤\+CS​\(t\)​AR​\(t\),\\dot\{C\}\_\{S\}\(t\)=A\_\{L\}\(t\)C\_\{S\}\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\+C\_\{S\}\(t\)A\_\{R\}\(t\),which proves the Sylvester coefficient dynamics\. ∎

#### Proof of Corollary[2\.3](https://arxiv.org/html/2607.03055#S2.Thmassumption3)\.

The corollary is not a statement about arbitrary matrix ODEs\. It applies to closed one\-sided coefficient equations obtained from the online projection setup above\. Suppose that, under the HiPPO closure condition and the orthonormalized normal equation, the ordinary coefficient trajectory satisfies

C˙​\(t\)=AL​\(t\)​C​\(t\)\+BL​\(t\)​f​\(t\)⊤\.\\dot\{C\}\(t\)=A\_\{L\}\(t\)C\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\.For any admissible right\-transport pathARA\_\{R\}, Definition[2\.1](https://arxiv.org/html/2607.03055#S2.Thmassumption1)uses the same left approximation family and source injection, while replacing the signal by its transported pathwise history in the normal equation\. Hence

CS​\(t\)=∫0tψ​\(t,τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑τ\.C\_\{S\}\(t\)=\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\tau\.Differentiating this representation at differentiability times gives the same left\-memory contribution and boundary source as in the one\-sided equation, while the right transport contributes

∫0tψ​\(t,τ\)​f​\(τ\)⊤​P​\(t,τ\)​AR​\(t\)​𝑑τ=CS​\(t\)​AR​\(t\)\.\\int\_\{0\}^\{t\}\\psi\(t,\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)A\_\{R\}\(t\)\\,d\\tau=C\_\{S\}\(t\)A\_\{R\}\(t\)\.Therefore

C˙S​\(t\)=AL​\(t\)​CS​\(t\)\+BL​\(t\)​f​\(t\)⊤\+CS​\(t\)​AR​\(t\)\.\\dot\{C\}\_\{S\}\(t\)=A\_\{L\}\(t\)C\_\{S\}\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\+C\_\{S\}\(t\)A\_\{R\}\(t\)\.Thus the transported lift preserves the left online\-memory operator inherited from the projection problem and adds the chosen right\-action transport as an external modeling path\. Architectural restrictions onARA\_\{R\}, such as group\-local or controller\-compatible forms, enter only in the scan\-compatible realization of Section[3](https://arxiv.org/html/2607.03055#S3)\.

#### Non\-orthonormal bases\.

IfG​\(t\)≠ING\(t\)\\neq I\_\{N\}, define

bS\(t\):=∫0tΦt\(τ\)f\(τ\)⊤P\(t,τ\)dμt\(τ\)\.b\_\{S\}\(t\):=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.Then

CS​\(t\)=G​\(t\)−1​bS​\(t\),C\_\{S\}\(t\)=G\(t\)^\{\-1\}b\_\{S\}\(t\),and differentiating gives

C˙S​\(t\)=G​\(t\)−1​b˙S​\(t\)−G​\(t\)−1​G˙​\(t\)​CS​\(t\)\.\\dot\{C\}\_\{S\}\(t\)=G\(t\)^\{\-1\}\\dot\{b\}\_\{S\}\(t\)\-G\(t\)^\{\-1\}\\dot\{G\}\(t\)C\_\{S\}\(t\)\.Thus the clean Sylvester form in the main text is the orthonormalizedG​\(t\)=ING\(t\)=I\_\{N\}case, matching the usual HiPPO presentation\.

### A\.6Metric\-only modifications do not generally close

This subsection justifies the coupled transport used in Definition[2\.1](https://arxiv.org/html/2607.03055#S2.Thmassumption1)by analyzing a nearby but different construction in which only the channel metric is changed\. This construction is not part of SHiPPO itself and, in general, does not preserve a finite HiPPO\-style coefficient closure\.

Keep the ordinary HiPPO family

𝒢tH=\{τ↦C⊤​Φt​\(τ\):C∈ℝN×d\},\\mathcal\{G\}\_\{t\}^\{\\mathrm\{H\}\}=\\\{\\,\\tau\\mapsto C^\{\\top\}\\Phi\_\{t\}\(\\tau\):C\\in\\mathbb\{R\}^\{N\\times d\}\\,\\\},and replace the Euclidean channel metric by a possiblytt\- andτ\\tau\-dependent SPD matrix

M​\(t,τ\)∈𝕊\+\+d\.M\(t,\\tau\)\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}\.Define the metric\-only objective

JtM​\(C\):=∫0t\(f​\(τ\)−C⊤​Φt​\(τ\)\)⊤​M​\(t,τ\)​\(f​\(τ\)−C⊤​Φt​\(τ\)\)​𝑑μt​\(τ\)\.J\_\{t\}^\{M\}\(C\):=\\int\_\{0\}^\{t\}\\bigl\(f\(\\tau\)\-C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\\bigr\)^\{\\top\}M\(t,\\tau\)\\bigl\(f\(\\tau\)\-C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\\bigr\)\\,d\\mu\_\{t\}\(\\tau\)\.
###### Proposition A\.5\(Stationary condition for the metric\-only extension\)\.

Any minimizerCM​\(t\)C\_\{M\}\(t\)ofJtMJ\_\{t\}^\{M\}satisfies

∫0tΦt​\(τ\)​\(M​\(t,τ\)​\(f​\(τ\)−CM​\(t\)⊤​Φt​\(τ\)\)\)⊤​𝑑μt​\(τ\)=0\.\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)\\left\(M\(t,\\tau\)\\bigl\(f\(\\tau\)\-C\_\{M\}\(t\)^\{\\top\}\\Phi\_\{t\}\(\\tau\)\\bigr\)\\right\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)=0\.Equivalently,

∫0tΦt​\(τ\)​f​\(τ\)⊤​M​\(t,τ\)​𝑑μt​\(τ\)=∫0tΦt​\(τ\)​Φt​\(τ\)⊤​CM​\(t\)​M​\(t,τ\)​𝑑μt​\(τ\)\.\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}M\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)\\Phi\_\{t\}\(\\tau\)^\{\\top\}C\_\{M\}\(t\)M\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.

###### Proof\.

Fixtt, abbreviate

Φ​\(τ\)=Φt​\(τ\),M​\(τ\)=M​\(t,τ\),\\Phi\(\\tau\)=\\Phi\_\{t\}\(\\tau\),\\qquad M\(\\tau\)=M\(t,\\tau\),and define

e​\(τ;C\):=f​\(τ\)−C⊤​Φ​\(τ\)\.e\(\\tau;C\):=f\(\\tau\)\-C^\{\\top\}\\Phi\(\\tau\)\.ForC↦C\+ε​ΔC\\mapsto C\+\\varepsilon\\Delta,

e​\(τ;C\+ε​Δ\)=e​\(τ;C\)−ε​Δ⊤​Φ​\(τ\)\.e\(\\tau;C\+\\varepsilon\\Delta\)=e\(\\tau;C\)\-\\varepsilon\\,\\Delta^\{\\top\}\\Phi\(\\tau\)\.SinceM​\(τ\)M\(\\tau\)is symmetric,

δ​JtM​\(C;Δ\)\\displaystyle\\delta J\_\{t\}^\{M\}\(C;\\Delta\)=2​∫0t\(δ​e​\(τ;C\)\)⊤​M​\(τ\)​e​\(τ;C\)​𝑑μt​\(τ\)\\displaystyle=2\\int\_\{0\}^\{t\}\\bigl\(\\delta e\(\\tau;C\)\\bigr\)^\{\\top\}M\(\\tau\)e\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\)=−2​∫0t\(Δ⊤​Φ​\(τ\)\)⊤​M​\(τ\)​e​\(τ;C\)​𝑑μt​\(τ\)\.\\displaystyle=\-2\\int\_\{0\}^\{t\}\\bigl\(\\Delta^\{\\top\}\\Phi\(\\tau\)\\bigr\)^\{\\top\}M\(\\tau\)e\(\\tau;C\)\\,d\\mu\_\{t\}\(\\tau\)\.Using

\(Δ⊤​Φ\)⊤​M​e=tr​\(Δ⊤​Φ​\(M​e\)⊤\),\\bigl\(\\Delta^\{\\top\}\\Phi\\bigr\)^\{\\top\}Me=\\mathrm\{tr\}\\\!\\left\(\\Delta^\{\\top\}\\Phi\(Me\)^\{\\top\}\\right\),we get

δ​JtM​\(C;Δ\)=−2​tr​\[Δ⊤​∫0tΦ​\(τ\)​\(M​\(τ\)​e​\(τ;C\)\)⊤​𝑑μt​\(τ\)\]\.\\delta J\_\{t\}^\{M\}\(C;\\Delta\)=\-2\\,\\mathrm\{tr\}\\\!\\left\[\\Delta^\{\\top\}\\int\_\{0\}^\{t\}\\Phi\(\\tau\)\\bigl\(M\(\\tau\)e\(\\tau;C\)\\bigr\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\\right\]\.Stationarity for allΔ\\Deltayields

∫0tΦ​\(τ\)​\(M​\(τ\)​e​\(τ;C\)\)⊤​𝑑μt​\(τ\)=0\.\\int\_\{0\}^\{t\}\\Phi\(\\tau\)\\bigl\(M\(\\tau\)e\(\\tau;C\)\\bigr\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)=0\.Expandinge​\(τ;C\)=f​\(τ\)−C⊤​Φ​\(τ\)e\(\\tau;C\)=f\(\\tau\)\-C^\{\\top\}\\Phi\(\\tau\)gives

∫0tΦ​\(τ\)​f​\(τ\)⊤​M​\(τ\)​𝑑μt​\(τ\)=∫0tΦ​\(τ\)​Φ​\(τ\)⊤​C​M​\(τ\)​𝑑μt​\(τ\)\.\\int\_\{0\}^\{t\}\\Phi\(\\tau\)f\(\\tau\)^\{\\top\}M\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\)=\\int\_\{0\}^\{t\}\\Phi\(\\tau\)\\Phi\(\\tau\)^\{\\top\}CM\(\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.SubstitutingC=CM​\(t\)C=C\_\{M\}\(t\)proves the claim\. ∎

It is convenient to write this stationary equation as a linear operator equation\. Define

𝒦M,t​\[C\]:=∫0tΦt​\(τ\)​Φt​\(τ\)⊤​C​M​\(t,τ\)​𝑑μt​\(τ\),\\mathcal\{K\}\_\{M,t\}\[C\]:=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)\\Phi\_\{t\}\(\\tau\)^\{\\top\}CM\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\),and

bM,t:=∫0tΦt​\(τ\)​f​\(τ\)⊤​M​\(t,τ\)​𝑑μt​\(τ\)\.b\_\{M,t\}:=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}M\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\)\.Then

𝒦M,t​\[CM​\(t\)\]=bM,t\.\\mathcal\{K\}\_\{M,t\}\[C\_\{M\}\(t\)\]=b\_\{M,t\}\.
###### Proposition A\.6\(τ\\tau\-independent metrics reduce to ordinary HiPPO\)\.

If

M​\(t,τ\)=M¯​\(t\)∈𝕊\+\+dM\(t,\\tau\)=\\overline\{M\}\(t\)\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}is independent ofτ\\tau, then the metric\-only stationary equation reduces to

G​\(t\)​CM​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​𝑑μt​\(τ\),G\(t\)C\_\{M\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\),the same normal equation as ordinary vector\-valued HiPPO\.

###### Proof\.

WhenM​\(t,τ\)=M¯​\(t\)M\(t,\\tau\)=\\overline\{M\}\(t\), the operator equation becomes

\(∫0tΦt​\(τ\)​Φt​\(τ\)⊤​𝑑μt​\(τ\)\)​CM​\(t\)​M¯​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​M¯​\(t\)​𝑑μt​\(τ\)\.\\left\(\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)\\Phi\_\{t\}\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\\right\)C\_\{M\}\(t\)\\overline\{M\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}\\overline\{M\}\(t\)\\,d\\mu\_\{t\}\(\\tau\)\.SinceM¯​\(t\)\\overline\{M\}\(t\)is invertible, right\-multiplying byM¯​\(t\)−1\\overline\{M\}\(t\)^\{\-1\}gives

G​\(t\)​CM​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​𝑑μt​\(τ\)\.G\(t\)C\_\{M\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\.∎

#### Whyτ\\tau\-dependent metrics are obstructive\.

WhenM​\(t,τ\)M\(t,\\tau\)genuinely depends onτ\\tau, the operator𝒦M,t\\mathcal\{K\}\_\{M,t\}does not, in general, factor as

C↦G​\(t\)​C​M¯​\(t\)C\\mapsto G\(t\)C\\overline\{M\}\(t\)for some matrixM¯​\(t\)\\overline\{M\}\(t\)\. Consequently,CM​\(t\)C\_\{M\}\(t\)is not generally given by a simple HiPPO\-style normal equation\. Recovering an online closed ODE forCM​\(t\)C\_\{M\}\(t\)typically requires either solving a time\-varying linear operator equation at eachtt, or augmenting the state with additional metric\-dependent moments\.

SHiPPO avoids this generic obstruction by coupling the metric with the approximation family\. In the transported construction,

MP​\(t,τ\)=P​\(t,τ\)​P​\(t,τ\)⊤M\_\{P\}\(t,\\tau\)=P\(t,\\tau\)P\(t,\\tau\)^\{\\top\}is paired with the transported family

τ↦P​\(t,τ\)−⊤​C⊤​Φt​\(τ\)\.\\tau\\mapsto P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\.This coupling is exactly what produces the stationarity identity

G​\(t\)​CS​\(t\)=∫0tΦt​\(τ\)​f​\(τ\)⊤​P​\(t,τ\)​𝑑μt​\(τ\),G\(t\)C\_\{S\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)f\(\\tau\)^\{\\top\}P\(t,\\tau\)\\,d\\mu\_\{t\}\(\\tau\),and hence the Sylvester coefficient dynamics of Theorem[2\.2](https://arxiv.org/html/2607.03055#S2.Thmassumption2)\.

## Appendix BReductions, Identity Metrics, and Moving Frames

This appendix supports the reduction, identity\-metric, and moving\-frame claims used at the end of Section[2](https://arxiv.org/html/2607.03055#S2)\. Fix an admissible right\-generator pathAR∈L1​\(\[0,T\];ℝd×d\)A\_\{R\}\\in L^\{1\}\(\[0,T\];\\mathbb\{R\}^\{d\\times d\}\)\. The results below are exact for such a chosen path, and apply pathwise whenARA\_\{R\}is generated by a causal controller\. This appendix is not about optimizing overARA\_\{R\}, but about the algebraic consequences of the transported approximation problem once a transport path has been specified\. SinceARA\_\{R\}is only assumed integrable, differential identities involvingAR​\(t\)A\_\{R\}\(t\)hold almost everywhere; statements such asAR≡0A\_\{R\}\\equiv 0orAR\+AR⊤≡0A\_\{R\}\+A\_\{R\}^\{\\top\}\\equiv 0are interpreted in this a\.e\. sense unless additional continuity is assumed\.

### B\.1Transport basics

Throughout this appendix, assume

AR:\[0,T\]→ℝd×dA\_\{R\}:\[0,T\]\\to\\mathbb\{R\}^\{d\\times d\}is measurable and integrable\. For each fixedτ∈\[0,T\]\\tau\\in\[0,T\], defineP​\(⋅,τ\)P\(\\cdot,\\tau\)as the unique absolutely continuous solution of

P​\(t,τ\)=Id\+∫τtP​\(s,τ\)​AR​\(s\)​𝑑s,0≤τ≤t≤T\.P\(t,\\tau\)=I\_\{d\}\+\\int\_\{\\tau\}^\{t\}P\(s,\\tau\)A\_\{R\}\(s\)\\,ds,\\qquad 0\\leq\\tau\\leq t\\leq T\.\(8\)Equivalently,

∂tP​\(t,τ\)=P​\(t,τ\)​AR​\(t\)for a\.e\.​t∈\[τ,T\],P​\(τ,τ\)=Id\.\\partial\_\{t\}P\(t,\\tau\)=P\(t,\\tau\)A\_\{R\}\(t\)\\quad\\text\{for a\.e\. \}t\\in\[\\tau,T\],\\qquad P\(\\tau,\\tau\)=I\_\{d\}\.\(9\)
###### Lemma B\.1\(Invertibility and inverse dynamics\)\.

For every0≤τ≤t≤T0\\leq\\tau\\leq t\\leq T,P​\(t,τ\)∈G​L​\(d\)P\(t,\\tau\)\\in GL\(d\)\. Moreover,

∂tP​\(t,τ\)−1=−AR​\(t\)​P​\(t,τ\)−1for a\.e\.​t∈\[τ,T\],P​\(τ,τ\)−1=Id\.\\partial\_\{t\}P\(t,\\tau\)^\{\-1\}=\-A\_\{R\}\(t\)P\(t,\\tau\)^\{\-1\}\\quad\\text\{for a\.e\. \}t\\in\[\\tau,T\],\\qquad P\(\\tau,\\tau\)^\{\-1\}=I\_\{d\}\.The determinant satisfies Liouville’s formula

detP​\(t,τ\)=exp⁡\(∫τttr​AR​\(s\)​𝑑s\)≠0\.\\det P\(t,\\tau\)=\\exp\\\!\\left\(\\int\_\{\\tau\}^\{t\}\\mathrm\{tr\}\\,A\_\{R\}\(s\)\\,ds\\right\)\\neq 0\.

###### Proof\.

DefineY​\(⋅,τ\)Y\(\\cdot,\\tau\)as the absolutely continuous solution of

∂tY​\(t,τ\)=−AR​\(t\)​Y​\(t,τ\)for a\.e\.​t∈\[τ,T\],Y​\(τ,τ\)=Id\.\\partial\_\{t\}Y\(t,\\tau\)=\-A\_\{R\}\(t\)Y\(t,\\tau\)\\quad\\text\{for a\.e\. \}t\\in\[\\tau,T\],\\qquad Y\(\\tau,\\tau\)=I\_\{d\}\.Then, using∂tP=P​AR\\partial\_\{t\}P=PA\_\{R\},

∂t\(P​\(t,τ\)​Y​\(t,τ\)\)=P​\(t,τ\)​AR​\(t\)​Y​\(t,τ\)−P​\(t,τ\)​AR​\(t\)​Y​\(t,τ\)=0\\partial\_\{t\}\\bigl\(P\(t,\\tau\)Y\(t,\\tau\)\\bigr\)=P\(t,\\tau\)A\_\{R\}\(t\)Y\(t,\\tau\)\-P\(t,\\tau\)A\_\{R\}\(t\)Y\(t,\\tau\)=0for a\.e\.tt\. SinceP​\(τ,τ\)​Y​\(τ,τ\)=IdP\(\\tau,\\tau\)Y\(\\tau,\\tau\)=I\_\{d\}, we get

P​\(t,τ\)​Y​\(t,τ\)=IdP\(t,\\tau\)Y\(t,\\tau\)=I\_\{d\}for allt≥τt\\geq\\tau\. Because the matrices are square, the existence of this right inverse impliesP​\(t,τ\)∈G​L​\(d\)P\(t,\\tau\)\\in GL\(d\), and hence

Y​\(t,τ\)=P​\(t,τ\)−1\.Y\(t,\\tau\)=P\(t,\\tau\)^\{\-1\}\.This proves the inverse dynamics\.

For the determinant, sinceP​\(t,τ\)P\(t,\\tau\)is absolutely continuous and invertible, Jacobi’s formula gives, for a\.e\.tt,

dd​t​log​detP​\(t,τ\)=tr​\(P​\(t,τ\)−1​∂tP​\(t,τ\)\)=tr​\(P​\(t,τ\)−1​P​\(t,τ\)​AR​\(t\)\)=tr​AR​\(t\)\.\\frac\{d\}\{dt\}\\log\\det P\(t,\\tau\)=\\mathrm\{tr\}\\\!\\left\(P\(t,\\tau\)^\{\-1\}\\partial\_\{t\}P\(t,\\tau\)\\right\)=\\mathrm\{tr\}\\\!\\left\(P\(t,\\tau\)^\{\-1\}P\(t,\\tau\)A\_\{R\}\(t\)\\right\)=\\mathrm\{tr\}\\,A\_\{R\}\(t\)\.Integrating fromτ\\tautott, and usingdetP​\(τ,τ\)=1\\det P\(\\tau,\\tau\)=1, yields

detP​\(t,τ\)=exp⁡\(∫τttr​AR​\(s\)​𝑑s\)\.\\det P\(t,\\tau\)=\\exp\\\!\\left\(\\int\_\{\\tau\}^\{t\}\\mathrm\{tr\}\\,A\_\{R\}\(s\)\\,ds\\right\)\.The right\-hand side is nonzero, completing the proof\. ∎

###### Lemma B\.2\(Composition rule\)\.

For0≤τ≤σ≤t≤T0\\leq\\tau\\leq\\sigma\\leq t\\leq T,

P​\(t,τ\)=P​\(σ,τ\)​P​\(t,σ\)\.P\(t,\\tau\)=P\(\\sigma,\\tau\)P\(t,\\sigma\)\.

###### Proof\.

Fix0≤τ≤σ≤T0\\leq\\tau\\leq\\sigma\\leq T, and define

Q​\(t,τ\):=P​\(σ,τ\)​P​\(t,σ\),t∈\[σ,T\]\.Q\(t,\\tau\):=P\(\\sigma,\\tau\)P\(t,\\sigma\),\\qquad t\\in\[\\sigma,T\]\.Then

∂tQ​\(t,τ\)=P​\(σ,τ\)​P​\(t,σ\)​AR​\(t\)=Q​\(t,τ\)​AR​\(t\)\\partial\_\{t\}Q\(t,\\tau\)=P\(\\sigma,\\tau\)P\(t,\\sigma\)A\_\{R\}\(t\)=Q\(t,\\tau\)A\_\{R\}\(t\)for a\.e\.t∈\[σ,T\]t\\in\[\\sigma,T\], and

Q​\(σ,τ\)=P​\(σ,τ\)​P​\(σ,σ\)=P​\(σ,τ\)\.Q\(\\sigma,\\tau\)=P\(\\sigma,\\tau\)P\(\\sigma,\\sigma\)=P\(\\sigma,\\tau\)\.On the other hand,P​\(t,τ\)P\(t,\\tau\)restricted tot∈\[σ,T\]t\\in\[\\sigma,T\]satisfies the same differential equation and the same initial value att=σt=\\sigma\. By uniqueness,

P​\(t,τ\)=P​\(σ,τ\)​P​\(t,σ\)\.P\(t,\\tau\)=P\(\\sigma,\\tau\)P\(t,\\sigma\)\.∎

#### Order convention\.

The composition rule reflects our right\-action convention\. If a state evolves asH​\(t\)=H​\(τ\)​P​\(t,τ\)H\(t\)=H\(\\tau\)P\(t,\\tau\), then the segment fromτ\\tautottfactors as first applyingP​\(σ,τ\)P\(\\sigma,\\tau\), then applyingP​\(t,σ\)P\(t,\\sigma\):

H​\(t\)=H​\(τ\)​P​\(σ,τ\)​P​\(t,σ\)\.H\(t\)=H\(\\tau\)P\(\\sigma,\\tau\)P\(t,\\sigma\)\.

### B\.2Exact reduction to ordinary HiPPO

###### Proposition B\.3\(Exact reduction to ordinary HiPPO\)\.

The following are equivalent:

1. 1\.AR​\(t\)=0A\_\{R\}\(t\)=0for a\.e\.t∈\[0,T\]t\\in\[0,T\];
2. 2\.P​\(t,τ\)=IdP\(t,\\tau\)=I\_\{d\}for all0≤τ≤t≤T0\\leq\\tau\\leq t\\leq T\.

In this case,

MP​\(t,τ\)=P​\(t,τ\)​P​\(t,τ\)⊤=Id,M\_\{P\}\(t,\\tau\)=P\(t,\\tau\)P\(t,\\tau\)^\{\\top\}=I\_\{d\},the transported approximation family equals the ordinary HiPPO approximation family, and SHiPPO reduces exactly to ordinary vector\-valued HiPPO\.

###### Proof\.

IfAR​\(t\)=0A\_\{R\}\(t\)=0for a\.e\.tt, then the integral equation \([8](https://arxiv.org/html/2607.03055#A2.E8)\) gives

P​\(t,τ\)=Id\+∫τtP​\(s,τ\)​AR​\(s\)​𝑑s=IdP\(t,\\tau\)=I\_\{d\}\+\\int\_\{\\tau\}^\{t\}P\(s,\\tau\)A\_\{R\}\(s\)\\,ds=I\_\{d\}for allt≥τt\\geq\\tau\.

Conversely, supposeP​\(t,τ\)=IdP\(t,\\tau\)=I\_\{d\}for all0≤τ≤t≤T0\\leq\\tau\\leq t\\leq T\. Takingτ=0\\tau=0, we haveP​\(t,0\)=IdP\(t,0\)=I\_\{d\}for allt∈\[0,T\]t\\in\[0,T\]\. Since∂tP​\(t,0\)=P​\(t,0\)​AR​\(t\)\\partial\_\{t\}P\(t,0\)=P\(t,0\)A\_\{R\}\(t\)for a\.e\.tt, we obtain

0=∂tP​\(t,0\)=AR​\(t\)0=\\partial\_\{t\}P\(t,0\)=A\_\{R\}\(t\)for a\.e\.t∈\[0,T\]t\\in\[0,T\]\.

IfP​\(t,τ\)=IdP\(t,\\tau\)=I\_\{d\}, thenMP​\(t,τ\)=IdM\_\{P\}\(t,\\tau\)=I\_\{d\}and

P​\(t,τ\)−⊤​C⊤​Φt​\(τ\)=C⊤​Φt​\(τ\)\.P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\)=C^\{\\top\}\\Phi\_\{t\}\(\\tau\)\.Thus the SHiPPO metric and approximation family are exactly the ordinary Euclidean\-channel HiPPO metric and approximation family\. ∎

### B\.3Identity metric versus identity transport

Recall the transport\-induced metric

MP​\(t,τ\):=P​\(t,τ\)​P​\(t,τ\)⊤\.M\_\{P\}\(t,\\tau\):=P\(t,\\tau\)P\(t,\\tau\)^\{\\top\}\.The conditionMP​\(t,τ\)=IdM\_\{P\}\(t,\\tau\)=I\_\{d\}does not implyP​\(t,τ\)=IdP\(t,\\tau\)=I\_\{d\}\. It only constrains the symmetric part of the right\-transport generator\.

###### Proposition B\.4\(Identity metric iff skew transport generator\)\.

The following are equivalent:

1. 1\.MP​\(t,τ\)=IdM\_\{P\}\(t,\\tau\)=I\_\{d\}for all0≤τ≤t≤T0\\leq\\tau\\leq t\\leq T;
2. 2\.AR​\(t\)\+AR​\(t\)⊤=0A\_\{R\}\(t\)\+A\_\{R\}\(t\)^\{\\top\}=0for a\.e\.t∈\[0,T\]t\\in\[0,T\]\.

Consequently,MP≡IdM\_\{P\}\\equiv I\_\{d\}does not implyP≡IdP\\equiv I\_\{d\}unless the skew transport is also trivial\.

###### Proof\.

Fixτ\\tau\. SinceP​\(⋅,τ\)P\(\\cdot,\\tau\)is absolutely continuous,MP​\(⋅,τ\)M\_\{P\}\(\\cdot,\\tau\)is absolutely continuous, and for a\.e\.t≥τt\\geq\\tau,

∂tMP​\(t,τ\)\\displaystyle\\partial\_\{t\}M\_\{P\}\(t,\\tau\)=\(∂tP​\(t,τ\)\)​P​\(t,τ\)⊤\+P​\(t,τ\)​\(∂tP​\(t,τ\)\)⊤\\displaystyle=\(\\partial\_\{t\}P\(t,\\tau\)\)P\(t,\\tau\)^\{\\top\}\+P\(t,\\tau\)\(\\partial\_\{t\}P\(t,\\tau\)\)^\{\\top\}=P​\(t,τ\)​AR​\(t\)​P​\(t,τ\)⊤\+P​\(t,τ\)​AR​\(t\)⊤​P​\(t,τ\)⊤\\displaystyle=P\(t,\\tau\)A\_\{R\}\(t\)P\(t,\\tau\)^\{\\top\}\+P\(t,\\tau\)A\_\{R\}\(t\)^\{\\top\}P\(t,\\tau\)^\{\\top\}=P​\(t,τ\)​\(AR​\(t\)\+AR​\(t\)⊤\)​P​\(t,τ\)⊤\.\\displaystyle=P\(t,\\tau\)\\bigl\(A\_\{R\}\(t\)\+A\_\{R\}\(t\)^\{\\top\}\\bigr\)P\(t,\\tau\)^\{\\top\}\.
IfAR​\(t\)\+AR​\(t\)⊤=0A\_\{R\}\(t\)\+A\_\{R\}\(t\)^\{\\top\}=0for a\.e\.tt, then

∂tMP​\(t,τ\)=0\\partial\_\{t\}M\_\{P\}\(t,\\tau\)=0for a\.e\.t≥τt\\geq\\tau\. SinceMP​\(τ,τ\)=IdM\_\{P\}\(\\tau,\\tau\)=I\_\{d\}, it follows that

MP​\(t,τ\)=IdM\_\{P\}\(t,\\tau\)=I\_\{d\}for allt≥τt\\geq\\tau\.

Conversely, supposeMP​\(t,τ\)=IdM\_\{P\}\(t,\\tau\)=I\_\{d\}for all0≤τ≤t≤T0\\leq\\tau\\leq t\\leq T\. Takingτ=0\\tau=0, we haveMP​\(t,0\)=IdM\_\{P\}\(t,0\)=I\_\{d\}for alltt\. Hence

0=∂tMP​\(t,0\)=P​\(t,0\)​\(AR​\(t\)\+AR​\(t\)⊤\)​P​\(t,0\)⊤0=\\partial\_\{t\}M\_\{P\}\(t,0\)=P\(t,0\)\\bigl\(A\_\{R\}\(t\)\+A\_\{R\}\(t\)^\{\\top\}\\bigr\)P\(t,0\)^\{\\top\}for a\.e\.tt\. By Lemma[B\.1](https://arxiv.org/html/2607.03055#A2.Thmassumption1),P​\(t,0\)P\(t,0\)is invertible, so

AR​\(t\)\+AR​\(t\)⊤=0A\_\{R\}\(t\)\+A\_\{R\}\(t\)^\{\\top\}=0for a\.e\.tt\. ∎

#### Implication for the projection problem\.

Even whenMP≡IdM\_\{P\}\\equiv I\_\{d\}, SHiPPO need not reduce to ordinary HiPPO because the approximation family is still transported:

𝒢tSH=\{τ↦P​\(t,τ\)−⊤​C⊤​Φt​\(τ\):C∈ℝN×d\}\.\\mathcal\{G\}\_\{t\}^\{\\mathrm\{SH\}\}=\\\{\\,\\tau\\mapsto P\(t,\\tau\)^\{\-\\top\}C^\{\\top\}\\Phi\_\{t\}\(\\tau\):C\\in\\mathbb\{R\}^\{N\\times d\}\\,\\\}\.Equivalently, by Proposition[A\.4](https://arxiv.org/html/2607.03055#A1.Thmassumption4),

shippot⁡\(f\)=hippot⁡\(𝒯t​f\),\(𝒯t​f\)​\(τ\)=P​\(t,τ\)⊤​f​\(τ\)\.\\operatorname\{shippo\}\_\{t\}\(f\)=\\operatorname\{hippo\}\_\{t\}\(\\mathcal\{T\}\_\{t\}f\),\\qquad\(\\mathcal\{T\}\_\{t\}f\)\(\\tau\)=P\(t,\\tau\)^\{\\top\}f\(\\tau\)\.Thus, underMP≡IdM\_\{P\}\\equiv I\_\{d\}, SHiPPO is ordinary HiPPO applied to a moving\-frame history, not ordinary HiPPO applied to the original history\.

###### Example B\.5\(Orthogonal transport with nontrivial moving frame\)\.

Letd=2d=2and set

AR​\(t\)=ω​\(0−110\),ω≠0\.A\_\{R\}\(t\)=\\omega\\begin\{pmatrix\}0&\-1\\\\ 1&0\\end\{pmatrix\},\\qquad\\omega\\neq 0\.Then

AR​\(t\)⊤=−AR​\(t\),A\_\{R\}\(t\)^\{\\top\}=\-A\_\{R\}\(t\),so Proposition[B\.4](https://arxiv.org/html/2607.03055#A2.Thmassumption4)gives

MP​\(t,τ\)=I2\.M\_\{P\}\(t,\\tau\)=I\_\{2\}\.However,

P​\(t,τ\)=exp⁡\(\(t−τ\)​AR\)=\(cos⁡\(ω​\(t−τ\)\)−sin⁡\(ω​\(t−τ\)\)sin⁡\(ω​\(t−τ\)\)cos⁡\(ω​\(t−τ\)\)\),P\(t,\\tau\)=\\exp\\\!\\bigl\(\(t\-\\tau\)A\_\{R\}\\bigr\)=\\begin\{pmatrix\}\\cos\(\\omega\(t\-\\tau\)\)&\-\\sin\(\\omega\(t\-\\tau\)\)\\\\ \\sin\(\\omega\(t\-\\tau\)\)&\\cos\(\\omega\(t\-\\tau\)\)\\end\{pmatrix\},which is not equal toI2I\_\{2\}for generict≠τt\\neq\\tau\. Therefore SHiPPO does not reduce to ordinary HiPPO\. The right\-action term

CS​\(t\)​AR​\(t\)C\_\{S\}\(t\)A\_\{R\}\(t\)in Theorem[2\.2](https://arxiv.org/html/2607.03055#S2.Thmassumption2)remains present in the coefficient dynamics and is generically nonzero\.

### B\.4Moving\-frame factorization for a chosen transport path

This subsection records the exact moving\-frame factorization associated with a chosen admissible transport pathARA\_\{R\}\. When the transport is controller\-generated, the same factorization applies pathwise to the realized trajectory\. Because the frame then depends on the input trajectory and model parameters, the factorization should not be interpreted as an input\-independent reduction of SHiPPO to ordinary HiPPO\.

###### Definition B\.6\(Fundamental matrix and moving channel frame\)\.

LetV:\[0,T\]→G​L​\(d\)V:\[0,T\]\\to GL\(d\)be the fundamental matrix solving

V˙​\(t\)=V​\(t\)​AR​\(t\)for a\.e\.​t∈\[0,T\],V​\(0\)=Id\.\\dot\{V\}\(t\)=V\(t\)A\_\{R\}\(t\)\\quad\\text\{for a\.e\. \}t\\in\[0,T\],\\qquad V\(0\)=I\_\{d\}\.Define the history encoder and coefficient decoder by

\(ℰV​f\)​\(τ\):=V​\(τ\)−T​f​\(τ\),𝒟V,t​\(C\):=C​V​\(t\)\.\(\\mathcal\{E\}\_\{V\}f\)\(\\tau\):=V\(\\tau\)^\{\-T\}f\(\\tau\),\\qquad\\mathcal\{D\}\_\{V,t\}\(C\):=C\\,V\(t\)\.

###### Lemma B\.7\(Frame representation of the transport\)\.

For all0≤τ≤t≤T0\\leq\\tau\\leq t\\leq T,

P​\(t,τ\)=V​\(τ\)−1​V​\(t\),P​\(t,τ\)⊤=V​\(t\)⊤​V​\(τ\)−T\.P\(t,\\tau\)=V\(\\tau\)^\{\-1\}V\(t\),\\qquad P\(t,\\tau\)^\{\\top\}=V\(t\)^\{\\top\}V\(\\tau\)^\{\-T\}\.

###### Proof\.

Fixτ\\tau, and define

Q​\(t,τ\):=V​\(τ\)−1​V​\(t\),t∈\[τ,T\]\.Q\(t,\\tau\):=V\(\\tau\)^\{\-1\}V\(t\),\\qquad t\\in\[\\tau,T\]\.Then

∂tQ​\(t,τ\)=V​\(τ\)−1​V˙​\(t\)=V​\(τ\)−1​V​\(t\)​AR​\(t\)=Q​\(t,τ\)​AR​\(t\)\\partial\_\{t\}Q\(t,\\tau\)=V\(\\tau\)^\{\-1\}\\dot\{V\}\(t\)=V\(\\tau\)^\{\-1\}V\(t\)A\_\{R\}\(t\)=Q\(t,\\tau\)A\_\{R\}\(t\)for a\.e\.t≥τt\\geq\\tau, and

Q​\(τ,τ\)=Id\.Q\(\\tau,\\tau\)=I\_\{d\}\.By uniqueness of solutions to \([9](https://arxiv.org/html/2607.03055#A2.E9)\),

Q​\(t,τ\)=P​\(t,τ\)\.Q\(t,\\tau\)=P\(t,\\tau\)\.Taking transposes gives

P​\(t,τ\)⊤=V​\(t\)⊤​V​\(τ\)−T\.P\(t,\\tau\)^\{\\top\}=V\(t\)^\{\\top\}V\(\\tau\)^\{\-T\}\.∎

###### Lemma B\.8\(HiPPO equivariance to fixed channel mixing\)\.

FixQ∈G​L​\(d\)Q\\in GL\(d\)independent ofτ\\tau, and define

\(ℳQ,t​u\)​\(τ\):=Q⊤​u​\(τ\)\.\(\\mathcal\{M\}\_\{Q,t\}u\)\(\\tau\):=Q^\{\\top\}u\(\\tau\)\.Then

hippot⁡\(ℳQ,t​u\)=hippot⁡\(u\)​Q\.\\operatorname\{hippo\}\_\{t\}\(\\mathcal\{M\}\_\{Q,t\}u\)=\\operatorname\{hippo\}\_\{t\}\(u\)\\,Q\.

###### Proof\.

LetCu​\(t\)=hippot⁡\(u\)C\_\{u\}\(t\)=\\operatorname\{hippo\}\_\{t\}\(u\)\. By the ordinary HiPPO normal equation,

G​\(t\)​Cu​\(t\)=∫0tΦt​\(τ\)​u​\(τ\)⊤​𝑑μt​\(τ\)\.G\(t\)C\_\{u\}\(t\)=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)u\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)\.ForℳQ,t​u\\mathcal\{M\}\_\{Q,t\}u,

G​\(t\)​CℳQ​u​\(t\)\\displaystyle G\(t\)C\_\{\\mathcal\{M\}\_\{Q\}u\}\(t\)=∫0tΦt​\(τ\)​\(ℳQ,t​u\)​\(τ\)⊤​𝑑μt​\(τ\)\\displaystyle=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)\(\\mathcal\{M\}\_\{Q,t\}u\)\(\\tau\)^\{\\top\}\\,d\\mu\_\{t\}\(\\tau\)=∫0tΦt​\(τ\)​u​\(τ\)⊤​Q​𝑑μt​\(τ\)\\displaystyle=\\int\_\{0\}^\{t\}\\Phi\_\{t\}\(\\tau\)u\(\\tau\)^\{\\top\}Q\\,d\\mu\_\{t\}\(\\tau\)=G​\(t\)​Cu​\(t\)​Q\.\\displaystyle=G\(t\)C\_\{u\}\(t\)Q\.SinceG​\(t\)G\(t\)is invertible,

CℳQ​u​\(t\)=Cu​\(t\)​Q\.C\_\{\\mathcal\{M\}\_\{Q\}u\}\(t\)=C\_\{u\}\(t\)Q\.∎

###### Proposition B\.9\(Moving\-frame factorization of SHiPPO\)\.

For a chosen transport pathARA\_\{R\}and its fundamental matrixVV, SHiPPO admits the factorization

shippot=𝒟V,t∘hippot∘ℰV\.\\operatorname\{shippo\}\_\{t\}=\\mathcal\{D\}\_\{V,t\}\\circ\\operatorname\{hippo\}\_\{t\}\\circ\\mathcal\{E\}\_\{V\}\.Equivalently, for every signalff,

shippot⁡\(f\)=hippot⁡\(ℰV​f\)​V​\(t\)\.\\operatorname\{shippo\}\_\{t\}\(f\)=\\operatorname\{hippo\}\_\{t\}\(\\mathcal\{E\}\_\{V\}f\)\\,V\(t\)\.

###### Proof\.

By Lemma[B\.7](https://arxiv.org/html/2607.03055#A2.Thmassumption7),

\(𝒯t​f\)​\(τ\)=P​\(t,τ\)⊤​f​\(τ\)=V​\(t\)⊤​V​\(τ\)−T​f​\(τ\)=V​\(t\)⊤​\(ℰV​f\)​\(τ\)\.\(\\mathcal\{T\}\_\{t\}f\)\(\\tau\)=P\(t,\\tau\)^\{\\top\}f\(\\tau\)=V\(t\)^\{\\top\}V\(\\tau\)^\{\-T\}f\(\\tau\)=V\(t\)^\{\\top\}\(\\mathcal\{E\}\_\{V\}f\)\(\\tau\)\.Thus

𝒯t=ℳV​\(t\),t∘ℰV\.\\mathcal\{T\}\_\{t\}=\\mathcal\{M\}\_\{V\(t\),t\}\\circ\\mathcal\{E\}\_\{V\}\.Using the SHiPPO–HiPPO conjugacy

shippot=hippot∘𝒯t\\operatorname\{shippo\}\_\{t\}=\\operatorname\{hippo\}\_\{t\}\\circ\\mathcal\{T\}\_\{t\}from Proposition[A\.4](https://arxiv.org/html/2607.03055#A1.Thmassumption4), and applying Lemma[B\.8](https://arxiv.org/html/2607.03055#A2.Thmassumption8)withQ=V​\(t\)Q=V\(t\), we obtain

shippot⁡\(f\)\\displaystyle\\operatorname\{shippo\}\_\{t\}\(f\)=hippot⁡\(ℳV​\(t\),t​\(ℰV​f\)\)\\displaystyle=\\operatorname\{hippo\}\_\{t\}\\bigl\(\\mathcal\{M\}\_\{V\(t\),t\}\(\\mathcal\{E\}\_\{V\}f\)\\bigr\)=hippot⁡\(ℰV​f\)​V​\(t\)\\displaystyle=\\operatorname\{hippo\}\_\{t\}\(\\mathcal\{E\}\_\{V\}f\)\\,V\(t\)=𝒟V,t​\(hippot⁡\(ℰV​f\)\)\.\\displaystyle=\\mathcal\{D\}\_\{V,t\}\\bigl\(\\operatorname\{hippo\}\_\{t\}\(\\mathcal\{E\}\_\{V\}f\)\\bigr\)\.∎

#### Interpretation\.

Proposition[B\.9](https://arxiv.org/html/2607.03055#A2.Thmassumption9)shows that, for a chosen transport path, SHiPPO can be viewed as ordinary HiPPO in the moving channel frameV​\(τ\)−TV\(\\tau\)^\{\-T\}, followed by decoding with the same frameV​\(t\)V\(t\)\. The same frame determines both the history encoder

\(ℰV​f\)​\(τ\)=V​\(τ\)−T​f​\(τ\)\(\\mathcal\{E\}\_\{V\}f\)\(\\tau\)=V\(\\tau\)^\{\-T\}f\(\\tau\)and the coefficient decoder

𝒟V,t​\(C\)=C​V​\(t\)\.\\mathcal\{D\}\_\{V,t\}\(C\)=CV\(t\)\.Thus this is a tied, time\-varying encoder–decoder factorization, not an equivalence to arbitrary independent channel projections\. When the transport is controller\-generated, the same factorization remains valid pathwise; because the frame then depends on the realized input trajectory and model parameters, it does not yield an input\-independent encoder–HiPPO–decoder decomposition\.

### B\.5Gauge equivalence of coefficient dynamics

The moving\-frame factorization also appears directly at the level of the coefficient ODEs\.

###### Proposition B\.10\(Gauge equivalence of coefficient dynamics\)\.

AssumeG​\(t\)=ING\(t\)=I\_\{N\}and the HiPPO closure condition holds\. For a chosen transport pathARA\_\{R\}, letVVbe the fundamental matrix from Definition[B\.6](https://arxiv.org/html/2607.03055#A2.Thmassumption6), and define

f~​\(t\):=V​\(t\)−T​f​\(t\)\.\\widetilde\{f\}\(t\):=V\(t\)^\{\-T\}f\(t\)\.LetC~​\(t\)\\widetilde\{C\}\(t\)be the ordinary HiPPO coefficient trajectory off~\\widetilde\{f\}:

C~˙​\(t\)=AL​\(t\)​C~​\(t\)\+BL​\(t\)​f~​\(t\)⊤\.\\dot\{\\widetilde\{C\}\}\(t\)=A\_\{L\}\(t\)\\widetilde\{C\}\(t\)\+B\_\{L\}\(t\)\\widetilde\{f\}\(t\)^\{\\top\}\.Define

CS​\(t\):=C~​\(t\)​V​\(t\)\.C\_\{S\}\(t\):=\\widetilde\{C\}\(t\)V\(t\)\.ThenCSC\_\{S\}satisfies the SHiPPO Sylvester dynamics

C˙S​\(t\)=AL​\(t\)​CS​\(t\)\+BL​\(t\)​f​\(t\)⊤\+CS​\(t\)​AR​\(t\)\.\\dot\{C\}\_\{S\}\(t\)=A\_\{L\}\(t\)C\_\{S\}\(t\)\+B\_\{L\}\(t\)f\(t\)^\{\\top\}\+C\_\{S\}\(t\)A\_\{R\}\(t\)\.Conversely, ifCSC\_\{S\}satisfies the Sylvester dynamics above, then

C~​\(t\):=CS​\(t\)​V​\(t\)−1\\widetilde\{C\}\(t\):=C\_\{S\}\(t\)V\(t\)^\{\-1\}satisfies the ordinary HiPPO dynamics driven byf~​\(t\)=V​\(t\)−T​f​\(t\)\\widetilde\{f\}\(t\)=V\(t\)^\{\-T\}f\(t\)\.

###### Proof\.

DifferentiateCS=C~​VC\_\{S\}=\\widetilde\{C\}V:

C˙S=C~˙​V\+C~​V˙\.\\dot\{C\}\_\{S\}=\\dot\{\\widetilde\{C\}\}V\+\\widetilde\{C\}\\dot\{V\}\.Substituting

C~˙=AL​C~\+BL​f~⊤,V˙=V​AR,\\dot\{\\widetilde\{C\}\}=A\_\{L\}\\widetilde\{C\}\+B\_\{L\}\\widetilde\{f\}^\{\\top\},\\qquad\\dot\{V\}=VA\_\{R\},gives

C˙S\\displaystyle\\dot\{C\}\_\{S\}=\(AL​C~\+BL​f~⊤\)​V\+C~​V​AR\\displaystyle=\(A\_\{L\}\\widetilde\{C\}\+B\_\{L\}\\widetilde\{f\}^\{\\top\}\)V\+\\widetilde\{C\}VA\_\{R\}=AL​CS\+BL​\(f~⊤​V\)\+CS​AR\.\\displaystyle=A\_\{L\}C\_\{S\}\+B\_\{L\}\(\\widetilde\{f\}^\{\\top\}V\)\+C\_\{S\}A\_\{R\}\.Since

f~⊤​V=\(V−T​f\)⊤​V=f⊤​V−1​V=f⊤,\\widetilde\{f\}^\{\\top\}V=\(V^\{\-T\}f\)^\{\\top\}V=f^\{\\top\}V^\{\-1\}V=f^\{\\top\},we obtain

C˙S=AL​CS\+BL​f⊤\+CS​AR\.\\dot\{C\}\_\{S\}=A\_\{L\}C\_\{S\}\+B\_\{L\}f^\{\\top\}\+C\_\{S\}A\_\{R\}\.
For the converse, defineC~=CS​V−1\\widetilde\{C\}=C\_\{S\}V^\{\-1\}\. Since

∂tV−1=−AR​V−1,\\partial\_\{t\}V^\{\-1\}=\-A\_\{R\}V^\{\-1\},we have

C~˙\\displaystyle\\dot\{\\widetilde\{C\}\}=C˙S​V−1\+CS​∂tV−1\\displaystyle=\\dot\{C\}\_\{S\}V^\{\-1\}\+C\_\{S\}\\partial\_\{t\}V^\{\-1\}=\(AL​CS\+BL​f⊤\+CS​AR\)​V−1−CS​AR​V−1\\displaystyle=\(A\_\{L\}C\_\{S\}\+B\_\{L\}f^\{\\top\}\+C\_\{S\}A\_\{R\}\)V^\{\-1\}\-C\_\{S\}A\_\{R\}V^\{\-1\}=AL​C~\+BL​f⊤​V−1\.\\displaystyle=A\_\{L\}\\widetilde\{C\}\+B\_\{L\}f^\{\\top\}V^\{\-1\}\.Finally,

f⊤​V−1=\(V−T​f\)⊤=f~⊤,f^\{\\top\}V^\{\-1\}=\(V^\{\-T\}f\)^\{\\top\}=\\widetilde\{f\}^\{\\top\},so

C~˙=AL​C~\+BL​f~⊤\.\\dot\{\\widetilde\{C\}\}=A\_\{L\}\\widetilde\{C\}\+B\_\{L\}\\widetilde\{f\}^\{\\top\}\.∎

#### What the equivalence does and does not say\.

Propositions[B\.9](https://arxiv.org/html/2607.03055#A2.Thmassumption9)and[B\.10](https://arxiv.org/html/2607.03055#A2.Thmassumption10)are exact for a chosen transport path, and pathwise for controller\-generated transport\. They show that the Sylvester right\-action term is precisely the gauge term induced by decoding ordinary HiPPO coefficients from a moving channel frame\.

They do not reduce SHiPPO to a single static channel mixer\. IfV​\(t\)≡V0V\(t\)\\equiv V\_\{0\}is constant, then

0=V˙​\(t\)=V0​AR​\(t\),0=\\dot\{V\}\(t\)=V\_\{0\}A\_\{R\}\(t\),and sinceV0∈G​L​\(d\)V\_\{0\}\\in GL\(d\), this impliesAR​\(t\)=0A\_\{R\}\(t\)=0almost everywhere\. Thus any nontrivial transport requires a genuinely time\-varying frame\.

Nor do they justify arbitrary independent encoder and decoder maps\. The factorization relies on the tied moving\-frame relation

\(ℰV​f\)​\(τ\)=V​\(τ\)−T​f​\(τ\),𝒟V,t​\(C\)=C​V​\(t\),\(\\mathcal\{E\}\_\{V\}f\)\(\\tau\)=V\(\\tau\)^\{\-T\}f\(\\tau\),\\qquad\\mathcal\{D\}\_\{V,t\}\(C\)=CV\(t\),and this tied structure is exactly what produces the Sylvester termCS​ARC\_\{S\}A\_\{R\}\. In the input\-dependent selective setting, the same calculation remains valid pathwise, but the frame itself depends on the realized input trajectory, so there is generally no fixed encoder–HiPPO–decoder decomposition independent of the input\.

## Appendix CProofs for the Scan\-Compatible SHiPPO Lift

This appendix supports Section[3](https://arxiv.org/html/2607.03055#S3)\. It proves the obstruction for the direct channelwise lift, the block\-affine scan closure of the restricted realization, the sufficient collapse criterion under fixed simultaneous block reduction, and the split\-flow and discretization claims used to construct the scan\-compatible SHiPPO cell\. It does not re\-derive the abstract SHiPPO online approximation operator of Section[2](https://arxiv.org/html/2607.03055#S2); that role is played by Appendices A and B\.

Unless otherwise stated, we work with a single transport group and omit the group index\. The group\-local memory state is

Ht∈ℝN×P,H\_\{t\}\\in\\mathbb\{R\}^\{N\\times P\},whereNNis the left state size andPPis the group\-local channel width\. The left factor is

Lt=Diag⁡\(αt\)∈ℝN×N,L\_\{t\}=\\operatorname\{Diag\}\(\\alpha\_\{t\}\)\\in\\mathbb\{R\}^\{N\\times N\},the group\-local right action is

Rt∈G​L​\(P\),R\_\{t\}\\in GL\(P\),and the additive source is

Ut∈ℝN×P\.U\_\{t\}\\in\\mathbb\{R\}^\{N\\times P\}\.All statements in this appendix concern the restricted scan\-compatible realization of Section[3](https://arxiv.org/html/2607.03055#S3)\. The group\-tied diagonal\-left restriction, controller\-compatible right transport, and group\-local right actions are computational restrictions of this realization, not part of the abstract SHiPPO definition\.

### C\.1Why the direct channelwise lift does not preserve a small block\-affine scan algebra

This subsection justifies the statement in Section[3](https://arxiv.org/html/2607.03055#S3)that a fully channelwise lift of a diagonal selective recurrence does not generally close in the small block\-affine scan algebra used by the restricted realization\. The direct groupwise lift would be

Htfull=\(A¯t⊙Ht−1full\)​Rt\+Ut,A¯t∈ℝN×P\.H^\{\\mathrm\{full\}\}\_\{t\}=\(\\bar\{A\}\_\{t\}\\odot H^\{\\mathrm\{full\}\}\_\{t\-1\}\)R\_\{t\}\+U\_\{t\},\\qquad\\bar\{A\}\_\{t\}\\in\\mathbb\{R\}^\{N\\times P\}\.\(10\)It keeps channel\-specific left decays, but nontrivial right transport generally destroys the shared factorized right summary used in Proposition[3\.2](https://arxiv.org/html/2607.03055#S3.Thmassumption2)\.

###### Proposition C\.1\(Failure of closure for the direct channelwise factorization\)\.

For the direct lift \([10](https://arxiv.org/html/2607.03055#A3.E10)\), writea¯t,n∈ℝP\\bar\{a\}\_\{t,n\}\\in\\mathbb\{R\}^\{P\}for thenn\-th row ofA¯t\\bar\{A\}\_\{t\}and set

Dt,n:=Diag⁡\(a¯t,n\)\.D\_\{t,n\}:=\\operatorname\{Diag\}\(\\bar\{a\}\_\{t,n\}\)\.Then thenn\-th row satisfies

ht,nfull=ht−1,nfull​Dt,n​Rt\+ut,n\.h^\{\\mathrm\{full\}\}\_\{t,n\}=h^\{\\mathrm\{full\}\}\_\{t\-1,n\}D\_\{t,n\}R\_\{t\}\+u\_\{t,n\}\.After two steps,

h2,nfull=h0,nfull​D1,n​R1​D2,n​R2\+u1,n​D2,n​R2\+u2,n\.h^\{\\mathrm\{full\}\}\_\{2,n\}=h^\{\\mathrm\{full\}\}\_\{0,n\}D\_\{1,n\}R\_\{1\}D\_\{2,n\}R\_\{2\}\+u\_\{1,n\}D\_\{2,n\}R\_\{2\}\+u\_\{2,n\}\.In general there need not exist a*shared*right factorR⋆∈G​L​\(P\)R\_\{\\star\}\\in GL\(P\)and rowwise diagonalsD~n\\widetilde\{D\}\_\{n\}such that

D1,n​R1​D2,n​R2=D~n​R⋆for all​n\.D\_\{1,n\}R\_\{1\}D\_\{2,n\}R\_\{2\}=\\widetilde\{D\}\_\{n\}R\_\{\\star\}\\qquad\\text\{for all \}n\.Hence the direct lift does not, in general, preserve the small block\-affine scan algebra used by the restricted realization\.

###### Proof\.

The rowwise formula is immediate from

\(A¯t⊙H\)n,:=Hn,:​Dt,n\.\(\\bar\{A\}\_\{t\}\\odot H\)\_\{n,:\}=H\_\{n,:\}D\_\{t,n\}\.Composing two steps gives

h2,nfull\\displaystyle h^\{\\mathrm\{full\}\}\_\{2,n\}=\(h1,nfull​D2,n\)​R2\+u2,n\\displaystyle=\\bigl\(h^\{\\mathrm\{full\}\}\_\{1,n\}D\_\{2,n\}\\bigr\)R\_\{2\}\+u\_\{2,n\}=\(h0,nfull​D1,n​R1\+u1,n\)​D2,n​R2\+u2,n\\displaystyle=\\bigl\(h^\{\\mathrm\{full\}\}\_\{0,n\}D\_\{1,n\}R\_\{1\}\+u\_\{1,n\}\\bigr\)D\_\{2,n\}R\_\{2\}\+u\_\{2,n\}=h0,nfull​D1,n​R1​D2,n​R2\+u1,n​D2,n​R2\+u2,n\.\\displaystyle=h^\{\\mathrm\{full\}\}\_\{0,n\}D\_\{1,n\}R\_\{1\}D\_\{2,n\}R\_\{2\}\+u\_\{1,n\}D\_\{2,n\}R\_\{2\}\+u\_\{2,n\}\.So closure in the same factorized family would require a commonR⋆R\_\{\\star\}and rowwise diagonalsD~n\\widetilde\{D\}\_\{n\}satisfying

D1,n​R1​D2,n​R2=D~n​R⋆for all​n\.D\_\{1,n\}R\_\{1\}D\_\{2,n\}R\_\{2\}=\\widetilde\{D\}\_\{n\}R\_\{\\star\}\\qquad\\text\{for all \}n\.This need not hold\. Consider the concrete exampleP=2P=2,N≥2N\\geq 2,

R1=\(111−1\),R2=I2,D1,n=I2,R\_\{1\}=\\begin\{pmatrix\}1&1\\\\ 1&\-1\\end\{pmatrix\},\\qquad R\_\{2\}=I\_\{2\},\\qquad D\_\{1,n\}=I\_\{2\},and choose

D2,1=Diag​\(1,1\),D2,2=Diag​\(1,2\)\.D\_\{2,1\}=\\mathrm\{Diag\}\(1,1\),\\qquad D\_\{2,2\}=\\mathrm\{Diag\}\(1,2\)\.Then

R1​D2,1=\(111−1\),R1​D2,2=\(121−2\)\.R\_\{1\}D\_\{2,1\}=\\begin\{pmatrix\}1&1\\\\ 1&\-1\\end\{pmatrix\},\\qquad R\_\{1\}D\_\{2,2\}=\\begin\{pmatrix\}1&2\\\\ 1&\-2\\end\{pmatrix\}\.If there were a sharedR⋆R\_\{\\star\}and diagonalsD~1,D~2\\widetilde\{D\}\_\{1\},\\widetilde\{D\}\_\{2\}with

R1​D2,j=D~j​R⋆,j∈\{1,2\},R\_\{1\}D\_\{2,j\}=\\widetilde\{D\}\_\{j\}R\_\{\\star\},\\qquad j\\in\\\{1,2\\\},then the first rows ofR1​D2,1R\_\{1\}D\_\{2,1\}andR1​D2,2R\_\{1\}D\_\{2,2\}would have to be scalar multiples of the same row ofR⋆R\_\{\\star\}\. But\(1,1\)\(1,1\)and\(1,2\)\(1,2\)are not scalar multiples, so no such commonR⋆R\_\{\\star\}exists\. ∎

#### Interpretation\.

Proposition[C\.1](https://arxiv.org/html/2607.03055#A3.Thmassumption1)does*not*say that the direct lift is invalid or unscannable\. Rather, it says that the small factorized scan algebra of the main text is lost\. Exact scan is still possible in larger summaries, for example by carrying rowwiseP×PP\\times Ptransition matrices\. The group\-tied diagonal\-left realization in Section[3](https://arxiv.org/html/2607.03055#S3)is introduced precisely to recover a lightweight factorized block\-affine scan\.

### C\.2Controller\-compatible block\-affine scan closure

We now prove Proposition[3\.2](https://arxiv.org/html/2607.03055#S3.Thmassumption2)\. The key point is that the right action may be token\-dependent, but it must be fixed conditional on a precomputed controller path and independent of the main memory state\.

###### Proof of Proposition[3\.2](https://arxiv.org/html/2607.03055#S3.Thmassumption2)\.

Conditional on a causal controller path computed independently of the main memory recurrence,LtL\_\{t\},RtR\_\{t\}, andUtU\_\{t\}are fixed with respect to the main memory variableHt−1H\_\{t\-1\}\. Hence each step map has the affine two\-sided form

Ft​\(H\)=Lt​H​Rt\+Ut\.F\_\{t\}\(H\)=L\_\{t\}HR\_\{t\}\+U\_\{t\}\.Consider two successive steps

F1​\(H\)=L1​H​R1\+U1,F2​\(H\)=L2​H​R2\+U2\.F\_\{1\}\(H\)=L\_\{1\}HR\_\{1\}\+U\_\{1\},\\qquad F\_\{2\}\(H\)=L\_\{2\}HR\_\{2\}\+U\_\{2\}\.Their composition is

F2​\(F1​\(H\)\)\\displaystyle F\_\{2\}\(F\_\{1\}\(H\)\)=L2​\(L1​H​R1\+U1\)​R2\+U2\\displaystyle=L\_\{2\}\(L\_\{1\}HR\_\{1\}\+U\_\{1\}\)R\_\{2\}\+U\_\{2\}=\(L2​L1\)​H​\(R1​R2\)\+L2​U1​R2\+U2\.\\displaystyle=\(L\_\{2\}L\_\{1\}\)H\(R\_\{1\}R\_\{2\}\)\+L\_\{2\}U\_\{1\}R\_\{2\}\+U\_\{2\}\.Therefore the composite update is again of the same affine two\-sided form, with summary

\(L2,R2,U2\)⋆\(L1,R1,U1\)=\(L2​L1,R1​R2,L2​U1​R2\+U2\)\.\(L\_\{2\},R\_\{2\},U\_\{2\}\)\\star\(L\_\{1\},R\_\{1\},U\_\{1\}\)=\(L\_\{2\}L\_\{1\},\\;R\_\{1\}R\_\{2\},\\;L\_\{2\}U\_\{1\}R\_\{2\}\+U\_\{2\}\)\.Associativity of⋆\\starfollows from associativity of function composition, or directly from associativity of matrix multiplication\. The resulting parallel execution is an associative prefix\-scan computation in the standard sense\[[5](https://arxiv.org/html/2607.03055#bib.bib72),[38](https://arxiv.org/html/2607.03055#bib.bib73)\]\. ∎

#### Why direct state coupling generically breaks the block\-affine algebra\.

Suppose instead that the right action depends directly on the main memory,

Ft​\(H\)=Lt​H​R​\(ξt,H\)\+Ut\.F\_\{t\}\(H\)=L\_\{t\}H\\,R\(\\xi\_\{t\},H\)\+U\_\{t\}\.Any affine mapFFsatisfies, for all scalarsss,

F​\(s​H\)−F​\(0\)=s​\(F​\(H\)−F​\(0\)\)\.F\(sH\)\-F\(0\)=s\\bigl\(F\(H\)\-F\(0\)\\bigr\)\.HereF​\(0\)=UtF\(0\)=U\_\{t\}, so affine\-ness would require

Lt​\(s​H\)​R​\(ξt,s​H\)=s​Lt​H​R​\(ξt,H\)for all​s\.L\_\{t\}\(sH\)R\(\\xi\_\{t\},sH\)=sL\_\{t\}HR\(\\xi\_\{t\},H\)\\qquad\\text\{for all \}s\.Equivalently,

Lt​H​R​\(ξt,s​H\)=Lt​H​R​\(ξt,H\)for all​s\.L\_\{t\}H\\,R\(\\xi\_\{t\},sH\)=L\_\{t\}H\\,R\(\\xi\_\{t\},H\)\\qquad\\text\{for all \}s\.If there exists anHHsuch thatLt​HL\_\{t\}Hhas full column rank, then right multiplication byRRis injective:

Lt​H​R1=Lt​H​R2⟹R1=R2\.L\_\{t\}HR\_\{1\}=L\_\{t\}HR\_\{2\}\\quad\\Longrightarrow\\quad R\_\{1\}=R\_\{2\}\.Thus affine\-ness forces

R​\(ξt,s​H\)=R​\(ξt,H\)for all​sR\(\\xi\_\{t\},sH\)=R\(\\xi\_\{t\},H\)\\qquad\\text\{for all \}salong that ray\. Therefore any genuine dependence onHH, meaning a dependence for which someHHwithrank​\(Lt​H\)=P\\mathrm\{rank\}\(L\_\{t\}H\)=Pand somes≠1s\\neq 1satisfy

R​\(ξt,s​H\)≠R​\(ξt,H\),R\(\\xi\_\{t\},sH\)\\neq R\(\\xi\_\{t\},H\),makes the step map non\-affine\. In the group\-tied diagonal\-left cell, this full\-rank condition holds generically whenever all diagonal entries ofLtL\_\{t\}are nonzero,N≥PN\\geq P, andHHhas full column rank\.

#### Concrete failure mode\.

LetW∈ℝN×PW\\in\\mathbb\{R\}^\{N\\times P\}and letM∈ℝP×PM\\in\\mathbb\{R\}^\{P\\times P\}be any nonzero nilpotent matrix, for exampleM=ei​ej⊤M=e\_\{i\}e\_\{j\}^\{\\top\}withi≠ji\\neq j\. Define

R​\(ξt,H\):=exp⁡\(κ​⟨W,H⟩F​M\)=IP\+κ​⟨W,H⟩F​M,κ≠0,R\(\\xi\_\{t\},H\):=\\exp\\\!\\bigl\(\\kappa\\langle W,H\\rangle\_\{F\}M\\bigr\)=I\_\{P\}\+\\kappa\\langle W,H\\rangle\_\{F\}M,\\qquad\\kappa\\neq 0,where the second equality usesM2=0M^\{2\}=0\. Then the update

Ht=Lt​Ht−1​R​\(ξt,Ht−1\)\+UtH\_\{t\}=L\_\{t\}H\_\{t\-1\}R\(\\xi\_\{t\},H\_\{t\-1\}\)\+U\_\{t\}contains the quadratic term

κ​⟨W,Ht−1⟩F​Lt​Ht−1​M,\\kappa\\langle W,H\_\{t\-1\}\\rangle\_\{F\}L\_\{t\}H\_\{t\-1\}M,so it cannot be represented by a finite block\-affine summary triple of the form\(L,R,U\)\(L,R,U\)\. This is a limitation of finite block\-affine scan summaries, not a limitation of SHiPPO as an online approximation framework: state\-coupled transport can be defined pathwise, but it does not generically preserve the finite summary algebra used by the scan\-compatible realization\.

### C\.3Collapse under fixed simultaneous block reduction

We prove Proposition[3\.3](https://arxiv.org/html/2607.03055#S3.Thmassumption3)\. The result is a sufficient algebraic collapse criterion: it shows that right\-generator families preserving a fixed common nontrivial block decomposition reduce to independent scalar or blockwise transported banks after a static change of channel basis\. It is not a complete classification of all possible degenerate transport families\.

###### Proof of Proposition[3\.3](https://arxiv.org/html/2607.03055#S3.Thmassumption3)\.

Assume there exists a fixedQ∈G​L​\(P\)Q\\in GL\(P\)such that

Gm=Q​Λm​Q−1\(m=1,…,M\)G\_\{m\}=Q\\Lambda\_\{m\}Q^\{\-1\}\\qquad\(m=1,\\dots,M\)for allmm, where eachΛm\\Lambda\_\{m\}is either diagonal or shares a common block\-diagonal structure with the same fixed nontrivial block partition\. Define

AR,t=∑m=1Mρt,m​Gm\.A\_\{R,t\}=\\sum\_\{m=1\}^\{M\}\\rho\_\{t,m\}G\_\{m\}\.Then

AR,t=Q​\(∑m=1Mρt,m​Λm\)​Q−1=Q​Λt​Q−1,Λt:=∑m=1Mρt,m​Λm\.A\_\{R,t\}=Q\\left\(\\sum\_\{m=1\}^\{M\}\\rho\_\{t,m\}\\Lambda\_\{m\}\\right\)Q^\{\-1\}=Q\\Lambda\_\{t\}Q^\{\-1\},\\qquad\\Lambda\_\{t\}:=\\sum\_\{m=1\}^\{M\}\\rho\_\{t,m\}\\Lambda\_\{m\}\.By construction,Λt\\Lambda\_\{t\}is diagonal, or has the same fixed block\-diagonal structure\.

For the dense exponential choice,

Rt=exp⁡\(Δt​AR,t\),R\_\{t\}=\\exp\(\\Delta\_\{t\}A\_\{R,t\}\),we obtain

Rt=Q​exp⁡\(Δt​Λt\)​Q−1\.R\_\{t\}=Q\\exp\(\\Delta\_\{t\}\\Lambda\_\{t\}\)Q^\{\-1\}\.The matrixexp⁡\(Δt​Λt\)\\exp\(\\Delta\_\{t\}\\Lambda\_\{t\}\)is diagonal in the diagonal case and block\-diagonal with the same fixed blocks in the block case\.

The same conclusion holds for any fixed\-order split product formed from the same generators\. Indeed, each factor satisfies

exp⁡\(Δt​ρt,m​Gm\)=Q​exp⁡\(Δt​ρt,m​Λm\)​Q−1\.\\exp\(\\Delta\_\{t\}\\rho\_\{t,m\}G\_\{m\}\)=Q\\exp\(\\Delta\_\{t\}\\rho\_\{t,m\}\\Lambda\_\{m\}\)Q^\{\-1\}\.Thus any fixed\-order product of such factors has the form

∏m=1Mexp⁡\(Δt​ρt,m​Gm\)=Q​\(∏m=1Mexp⁡\(Δt​ρt,m​Λm\)\)​Q−1,\\prod\_\{m=1\}^\{M\}\\exp\(\\Delta\_\{t\}\\rho\_\{t,m\}G\_\{m\}\)=Q\\left\(\\prod\_\{m=1\}^\{M\}\\exp\(\\Delta\_\{t\}\\rho\_\{t,m\}\\Lambda\_\{m\}\)\\right\)Q^\{\-1\},with the products taken in the implemented fixed order\. The middle product remains diagonal or block\-diagonal with the same fixed block structure\.

Now define the transformed state

H~t:=Ht​Q,U~t:=Ut​Q\.\\widetilde\{H\}\_\{t\}:=H\_\{t\}Q,\\qquad\\widetilde\{U\}\_\{t\}:=U\_\{t\}Q\.The recurrence

Ht=Lt​Ht−1​Rt\+UtH\_\{t\}=L\_\{t\}H\_\{t\-1\}R\_\{t\}\+U\_\{t\}then becomes

H~t=Lt​H~t−1​R~t\+U~t,R~t:=Q−1​Rt​Q,\\widetilde\{H\}\_\{t\}=L\_\{t\}\\widetilde\{H\}\_\{t\-1\}\\widetilde\{R\}\_\{t\}\+\\widetilde\{U\}\_\{t\},\\qquad\\widetilde\{R\}\_\{t\}:=Q^\{\-1\}R\_\{t\}Q,whereR~t\\widetilde\{R\}\_\{t\}is diagonal or block\-diagonal with the same fixed block partition\.

IfR~t\\widetilde\{R\}\_\{t\}is diagonal, the columns ofH~t\\widetilde\{H\}\_\{t\}evolve independently, so the recurrence decomposes into independent scalar transported banks\. IfR~t\\widetilde\{R\}\_\{t\}is block\-diagonal with a fixed partition, then columns evolve independently across blocks, with mixing only within each block, yielding independent blockwise transported banks\.

This proves the recurrence\-level collapse\. The fixed basis changeQQappears only as a static change of channel coordinates: it can be absorbed into surrounding static projections or treated as a one\-time change of channel basis\. In particular, if the source is factorized asUt=bt​xt⊤U\_\{t\}=b\_\{t\}x\_\{t\}^\{\\top\}, then

Ut​Q=bt​\(Q⊤​xt\)⊤,U\_\{t\}Q=b\_\{t\}\(Q^\{\\top\}x\_\{t\}\)^\{\\top\},so the change of basis is absorbed into the group\-local input coordinates\. Thus the recurrence collapses, after the static basis changeQQ, to independent scalar or fixed\-block transported banks\. This proves the sufficient collapse criterion in Proposition[3\.3](https://arxiv.org/html/2607.03055#S3.Thmassumption3): token\-dependent coefficients over a simultaneously reducible generator family do not yield genuinely coupled transported memory across the fixed blocks\. ∎

### C\.4Closed\-form right actions for split\-flow factors

This subsection records closed\-form exponentials and right\-actions for the structured factors used in the split\-flow generator \([2](https://arxiv.org/html/2607.03055#S3.E2)\)\. We use matrix exponentials in the standard matrix\-function sense\[[28](https://arxiv.org/html/2607.03055#bib.bib74)\]\. These formulas define the optional structured backend used to construct the per\-token actionRtR\_\{t\}\. The dense frozen\-ODE backend uses

Rtden=exp⁡\(Δt​AR,t\)R\_\{t\}^\{\\mathrm\{den\}\}=\\exp\(\\Delta\_\{t\}A\_\{R,t\}\)and is always invertible\. A split backend defines the implemented right actionRtsplitR\_\{t\}^\{\\mathrm\{split\}\}as a product of factor exponentials and also preserves invertibility because each factor below is invertible\.

For the dissipative diagonal component in \([2](https://arxiv.org/html/2607.03055#S3.E2)\), the diagonal factor is

Dtdiag=−Δt​Diag⁡\(dt\),dt∈ℝ\+P\.D\_\{t\}^\{\\mathrm\{diag\}\}=\-\\Delta\_\{t\}\\operatorname\{Diag\}\(d\_\{t\}\),\\qquad d\_\{t\}\\in\\mathbb\{R\}\_\{\+\}^\{P\}\.Hence

exp⁡\(Dtdiag\)=Diag⁡\(e−Δt​dt\),\\exp\(D\_\{t\}^\{\\mathrm\{diag\}\}\)=\\operatorname\{Diag\}\\bigl\(e^\{\-\\Delta\_\{t\}d\_\{t\}\}\\bigr\),whose entries lie in\(0,1\]\(0,1\]forΔt≥0\\Delta\_\{t\}\\geq 0\. This is the sense in which the diagonal component is dissipative\. The proposition below records the more general diagonal scaling formula used by the implementation\.

###### Proposition C\.2\(Closed forms and cheap right\-actions\)\.

LetH∈ℝN×PH\\in\\mathbb\{R\}^\{N\\times P\}\. The following right\-actions have closed forms\.

1. 1\.Diagonal scalings\.IfD=Diag​\(δ\)D=\\mathrm\{Diag\}\(\\delta\), then exp⁡\(D\)=Diag​\(exp⁡\(δ\)\)\.\\exp\(D\)=\\mathrm\{Diag\}\(\\exp\(\\delta\)\)\.ThusH​exp⁡\(D\)H\\exp\(D\)costsO​\(N​P\)O\(NP\), and the inverse isexp⁡\(−D\)\\exp\(\-D\)\.
2. 2\.Two\-plane rotations\.Fix distinct indicesi≠ji\\neq j, and define the skew\-symmetric generator Ωi​j:=ej​ei⊤−ei​ej⊤\.\\Omega\_\{ij\}:=e\_\{j\}e\_\{i\}^\{\\top\}\-e\_\{i\}e\_\{j\}^\{\\top\}\.ThenΩi​j⊤=−Ωi​j\\Omega\_\{ij\}^\{\\top\}=\-\\Omega\_\{ij\}, andexp⁡\(ϕ​Ωi​j\)\\exp\(\\phi\\Omega\_\{ij\}\)acts nontrivially only on columnsiiandjj: \(H:,iH:,j\)↦\(H:,iH:,j\)​\(cos⁡ϕ−sin⁡ϕsin⁡ϕcos⁡ϕ\)\.\\begin\{pmatrix\}H\_\{:,i\}&H\_\{:,j\}\\end\{pmatrix\}\\mapsto\\begin\{pmatrix\}H\_\{:,i\}&H\_\{:,j\}\\end\{pmatrix\}\\begin\{pmatrix\}\\cos\\phi&\-\\sin\\phi\\\\ \\sin\\phi&\\cos\\phi\\end\{pmatrix\}\.The right\-action costsO​\(N\)O\(N\)\. The inverse is exp\(−ϕΩi​j\)=exp\(ϕΩi​j\)⊤\.\\exp\(\-\\phi\\Omega\_\{ij\}\)=\\exp\(\\phi\\Omega\_\{ij\}\)^\{\\top\}\.
3. 3\.Nilpotent shears\.Fixi≠ji\\neq j, and define Ni​j:=ei​ej⊤\.N\_\{ij\}:=e\_\{i\}e\_\{j\}^\{\\top\}\.ThenNi​j2=0N\_\{ij\}^\{2\}=0, so exp⁡\(η​Ni​j\)=I\+η​Ni​j\.\\exp\(\\eta N\_\{ij\}\)=I\+\\eta N\_\{ij\}\.Moreover, H​exp⁡\(η​Ni​j\)=H\+η​H​Ni​j,H​Ni​j=H​ei​ej⊤,H\\exp\(\\eta N\_\{ij\}\)=H\+\\eta HN\_\{ij\},\\qquad HN\_\{ij\}=He\_\{i\}e\_\{j\}^\{\\top\},which addsη\\etatimes columniiinto columnjj\. The cost isO​\(N\)O\(N\), and the inverse isI−η​Ni​jI\-\\eta N\_\{ij\}\.
4. 4\.Rank\-one exponentials\.LetA=u​v⊤∈ℝP×PA=uv^\{\\top\}\\in\\mathbb\{R\}^\{P\\times P\}be rank one\. Since A2=\(v⊤​u\)​A,A^\{2\}=\(v^\{\\top\}u\)A,for any scalarss, exp⁡\(s​u​v⊤\)=I\+φ​\(s​v⊤​u\)​s​u​v⊤,\\exp\(suv^\{\\top\}\)=I\+\\varphi\(s\\,v^\{\\top\}u\)\\,suv^\{\\top\},where φ​\(κ\):=\{eκ−1κ,κ≠0,1,κ=0\.\\varphi\(\\kappa\):=\\begin\{cases\}\\dfrac\{e^\{\\kappa\}\-1\}\{\\kappa\},&\\kappa\\neq 0,\\\\\[7\.5pt\] 1,&\\kappa=0\.\\end\{cases\}Consequently, H​exp⁡\(s​u​v⊤\)=H\+\(φ​\(s​v⊤​u\)​s\)​\(H​u\)​v⊤\.H\\exp\(suv^\{\\top\}\)=H\+\\bigl\(\\varphi\(s\\,v^\{\\top\}u\)\\,s\\bigr\)\(Hu\)v^\{\\top\}\.The right\-action costsO​\(N​P\)O\(NP\)\. The inverse is obtained by replacingsswith−s\-s\.

###### Proof\.

The diagonal case is immediate from entrywise exponentiation\.

For the two\-plane rotation, the restriction ofΩi​j\\Omega\_\{ij\}to the\(i,j\)\(i,j\)\-plane is

\(0−110\),\\begin\{pmatrix\}0&\-1\\\\ 1&0\\end\{pmatrix\},whose exponential is

\(cos⁡ϕ−sin⁡ϕsin⁡ϕcos⁡ϕ\)\.\\begin\{pmatrix\}\\cos\\phi&\-\\sin\\phi\\\\ \\sin\\phi&\\cos\\phi\\end\{pmatrix\}\.All other coordinates are fixed\.

For the nilpotent shear,Ni​j2=0N\_\{ij\}^\{2\}=0, so the exponential series truncates:

exp⁡\(η​Ni​j\)=I\+η​Ni​j\.\\exp\(\\eta N\_\{ij\}\)=I\+\\eta N\_\{ij\}\.Right multiplication byNi​jN\_\{ij\}maps columniiinto columnjj, giving the stated update\.

For the rank\-one exponential, the identity

\(u​v⊤\)k=\(v⊤​u\)k−1​u​v⊤\(k≥1\)\(uv^\{\\top\}\)^\{k\}=\(v^\{\\top\}u\)^\{k\-1\}uv^\{\\top\}\\qquad\(k\\geq 1\)gives the desired formula\. Ifv⊤​u≠0v^\{\\top\}u\\neq 0, then

exp⁡\(s​u​v⊤\)=I\+∑k=1∞sk​\(v⊤​u\)k−1k\!​u​v⊤=I\+es​v⊤​u−1v⊤​u​u​v⊤=I\+φ​\(s​v⊤​u\)​s​u​v⊤\.\\exp\(suv^\{\\top\}\)=I\+\\sum\_\{k=1\}^\{\\infty\}\\frac\{s^\{k\}\(v^\{\\top\}u\)^\{k\-1\}\}\{k\!\}uv^\{\\top\}=I\+\\frac\{e^\{sv^\{\\top\}u\}\-1\}\{v^\{\\top\}u\}uv^\{\\top\}=I\+\\varphi\(sv^\{\\top\}u\)\\,suv^\{\\top\}\.Ifv⊤​u=0v^\{\\top\}u=0, then\(u​v⊤\)2=0\(uv^\{\\top\}\)^\{2\}=0, so

exp⁡\(s​u​v⊤\)=I\+s​u​v⊤,\\exp\(suv^\{\\top\}\)=I\+suv^\{\\top\},which is the same formula with the continuous extensionφ​\(0\)=1\\varphi\(0\)=1\. Right multiplyingHHgives the displayed formula\. ∎

#### Numerical note\.

For small\|κ\|\|\\kappa\|, the scalar

φ​\(κ\)=eκ−1κ\\varphi\(\\kappa\)=\\frac\{e^\{\\kappa\}\-1\}\{\\kappa\}should be evaluated with a numerically stableexpm1\\mathrm\{expm1\}\-style implementation\.

### C\.5Lie–Trotter split products

A structured backend may define the discrete right actionRtR\_\{t\}as a fixed\-order product of cheap factor exponentials\. This product is the actual right action used by the discrete recurrence \([3](https://arxiv.org/html/2607.03055#S3.E3)\)\. Hence the block\-affine scan algebra remains exact for that chosen recurrence\. The lemma below quantifies only the local approximation error of the split product relative to the dense exponentialexp⁡\(Δt​AR,t\)\\exp\(\\Delta\_\{t\}A\_\{R,t\}\), as in standard first\-order splitting methods for differential equations\[[25](https://arxiv.org/html/2607.03055#bib.bib75)\]\.

###### Lemma C\.3\(First\-order Lie–Trotter local error\)\.

Let

A=∑k=1KAk,A=\\sum\_\{k=1\}^\{K\}A\_\{k\},and define the fixed\-order product

R~​\(Δ\):=exp⁡\(Δ​A1\)​exp⁡\(Δ​A2\)​⋯​exp⁡\(Δ​AK\)\.\\widetilde\{R\}\(\\Delta\):=\\exp\(\\Delta A\_\{1\}\)\\exp\(\\Delta A\_\{2\}\)\\cdots\\exp\(\\Delta A\_\{K\}\)\.If‖Ak‖≤M\\\|A\_\{k\}\\\|\\leq Mfor allkk, then for sufficiently smallΔ\\Delta,

‖R~​\(Δ\)−exp⁡\(Δ​A\)‖≤C​\(K,M\)​Δ2,\\bigl\\\|\\widetilde\{R\}\(\\Delta\)\-\\exp\(\\Delta A\)\\bigr\\\|\\leq C\(K,M\)\\Delta^\{2\},whereC​\(K,M\)C\(K,M\)depends only onKKandMM\.

###### Proof\.

Use a submultiplicative matrix norm\. ForΔ≤1\\Delta\\leq 1, Taylor’s theorem gives

exp⁡\(Δ​Ak\)=I\+Δ​Ak\+Ek​\(Δ\),‖Ek​\(Δ\)‖≤c​\(M\)​Δ2\\exp\(\\Delta A\_\{k\}\)=I\+\\Delta A\_\{k\}\+E\_\{k\}\(\\Delta\),\\qquad\\\|E\_\{k\}\(\\Delta\)\\\|\\leq c\(M\)\\Delta^\{2\}for a constantc​\(M\)c\(M\)depending only onMM\. Multiplying theKKfactors in this fixed order, all first\-order terms add and all products containing at least two first\-order or remainder terms areOK,M​\(Δ2\)O\_\{K,M\}\(\\Delta^\{2\}\)\. Hence

R~​\(Δ\)=I\+Δ​∑k=1KAk\+Eprod​\(Δ\),‖Eprod​\(Δ\)‖≤c1​\(K,M\)​Δ2\.\\widetilde\{R\}\(\\Delta\)=I\+\\Delta\\sum\_\{k=1\}^\{K\}A\_\{k\}\+E\_\{\\mathrm\{prod\}\}\(\\Delta\),\\qquad\\\|E\_\{\\mathrm\{prod\}\}\(\\Delta\)\\\|\\leq c\_\{1\}\(K,M\)\\Delta^\{2\}\.Similarly,

exp⁡\(Δ​A\)=I\+Δ​A\+Eexp​\(Δ\),‖Eexp​\(Δ\)‖≤c2​\(K,M\)​Δ2,\\exp\(\\Delta A\)=I\+\\Delta A\+E\_\{\\mathrm\{exp\}\}\(\\Delta\),\\qquad\\\|E\_\{\\mathrm\{exp\}\}\(\\Delta\)\\\|\\leq c\_\{2\}\(K,M\)\\Delta^\{2\},because

‖A‖≤∑k=1K‖Ak‖≤K​M\.\\\|A\\\|\\leq\\sum\_\{k=1\}^\{K\}\\\|A\_\{k\}\\\|\\leq KM\.SinceA=∑kAkA=\\sum\_\{k\}A\_\{k\}, subtracting the two expansions gives

‖R~​\(Δ\)−exp⁡\(Δ​A\)‖≤\(c1​\(K,M\)\+c2​\(K,M\)\)​Δ2\.\\bigl\\\|\\widetilde\{R\}\(\\Delta\)\-\\exp\(\\Delta A\)\\bigr\\\|\\leq\\bigl\(c\_\{1\}\(K,M\)\+c\_\{2\}\(K,M\)\\bigr\)\\Delta^\{2\}\.TakingC​\(K,M\)=c1​\(K,M\)\+c2​\(K,M\)C\(K,M\)=c\_\{1\}\(K,M\)\+c\_\{2\}\(K,M\)proves the claim\. ∎

### C\.6Proof of Theorem[3\.4](https://arxiv.org/html/2607.03055#S3.Thmassumption4): exponential\-adjusted discrete cell

We now prove Theorem[3\.4](https://arxiv.org/html/2607.03055#S3.Thmassumption4)\. The derivation below establishes only the dense\-backend frozen\-ODE interpretation, corresponding to the choice

Rt=Rtden=exp⁡\(Δt​AR,t\)\.R\_\{t\}=R\_\{t\}^\{\\mathrm\{den\}\}=\\exp\(\\Delta\_\{t\}A\_\{R,t\}\)\.If a structured backend instead definesRt=RtsplitR\_\{t\}=R\_\{t\}^\{\\mathrm\{split\}\}by a Lie–Trotter product, the same algebraic update is exact for that chosen split\-flow recurrence, while Lemma[C\.3](https://arxiv.org/html/2607.03055#A3.Thmassumption3)describes only the product’s local approximation to the dense exponential flow\. A comparison of the split\-backend update with the dense frozen ODE must therefore add the split\-backend replacement error recorded in Remark[C\.5](https://arxiv.org/html/2607.03055#A3.Thmassumption5)\.

###### Proof of Theorem[3\.4](https://arxiv.org/html/2607.03055#S3.Thmassumption4)\.

Consider one step of lengthΔt\>0\\Delta\_\{t\}\>0\. For the local discretization analysis, freeze the left driftat∈ℝNa\_\{t\}\\in\\mathbb\{R\}^\{N\}and the right generatorAR,t∈ℝP×PA\_\{R,t\}\\in\\mathbb\{R\}^\{P\\times P\}on the interval\[t−Δt,t\]\[t\-\\Delta\_\{t\},t\]\. Let the source vary over the step:

U​\(τ\)∈ℝN×P\.U\(\\tau\)\\in\\mathbb\{R\}^\{N\\times P\}\.The step\-frozen restricted dynamics are

H˙​\(τ\)=Diag⁡\(at\)​H​\(τ\)\+U​\(τ\)\+H​\(τ\)​AR,t,τ∈\[t−Δt,t\]\.\\dot\{H\}\(\\tau\)=\\operatorname\{Diag\}\(a\_\{t\}\)H\(\\tau\)\+U\(\\tau\)\+H\(\\tau\)A\_\{R,t\},\\qquad\\tau\\in\[t\-\\Delta\_\{t\},t\]\.\(11\)Define

Lt:=exp⁡\(Δt​Diag⁡\(at\)\)=Diag⁡\(αt\),Rtden:=exp⁡\(Δt​AR,t\),L\_\{t\}:=\\exp\\\!\\bigl\(\\Delta\_\{t\}\\operatorname\{Diag\}\(a\_\{t\}\)\\bigr\)=\\operatorname\{Diag\}\(\\alpha\_\{t\}\),\\qquad R\_\{t\}^\{\\mathrm\{den\}\}:=\\exp\(\\Delta\_\{t\}A\_\{R,t\}\),whereαt:=exp⁡\(Δt​at\)\\alpha\_\{t\}:=\\exp\(\\Delta\_\{t\}a\_\{t\}\)entrywise\. By variation of constants for the two\-sided linear equation \([11](https://arxiv.org/html/2607.03055#A3.E11)\), the exact dense frozen\-ODE step is

Ht⋆=Lt​Ht−1​Rtden\+∫0Δtexp⁡\(\(Δt−s\)​Diag⁡\(at\)\)​U​\(t−Δt\+s\)​exp⁡\(\(Δt−s\)​AR,t\)​𝑑s,H\_\{t\}^\{\\star\}=L\_\{t\}H\_\{t\-1\}R\_\{t\}^\{\\mathrm\{den\}\}\+\\int\_\{0\}^\{\\Delta\_\{t\}\}\\exp\\\!\\bigl\(\(\\Delta\_\{t\}\-s\)\\operatorname\{Diag\}\(a\_\{t\}\)\\bigr\)U\(t\-\\Delta\_\{t\}\+s\)\\exp\\\!\\bigl\(\(\\Delta\_\{t\}\-s\)A\_\{R,t\}\\bigr\)\\,ds,\(12\)where

Ht−1:=H​\(t−Δt\),Ht⋆:=H​\(t\)\.H\_\{t\-1\}:=H\(t\-\\Delta\_\{t\}\),\\qquad H\_\{t\}^\{\\star\}:=H\(t\)\.Indeed, this follows by multiplying on the left byexp⁡\(−\(τ−\(t−Δt\)\)​Diag⁡\(at\)\)\\exp\(\-\(\\tau\-\(t\-\\Delta\_\{t\}\)\)\\operatorname\{Diag\}\(a\_\{t\}\)\)and on the right byexp⁡\(−\(τ−\(t−Δt\)\)​AR,t\)\\exp\(\-\(\\tau\-\(t\-\\Delta\_\{t\}\)\)A\_\{R,t\}\), differentiating the transformed quantity, and integrating over the interval\.

Define the Duhamel integrand

G​\(s\):=exp⁡\(\(Δt−s\)​Diag⁡\(at\)\)​U​\(t−Δt\+s\)​exp⁡\(\(Δt−s\)​AR,t\),s∈\[0,Δt\]\.G\(s\):=\\exp\\\!\\bigl\(\(\\Delta\_\{t\}\-s\)\\operatorname\{Diag\}\(a\_\{t\}\)\\bigr\)U\(t\-\\Delta\_\{t\}\+s\)\\exp\\\!\\bigl\(\(\\Delta\_\{t\}\-s\)A\_\{R,t\}\\bigr\),\\qquad s\\in\[0,\\Delta\_\{t\}\]\.Using endpoint source samples

Ut−1=U​\(t−Δt\),Ut=U​\(t\),U\_\{t\-1\}=U\(t\-\\Delta\_\{t\}\),\\qquad U\_\{t\}=U\(t\),or discrete approximations interpreted as these endpoint source values, we have

G​\(0\)=Lt​Ut−1​Rtden,G​\(Δt\)=Ut\.G\(0\)=L\_\{t\}U\_\{t\-1\}R\_\{t\}^\{\\mathrm\{den\}\},\\qquad G\(\\Delta\_\{t\}\)=U\_\{t\}\.Fort=1t=1withβ1≠0\\beta\_\{1\}\\neq 0, the left endpoint requires an explicit boundary source value representingU​\(0\)U\(0\); the implementation conventionU0:=0U\_\{0\}:=0corresponds to the assumptionU​\(0\)=0U\(0\)=0or to an intended zero boundary source\.

Approximate the integral in \([12](https://arxiv.org/html/2607.03055#A3.E12)\) by the two\-point family

∫0ΔtG​\(s\)​𝑑s≈\(1−λt\)​Δt​G​\(0\)\+λt​Δt​G​\(Δt\),λt∈\[0,1\]\.\\int\_\{0\}^\{\\Delta\_\{t\}\}G\(s\)\\,ds\\approx\(1\-\\lambda\_\{t\}\)\\Delta\_\{t\}G\(0\)\+\\lambda\_\{t\}\\Delta\_\{t\}G\(\\Delta\_\{t\}\),\\qquad\\lambda\_\{t\}\\in\[0,1\]\.This gives

∫0ΔtG​\(s\)​𝑑s≈\(1−λt\)​Δt​Lt​Ut−1​Rtden\+λt​Δt​Ut\.\\int\_\{0\}^\{\\Delta\_\{t\}\}G\(s\)\\,ds\\approx\(1\-\\lambda\_\{t\}\)\\Delta\_\{t\}L\_\{t\}U\_\{t\-1\}R\_\{t\}^\{\\mathrm\{den\}\}\+\\lambda\_\{t\}\\Delta\_\{t\}U\_\{t\}\.Define the dense\-backend source update

U^tden:=\(1−λt\)​Δt​Lt​Ut−1​Rtden\+λt​Δt​Ut\.\\widehat\{U\}\_\{t\}^\{\\mathrm\{den\}\}:=\(1\-\\lambda\_\{t\}\)\\Delta\_\{t\}L\_\{t\}U\_\{t\-1\}R\_\{t\}^\{\\mathrm\{den\}\}\+\\lambda\_\{t\}\\Delta\_\{t\}U\_\{t\}\.The corresponding dense discrete update is

Htden:=Lt​Ht−1​Rtden\+U^tden,H\_\{t\}^\{\\mathrm\{den\}\}:=L\_\{t\}H\_\{t\-1\}R\_\{t\}^\{\\mathrm\{den\}\}\+\\widehat\{U\}\_\{t\}^\{\\mathrm\{den\}\},which is exactly the dense\-backend instance of the discrete cell \([3](https://arxiv.org/html/2607.03055#S3.E3)\)\. For a split backend, the same algebraic cell is executed with the implemented right actionRtsplitR\_\{t\}^\{\\mathrm\{split\}\}, but that is a statement about the chosen discrete recurrence rather than the dense frozen\-ODE Duhamel step\.

Ifλt=1\\lambda\_\{t\}=1, then

U^tden=Δt​Ut,\\widehat\{U\}\_\{t\}^\{\\mathrm\{den\}\}=\\Delta\_\{t\}U\_\{t\},so the source is evaluated at the right endpoint\. This is the exponential\-Euler source rule\. Ifλt=12\+O​\(Δt\)\\lambda\_\{t\}=\\frac\{1\}\{2\}\+O\(\\Delta\_\{t\}\), the local quadrature behavior is the generalized exponential\-trapezoidal behavior stated in Corollary[C\.4](https://arxiv.org/html/2607.03055#A3.Thmassumption4)below\.

Finally, ifAR,t≡0A\_\{R,t\}\\equiv 0, thenRtden=IPR\_\{t\}^\{\\mathrm\{den\}\}=I\_\{P\}; any split backend formed from zero right\-generator factors also gives the identity right action\. The update becomes

Ht=Lt​Ht−1\+\(1−λt\)​Δt​Lt​Ut−1\+λt​Δt​Ut\.H\_\{t\}=L\_\{t\}H\_\{t\-1\}\+\(1\-\\lambda\_\{t\}\)\\Delta\_\{t\}L\_\{t\}U\_\{t\-1\}\+\\lambda\_\{t\}\\Delta\_\{t\}U\_\{t\}\.This is the corresponding one\-sided group\-tied diagonal\-left selective update\. If, in addition,P=1P=1andλt=1\\lambda\_\{t\}=1, then

Ht=Lt​Ht−1\+Δt​Ut,H\_\{t\}=L\_\{t\}H\_\{t\-1\}\+\\Delta\_\{t\}U\_\{t\},which is the one\-channel exponential\-Euler selective update associated with the diagonal\-left recurrence\. Forλt≠1\\lambda\_\{t\}\\neq 1, the neutral right\-action limit is still one\-sided, but it uses the two\-point source rule above rather than the standard right\-endpoint exponential\-Euler source rule\. ∎

### C\.7Local truncation behavior of the source quadrature

Theorem[3\.4](https://arxiv.org/html/2607.03055#S3.Thmassumption4)uses a two\-point quadrature rule for the dense Duhamel source integral\. This appendix\-level corollary records the local quadrature details omitted from Section[3\.4](https://arxiv.org/html/2607.03055#S3.SS4)for space\. It concerns source quadrature relative to the step\-frozen continuous dynamics, not the exact block\-affine scan of the resulting discrete recurrence\.

###### Corollary C\.4\(Dense\-backend local truncation behavior under frozen coefficients\)\.

AssumeRt=Rtden=exp⁡\(Δt​AR,t\)R\_\{t\}=R\_\{t\}^\{\\mathrm\{den\}\}=\\exp\(\\Delta\_\{t\}A\_\{R,t\}\), assumeata\_\{t\}andAR,tA\_\{R,t\}are frozen on\[t−Δt,t\]\[t\-\\Delta\_\{t\},t\], and assume the Duhamel integrandGGis sufficiently smooth\. Also assume that the discrete endpoint source samples represent the continuous endpoint values,

Ut−1=U​\(t−Δt\),Ut=U​\(t\)\.U\_\{t\-1\}=U\(t\-\\Delta\_\{t\}\),\\qquad U\_\{t\}=U\(t\)\.Fort=1t=1with hard\-wiredU0=0U\_\{0\}=0andβ1≠0\\beta\_\{1\}\\neq 0, this means assumingU​\(0\)=0U\(0\)=0or an explicit zero boundary source for the left endpoint\. LetHt⋆H\_\{t\}^\{\\star\}denote the exact step map \([12](https://arxiv.org/html/2607.03055#A3.E12)\), and letHtdenH\_\{t\}^\{\\mathrm\{den\}\}denote the dense discrete update from Theorem[3\.4](https://arxiv.org/html/2607.03055#S3.Thmassumption4)\. Then:

1. 1\.For anyλt∈\[0,1\]\\lambda\_\{t\}\\in\[0,1\], ‖Htden−Ht⋆‖≤Δt22​sups∈\[0,Δt\]‖G′​\(s\)‖\.\\\|H\_\{t\}^\{\\mathrm\{den\}\}\-H\_\{t\}^\{\\star\}\\\|\\leq\\frac\{\\Delta\_\{t\}^\{2\}\}\{2\}\\sup\_\{s\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\}\(s\)\\\|\.Thus the method is locally first\-order accurate inΔt\\Delta\_\{t\}\.
2. 2\.Ifλt=12\\lambda\_\{t\}=\\frac\{1\}\{2\}andG∈C2G\\in C^\{2\}, then ‖Htden−Ht⋆‖≤Δt312​sups∈\[0,Δt\]‖G′′​\(s\)‖\.\\\|H\_\{t\}^\{\\mathrm\{den\}\}\-H\_\{t\}^\{\\star\}\\\|\\leq\\frac\{\\Delta\_\{t\}^\{3\}\}\{12\}\\sup\_\{s\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\\prime\}\(s\)\\\|\.Thus the method is locally second\-order accurate inΔt\\Delta\_\{t\}\.
3. 3\.More generally, if \|λt−12\|≤c​Δt\\left\|\\lambda\_\{t\}\-\\frac\{1\}\{2\}\\right\|\\leq c\\Delta\_\{t\}for a constantcc, andG∈C2G\\in C^\{2\}, then ‖Htden−Ht⋆‖≤Δt3​\(112​sups∈\[0,Δt\]‖G′′​\(s\)‖\+c​sups∈\[0,Δt\]‖G′​\(s\)‖\)\.\\\|H\_\{t\}^\{\\mathrm\{den\}\}\-H\_\{t\}^\{\\star\}\\\|\\leq\\Delta\_\{t\}^\{3\}\\left\(\\frac\{1\}\{12\}\\sup\_\{s\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\\prime\}\(s\)\\\|\+c\\sup\_\{s\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\}\(s\)\\\|\\right\)\.Thusλt=12\+O​\(Δt\)\\lambda\_\{t\}=\\frac\{1\}\{2\}\+O\(\\Delta\_\{t\}\)preserves the same localO​\(Δt3\)O\(\\Delta\_\{t\}^\{3\}\)quadrature\-error order\.

###### Proof of Corollary[C\.4](https://arxiv.org/html/2607.03055#A3.Thmassumption4)\.

Let

I​\(G\):=∫0ΔtG​\(s\)​𝑑s,Qλ​\(G\):=\(1−λt\)​Δt​G​\(0\)\+λt​Δt​G​\(Δt\)\.I\(G\):=\\int\_\{0\}^\{\\Delta\_\{t\}\}G\(s\)\\,ds,\\qquad Q\_\{\\lambda\}\(G\):=\(1\-\\lambda\_\{t\}\)\\Delta\_\{t\}G\(0\)\+\\lambda\_\{t\}\\Delta\_\{t\}G\(\\Delta\_\{t\}\)\.By the definitions ofHt⋆H\_\{t\}^\{\\star\}andHtdenH\_\{t\}^\{\\mathrm\{den\}\},

Ht⋆−Htden=I​\(G\)−Qλ​\(G\)\.H\_\{t\}^\{\\star\}\-H\_\{t\}^\{\\mathrm\{den\}\}=I\(G\)\-Q\_\{\\lambda\}\(G\)\.
For the first claim, write

I​\(G\)−Qλ​\(G\)=\(1−λt\)​\(I​\(G\)−Δt​G​\(0\)\)\+λt​\(I​\(G\)−Δt​G​\(Δt\)\)\.I\(G\)\-Q\_\{\\lambda\}\(G\)=\(1\-\\lambda\_\{t\}\)\\bigl\(I\(G\)\-\\Delta\_\{t\}G\(0\)\\bigr\)\+\\lambda\_\{t\}\\bigl\(I\(G\)\-\\Delta\_\{t\}G\(\\Delta\_\{t\}\)\\bigr\)\.Using

G​\(s\)−G​\(0\)=∫0sG′​\(r\)​𝑑rG\(s\)\-G\(0\)=\\int\_\{0\}^\{s\}G^\{\\prime\}\(r\)\\,drgives

‖I​\(G\)−Δt​G​\(0\)‖≤∫0Δts​𝑑s​supr∈\[0,Δt\]‖G′​\(r\)‖=Δt22​supr∈\[0,Δt\]‖G′​\(r\)‖\.\\left\\\|I\(G\)\-\\Delta\_\{t\}G\(0\)\\right\\\|\\leq\\int\_\{0\}^\{\\Delta\_\{t\}\}s\\,ds\\,\\sup\_\{r\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\}\(r\)\\\|=\\frac\{\\Delta\_\{t\}^\{2\}\}\{2\}\\sup\_\{r\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\}\(r\)\\\|\.Similarly,

‖I​\(G\)−Δt​G​\(Δt\)‖≤Δt22​supr∈\[0,Δt\]‖G′​\(r\)‖\.\\left\\\|I\(G\)\-\\Delta\_\{t\}G\(\\Delta\_\{t\}\)\\right\\\|\\leq\\frac\{\\Delta\_\{t\}^\{2\}\}\{2\}\\sup\_\{r\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\}\(r\)\\\|\.Sinceλt∈\[0,1\]\\lambda\_\{t\}\\in\[0,1\], the first bound follows\.

Forλt=12\\lambda\_\{t\}=\\frac\{1\}\{2\},QλQ\_\{\\lambda\}is the trapezoidal rule\. The standard trapezoidal remainder for a Banach\-space\-valuedC2C^\{2\}function gives

‖I​\(G\)−Q1/2​\(G\)‖≤Δt312​sups∈\[0,Δt\]‖G′′​\(s\)‖\.\\\|I\(G\)\-Q\_\{1/2\}\(G\)\\\|\\leq\\frac\{\\Delta\_\{t\}^\{3\}\}\{12\}\\sup\_\{s\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\\prime\}\(s\)\\\|\.
For the final claim, write

Qλt​\(G\)−Q1/2​\(G\)=\(λt−12\)​Δt​\(G​\(Δt\)−G​\(0\)\)\.Q\_\{\\lambda\_\{t\}\}\(G\)\-Q\_\{1/2\}\(G\)=\\left\(\\lambda\_\{t\}\-\\frac\{1\}\{2\}\\right\)\\Delta\_\{t\}\\bigl\(G\(\\Delta\_\{t\}\)\-G\(0\)\\bigr\)\.Therefore, if

\|λt−12\|≤c​Δt,\\left\|\\lambda\_\{t\}\-\\frac\{1\}\{2\}\\right\|\\leq c\\Delta\_\{t\},then

‖Qλt​\(G\)−Q1/2​\(G\)‖≤c​Δt2​∫0Δt‖G′​\(s\)‖​𝑑s≤c​Δt3​sups∈\[0,Δt\]‖G′​\(s\)‖\.\\\|Q\_\{\\lambda\_\{t\}\}\(G\)\-Q\_\{1/2\}\(G\)\\\|\\leq c\\Delta\_\{t\}^\{2\}\\int\_\{0\}^\{\\Delta\_\{t\}\}\\\|G^\{\\prime\}\(s\)\\\|\\,ds\\leq c\\Delta\_\{t\}^\{3\}\\sup\_\{s\\in\[0,\\Delta\_\{t\}\]\}\\\|G^\{\\prime\}\(s\)\\\|\.Combining this with the trapezoidal bound proves the result\. ∎

## Appendix DFormal Diagnostics and Experimental Protocol

This appendix specifies the controlled diagnostics used in Section[4](https://arxiv.org/html/2607.03055#S4)\. The purpose of these experiments is narrow: to separate high\-rank source/write capacity from future right transport of memory that has already been written\. The diagnostics are synthetic by design, but they remove direct source\-injection shortcuts and expose the mechanism that is central to the paper\.

### D\.1Diagnostic scope

The paper\-facing diagnostics have four roles\. The paired noncommutative transport diagnostic is the main separation test: it isolates whether high\-rank current\-step source writes can replace future right transport of already\-written memory\. The right\-transport parameterization/intervention study is also main evidence: it tests oracle, learned, and selective right transports together with theRt→IR\_\{t\}\\\!\\to Ievaluation\-time intervention\. The transported\-projection prediction task is auxiliary, because it checks a related transported\-projection quantity but does not include the paired\-difference intervention metric\. The group\-local audit is a limitation diagnostic, used only to clarify how the scan\-compatible group\-local restriction differs from full right transport\.

### D\.2Paired noncommutative transport task

The primary diagnostic tests whether high\-rank current\-step source writes can replace future right transport of memory written earlier\. Each example first writes a payload vectorv∈ℝdv\\in\\mathbb\{R\}^\{d\}, then applies two operation tokens\. The paired examples share the same payload and operation parameters but reverse the operation order,a​babversusb​aba\. In the minimal setting used for the main diagnostic, the two right actions are generated by the elementary shears

Ra​\(α\)=I\+α​e1​e2⊤,Rb​\(β\)=I\+β​e2​e3⊤,R\_\{a\}\(\\alpha\)=I\+\\alpha e\_\{1\}e\_\{2\}^\{\\top\},\\qquad R\_\{b\}\(\\beta\)=I\+\\beta e\_\{2\}e\_\{3\}^\{\\top\},\(13\)so that

Ra​\(α\)​Rb​\(β\)−Rb​\(β\)​Ra​\(α\)=α​β​e1​e3⊤\.R\_\{a\}\(\\alpha\)R\_\{b\}\(\\beta\)\-R\_\{b\}\(\\beta\)R\_\{a\}\(\\alpha\)=\\alpha\\beta e\_\{1\}e\_\{3\}^\{\\top\}\.\(14\)The paired target is

Δtrue=v⊤​\(Ra​Rb−Rb​Ra\),\\Delta\_\{\\mathrm\{true\}\}=v^\{\\top\}\\bigl\(R\_\{a\}R\_\{b\}\-R\_\{b\}R\_\{a\}\\bigr\),\(15\)and the model predictionΔpred\\Delta\_\{\\mathrm\{pred\}\}is the difference between its outputs on the two paired examples\. The main metric is

Pair⁡Δ​NMSE=∥Δpred−Δtrue∥22max⁡\{∥Δtrue∥22,ϵden\},ϵden=10−8\.\\operatorname\{Pair\}\\Delta\\operatorname\{NMSE\}=\\frac\{\\lVert\\Delta\_\{\\mathrm\{pred\}\}\-\\Delta\_\{\\mathrm\{true\}\}\\rVert\_\{2\}^\{2\}\}\{\\max\\\{\\lVert\\Delta\_\{\\mathrm\{true\}\}\\rVert\_\{2\}^\{2\},\\epsilon\_\{\\mathrm\{den\}\}\\\}\},\\qquad\\epsilon\_\{\\mathrm\{den\}\}=10^\{\-8\}\.\(16\)A value near one means that the model predicts essentially no paired noncommutative difference; a value near zero means that it recovers the transport\-order difference\. The denominator floor is fixed before training and is used only to avoid numerical instability on near\-zero paired differences\.

The key constraint is that source injection is allowed only at WRITE tokens\. Operation tokens do not inject payload\-dependent source\. Thus a no\-right rank\-rrsource/write model can increase the rank of what it writes at the current step, but it cannot implement a later operation that right\-multiplies already\-written channel coordinates\. This is the intended separation from SHiPPO\-style right transport\.

The task schematic and aggregate results are shown in Figure[2](https://arxiv.org/html/2607.03055#S4.F2); this appendix gives the formal task definition and sampling details\.

#### Sampling\.

For the paired\-transport runs,d=4d=4, the state size isN=16N=16, and the operation family hasK=2K=2generators\. Payloads are sampled asv∼𝒩​\(0,Id/d\)v\\sim\\mathcal\{N\}\(0,I\_\{d\}/d\), and operation coefficients are sampled uniformly from\[−0\.5,0\.5\]\[\-0\.5,0\.5\]\. The paired evaluation uses the samev,α,βv,\\alpha,\\betafor the two orders, so that the measured difference isolates the order\-dependent commutator term\.

### D\.3Model families and interventions

All formal diagnostics use a controlled matrix\-state family with memory stateHt∈ℝN×dH\_\{t\}\\in\\mathbb\{R\}^\{N\\times d\}\. The no\-right rank\-rrsource/write baseline uses

Ht=Lt​Ht−1\+U^t\(r\),U^t\(r\)=1r​Bt​Xt⊤,H\_\{t\}=L\_\{t\}H\_\{t\-1\}\+\\widehat\{U\}\_\{t\}^\{\(r\)\},\\qquad\\widehat\{U\}\_\{t\}^\{\(r\)\}=\\frac\{1\}\{\\sqrt\{r\}\}B\_\{t\}X\_\{t\}^\{\\top\},\(17\)where increasingrrincreases the rank of the current source update\. This is the controlled high\-rank write/read baseline\.

SHiPPO\-style variants add a right transport:

Ht=Lt​Ht−1​Rt\+U^t\.H\_\{t\}=L\_\{t\}H\_\{t\-1\}R\_\{t\}\+\\widehat\{U\}\_\{t\}\.\(18\)The right action is either oracle\-provided, learned as a Lie\-style generator parameterization, or generated by a selective controller\. We evaluate oracle\-RR, learned\-RRinitialized from true, zero, or random generators, and selective\-RR\. For right\-transport models we also report an evaluation\-timeRt→IR\_\{t\}\\\!\\to\\\!Iintervention: the right action is replaced by identity while all other learned weights remain fixed\. If the paired\-difference signal disappears under this intervention, the behavior is mediated by right transport rather than by a static readout or source\-only shortcut\.

### D\.4Training and evaluation protocol

All reported values are means and sample standard deviations over independent training seeds\. Training uses AdamW\[[37](https://arxiv.org/html/2607.03055#bib.bib78)\]with learning rate3×10−43\\times 10^\{\-4\}, weight decay10−210^\{\-2\}, gradient clipping at1\.01\.0, normalized MSE loss, and evaluation every 1000 steps\. We report the final training step; no validation\-based checkpoint selection is used for the paper\-facing tables\. The transported\-projection prediction task uses generator seed 321, coefficient range 0\.15, andεproj=0\.05\\varepsilon\_\{\\mathrm\{proj\}\}=0\.05in the task generator\. The paired\-transport diagnostics use coefficient range 0\.5\. Table[2](https://arxiv.org/html/2607.03055#A4.T2)records the main settings\.

Table 2:Training and evaluation settings for the formal diagnostics\. All runs use20,00020\{,\}000training steps\. We usen=5n=5seeds except for the group\-local audit, which usesn=3n=3\. The “Eval batches” column records the number of evaluation batches\.DiagnosticBatchEvalbatchesKey settingsPaired noncommutative transport12816d=4d=4,N=16N=16,K=2K=2, depth 2; WRITE\-only source gate; pair\-loss weight 5\.0; no\-right ranksr∈\{1,2,4,8\}r\\in\\\{1,2,4,8\\\}\.Right\-transport parameterizations/intervention12816Same paired\-transport task; compares no\-rightr=8r=8, oracle\-RR, learned\-RR, and selective\-RR\.Transported\-projection prediction328d=16d=16,N=16N=16,K=16K=16, length 256; skew–nilpotent generator family; source rankr=4r=4, fixed left operator\.Group\-local audit1288Paired\-transport task with group sizes 2 and 4; oracle\-RRand learned\-RRtrue initialization; audit batch size 512\.
### D\.5Full numerical summaries

Tables[3](https://arxiv.org/html/2607.03055#A4.T3)–[5](https://arxiv.org/html/2607.03055#A4.T5)give the numerical values underlying Figure[2](https://arxiv.org/html/2607.03055#S4.F2)and the auxiliary transported\-projection result\. All entries in these three tables are means±\\pmsample standard deviations overn=5n=5independent training seeds\.

Table 3:Paired noncommutative transport diagnostic\. Entries are means±\\pmsample standard deviations overn=5n=5seeds\.ModelEvalNMSE↓\\downarrowPairΔ\\DeltaNMSE↓\\downarrowPairΔ\\DeltaR2R^\{2\}↑\\uparrowR→IR\\\!\\to\\\!IPairΔ\\DeltaNMSE↓\\downarrowOracle solver0±0\\pm\\,00±0\\pm\\,011±0\\pm\\,0–No\-rightr=1r=14\.145×10−44\.145\\\!\\times\\\!10^\{\-4\}±5\.33×10−5\\pm\\,5\.33\\\!\\times\\\!10^\{\-5\}11±4\.07×10−7\\pm\\,4\.07\\\!\\times\\\!10^\{\-7\}−1\.52×10−3\-1\.52\\\!\\times\\\!10^\{\-3\}±2\.165×10−3\\pm\\,2\.165\\\!\\times\\\!10^\{\-3\}–No\-rightr=2r=23\.568×10−43\.568\\\!\\times\\\!10^\{\-4\}±4\.38×10−5\\pm\\,4\.38\\\!\\times\\\!10^\{\-5\}11±3\.22×10−7\\pm\\,3\.22\\\!\\times\\\!10^\{\-7\}−1\.52×10−3\-1\.52\\\!\\times\\\!10^\{\-3\}±2\.166×10−3\\pm\\,2\.166\\\!\\times\\\!10^\{\-3\}–No\-rightr=4r=43\.389×10−43\.389\\\!\\times\\\!10^\{\-4\}±4\.43×10−5\\pm\\,4\.43\\\!\\times\\\!10^\{\-5\}11±3\.11×10−7\\pm\\,3\.11\\\!\\times\\\!10^\{\-7\}−1\.52×10−3\-1\.52\\\!\\times\\\!10^\{\-3\}±2\.165×10−3\\pm\\,2\.165\\\!\\times\\\!10^\{\-3\}–No\-rightr=8r=82\.797×10−42\.797\\\!\\times\\\!10^\{\-4\}±2\.31×10−5\\pm\\,2\.31\\\!\\times\\\!10^\{\-5\}11±2\.02×10−7\\pm\\,2\.02\\\!\\times\\\!10^\{\-7\}−1\.52×10−3\-1\.52\\\!\\times\\\!10^\{\-3\}±2\.165×10−3\\pm\\,2\.165\\\!\\times\\\!10^\{\-3\}–SHiPPO oracle\-RR4\.377×10−34\.377\\\!\\times\\\!10^\{\-3\}±9\.788×10−3\\pm\\,9\.788\\\!\\times\\\!10^\{\-3\}2\.44×10−72\.44\\\!\\times\\\!10^\{\-7\}±4\.61×10−7\\pm\\,4\.61\\\!\\times\\\!10^\{\-7\}11±4\.66×10−7\\pm\\,4\.66\\\!\\times\\\!10^\{\-7\}11±0\\pm\\,0SHiPPO learned\-RR\(true init\)7\.019×10−47\.019\\\!\\times\\\!10^\{\-4\}±1\.18×10−3\\pm\\,1\.18\\\!\\times\\\!10^\{\-3\}2\.11×10−52\.11\\\!\\times\\\!10^\{\-5\}±2\.86×10−5\\pm\\,2\.86\\\!\\times\\\!10^\{\-5\}11±2\.86×10−5\\pm\\,2\.86\\\!\\times\\\!10^\{\-5\}11±0\\pm\\,0Table 4:Right\-transport parameterizations and evaluation\-time intervention\. Entries are means±\\pmsample standard deviations overn=5n=5seeds\. Rows marked†\\daggerare repeated from Table[3](https://arxiv.org/html/2607.03055#A4.T3)\.ModelEvalNMSE↓\\downarrowPairΔ\\DeltaNMSE↓\\downarrowPairΔ\\DeltaR2R^\{2\}↑\\uparrowR→IR\\\!\\to\\\!IPairΔ\\DeltaNMSE↓\\downarrowNo\-rightr=8r=8†\\dagger2\.797×10−42\.797\\\!\\times\\\!10^\{\-4\}±2\.31×10−5\\pm\\,2\.31\\\!\\times\\\!10^\{\-5\}11±2\.02×10−7\\pm\\,2\.02\\\!\\times\\\!10^\{\-7\}−1\.52×10−3\-1\.52\\\!\\times\\\!10^\{\-3\}±2\.165×10−3\\pm\\,2\.165\\\!\\times\\\!10^\{\-3\}–SHiPPO oracle\-RR†\\dagger4\.377×10−34\.377\\\!\\times\\\!10^\{\-3\}±9\.788×10−3\\pm\\,9\.788\\\!\\times\\\!10^\{\-3\}2\.44×10−72\.44\\\!\\times\\\!10^\{\-7\}±4\.61×10−7\\pm\\,4\.61\\\!\\times\\\!10^\{\-7\}11±4\.66×10−7\\pm\\,4\.66\\\!\\times\\\!10^\{\-7\}11±0\\pm\\,0SHiPPO learned\-RR\(true init\)†\\dagger7\.019×10−47\.019\\\!\\times\\\!10^\{\-4\}±1\.18×10−3\\pm\\,1\.18\\\!\\times\\\!10^\{\-3\}2\.11×10−52\.11\\\!\\times\\\!10^\{\-5\}±2\.86×10−5\\pm\\,2\.86\\\!\\times\\\!10^\{\-5\}11±2\.86×10−5\\pm\\,2\.86\\\!\\times\\\!10^\{\-5\}11±0\\pm\\,0SHiPPO learned\-RR\(zero init\)3\.65×10−53\.65\\\!\\times\\\!10^\{\-5\}±7\.95×10−5\\pm\\,7\.95\\\!\\times\\\!10^\{\-5\}1\.87×10−61\.87\\\!\\times\\\!10^\{\-6\}±2\.57×10−6\\pm\\,2\.57\\\!\\times\\\!10^\{\-6\}11±2\.60×10−6\\pm\\,2\.60\\\!\\times\\\!10^\{\-6\}11±0\\pm\\,0SHiPPO learned\-RR\(random init\)4\.260×10−44\.260\\\!\\times\\\!10^\{\-4\}±5\.462×10−4\\pm\\,5\.462\\\!\\times\\\!10^\{\-4\}6\.58×10−66\.58\\\!\\times\\\!10^\{\-6\}±8\.08×10−6\\pm\\,8\.08\\\!\\times\\\!10^\{\-6\}11±8\.11×10−6\\pm\\,8\.11\\\!\\times\\\!10^\{\-6\}11±0\\pm\\,0SHiPPO selective\-RR1\.235×10−31\.235\\\!\\times\\\!10^\{\-3\}±1\.713×10−3\\pm\\,1\.713\\\!\\times\\\!10^\{\-3\}5\.58×10−55\.58\\\!\\times\\\!10^\{\-5\}±8\.12×10−5\\pm\\,8\.12\\\!\\times\\\!10^\{\-5\}0\.99990\.9999±8\.13×10−5\\pm\\,8\.13\\\!\\times\\\!10^\{\-5\}11±0\\pm\\,0
TheRt→IR\_\{t\}\\\!\\to\\\!Iintervention is applicable only to right\-transport models; “–” denotes not applicable\.

Table 5:Transported\-projection prediction auxiliary result\. Entries are means±\\pmsample standard deviations overn=5n=5seeds\.ModelEval NMSE↓\\downarrowEvalR2R^\{2\}↑\\uparrowNo\-rightr=4r=41\.073×10−3±1\.471×10−41\.073\\\!\\times\\\!10^\{\-3\}\\pm 1\.471\\\!\\times\\\!10^\{\-4\}0\.9989±1\.471×10−40\.9989\\pm 1\.471\\\!\\times\\\!10^\{\-4\}SHiPPO learned\-RR\(true init\)5\.82×10−5±6\.45×10−55\.82\\\!\\times\\\!10^\{\-5\}\\pm 6\.45\\\!\\times\\\!10^\{\-5\}0\.9999±6\.45×10−50\.9999\\pm 6\.45\\\!\\times\\\!10^\{\-5\}SHiPPO oracle\-RR6\.571×10−4±3\.808×10−46\.571\\\!\\times\\\!10^\{\-4\}\\pm 3\.808\\\!\\times\\\!10^\{\-4\}0\.9993±3\.808×10−40\.9993\\pm 3\.808\\\!\\times\\\!10^\{\-4\}
### D\.6Group\-local audit and limitation

The main separation claim is made for full right transport\. The scan\-compatible realization in Section 3 uses group\-local right transport as a computational restriction, so we audit grouping in the minimal paired\-transport setting\. Sinced=4d=4, group size 4 is equivalent to full right transport\. Group size 2 forces the effective update to use block\-local slices of the generated right action\.

Table[6](https://arxiv.org/html/2607.03055#A4.T6)reports the group\-local audit; this is a limitation diagnostic rather than part of the main full\-transport separation claim\.

Table 6:Group\-local audit\. Entries usen=3n=3seeds\. PairΔ\\DeltaNMSE is averaged over seeds; raw off\-block ratio is mean±\\pmsample standard deviation\. Group size 4 corresponds to full right transport for thisd=4d=4diagnostic\.ModelGroupPairΔ\\DeltaNMSE↓\\downarrowRaw off\-blockratioSHiPPO learned\-RR21\.60×10−51\.60\\\!\\times\\\!10^\{\-5\}0\.1305±0\.02770\.1305\\pm 0\.0277SHiPPO learned\-RR43\.78×10−53\.78\\\!\\times\\\!10^\{\-5\}0±00\\pm 0SHiPPO oracle\-RR2110\.1015±0\.00060\.1015\\pm 0\.0006SHiPPO oracle\-RR46\.70×10−86\.70\\\!\\times\\\!10^\{\-8\}0±00\\pm 0This audit should not be interpreted as evidence that group\-local transport faithfully implements full transport\. Rather, it shows that learned front\-end, source, and readout maps may partially recode the diagnostic under a mismatched grouping\. The formal separation claim is therefore made for the full\-transport setting; group\-local transport is a computational realization whose expressivity depends on the grouping\.

### D\.7Excluded exploratory diagnostics

We also explored broader associative\-recall diagnostics, span\-capacity pilots, and program\-like wrappers\. We do not use these runs as evidence for the mechanistic separation studied here: broad associative\-recall tasks can admit shortcuts based on high\-rank current\-step writes, span\-capacity pilots do not include paired\-difference or intervention metrics, and the current program\-like wrappers introduce additional variable\-tracking difficulty beyond the transport mechanism\. These experiments motivated the paired noncommutative transport diagnostic but are not part of the paper\-facing evidence\.

## Appendix ETransport\-MQAR Details

This appendix records the Transport\-MQAR generator, metrics, model geometry, training protocol, full result tables, and controller counterfactuals used for Section[4\.3](https://arxiv.org/html/2607.03055#S4.SS3)\. Transport\-MQAR and the controller\-suffix intervention provide diagnostic evidence for learned right\-action pathways in an autoregressive transported\-recall setting\.

### E\.1Task, metrics, and generator

Transport\-MQAR is a finite\-field, right\-transport variant of multi\-query associative recall \(MQAR\)\[[2](https://arxiv.org/html/2607.03055#bib.bib76)\]\. Each example is generated over𝔽31\\mathbb\{F\}\_\{31\}with 256 keys and four\-coordinate values\. A binding event stores a key–value pair; an operation event applies an invertible right action to all stored values; and a query event asks for the currently transported value of a key\. The configured training length is 512, and final evaluations use lengths128,512,2048,4096128,512,2048,4096\. We report coordinate accuracy, the fraction of target coordinates predicted correctly, and exact accuracy, the fraction of queries for which all four coordinates are correct\. The main runs use a nonreducible library of 13 finite\-field operations\.

Table 7:Transport\-MQAR generator configuration for the reported primary runs\.FieldValueFinite field𝔽31\\mathbb\{F\}\_\{31\}Target coordinatesp=4p=4Number of keys256Configured training length512Evaluation lengths128, 512, 2048, 4096Generator modenonreducibleEvent probabilitiespop=0\.50,pbind=0\.22,pquery=0\.28p\_\{\\mathrm\{op\}\}=0\.50,\\ p\_\{\\mathrm\{bind\}\}=0\.22,\\ p\_\{\\mathrm\{query\}\}=0\.28Number of operation generators13Evaluation examples640 per length and seed unless noted otherwiseTraining losscoordinate\-wise cross\-entropy at query positions
### E\.2Controls and realization geometry

GRU\[[8](https://arxiv.org/html/2607.03055#bib.bib77)\]and Transformer\[[64](https://arxiv.org/html/2607.03055#bib.bib79)\]are same\-width sequence baselines\. Free enc/dec adds static input/output basis flexibility around a no\-right SHiPPO\-style memory\. No\-right preserves the matrix\-state geometry but fixesRt=IR\_\{t\}=I\. DirectGen\-SingleExp is an intra\-family right\-action realization: it keeps the group\-local matrix\-state recurrence

Ht,g=Lt,g​Ht−1,g​Rt,g\+U^t,g,H\_\{t,g\}=L\_\{t,g\}H\_\{t\-1,g\}R\_\{t,g\}\+\\widehat\{U\}\_\{t,g\},but directly emits a group\-local generatorAt,g∈ℝPg×PgA\_\{t,g\}\\in\\mathbb\{R\}^\{P\_\{g\}\\times P\_\{g\}\}and sets

Rt,g=exp⁡\(Δt,g​At,g\)\.R\_\{t,g\}=\\exp\(\\Delta\_\{t,g\}A\_\{t,g\}\)\.StructGen\-Split uses a structured right\-generator library and a split product of factor exponentials\. Thus the DirectGen\-SingleExp comparison is an intra\-family comparison between right\-action realizations, and it changes both the generator family and the discrete right\-action backend\.

Table 8:Model geometry and parameter counts in the primary Transport\-MQAR comparison\. All models usedmodel=128d\_\{\\rm model\}=128\.ModeleeDcellD\_\{\\rm cell\}NNPgP\_\{g\}GGRoleParamsGRU–––––recurrent baseline0\.50MTransformer–––––attention baseline0\.89MFree enc/dec225632464static\-basis control5\.11MNo\-right225632464right\-action ablation4\.98MDirectGen\-SingleExp112832432direct\-generator right action1\.52MStructGen\-Split225632464structured split right action6\.03M

### E\.3Training protocol

Final out\-of\-distribution evaluation lengths are not used for model selection\. For SHiPPO\-family variants, hyperparameters are selected using validation data at the configured training length only\. All main Transport\-MQAR results are means and sample standard deviations overn=3n=3independent training seeds\. Training uses AdamW\[[37](https://arxiv.org/html/2607.03055#bib.bib78)\]with learning rate5×10−45\\times 10^\{\-4\}, weight decay 0\.01, batch size 16, 5000 steps, evaluation every 250 steps, and gradient clipping at 1\.0\. Matched\-capacity controls use a separate shared stress\-test budget and are capacity checks rather than the primary same\-width comparison\.

### E\.4Full primary results

Table[9](https://arxiv.org/html/2607.03055#A5.T9)reports the full length sweep behind Table[1](https://arxiv.org/html/2607.03055#S4.T1)\. The main\-text interpretation is based on this table: StructGen\-Split improves coordinate accuracy relative to no\-right and static\-basis ablations, while DirectGen\-SingleExp remains stronger on exact accuracy\. Thus the results support the usefulness of a right\-action pathway while leaving the preferred discrete right\-action realization open\.

Table 9:Primary Transport\-MQAR length sweep\. Entries are means±\\pmsample standard deviations overn=3n=3seeds\.Coordinate accuracyModel12851220484096GRU0\.18410\.1841±0\.0012\\pm\\,0\.00120\.11500\.1150±0\.0021\\pm\\,0\.00210\.09820\.0982±0\.0011\\pm\\,0\.00110\.09520\.0952±0\.0017\\pm\\,0\.0017Transformer0\.23570\.2357±0\.0145\\pm\\,0\.01450\.09860\.0986±0\.0099\\pm\\,0\.00990\.05060\.0506±0\.0032\\pm\\,0\.00320\.04160\.0416±0\.0012\\pm\\,0\.0012Free enc/dec0\.18970\.1897±0\.0030\\pm\\,0\.00300\.11940\.1194±0\.0003\\pm\\,0\.00030\.09900\.0990±0\.0041\\pm\\,0\.00410\.09160\.0916±0\.0100\\pm\\,0\.0100No\-right0\.19500\.1950±0\.0030\\pm\\,0\.00300\.12350\.1235±0\.0008\\pm\\,0\.00080\.10530\.1053±0\.0004\\pm\\,0\.00040\.10190\.1019±0\.0000\\pm\\,0\.0000DirectGen\-SingleExp0\.19660\.1966±0\.0041\\pm\\,0\.00410\.12480\.1248±0\.0009\\pm\\,0\.00090\.10660\.1066±0\.0006\\pm\\,0\.00060\.10320\.1032±0\.0007\\pm\\,0\.0007StructGen\-Split0\.21150\.2115±0\.0028\\pm\\,0\.00280\.13460\.1346±0\.0007\\pm\\,0\.00070\.11400\.1140±0\.0002\\pm\\,0\.00020\.11040\.1104±0\.0010\\pm\\,0\.0010
Exact accuracyModel12851220484096GRU0\.05350\.0535±0\.0021\\pm\\,0\.00210\.02390\.0239±0\.0026\\pm\\,0\.00260\.01670\.0167±0\.0021\\pm\\,0\.00210\.01590\.0159±0\.0024\\pm\\,0\.0024Transformer0\.06480\.0648±0\.0120\\pm\\,0\.01200\.01660\.0166±0\.0035\\pm\\,0\.00350\.00430\.0043±0\.0010\\pm\\,0\.00100\.00210\.0021±0\.0006\\pm\\,0\.0006Free enc/dec0\.04460\.0446±0\.0023\\pm\\,0\.00230\.02070\.0207±0\.0003\\pm\\,0\.00030\.01200\.0120±0\.0023\\pm\\,0\.00230\.00940\.0094±0\.0042\\pm\\,0\.0042No\-right0\.05010\.0501±0\.0022\\pm\\,0\.00220\.02420\.0242±0\.0001\\pm\\,0\.00010\.01700\.0170±0\.0004\\pm\\,0\.00040\.01580\.0158±0\.0003\\pm\\,0\.0003DirectGen\-SingleExp0\.08510\.0851±0\.0020\\pm\\,0\.00200\.04680\.0468±0\.0017\\pm\\,0\.00170\.03750\.0375±0\.0019\\pm\\,0\.00190\.03550\.0355±0\.0005\\pm\\,0\.0005StructGen\-Split0\.07950\.0795±0\.0021\\pm\\,0\.00210\.04390\.0439±0\.0016\\pm\\,0\.00160\.03420\.0342±0\.0018\\pm\\,0\.00180\.03290\.0329±0\.0013\\pm\\,0\.0013

### E\.5Controller\-suffix counterfactual

For StructGen\-Split, the controller emits an additional 1024\-dimensional suffix associated with the split\-flow right\-transport coordinates\. We zero this suffix at evaluation time while keeping all trained weights fixed\. Table[10](https://arxiv.org/html/2607.03055#A5.T10)shows that the trained predictor depends on these additional coordinates\. This counterfactual does not identify the learned transport geometry or prove that the model implements a specific noncommutative algorithm\.

Table 10:Evaluation\-time counterfactual for the StructGen\-Split controller suffix\. Means over three seeds\.Coordinate accuracyExact accuracyLengthnormalzeroedΔ\\DeltanormalzeroedΔ\\Delta1280\.21150\.1936\-0\.01790\.07950\.0436\-0\.03595120\.13460\.1250\-0\.00960\.04390\.0239\-0\.020020480\.11400\.1074\-0\.00660\.03420\.0185\-0\.015740960\.11040\.1044\-0\.00600\.03290\.0176\-0\.0154

## Appendix FAdditional Related Work and Positioning

We organize related work by the source of the recurrent dynamics and by where channel or matrix\-valued interaction enters the memory\. In sequence modeling, choosing an ODE or recurrence family is itself a restrictive prior: it biases which finite\-dimensional causal memory operators are representable, which stability constraints are natural, and which execution schemes are compatible with the model\. Parameterization, initialization, and discretization then determine how a trainable layer explores and implements that family\. SHiPPO contributes at the dynamics\-selection level: it derives a channel\-interacting Sylvester ODE from a transported online approximation problem, and then studies scan\-compatible parameterizations of that ODE\.

Relative to nearby recurrent, SSM, and matrix\-state model families, SHiPPO’s main distinction is that channel interaction is tied to transported online\-projection semantics\. It differs from ODE/CDE and stable RNN families by deriving the ODE from projection coefficients rather than choosing a generic or stability\-motivated vector field\[[7](https://arxiv.org/html/2607.03055#bib.bib32),[31](https://arxiv.org/html/2607.03055#bib.bib34),[6](https://arxiv.org/html/2607.03055#bib.bib35),[13](https://arxiv.org/html/2607.03055#bib.bib37)\]\. It differs from objective\-derived recurrences by transporting the polynomial projection objective, yielding closed Sylvester coefficient dynamics\[[65](https://arxiv.org/html/2607.03055#bib.bib41),[19](https://arxiv.org/html/2607.03055#bib.bib1),[23](https://arxiv.org/html/2607.03055#bib.bib3),[35](https://arxiv.org/html/2607.03055#bib.bib42)\]\. It differs from structured and selective SSMs by changing the memory ODE induced by the online\-memory prior before choosing a scan\-compatible parameterization\[[21](https://arxiv.org/html/2607.03055#bib.bib2),[22](https://arxiv.org/html/2607.03055#bib.bib5),[57](https://arxiv.org/html/2607.03055#bib.bib6),[20](https://arxiv.org/html/2607.03055#bib.bib8),[11](https://arxiv.org/html/2607.03055#bib.bib9),[33](https://arxiv.org/html/2607.03055#bib.bib20)\]\. It differs from linear attention, DeltaNet, GLA, GDN, and matrix\-memory RNNs by assigning projection\-coefficient and transported\-channel\-frame semantics to the two matrix axes\[[50](https://arxiv.org/html/2607.03055#bib.bib46),[73](https://arxiv.org/html/2607.03055#bib.bib47),[72](https://arxiv.org/html/2607.03055#bib.bib48),[71](https://arxiv.org/html/2607.03055#bib.bib16),[3](https://arxiv.org/html/2607.03055#bib.bib50),[45](https://arxiv.org/html/2607.03055#bib.bib51),[40](https://arxiv.org/html/2607.03055#bib.bib52)\]\. Finally, Proposition[3\.3](https://arxiv.org/html/2607.03055#S3.Thmassumption3)provides an internal collapse criterion: in the scan\-compatible group\-local realization, simultaneously reducible right transports are equivalent, after a fixed channel\-basis change, to independent scalar or blockwise transported banks\. This criterion connects SHiPPO to structured\-transition and state\-tracking work that studies expressivity and efficient execution under restricted transition families\[[39](https://arxiv.org/html/2607.03055#bib.bib53),[61](https://arxiv.org/html/2607.03055#bib.bib54),[55](https://arxiv.org/html/2607.03055#bib.bib57),[68](https://arxiv.org/html/2607.03055#bib.bib58)\]\.

#### Dynamics families as restrictive priors\.

Continuous\-time sequence models use differential equations as modeling primitives\. Neural ODEs parameterize the derivative of the hidden state by a neural network and compute outputs through ODE solvers, while Latent ODE and Neural CDE models extend this viewpoint to irregularly sampled or partially observed time series\[[7](https://arxiv.org/html/2607.03055#bib.bib32),[46](https://arxiv.org/html/2607.03055#bib.bib33),[31](https://arxiv.org/html/2607.03055#bib.bib34)\]\. Other recurrent architectures choose more structured ODE forms to impose stability, bounded\-gradient, oscillatory, or timescale priors: AntisymmetricRNNs, coRNNs, Lipschitz RNNs, Liquid Time\-constant Networks, Liquid\-S4, and LinOSS are examples of this dynamics\-as\-prior viewpoint\[[6](https://arxiv.org/html/2607.03055#bib.bib35),[47](https://arxiv.org/html/2607.03055#bib.bib36),[13](https://arxiv.org/html/2607.03055#bib.bib37),[26](https://arxiv.org/html/2607.03055#bib.bib38),[27](https://arxiv.org/html/2607.03055#bib.bib39),[48](https://arxiv.org/html/2607.03055#bib.bib61)\]\. Layer\-wise nonlinear SSMs and stable SSM parameterizations further show that expressivity and memory behavior depend on the chosen dynamics and its stable parameterization\[[70](https://arxiv.org/html/2607.03055#bib.bib59),[69](https://arxiv.org/html/2607.03055#bib.bib60)\]\. Recent dynamical\-systems frameworks also compare attention, SSMs, and RNNs in a common representation\[[53](https://arxiv.org/html/2607.03055#bib.bib40)\]\. SHiPPO is related at the level of treating dynamics as prior, but differs in the source of the dynamics: its ODE is not a generic learned vector field or a stability\-motivated continuous\-time RNN, but the coefficient dynamics induced by a transported online projection problem\.

#### Analytically and objectively derived memory dynamics\.

SHiPPO is closest in spirit to models whose recurrent updates are derived from an objective or analytic memory principle rather than postulated as an architecture\. The Legendre Memory Unit is an important precursor: it derives a continuous\-time recurrent memory cell whose ODE state represents a sliding history window in a Legendre basis\[[65](https://arxiv.org/html/2607.03055#bib.bib41)\]\. HiPPO generalizes this projection\-memory viewpoint into an online polynomial projection framework, deriving coefficient dynamics for compressing the revealed history\[[19](https://arxiv.org/html/2607.03055#bib.bib1),[23](https://arxiv.org/html/2607.03055#bib.bib3)\]\. Recent work such as HiPPO Zoo revisits this line by making orthogonal\-polynomial memory mechanisms explicit and interpretable\[[16](https://arxiv.org/html/2607.03055#bib.bib19)\]\. SHiPPO is complementary: it does not add a collection of explicit polynomial\-basis mechanisms, but transports the channel frame of the online approximation problem itself, yielding Sylvester right\-action coefficient dynamics\. Other recent work derives recurrent layers from online\-learning or test\-time learning objectives: Longhorn views SSMs as amortized online learners, TTT layers update hidden states through self\-supervised learning steps, and MesaNet derives recurrent layers from locally optimal in\-context regression objectives\[[35](https://arxiv.org/html/2607.03055#bib.bib42),[59](https://arxiv.org/html/2607.03055#bib.bib43),[66](https://arxiv.org/html/2607.03055#bib.bib44)\]\. SHiPPO shares the objective\-derived design philosophy, but its state is a transported approximation\-coefficient matrix, and its right\-action term is the transport required by the chosen approximation objective\.

#### Structured and selective SSMs: recurrence family versus parameterization\.

Deep SSMs illustrate that similar state\-equation templates can yield different models depending on parameterization, initialization, and discretization\. S4 uses a structured parameterization of continuous\-time state matrices for long\-sequence modeling, while DSS and S4D simplify this structure to diagonal state spaces while retaining much of the empirical benefit through appropriate parameterization and initialization\[[21](https://arxiv.org/html/2607.03055#bib.bib2),[24](https://arxiv.org/html/2607.03055#bib.bib4),[22](https://arxiv.org/html/2607.03055#bib.bib5)\]\. LRU similarly shows that linearization, diagonalization, initialization, and normalization can recover much of the performance of deep SSMs in an RNN block\[[44](https://arxiv.org/html/2607.03055#bib.bib45)\]\. S5 moves from many independent SISO systems to a MIMO SSM layer, and H3 introduces multiplicative interactions between SSM outputs and input projections to address recall and comparison capabilities in language modeling\[[57](https://arxiv.org/html/2607.03055#bib.bib6),[15](https://arxiv.org/html/2607.03055#bib.bib7)\]\. Selective SSMs such as Mamba make SSM parameters input\-dependent and design hardware\-aware scans; Mamba\-2/SSD clarifies the semiseparable structure behind such recurrences; Mamba\-3 further modifies the recurrence through discretization, complex dynamics, and MIMO updates; and recent analyses study the role of input selectivity in approximation, memorization, and associative recall\[[20](https://arxiv.org/html/2607.03055#bib.bib8),[11](https://arxiv.org/html/2607.03055#bib.bib9),[33](https://arxiv.org/html/2607.03055#bib.bib20),[29](https://arxiv.org/html/2607.03055#bib.bib67)\]\. Other work studies preferential or procedural biases within a chosen SSM family, including perturb\-then\-diagonalize, Hankel/Markov parameterization, spectral or frequency\-bias tuning, uncertainty\-aware initialization, autocorrelation\-aware initialization, and mimetic initialization\[[76](https://arxiv.org/html/2607.03055#bib.bib10),[75](https://arxiv.org/html/2607.03055#bib.bib13),[74](https://arxiv.org/html/2607.03055#bib.bib14),[58](https://arxiv.org/html/2607.03055#bib.bib17),[34](https://arxiv.org/html/2607.03055#bib.bib18),[36](https://arxiv.org/html/2607.03055#bib.bib12),[62](https://arxiv.org/html/2607.03055#bib.bib15)\]\. These works modify how a chosen dynamics family is parameterized, initialized, or trained\. SHiPPO instead changes the memory ODE selected by the online\-memory prior before choosing a scan\-compatible parameterization\.

#### Matrix memories, fast weights, and gated linear recurrences\.

Many efficient recurrent models introduce channel interaction through architectural or algorithmic mechanisms\. Fast\-weight and delta\-rule views of linear attention interpret matrix\-valued recurrent states as associative memories programmed by outer\-product or correction updates\[[50](https://arxiv.org/html/2607.03055#bib.bib46),[73](https://arxiv.org/html/2607.03055#bib.bib47)\]\. Gated Linear Attention and implicit\-attention views of gated\-linear RNNs maintain two\-dimensional associative memories and introduce data\-dependent gates\[[72](https://arxiv.org/html/2607.03055#bib.bib48),[77](https://arxiv.org/html/2607.03055#bib.bib68)\]\. Gated DeltaNet combines Mamba\-style gating with the delta rule\[[71](https://arxiv.org/html/2607.03055#bib.bib16)\], and Kimi Linear introduces Kimi Delta Attention \(KDA\), an extension of Gated DeltaNet with finer\-grained gating and efficient chunkwise execution\[[32](https://arxiv.org/html/2607.03055#bib.bib49)\]\. Matrix\-memory recurrent models such as xLSTM/mLSTM and HGRN2 increase memory capacity through exponential gating, covariance\-style memory, or outer\-product state expansion\[[3](https://arxiv.org/html/2607.03055#bib.bib50),[45](https://arxiv.org/html/2607.03055#bib.bib51)\]\. Recent matrix\-to\-matrix and weight\-space RNNs further explore matrix\-valued or weight\-valued hidden states\[[40](https://arxiv.org/html/2607.03055#bib.bib52),[43](https://arxiv.org/html/2607.03055#bib.bib71)\]\. SHiPPO is close to these works at the level of state shape, but the axes have different semantics: the left axis indexes online approximation coefficients, while the right axis is a transported channel frame\. The two\-sided update is therefore not introduced as an untied matrix\-state recurrence; it is the discrete realization of a transported coefficient ODE\.

#### Structured transitions and state\-tracking expressivity\.

A separate line of work studies which transition structures preserve expressivity while remaining parallelizable\. Formal\-language and state\-tracking analyses identify both strengths and limitations of common SSMs, selective SSMs, diagonal or positive\-eigenvalue recurrences, and gated variants\[[49](https://arxiv.org/html/2607.03055#bib.bib62),[39](https://arxiv.org/html/2607.03055#bib.bib53),[9](https://arxiv.org/html/2607.03055#bib.bib11),[60](https://arxiv.org/html/2607.03055#bib.bib63),[18](https://arxiv.org/html/2607.03055#bib.bib56),[51](https://arxiv.org/html/2607.03055#bib.bib55),[1](https://arxiv.org/html/2607.03055#bib.bib69)\]\. Several architectural responses enrich transition structure while trying to preserve efficient execution: fixed\-point RNNs interpolate from diagonal to dense recurrences, bilinear state transitions use hidden–input multiplicative updates, adaptive unitary SSMs use input\-dependent skew/unitary dynamics, DeltaProduct uses products of generalized Householder transformations, SLiCE studies structured input\-dependent transition matrices, and PD\-SSM uses structured sparse transitions for FSA state tracking\[[42](https://arxiv.org/html/2607.03055#bib.bib64),[12](https://arxiv.org/html/2607.03055#bib.bib65),[30](https://arxiv.org/html/2607.03055#bib.bib66),[55](https://arxiv.org/html/2607.03055#bib.bib57),[68](https://arxiv.org/html/2607.03055#bib.bib58),[61](https://arxiv.org/html/2607.03055#bib.bib54)\]\. Recent coefficient\-dynamics views also analyze sequence models through the dynamics of coefficients used to combine past value vectors\[[54](https://arxiv.org/html/2607.03055#bib.bib70)\]\. SHiPPO is complementary to this literature\. It does not propose a general structured\-transition framework; it identifies the two\-sided transition induced by transported online\-memory semantics\. Our collapse result shows that, in the scan\-compatible group\-local realization, simultaneously reducible right transports are functionally equivalent to static mixing followed by independent scalar or blockwise transported banks, motivating non\-reducible right\-transport families\.

#### Moving\-frame interpretation and non\-reduction\.

For a realized transport path, SHiPPO admits an exact moving\-frame HiPPO factorization, but not an arbitrary encoder–decoder reduction: the history encoder, coefficient decoder, metric, and Sylvester gauge term are tied by the same transport ODE\. With input\-dependent controllers, this interpretation is pathwise rather than a fixed input\-independent factorization\.