Finite-Lag Operator Geometry of Recurrent Representations

# Finite-Lag Operator Geometry of Recurrent Representations
Source: [https://arxiv.org/html/2607.01746](https://arxiv.org/html/2607.01746)
Kanishka Reddy Department of Applied Mathematics University of Washington Seattle, WA 98105 kani@uw\.edu

###### Abstract

Recurrent representations are trajectories, but representation geometry is often measured from static snapshots\. We develop finite\-lag operator geometry for recurrent hidden states from observed source–successor pairs\(Xt,Xt\+Δ\)\(X\_\{t\},X\_\{t\+\\Delta\}\)\. The primitive is the conditional transport lawQΔ​\(d​y∣x\)Q\_\{\\Delta\}\(dy\\mid x\), estimated by a dense Gaussian source\-smoothing operator\. From this directed finite\-lag law we derive a source\-centered transport tensorGΔG\_\{\\Delta\}, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}, which summarizes directed lagged flow\. We prove affine covariance with explicit metric dependence of scalar summaries, dense estimator stability on bounded trajectory clouds, and a finite\-lag separation result showing that source\-centered transport detects deterministic recurrent motion not recorded by infinitesimal carré\-du\-champ geometry\. A linear\-Gaussian closed form calibrates the quantities in terms of the updateAΔA\_\{\\Delta\}, source covariance, and innovation covariance\. Controlled experiments validate the decomposition, circulation, covariance, and stability predictions\. In performance matched repeat\-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace, while coherent displacement fraction is metric and resolution dependent\.

## 1Introduction

Neural representations are often analyzed as point clouds, with geometry recovered through similarity measures, neighborhood graphs, spectral summaries, or diffusion operators\[[35](https://arxiv.org/html/2607.01746#bib.bib7),[22](https://arxiv.org/html/2607.01746#bib.bib8),[2](https://arxiv.org/html/2607.01746#bib.bib5),[9](https://arxiv.org/html/2607.01746#bib.bib6),[33](https://arxiv.org/html/2607.01746#bib.bib9),[24](https://arxiv.org/html/2607.01746#bib.bib56),[1](https://arxiv.org/html/2607.01746#bib.bib57),[6](https://arxiv.org/html/2607.01746#bib.bib11),[11](https://arxiv.org/html/2607.01746#bib.bib12),[36](https://arxiv.org/html/2607.01746#bib.bib1)\]\. In static feedforward settings, an operator\-first view replaces hard graph constructions with a smooth Markov operator on the feature cloud and derives geometric observables from that operator\. This is natural when a layer is a snapshot\. Recurrent representations are different, since a hidden statehth\_\{t\}is meaningful not only for where it lies, but for where the recurrent computation sends it\.

The natural primitive is therefore a finite\-lag law on observed source–successor pairs, not a symmetric kernel on one cloud\. We define

QΔ​\(d​y∣x\)=ℒ​\(Xt\+Δ∈d​y∣Xt=x\),Q\_\{\\Delta\}\(dy\\mid x\)=\\mathcal\{L\}\(X\_\{t\+\\Delta\}\\in dy\\mid X\_\{t\}=x\),where the law is induced jointly by the recurrent update and the sequence/input distribution\. The empirical estimator smooths over nearby source states and transports to their attached successors\. This gives a directed finite\-lag operator even when the hidden state alone is not an autonomous Markov state\.

The central symmetric observable is the source\-centered transport tensor

GΔ​\(x\)=12​τ​∫\(y−x\)​\(y−x\)⊤​QΔ​\(d​y∣x\),τ=Δ​d​t\.G\_\{\\Delta\}\(x\)=\\frac\{1\}\{2\\tau\}\\int\(y\-x\)\(y\-x\)^\{\\top\}\\,Q\_\{\\Delta\}\(dy\\mid x\),\\qquad\\tau=\\Delta\\,dt\.Source\-centering is the key choice\. If

mΔ​\(x\)=τ−1​𝔼​\[Y−X∣X=x\],CΔ​\(x\)=τ−1​Cov⁡\(Y−X∣X=x\),m\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\mathbb\{E\}\[Y\-X\\mid X=x\],\\qquad C\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\operatorname\{Cov\}\(Y\-X\\mid X=x\),then

2​GΔ​\(x\)=CΔ​\(x\)\+τ​mΔ​\(x\)​mΔ​\(x\)⊤\.2G\_\{\\Delta\}\(x\)=C\_\{\\Delta\}\(x\)\+\\tau\\,m\_\{\\Delta\}\(x\)m\_\{\\Delta\}\(x\)^\{\\top\}\.Thus finite\-lag transport decomposes exactly into conditional spread and coherent displacement\. Centering at the conditional mean would remove the second term, while source\-centering keeps deterministic recurrent motion as part of the geometry\. We also define an antisymmetric coordinate circulation𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}, a lagged source–successor cross\-moment that summarizes directed flow in representation coordinates\.

These objects are related to transfer\-operator, Koopman, TICA, and VAMP methods\[[38](https://arxiv.org/html/2607.01746#bib.bib23),[45](https://arxiv.org/html/2607.01746#bib.bib24),[34](https://arxiv.org/html/2607.01746#bib.bib59),[39](https://arxiv.org/html/2607.01746#bib.bib60),[47](https://arxiv.org/html/2607.01746#bib.bib28),[28](https://arxiv.org/html/2607.01746#bib.bib29)\], but the use is different\. Those methods estimate predictive operators or extract slow modes\. Here the finite\-lag law is used geometrically, with its second moments describing transport scale, conditional spread, coherent displacement, and directed circulation in hidden\-state space\.

#### Contributions\.

We make four contributions\.

1. 1\.We define finite\-lag operator geometry for recurrent representations through the conditional transport lawQΔQ\_\{\\Delta\}, the source\-centered transport tensorGΔG\_\{\\Delta\}, its exact spread/displacement decomposition, and the coordinate circulation𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\.
2. 2\.We prove structural results: affine covariance with metric\-dependent scalar summaries, Lipschitz stability of the dense Gaussian estimator on bounded trajectory clouds, and a finite\-lag separation theorem showing that deterministic recurrent motion can be visible toGΔG\_\{\\Delta\}even when the infinitesimal carré du champ of a first\-order deterministic generator is zero\.
3. 3\.We derive a linear\-Gaussian closed form forG¯Δ\\bar\{G\}\_\{\\Delta\}and𝒲Δ\\mathcal\{W\}\_\{\\Delta\}, giving an analytic calibration parallel to the Gaussian bridge in static feedforward operator geometry\.
4. 4\.We validate the formal quantities on controlled systems and illustrate them on performance\-matched repeat\-copy networks with capacity and memory\-horizon controls\.

The framework is finite\-lag by construction\. It does not require hidden state alone to be autonomously Markov, and it does not reduce recurrent computation to a linear\-Gaussian model\. The linear\-Gaussian analysis is a calibration\. The trained\-network experiments report the framework’s observables under explicit bandwidth and normalization choices\.

## 2Related work

#### Operator geometry of representations\.

Diffusion geometry and Bakry–ÉmeryΓ\\Gamma\-calculus build geometric objects from Markov operators rather than hard neighborhood graphs\[[3](https://arxiv.org/html/2607.01746#bib.bib17),[4](https://arxiv.org/html/2607.01746#bib.bib18),[6](https://arxiv.org/html/2607.01746#bib.bib11),[11](https://arxiv.org/html/2607.01746#bib.bib12),[31](https://arxiv.org/html/2607.01746#bib.bib58),[17](https://arxiv.org/html/2607.01746#bib.bib3),[18](https://arxiv.org/html/2607.01746#bib.bib4),[16](https://arxiv.org/html/2607.01746#bib.bib2),[24](https://arxiv.org/html/2607.01746#bib.bib56),[1](https://arxiv.org/html/2607.01746#bib.bib57)\]\. In feedforward representations, a fixed\-layer feature cloud induces a Gaussian\-kernel diffusion operator whose transport, spectral, label\-boundary, and local\-scale observables can be studied directly\[[36](https://arxiv.org/html/2607.01746#bib.bib1)\]\. We keep the operator\-first principle but replace the static symmetric diffusion operator with a directed finite\-lag transport law on source–successor hidden\-state pairs\.

#### Transfer operators and lagged representation learning\.

Koopman and Perron–Frobenius methods, dynamic mode decomposition, EDMD, kernel EDMD, TICA, VAMP, and transfer\-operator learning all use lagged operators to model dynamical data\[[38](https://arxiv.org/html/2607.01746#bib.bib23),[45](https://arxiv.org/html/2607.01746#bib.bib24),[46](https://arxiv.org/html/2607.01746#bib.bib25),[20](https://arxiv.org/html/2607.01746#bib.bib26),[21](https://arxiv.org/html/2607.01746#bib.bib27),[34](https://arxiv.org/html/2607.01746#bib.bib59),[39](https://arxiv.org/html/2607.01746#bib.bib60),[47](https://arxiv.org/html/2607.01746#bib.bib28),[28](https://arxiv.org/html/2607.01746#bib.bib29),[13](https://arxiv.org/html/2607.01746#bib.bib21),[14](https://arxiv.org/html/2607.01746#bib.bib22),[10](https://arxiv.org/html/2607.01746#bib.bib19),[29](https://arxiv.org/html/2607.01746#bib.bib20)\]\. Their main goals are prediction, coherent mode discovery, or variational scoring\. Our use is geometric, summarizing the finite\-lag conditional law by source\-centered symmetric moments and antisymmetric lagged cross\-moments\.

#### Recurrent network analysis\.

RNN dynamics have been studied through fixed points, slow points, local linearization, low\-rank structure, transient dynamics, and gating mechanisms\[[41](https://arxiv.org/html/2607.01746#bib.bib30),[42](https://arxiv.org/html/2607.01746#bib.bib31),[26](https://arxiv.org/html/2607.01746#bib.bib32),[27](https://arxiv.org/html/2607.01746#bib.bib33),[37](https://arxiv.org/html/2607.01746#bib.bib34),[40](https://arxiv.org/html/2607.01746#bib.bib35),[30](https://arxiv.org/html/2607.01746#bib.bib36),[5](https://arxiv.org/html/2607.01746#bib.bib37),[44](https://arxiv.org/html/2607.01746#bib.bib38),[32](https://arxiv.org/html/2607.01746#bib.bib39),[25](https://arxiv.org/html/2607.01746#bib.bib40),[23](https://arxiv.org/html/2607.01746#bib.bib41),[19](https://arxiv.org/html/2607.01746#bib.bib42),[12](https://arxiv.org/html/2607.01746#bib.bib43),[15](https://arxiv.org/html/2607.01746#bib.bib55),[8](https://arxiv.org/html/2607.01746#bib.bib54)\]\. These approaches expose dynamical skeletons or reduced\-order mechanisms, often through explicit modeling assumptions\. Finite\-lag operator geometry is complementary, summarizing the observed hidden\-state transport at a chosen lag without fixed\-point discovery, local linearization, or low\-rank parametrization\.

#### Stability versus hard graphs\.

Hardkk\-nearest\-neighbor adjacencies are not Lipschitz functions of the point cloud\. Tied or near\-tied neighbor distances can change the selected adjacency under arbitrarily small perturbations\[[7](https://arxiv.org/html/2607.01746#bib.bib62),[43](https://arxiv.org/html/2607.01746#bib.bib61),[36](https://arxiv.org/html/2607.01746#bib.bib1)\]\. Smooth Gaussian\-kernel operators avoid this discontinuity\. We prove the corresponding dense\-estimator stability result for finite\-lag recurrent transport and use the dense estimator as the object matched to the theory\.

## 3Finite\-lag operator geometry

We formalize recurrent representation geometry through a conditional finite\-lag transport law\. Rather than asking only how hidden states are arranged as a cloud, we ask where the recurrent computation sends them after a fixed lagΔ\\Delta\. This yields a source\-centered transport tensor for finite\-step motion and an antisymmetric circulation statistic for directed lagged flow\.

Letht\(s\)∈ℝdh\_\{t\}^\{\(s\)\}\\in\\mathbb\{R\}^\{d\}be the hidden state for sequencessat timett\. Fix a lagΔ≥1\\Delta\\geq 1, setτ=Δ​d​t\\tau=\\Delta\\,dt, and form

xi=ht\(s\),yi=ht\+Δ\(s\)x\_\{i\}=h\_\{t\}^\{\(s\)\},\\qquad y\_\{i\}=h\_\{t\+\\Delta\}^\{\(s\)\}over all indices for which both states are observed\.

### 3\.1Transport operator

###### Definition 1\(Finite\-lag transport law\)\.

The finite\-lag conditional transport law is

QΔ​\(d​y∣x\)=ℒ​\(Xt\+Δ∈d​y∣Xt=x\),Q\_\{\\Delta\}\(dy\\mid x\)=\\mathcal\{L\}\(X\_\{t\+\\Delta\}\\in dy\\mid X\_\{t\}=x\),where the law is taken with respect to the joint sequence\-and\-input distribution\. The associated operator on bounded observables is

\(PΔ​f\)​\(x\)=∫f​\(y\)​QΔ​\(d​y∣x\)\.\(P\_\{\\Delta\}f\)\(x\)=\\int f\(y\)\\,Q\_\{\\Delta\}\(dy\\mid x\)\.

We do not assume that hidden state alone is an autonomous Markov state\. For input\-driven recurrent networks,PΔP\_\{\\Delta\}is a conditional transport operator induced jointly by the recurrence and the input distribution\.

We estimateQΔQ\_\{\\Delta\}by Gaussian smoothing in source space\. For bandwidthε\>0\\varepsilon\>0,

wi​\(xq\)=exp⁡\(−‖xq−xi‖2/4​ε\)∑jexp⁡\(−‖xq−xj‖2/4​ε\),Q^Δ​\(d​y∣xq\)=∑iwi​\(xq\)​δyi​\(d​y\)\.w\_\{i\}\(x\_\{q\}\)=\\frac\{\\exp\(\-\\\|x\_\{q\}\-x\_\{i\}\\\|^\{2\}/4\\varepsilon\)\}\{\\sum\_\{j\}\\exp\(\-\\\|x\_\{q\}\-x\_\{j\}\\\|^\{2\}/4\\varepsilon\)\},\\qquad\\widehat\{Q\}\_\{\\Delta\}\(dy\\mid x\_\{q\}\)=\\sum\_\{i\}w\_\{i\}\(x\_\{q\}\)\\delta\_\{y\_\{i\}\}\(dy\)\.\(1\)The indexiidenotes a neighboring source whose attached successoryiy\_\{i\}contributes to the conditional law at query sourcexqx\_\{q\}\. Thus the dense estimator smooths in source space but transports to successor coordinates\. It is a source\-smoothing operator for successor\-valued transport, not a transition matrix whose column index is itself the successor state\.

At finite bandwidth,Q^Δ\\widehat\{Q\}\_\{\\Delta\}estimates a resolution\-smoothed conditional law\. Consequently,C^Δ\\widehat\{C\}\_\{\\Delta\}contains both genuine conditional variation and variation induced by smoothing over nearby source states\. This is the resolution at which the empirical transport law is queried\. The bandwidth sweeps in Section[5\.2](https://arxiv.org/html/2607.01746#S5.SS2)and Appendix[F\.6](https://arxiv.org/html/2607.01746#A6.SS6)report this dependence explicitly\. For controlled linear\-Gaussian calibration, where the population conditional moments are known, we compute the population moments directly\.

Akk\-nearest\-neighbor approximation is used only where stated\. Unlike the dense operator, hard neighbor selection is not Lipschitz in the source cloud \(Appendix[A\.6](https://arxiv.org/html/2607.01746#A1.SS6)\)\. Euclidean scalar summaries are reported after center\-RMS normalization \(Appendix[A\.5](https://arxiv.org/html/2607.01746#A1.SS5)\)\.

### 3\.2Source\-centered transport tensor

###### Definition 2\(Source\-centered transport tensor\)\.

For sourcex∈ℝdx\\in\\mathbb\{R\}^\{d\},

GΔ​\(x\)=12​τ​∫\(y−x\)​\(y−x\)⊤​QΔ​\(d​y∣x\),G¯Δρ=∫GΔ​\(x\)​ρ​\(d​x\)\.G\_\{\\Delta\}\(x\)=\\frac\{1\}\{2\\tau\}\\int\(y\-x\)\(y\-x\)^\{\\top\}\\,Q\_\{\\Delta\}\(dy\\mid x\),\\qquad\\bar\{G\}\_\{\\Delta\}^\{\\rho\}=\\int G\_\{\\Delta\}\(x\)\\,\\rho\(dx\)\.

GΔ​\(x\)G\_\{\\Delta\}\(x\)is a positive semidefinite finite\-lag quadratic form on coordinate functions\. It is centered at the source state rather than at the conditional mean successor, so it retains coherent motion at the chosen lag\.

Define

mΔ​\(x\)=τ−1​𝔼​\[Y−X∣X=x\],CΔ​\(x\)=τ−1​Cov⁡\(Y−X∣X=x\)\.m\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\mathbb\{E\}\[Y\-X\\mid X=x\],\\qquad C\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\operatorname\{Cov\}\(Y\-X\\mid X=x\)\.
###### Proposition 1\(Source\-centered decomposition\)\.

Whenever the conditional second moment exists,

2​GΔ​\(x\)=CΔ​\(x\)\+τ​mΔ​\(x\)​mΔ​\(x\)⊤\.2G\_\{\\Delta\}\(x\)=C\_\{\\Delta\}\(x\)\+\\tau\\,m\_\{\\Delta\}\(x\)m\_\{\\Delta\}\(x\)^\{\\top\}\.

The identity is the second\-moment decomposition forD=Y−XD=Y\-X\. Its role is conceptual,CΔC\_\{\\Delta\}measuring conditional spread, whileτ​mΔ​mΔ⊤\\tau m\_\{\\Delta\}m\_\{\\Delta\}^\{\\top\}measures coherent displacement that would be removed by conditional\-mean centering\.

Averaging and taking traces gives

2​tr⁡\(G¯Δρ\)=∫tr⁡\(CΔ​\(x\)\)​ρ​\(d​x\)⏟conditional spread trace\+τ​∫‖mΔ​\(x\)‖2​ρ​\(d​x\)⏟coherent displacement trace\.2\\,\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}^\{\\rho\}\)=\\underbrace\{\\int\\operatorname\{tr\}\(C\_\{\\Delta\}\(x\)\)\\,\\rho\(dx\)\}\_\{\\text\{conditional spread trace\}\}\+\\underbrace\{\\tau\\int\\\|m\_\{\\Delta\}\(x\)\\\|^\{2\}\\,\\rho\(dx\)\}\_\{\\text\{coherent displacement trace\}\}\.\(2\)The coherent displacement fractionFΔρF\_\{\\Delta\}^\{\\rho\}is the fraction of the right\-hand side accounted for by the second term\. For trained input\-driven networks, conditional spread includes variation from input continuations and finite\-resolution source smoothing\.

### 3\.3Coordinate circulation

The directed part of the finite\-lag law is summarized by the antisymmetric lagged cross\-moment

𝒲Δρ=τ−1​\(𝔼ρ​\[X~​Y~⊤\]−𝔼ρ​\[Y~​X~⊤\]\),\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}=\\tau^\{\-1\}\\left\(\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{X\}\\,\\widetilde\{Y\}^\{\\top\}\]\-\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{Y\}\\,\\widetilde\{X\}^\{\\top\}\]\\right\),\(3\)whereX∼ρX\\sim\\rho,Y∼QΔ\(⋅∣X\)Y\\sim Q\_\{\\Delta\}\(\\cdot\\mid X\), andX~,Y~\\widetilde\{X\},\\widetilde\{Y\}denote centered coordinates in the chosen coordinate system\. In experiments we use the center\-RMS normalized coordinates of Appendix[A\.5](https://arxiv.org/html/2607.01746#A1.SS5)\. The matrix𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}is real and skew\-symmetric, so its eigenvalues are purely imaginary in conjugate pairs\. For a stationary reversible Markov law on a common state space,𝒲Δρ=0\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}=0\.

We report‖𝒲Δρ‖F\\\|\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\}, the largest imaginary eigenvalue magnitudeωmax\\omega\_\{\\max\}, and the relative circulation

rcirc=‖𝒲Δρ‖Ftr⁡\(G¯Δρ\)\.r\_\{\\rm circ\}=\\frac\{\\\|\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\}\}\{\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}^\{\\rho\}\)\}\.Together,GΔG\_\{\\Delta\}and𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}give the symmetric and antisymmetric second\-moment summaries of finite\-lag hidden\-state transport\.

Table 1:Finite\-lag observables derived fromQΔQ\_\{\\Delta\}\. Trace quantities depend on the chosen metric\. Experiments report center\-RMS Euclidean summaries unless stated otherwise\.The quantities in Table[1](https://arxiv.org/html/2607.01746#S3.T1)are intended to be read jointly\. Largetr⁡\(G¯Δρ\)\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}^\{\\rho\}\)indicates large finite\-lag motion, but not whether that motion is coherent or spread across possible successors\. Equation \([2](https://arxiv.org/html/2607.01746#S3.E2)\) separates these cases\. The circulation statistic captures a different axis, an antisymmetric lagged source–successor structure\. A representation can therefore have large transport and zero circulation, as in isotropic contraction, or nonzero circulation with comparable transport scale, as in oriented rotation\.

## 4Structural Properties of Finite\-Lag Geometry

We establish four structural properties of the finite\-lag construction\. First, the tensorial observables are affine\-covariant, while scalar summaries depend on the metric used to contract tensors\. Second, the dense Gaussian estimator is Lipschitz on bounded trajectory clouds at fixed bandwidth\. Third, finite\-lag source\-centered transport separates from infinitesimal carré\-du\-champ geometry by retaining deterministic displacement at the chosen lag\. Fourth, a linear\-Gaussian model gives a closed\-form calibration of spread, coherent displacement, and circulation\. Proofs appear in Appendices[B](https://arxiv.org/html/2607.01746#A2)–[D](https://arxiv.org/html/2607.01746#A4)\.

### 4\.1Affine covariance

###### Theorem 1\(Affine covariance and metric dependence\)\.

Letϕ​\(x\)=A​x\+b\\phi\(x\)=Ax\+bbe an invertible affine map, and letQΔ′Q^\{\\prime\}\_\{\\Delta\}be the pushed\-forward conditional law\. For the unnormalized tensorial quantities, and for centered coordinates in𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\},

mΔ′=A​mΔ,CΔ′=A​CΔ​A⊤,GΔ′=A​GΔ​A⊤,m^\{\\prime\}\_\{\\Delta\}=A\\,m\_\{\\Delta\},\\qquad C^\{\\prime\}\_\{\\Delta\}=A\\,C\_\{\\Delta\}A^\{\\top\},\\qquad G^\{\\prime\}\_\{\\Delta\}=A\\,G\_\{\\Delta\}A^\{\\top\},and

\(𝒲Δ′\)ρ′=A​𝒲Δρ​A⊤\.\(\\mathcal\{W\}^\{\\prime\}\_\{\\Delta\}\)^\{\\rho^\{\\prime\}\}=A\\,\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}A^\{\\top\}\.ForM≻0M\\succ 0, define

SCM=∫tr⁡\(M​CΔ​\(x\)\)​ρ​\(d​x\),SmM=τ​∫mΔ​\(x\)⊤​M​mΔ​\(x\)​ρ​\(d​x\),S\_\{C\}^\{M\}=\\int\\operatorname\{tr\}\(MC\_\{\\Delta\}\(x\)\)\\,\\rho\(dx\),\\qquad S\_\{m\}^\{M\}=\\tau\\int m\_\{\\Delta\}\(x\)^\{\\top\}Mm\_\{\\Delta\}\(x\)\\,\\rho\(dx\),FM=SmMSCM\+SmM\.F^\{M\}=\\frac\{S\_\{m\}^\{M\}\}\{S\_\{C\}^\{M\}\+S\_\{m\}^\{M\}\}\.UnderM′=A−⊤​M​A−1M^\{\\prime\}=A^\{\-\\top\}MA^\{\-1\},

SC′M′=SCM,Sm′M′=SmM,FM′=FM\.S\_\{C^\{\\prime\}\}^\{M^\{\\prime\}\}=S\_\{C\}^\{M\},\\qquad S\_\{m^\{\\prime\}\}^\{M^\{\\prime\}\}=S\_\{m\}^\{M\},\\qquad F^\{M^\{\\prime\}\}=F^\{M\}\.

Thus the tensor observables are affine\-covariant, while scalar summaries require a metric\. Euclidean traces are preserved by orthogonal changes of coordinates\. Center\-RMS normalization also removes translations and global scalar rescalings\. Anisotropic reparameterizations require the metric correction above\.

### 4\.2Dense stability

For paired clouds𝒵=\{\(xi,yi\)\}i=1n\\mathcal\{Z\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}, write

‖𝒵−𝒵~‖∞=maxi⁡max⁡\{‖xi−x~i‖,‖yi−y~i‖\}\.\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}=\\max\_\{i\}\\max\\\{\\\|x\_\{i\}\-\\widetilde\{x\}\_\{i\}\\\|,\\\|y\_\{i\}\-\\widetilde\{y\}\_\{i\}\\\|\\\}\.
###### Theorem 2\(Stability of the dense empirical operator\)\.

Fixε\>0\\varepsilon\>0andR\>0R\>0\. On the set of paired clouds satisfyingmaxi⁡\{‖xi‖,‖yi‖\}≤R\\max\_\{i\}\\\{\\\|x\_\{i\}\\\|,\\\|y\_\{i\}\\\|\\\}\\leq R,

‖P^Δ​\(𝒵\)−P^Δ​\(𝒵~\)‖∞→∞≤4​Rε​exp⁡\(R2/ε\)​‖𝒵−𝒵~‖∞\.\\left\\\|\\widehat\{P\}\_\{\\Delta\}\(\\mathcal\{Z\}\)\-\\widehat\{P\}\_\{\\Delta\}\(\\widetilde\{\\mathcal\{Z\}\}\)\\right\\\|\_\{\\infty\\to\\infty\}\\leq\\frac\{4R\}\{\\varepsilon\}\\exp\(R^\{2\}/\\varepsilon\)\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}\.The derived empirical observablesG^Δ,m^Δ,C^Δ\\widehat\{G\}\_\{\\Delta\},\\widehat\{m\}\_\{\\Delta\},\\widehat\{C\}\_\{\\Delta\}, their trace summaries, and𝒲^Δρ\\widehat\{\\mathcal\{W\}\}\_\{\\Delta\}^\{\\rho\}are Lipschitz on bounded sets at fixedε\\varepsilon\.

The proof bounds squared\-distance perturbations, transfers the bound through the Gaussian kernel, and then controls row normalization using the positive kernel lower boundexp⁡\(−R2/ε\)\\exp\(\-R^\{2\}/\\varepsilon\)\. Narrower kernels resolve more local structure but worsen the stability constant\. After center\-RMS normalization, the same conclusion holds when the RMS scale is bounded away from zero\.

### 4\.3Finite\-lag separation

For an Itô diffusion, infinitesimal carré du champ records the second\-order diffusion tensor and cancels first\-order drift\. Finite\-lag source\-centered transport keeps the displacement at the chosen lag\.

###### Theorem 3\(Finite\-lag detection of deterministic motion\)\.

FixΔ≥1\\Delta\\geq 1, letτ=Δ​d​t\\tau=\\Delta\\,dt, and letT:ℝd→ℝdT:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}be measurable\. Assume

∫‖TΔ​\(x\)−x‖2​ρ​\(d​x\)<∞\\int\\\|T^\{\\Delta\}\(x\)\-x\\\|^\{2\}\\,\\rho\(dx\)<\\inftyand supposeTΔ​\(x\)≠xT^\{\\Delta\}\(x\)\\neq xon aρ\\rho\-positive set\. IfXt\+Δ=TΔ​\(Xt\)X\_\{t\+\\Delta\}=T^\{\\Delta\}\(X\_\{t\}\)deterministically, then

G¯Δρ=12​τ​∫\(TΔ​\(x\)−x\)​\(TΔ​\(x\)−x\)⊤​ρ​\(d​x\),\\bar\{G\}\_\{\\Delta\}^\{\\rho\}=\\frac\{1\}\{2\\tau\}\\int\(T^\{\\Delta\}\(x\)\-x\)\(T^\{\\Delta\}\(x\)\-x\)^\{\\top\}\\,\\rho\(dx\),and

tr⁡\(G¯Δρ\)\>0\.\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}^\{\\rho\}\)\>0\.For an oriented cyclic shiftTΔ=ΠT^\{\\Delta\}=\\Piin centered isotropic coordinates with covarianceΣ=σx2​I\\Sigma=\\sigma\_\{x\}^\{2\}I,

‖𝒲Δρ‖F=σx2τ​‖Π⊤−Π‖F\>0\\\|\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\}=\\frac\{\\sigma\_\{x\}^\{2\}\}\{\\tau\}\\\|\\Pi^\{\\top\}\-\\Pi\\\|\_\{F\}\>0wheneverΠ\\Piis not symmetric\. By contrast, a deterministic continuous\-time flow represented by the first\-order generatorL=b⋅∇L=b\\cdot\\nablahasΓL≡0\\Gamma\_\{L\}\\equiv 0\.

The theorem isolates the reason for using finite lag in recurrent representations\. A hidden state is geometrically meaningful through its finite\-step successor, andGΔG\_\{\\Delta\}records that coherent displacement directly\. In short,GΔG\_\{\\Delta\}captures finite\-step motion at the operating lag, including deterministic recurrent motion that an infinitesimal carré\-du\-champ limit removes from its second\-order geometry\.

### 4\.4Linear\-Gaussian closed form

The next theorem gives a parametric calibration in the simplest case where the update is linear and the innovation is additive Gaussian noise\. The closed form shows which terms are responsible for conditional spread, coherent displacement, and directed circulation\.

###### Theorem 4\(Linear\-Gaussian closed form\)\.

Let

Xt\+Δ=AΔ​Xt\+ξt,X\_\{t\+\\Delta\}=A\_\{\\Delta\}X\_\{t\}\+\\xi\_\{t\},whereXtX\_\{t\}is mean zero with covarianceΣ\\Sigma, andξt\\xi\_\{t\}is independent ofXtX\_\{t\}, mean zero, with covarianceΣξ\\Sigma\_\{\\xi\}\. Then

mΔ​\(x\)=τ−1​\(AΔ−I\)​x,CΔ​\(x\)=τ−1​Σξ,m\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\(A\_\{\\Delta\}\-I\)x,\\qquad C\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\Sigma\_\{\\xi\},G¯Δ=12​τ​\[Σξ\+\(AΔ−I\)​Σ​\(AΔ−I\)⊤\],\\bar\{G\}\_\{\\Delta\}=\\frac\{1\}\{2\\tau\}\\left\[\\Sigma\_\{\\xi\}\+\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\}\\right\],and

𝒲Δ=τ−1​\(Σ​AΔ⊤−AΔ​Σ\)\.\\mathcal\{W\}\_\{\\Delta\}=\\tau^\{\-1\}\\left\(\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigma\\right\)\.Consequently,

2​tr⁡\(G¯Δ\)=τ−1​tr⁡\(Σξ\)\+τ−1​tr⁡\(\(AΔ−I\)​Σ​\(AΔ−I\)⊤\)\.2\\,\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\)=\\tau^\{\-1\}\\operatorname\{tr\}\(\\Sigma\_\{\\xi\}\)\+\\tau^\{\-1\}\\operatorname\{tr\}\\\!\\left\(\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\}\\right\)\.

The closed form separates the three mechanisms measured by the framework\. The innovation covarianceΣξ\\Sigma\_\{\\xi\}contributes conditional spread\. The update offsetAΔ−IA\_\{\\Delta\}\-Icontributes coherent displacement through the source\-covariance\-weighted form

\(AΔ−I\)​Σ​\(AΔ−I\)⊤\.\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\}\.The antisymmetric mismatch

Σ​AΔ⊤−AΔ​Σ\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigmacontributes directed coordinate circulation\.

This parallels the Gaussian bridge in static feedforward operator geometry\[[36](https://arxiv.org/html/2607.01746#bib.bib1)\]\. There, class offsetsμa−μb\\mu\_\{a\}\-\\mu\_\{b\}enter through the bandwidth\-regularized inverse\-covariance form

cε\(a,b\)=14​\(μa−μb\)⊤​\(ε​I\+Σ\)−1​\(μa−μb\)\.c\_\{\\varepsilon\}^\{\(a,b\)\}=\\tfrac\{1\}\{4\}\(\\mu\_\{a\}\-\\mu\_\{b\}\)^\{\\top\}\(\\varepsilon I\+\\Sigma\)^\{\-1\}\(\\mu\_\{a\}\-\\mu\_\{b\}\)\.Here, recurrent update offsets replace class\-mean offsets, and the finite\-lag law contributes an additional directed component\. The population closed form contains no bandwidth parameter, as bandwidth enters through the empirical source\-smoothing estimator\.

## 5Empirical study of the formal quantities

The experiments calibrate the quantities from Sections[3](https://arxiv.org/html/2607.01746#S3)–[4](https://arxiv.org/html/2607.01746#S4)\. Controlled systems test the decomposition, circulation, affine covariance, and dense stability results\. We then use repeat\-copy networks as a compact recurrent case study\. Unless otherwise stated, empirical operators use the dense Gaussian source\-smoothing estimator, center\-RMS normalization, lagΔ=1\\Delta=1, and the median\-heuristic bandwidth\.

### 5\.1Controlled calibration and stability

We first validate the linear\-Gaussian closed form\. We sample

Y=A​X\+ξ,X∼𝒩​\(0,I\),ξ∼𝒩​\(0,σ2​I\),Y=AX\+\\xi,\\qquad X\\sim\\mathcal\{N\}\(0,I\),\\qquad\\xi\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\),withA=I\+β​BA=I\+\\beta Bfor fixed Frobenius\-normalizedBB\. We sweepβ∈\{0,0\.25,0\.5,0\.75,1\.0\}\\beta\\in\\\{0,0\.25,0\.5,0\.75,1\.0\\\}andσ∈\{0,0\.1,0\.25,0\.5\}\\sigma\\in\\\{0,0\.1,0\.25,0\.5\\\}atd=16d=16\. Because this experiment tests the population closed form, spread and coherent displacement are computed from the known conditional moments\. As predicted,

coherent trace=β2,spread trace=d​σ2\.\\text\{coherent trace\}=\\beta^\{2\},\\qquad\\text\{spread trace\}=d\\sigma^\{2\}\.Across all settings, the maximum relative error is below0\.7%0\.7\\%\. A separate two\-dimensional sweep withA=α​I\+γ​JA=\\alpha I\+\\gamma J, whereJJis antisymmetric, validates the directed statistic, showing that‖𝒲^Δ‖F\\\|\\widehat\{\\mathcal\{W\}\}\_\{\\Delta\}\\\|\_\{F\}grows approximately linearly withγ\\gammaand vanishes atγ=0\\gamma=0, matching the linear\-Gaussian formula\.

We next test the structural theorems\. For affine covariance, we apply translations, orthogonal maps, scalar dilations, and anisotropic diagonal maps to a cyclic\-shift trajectory cloud\. In the push\-forward experiment, the conditional weights are held fixed and the coordinates are transformed\. Translations and orthogonal maps preserve Euclidean trace summaries, scalar dilations scale traces by the squared scale, and the metric correctionM′=A−⊤​M​A−1M^\{\\prime\}=A^\{\-\\top\}MA^\{\-1\}restores the original trace summaries to numerical precision\. If the Gaussian kernel is rebuilt after an anisotropic transformation, the estimator itself changes, as expected; this sensitivity is reported separately in Appendix[F\.2](https://arxiv.org/html/2607.01746#A6.SS2)\.

For dense stability, we perturb a fixed cyclic\-shift cloud by Gaussian noise of scaleσp\\sigma\_\{p\}, keep the bandwidth fixed, and measure absolute changes intr⁡\(G¯Δ\)\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\), coherent displacement trace, and‖𝒲^Δ‖F\\\|\\widehat\{\\mathcal\{W\}\}\_\{\\Delta\}\\\|\_\{F\}, averaged over eight perturbations per scale\. The observed slopes are close to one, consistent with Theorem[2](https://arxiv.org/html/2607.01746#Thmtheorem2)\. Sparsekk\-NN approximations are tested in Appendix[F\.6](https://arxiv.org/html/2607.01746#A6.SS6)\. They are useful computationally, but change the local resolution and do not share the dense operator’s Lipschitz guarantee\.

![Refer to caption](https://arxiv.org/html/2607.01746v1/x1.png)Figure 1:Controlled calibration and stability of the finite\-lag observables\. \(A\) At fixed innovation noiseσ=0\.1\\sigma=0\.1, the trace decomposition separates conditional spread from coherent displacement as the update scaleβ\\betavaries\. \(B\) The coherent displacement trace scales with update strengthβ\\betaand is independent ofσ\\sigmaup to Monte Carlo error\. \(C\) The conditional spread trace scales with innovation noiseσ\\sigmaand is independent ofβ\\betaup to Monte Carlo error\. \(D\) Dense finite\-lag stability: mean absolute metric change versus trajectory perturbation scale on log\-log axes\. Empirical slopes are1\.161\.16,1\.131\.13, and1\.071\.07for transport scale, coherent displacement trace, and circulation, respectively\.
### 5\.2Repeat\-copy recurrent case study

We train Elman, GRU, and LSTM networks on repeat\-copy with feature dimension44, using a recall\-window\-weighted loss\. The five\-seed performance\-matched run uses copy length1010; the capacity\-control, memory\-horizon, and phase\-profile experiments use copy length88in the expanded grids \(Appendix[F\.4](https://arxiv.org/html/2607.01746#A6.SS4)\)\. All main\-table runs solve the task: recall sign accuracy is1\.0001\.000for Elman and GRU, and0\.99990\.9999for LSTM\. Table[2](https://arxiv.org/html/2607.01746#S5.T2)reports dense finite\-lag quantities on the hidden statehth\_\{t\}across five seeds\.

Table 2:Performance\-matched repeat\-copy case study\. Dense estimator,Δ=1\\Delta=1, median\-heuristic bandwidth, center\-RMS normalization\. Static effective rank is included as a snapshot baseline\. Mean±\\pmstandard deviation over five seeds\.The strongest architecture\-dependent differences are in total transport scale and coherent displacement trace\. Elman has largertr⁡\(G¯Δ\)\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\)than GRU and LSTM by roughly15%15\\%, and its coherent trace is larger by1616–26%26\\%\. In contrast, the coherent displacement fraction is not a stable architecture identifier: at the median bandwidth it lies near0\.450\.45–0\.500\.50, and the sensitivity sweeps below show that it changes with kernel resolution and coordinate normalization\.

Finally, phase\-resolved analysis localizes where the transport differences arise\. In a solved repeat\-copy setting with delayD=4D=4, we split source–successor pairs into write, cue, delay, and recall phases, apply the same global center\-RMS normalization, and compute phase\-local dense operators\. Figure[2](https://arxiv.org/html/2607.01746#S5.F2)shows that Elman has substantially larger transport and coherent displacement during write and recall, while GRU and LSTM maintain lower transport through write, cue, and delay and increase at recall\. Static effective rank follows a different pattern, showing that finite\-lag geometry captures when hidden states move, not only where they sit\. Full tables are in Appendix[F\.9](https://arxiv.org/html/2607.01746#A6.SS9)\.

![Refer to caption](https://arxiv.org/html/2607.01746v1/x2.png)Figure 2:Phase\-resolved finite\-lag geometry on repeat\-copy with delayD=4D=4\. Metrics are computed separately on source–successor pairs from the write, cue, delay, and recall phases after global center\-RMS normalization\. Elman shows larger transport and coherent displacement during write and recall, while GRU and LSTM maintain lower transport through write, cue, and delay and increase at recall\. Static effective rank gives a different snapshot view, illustrating that finite\-lag geometry localizes when hidden states move during the computation\.#### Capacity controls and sensitivity\.

Parameter\-count controls preserve the transport\-scale gap\. At nearly matched parameter counts, Elman\-64 has4,8044\{,\}804parameters andtr⁡\(G¯Δ\)=1\.013±0\.001\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\)=1\.013\\pm 0\.001, while GRU\-36 has4,7924\{,\}792parameters andtr⁡\(G¯Δ\)=0\.858±0\.067\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\)=0\.858\\pm 0\.067\. Across the controlled grid, Elman configurations remain near1\.011\.01, while gated configurations lie between0\.860\.86and0\.890\.89\.

Resolution and metric sweeps explain which scalars are robust\. For dense center\-RMS operators, narrowing the bandwidth fromεmed\\varepsilon\_\{\\rm med\}to0\.1​εmed0\.1\\varepsilon\_\{\\rm med\}increases coherent displacement fraction from about0\.500\.50to0\.620\.62for Elman,0\.450\.45to0\.530\.53for GRU, and0\.500\.50to0\.580\.58for LSTM\. Whitening changes absolute trace scale and shifts the fraction, as predicted by the metric dependence in Theorem[1](https://arxiv.org/html/2607.01746#Thmtheorem1)\. Sparsekk\-NN approximations produce still more local fractions and larger circulation values\. Thus the robust learned\-network finding is the transport\-scale and coherent\-trace separation\. Fractions and circulation norms must be read with their metric and resolution choices\.

A small\-delay memory\-horizon sweep gives the same qualitative picture along a task axis\. We insert blank delaysD∈\{0,2,4,6,8\}D\\in\\\{0,2,4,6,8\\\}between delimiter and recall\. GRU and LSTM solve all delays across three seeds\. Elman solves all seeds throughD=6D=6and two of three seeds atD=8D=8\. For GRU,tr⁡\(G¯Δ\)\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\)decreases from0\.8750\.875atD=0D=0to0\.7160\.716atD=8D=8, and coherent trace decreases from0\.8260\.826to0\.5630\.563\. LSTM shows the same trend, with transport decreasing from0\.8800\.880to0\.7280\.728and coherent trace from0\.8420\.842to0\.6400\.640\. Circulation increases from near zero to a larger plateau for both gated architectures\. Thus the finite\-lag observables respond systematically to memory horizon on solved runs\. Full tables are in Appendix[F\.10](https://arxiv.org/html/2607.01746#A6.SS10)\.

## 6Discussion

We introduced finite\-lag operator geometry as a trajectory\-directed counterpart of static diffusion operator geometry\. The primitive is a directed empirical transport law on observed source–successor pairs\. From this law we derive a source\-centered transport tensorGΔG\_\{\\Delta\}, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}, which summarizes directed finite\-lag flow\. The framework is built at the lag where the recurrent computation is observed\. It does not require estimating an underlying continuous\-time generator or assuming that hidden state alone is autonomously Markov\.

The structural results clarify what is intrinsic and what is conventional\. Affine covariance shows that the tensorial quantities transform naturally, while scalar summaries require a metric choice\. Dense stability justifies the smooth Gaussian source\-smoothing estimator over hardkk\-NN sparsification\. Finite\-lag separation explains why deterministic recurrent motion can be visible at finite lag even when infinitesimal carré\-du\-champ geometry is zero\. The linear\-Gaussian closed form provides an analytical calibration, showing transport scale comes from innovation spread and coherent update displacement, while circulation comes from the antisymmetric mismatch of lagged cross\-covariances\.

Empirically, controlled systems validate the formal quantities, and a performance\-matched repeat\-copy case study illustrates their behavior on trained recurrent networks\. The most robust finite\-lag differences in this case study are total transport scale and coherent displacement trace\. In contrast, coherent displacement fraction and circulation norms depend on the chosen metric and kernel resolution, as predicted by the theory\. Capacity controls preserve the transport\-scale gap, and memory\-horizon experiments show that the quantities respond to task demand on solved runs\.

These experiments are not intended as universal architecture claims\. They demonstrate how finite\-lag operator geometry can be used, by calibrating the formal quantities in controlled systems, then report transport scale, conditional spread, coherent displacement, and directed circulation under explicit normalization and bandwidth choices\. The broader message is that recurrent representations should be analyzed not only as clouds of states, but as finite\-lag transport laws describing where the computation sends those states\.

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## Appendix AEmpirical operator construction

Here we give the operational details of the empirical finite\-lag operatorP^Δ\\widehat\{P\}\_\{\\Delta\}used throughout the paper\. The dense Gaussian source\-smoothing estimator of Equation \([1](https://arxiv.org/html/2607.01746#S3.E1)\) is the object used in the stability theory\. Thekk\-nearest\-neighbor version is a scalable approximation and is used only where explicitly stated\.

### A\.1Lagged\-pair pooling and notation

Hidden trajectories are writtenht\(s\)∈ℝdh\_\{t\}^\{\(s\)\}\\in\\mathbb\{R\}^\{d\}, indexed by sequence or trialssand timett\. For a fixed integer lagΔ≥1\\Delta\\geq 1and physical lagτ=Δ​d​t\\tau=\\Delta\\,dt, define the pooled index set

𝒳=\{\(s,t\):ht\(s\)​is observed\},xp=ht\(s\)for​p=\(s,t\)\.\\mathcal\{X\}=\\\{\(s,t\):h\_\{t\}^\{\(s\)\}\\text\{ is observed\}\\\},\\qquad x\_\{p\}=h\_\{t\}^\{\(s\)\}\\quad\\text\{for \}p=\(s,t\)\.The valid source set is

𝒳−=\{p=\(s,t\)∈𝒳:\(s,t\+Δ\)∈𝒳\}\.\\mathcal\{X\}^\{\-\}=\\\{p=\(s,t\)\\in\\mathcal\{X\}:\(s,t\+\\Delta\)\\in\\mathcal\{X\}\\\}\.Forp∈𝒳−p\\in\\mathcal\{X\}^\{\-\}, writep\+Δ=\(s,t\+Δ\)p\+\\Delta=\(s,t\+\\Delta\), define the lagged successor mapTΔ​\(p\)=p\+ΔT\_\{\\Delta\}\(p\)=p\+\\Delta, and set

yp=xTΔ​\(p\)\.y\_\{p\}=x\_\{T\_\{\\Delta\}\(p\)\}\.The empirical lagged pair cloud is

\{\(xp,yp\):p∈𝒳−\}\.\\\{\(x\_\{p\},y\_\{p\}\):p\\in\\mathcal\{X\}^\{\-\}\\\}\.We writen=\|𝒳−\|n=\|\\mathcal\{X\}^\{\-\}\|for the number of valid pairs andddfor the hidden dimension\. Unless otherwise stated, the empirical source measure is uniform on𝒳−\\mathcal\{X\}^\{\-\}, soρp=1/n\\rho\_\{p\}=1/n\.

Throughout the construction, query indicesqqand neighbor indicesiirange over𝒳−\\mathcal\{X\}^\{\-\}\. Successor coordinatesyiy\_\{i\}may correspond to indices in𝒳∖𝒳−\\mathcal\{X\}\\setminus\\mathcal\{X\}^\{\-\}, but they are always attached to valid source indicesi∈𝒳−i\\in\\mathcal\{X\}^\{\-\}\.

### A\.2Dense Gaussian source\-smoothing operator

For bandwidthε\>0\\varepsilon\>0, define the Gaussian source kernel

Kq​i=kε​\(xq,xi\)=exp⁡\(−‖xq−xi‖24​ε\),Dq=∑i∈𝒳−Kq​i\.K\_\{qi\}=k\_\{\\varepsilon\}\(x\_\{q\},x\_\{i\}\)=\\exp\\\!\\left\(\-\\frac\{\\\|x\_\{q\}\-x\_\{i\}\\\|^\{2\}\}\{4\\varepsilon\}\\right\),\\qquad D\_\{q\}=\\sum\_\{i\\in\\mathcal\{X\}^\{\-\}\}K\_\{qi\}\.Sinceq∈𝒳−q\\in\\mathcal\{X\}^\{\-\}andKq​q=1K\_\{qq\}=1, eachDqD\_\{q\}is strictly positive\. The dense empirical source\-smoothing operator has normalized weights

wq​i=Kq​iDq,\(P^Δ\)q​i=wq​i,w\_\{qi\}=\\frac\{K\_\{qi\}\}\{D\_\{q\}\},\\qquad\(\\widehat\{P\}\_\{\\Delta\}\)\_\{qi\}=w\_\{qi\},and the corresponding empirical conditional transport law is

Q^Δ​\(d​y∣xq\)=∑i∈𝒳−wq​i​δyi​\(d​y\)\.\\widehat\{Q\}\_\{\\Delta\}\(dy\\mid x\_\{q\}\)=\\sum\_\{i\\in\\mathcal\{X\}^\{\-\}\}w\_\{qi\}\\,\\delta\_\{y\_\{i\}\}\(dy\)\.
The indexiiin\(P^Δ\)q​i\(\\widehat\{P\}\_\{\\Delta\}\)\_\{qi\}denotes a neighboring source point whose observed successoryiy\_\{i\}contributes to the conditional law at the query sourcexqx\_\{q\}\. ThusP^Δ\\widehat\{P\}\_\{\\Delta\}is a source\-smoothing operator for a successor\-valued empirical law\.

We use the Gaussian factor of44in the bandwidth following the convention used in diffusion\-map constructions\[[11](https://arxiv.org/html/2607.01746#bib.bib12)\]\. This matches the median heuristic bandwidth

εmed=14​mediani<j⁡‖xi−xj‖2,\\varepsilon\_\{\\rm med\}=\\frac\{1\}\{4\}\\,\\operatorname\{median\}\_\{i<j\}\\\|x\_\{i\}\-x\_\{j\}\\\|^\{2\},under which the median pairwise affinity ise−1e^\{\-1\}\.

### A\.3Empirical observables

Let

y¯q=∑iwq​i​yi\.\\bar\{y\}\_\{q\}=\\sum\_\{i\}w\_\{qi\}y\_\{i\}\.The empirical squared transport scale, finite\-lag drift, source\-centered transport tensor, and conditional spread tensor are

e^q2=∑iwq​i​‖yi−xq‖2,m^Δ​\(xq\)=τ−1​\(y¯q−xq\),\\widehat\{e\}\_\{q\}^\{\\,2\}=\\sum\_\{i\}w\_\{qi\}\\\|y\_\{i\}\-x\_\{q\}\\\|^\{2\},\\qquad\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)=\\tau^\{\-1\}\(\\bar\{y\}\_\{q\}\-x\_\{q\}\),G^Δ​\(xq\)=12​τ​∑iwq​i​\(yi−xq\)​\(yi−xq\)⊤,\\widehat\{G\}\_\{\\Delta\}\(x\_\{q\}\)=\\frac\{1\}\{2\\tau\}\\sum\_\{i\}w\_\{qi\}\(y\_\{i\}\-x\_\{q\}\)\(y\_\{i\}\-x\_\{q\}\)^\{\\top\},and

C^Δ​\(xq\)=τ−1​∑iwq​i​\[\(yi−xq\)−τ​m^Δ​\(xq\)\]​\[\(yi−xq\)−τ​m^Δ​\(xq\)\]⊤\.\\widehat\{C\}\_\{\\Delta\}\(x\_\{q\}\)=\\tau^\{\-1\}\\sum\_\{i\}w\_\{qi\}\\bigl\[\(y\_\{i\}\-x\_\{q\}\)\-\\tau\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)\\bigr\]\\bigl\[\(y\_\{i\}\-x\_\{q\}\)\-\\tau\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)\\bigr\]^\{\\top\}\.Equivalently, sinceτ​m^Δ​\(xq\)=y¯q−xq\\tau\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)=\\bar\{y\}\_\{q\}\-x\_\{q\},

C^Δ​\(xq\)=τ−1​∑iwq​i​\(yi−y¯q\)​\(yi−y¯q\)⊤=τ−1​\(∑iwq​i​yi​yi⊤−y¯q​y¯q⊤\)\.\\widehat\{C\}\_\{\\Delta\}\(x\_\{q\}\)=\\tau^\{\-1\}\\sum\_\{i\}w\_\{qi\}\(y\_\{i\}\-\\bar\{y\}\_\{q\}\)\(y\_\{i\}\-\\bar\{y\}\_\{q\}\)^\{\\top\}=\\tau^\{\-1\}\\left\(\\sum\_\{i\}w\_\{qi\}y\_\{i\}y\_\{i\}^\{\\top\}\-\\bar\{y\}\_\{q\}\\bar\{y\}\_\{q\}^\{\\top\}\\right\)\.Thus the conditional spread term is the covariance of the smoothed successor cloud attached to the query source\.

The pointwise decomposition

2​G^Δ​\(xq\)=C^Δ​\(xq\)\+τ​m^Δ​\(xq\)​m^Δ​\(xq\)⊤2\\,\\widehat\{G\}\_\{\\Delta\}\(x\_\{q\}\)=\\widehat\{C\}\_\{\\Delta\}\(x\_\{q\}\)\+\\tau\\,\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)^\{\\top\}holds exactly by construction\. Averaging over a source measureρ\\rhogives

G¯^Δρ=∑qρq​G^Δ​\(xq\),C¯^Δρ=∑qρq​C^Δ​\(xq\),\\widehat\{\\bar\{G\}\}\_\{\\Delta\}^\{\\rho\}=\\sum\_\{q\}\\rho\_\{q\}\\,\\widehat\{G\}\_\{\\Delta\}\(x\_\{q\}\),\\qquad\\widehat\{\\bar\{C\}\}\_\{\\Delta\}^\{\\rho\}=\\sum\_\{q\}\\rho\_\{q\}\\,\\widehat\{C\}\_\{\\Delta\}\(x\_\{q\}\),and hence

2​tr⁡\(G¯^Δρ\)=tr⁡\(C¯^Δρ\)\+τ​∑qρq​‖m^Δ​\(xq\)‖2\.2\\,\\operatorname\{tr\}\(\\widehat\{\\bar\{G\}\}\_\{\\Delta\}^\{\\rho\}\)=\\operatorname\{tr\}\(\\widehat\{\\bar\{C\}\}\_\{\\Delta\}^\{\\rho\}\)\+\\tau\\sum\_\{q\}\\rho\_\{q\}\\\|\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)\\\|^\{2\}\.We refer to the two terms on the right\-hand side as the conditional spread trace and coherent displacement trace, respectively\.

For efficient computation, local moments are evaluated chunk\-wise to avoid materializing the full\(chunk size,n,d\)\(\\text\{chunk size\},n,d\)displacement tensor\. The squared\-distance term is computed using

‖yi−xq‖2=‖yi‖2\+‖xq‖2−2​yi⊤​xq\.\\\|y\_\{i\}\-x\_\{q\}\\\|^\{2\}=\\\|y\_\{i\}\\\|^\{2\}\+\\\|x\_\{q\}\\\|^\{2\}\-2y\_\{i\}^\{\\top\}x\_\{q\}\.For the full transport tensor, we use the identity

∑iwq​i​\(yi−xq\)​\(yi−xq\)⊤=∑iwq​i​yi​yi⊤−xq​y¯q⊤−y¯q​xq⊤\+xq​xq⊤\.\\sum\_\{i\}w\_\{qi\}\(y\_\{i\}\-x\_\{q\}\)\(y\_\{i\}\-x\_\{q\}\)^\{\\top\}=\\sum\_\{i\}w\_\{qi\}y\_\{i\}y\_\{i\}^\{\\top\}\-x\_\{q\}\\bar\{y\}\_\{q\}^\{\\top\}\-\\bar\{y\}\_\{q\}x\_\{q\}^\{\\top\}\+x\_\{q\}x\_\{q\}^\{\\top\}\.The conditional spread tensor is computed using the covariance identity above\.

Numerical roundoff in chunk\-wise computation can produce a small negative residual in

e^q2−τ2​‖m^Δ​\(xq\)‖2\\widehat\{e\}\_\{q\}^\{\\,2\}\-\\tau^\{2\}\\\|\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)\\\|^\{2\}at machine precision\. We clamp this scalar residual to zero before reporting scalar spread traces, the tensor formulas above are unchanged\.

### A\.4Empirical coordinate circulation

The antisymmetric coordinate circulation in the main text is the lagged source–successor cross\-moment

𝒲Δρ=τ−1​\(𝔼ρ​\[X~​Y~⊤\]−𝔼ρ​\[Y~​X~⊤\]\),\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}=\\tau^\{\-1\}\\left\(\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{X\}\\widetilde\{Y\}^\{\\top\}\]\-\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{Y\}\\widetilde\{X\}^\{\\top\}\]\\right\),whereX~,Y~\\widetilde\{X\},\\widetilde\{Y\}denote the chosen normalized coordinates\. The default empirical operator estimate uses the same smoothed conditional lawQ^Δ\\widehat\{Q\}\_\{\\Delta\}as the transport tensor:

𝒲^Δ,smoothρ=τ−1​∑qρq​∑iwq​i​\(x~q​y~i⊤−y~i​x~q⊤\)\.\\widehat\{\\mathcal\{W\}\}\_\{\\Delta,\\mathrm\{smooth\}\}^\{\\rho\}=\\tau^\{\-1\}\\sum\_\{q\}\\rho\_\{q\}\\sum\_\{i\}w\_\{qi\}\\left\(\\widetilde\{x\}\_\{q\}\\widetilde\{y\}\_\{i\}^\{\\top\}\-\\widetilde\{y\}\_\{i\}\\widetilde\{x\}\_\{q\}^\{\\top\}\\right\)\.This matrix is skew\-symmetric by construction\.

For comparison with population joint\-moment calculations, we sometimes also report the raw lagged\-pair estimator

𝒲^Δ,rawρ=τ−1​∑qρq​\(x~q​y~q⊤−y~q​x~q⊤\)\.\\widehat\{\\mathcal\{W\}\}\_\{\\Delta,\\mathrm\{raw\}\}^\{\\rho\}=\\tau^\{\-1\}\\sum\_\{q\}\\rho\_\{q\}\\left\(\\widetilde\{x\}\_\{q\}\\widetilde\{y\}\_\{q\}^\{\\top\}\-\\widetilde\{y\}\_\{q\}\\widetilde\{x\}\_\{q\}^\{\\top\}\\right\)\.This is the special case of the smoothed formula obtained by replacingwq​iw\_\{qi\}with𝟏​\{i=q\}\\mathbf\{1\}\\\{i=q\\\}\. Unless explicitly described as raw, reported circulation values use the smoothed operator estimate\.

We summarize circulation by the Frobenius norm‖𝒲^Δρ‖F\\\|\\widehat\{\\mathcal\{W\}\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\}, the largest imaginary eigenvalue magnitudeωmax\\omega\_\{\\max\}, and the relative circulation

rcirc=‖𝒲^Δρ‖Ftr⁡\(G¯^Δρ\)\.r\_\{\\rm circ\}=\\frac\{\\\|\\widehat\{\\mathcal\{W\}\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\}\}\{\\operatorname\{tr\}\(\\widehat\{\\bar\{G\}\}\_\{\\Delta\}^\{\\rho\}\)\}\.The denominator is the total source\-centered transport trace in the same coordinate system\. When this trace is numerically zero, we leavercircr\_\{\\rm circ\}undefined rather than reporting an unstable ratio\.

### A\.5Center\-RMS coordinate normalization

Tensor quantities transform predictably under affine coordinate changes, but Euclidean scalar traces are not invariant under arbitrary linear reparameterizations\. Theorem[1](https://arxiv.org/html/2607.01746#Thmtheorem1)characterizes the exact transformation law for the unnormalized tensorial quantities\. In experiments, we adopt center\-RMS normalization as a reproducible reporting convention\.

Given pooled hidden states\{xp\}p∈𝒳\\\{x\_\{p\}\\\}\_\{p\\in\\mathcal\{X\}\}, define

x¯=\|𝒳\|−1​∑p∈𝒳xp,σRMS=\(\|𝒳\|−1​∑p∈𝒳‖xp−x¯‖2\)1/2,\\bar\{x\}=\|\\mathcal\{X\}\|^\{\-1\}\\sum\_\{p\\in\\mathcal\{X\}\}x\_\{p\},\\qquad\\sigma\_\{\\rm RMS\}=\\left\(\|\\mathcal\{X\}\|^\{\-1\}\\sum\_\{p\\in\\mathcal\{X\}\}\\\|x\_\{p\}\-\\bar\{x\}\\\|^\{2\}\\right\)^\{1/2\},and

x~p=xp−x¯σRMS\.\\widetilde\{x\}\_\{p\}=\\frac\{x\_\{p\}\-\\bar\{x\}\}\{\\sigma\_\{\\rm RMS\}\}\.The same affine map is applied to source and successor coordinates:

y~p=yp−x¯σRMS\.\\widetilde\{y\}\_\{p\}=\\frac\{y\_\{p\}\-\\bar\{x\}\}\{\\sigma\_\{\\rm RMS\}\}\.Using the same normalization for sources and successors is essential forGΔG\_\{\\Delta\},mΔm\_\{\\Delta\}, andCΔC\_\{\\Delta\}, since these quantities depend on displacementsy−xy\-x\.

The pooled normalized cloud has empirical mean zero and RMS norm one with respect to the measure used to computex¯\\bar\{x\}andσRMS\\sigma\_\{\\rm RMS\}\. When normalization is computed over all pooled hidden states𝒳\\mathcal\{X\}, the valid\-source subset𝒳−\\mathcal\{X\}^\{\-\}need not have exactly zero mean underρ\\rho\. This discrepancy is usually small in the trajectory windows we study, and the same affine coordinate system is applied consistently to sources and successors\.

For the circulation statistic, centering affects the decomposition between mean transport and rotational lagged cross\-covariance\. When the empirical source and successor marginals differ,𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}should be interpreted as an antisymmetric lagged cross\-moment in the chosen normalized coordinate system\. It may include finite horizon source–successor marginal imbalance in addition to cyclic flow\. In stationary or approximately stationary trajectory windows, this coincides with the usual centered lagged cross\-covariance interpretation\.

Center\-RMS normalization preserves Euclidean trace summaries under translations, orthogonal transformations, and global scalar rescalings\. It does not preserve them under anisotropic coordinate rescalings\. Those require the metric correction of Theorem[1](https://arxiv.org/html/2607.01746#Thmtheorem1)\. We use center\-RMS normalization in all main\-text experiments\.

IfσRMS\\sigma\_\{\\rm RMS\}is numerically zero, the trajectory cloud has no nontrivial empirical scale in the chosen window\. In that degenerate case the normalized Euclidean summaries are undefined\.

### A\.6kk\-nearest\-neighbor approximation

For larger trajectory clouds, the dense operator becomes computationally expensive\. Thekk\-nearest\-neighbor approximation replaces the dense weights with sparse weights restricted to nearby source points\.

For a queryxqx\_\{q\}and neighborhood sizekk, let𝒩k​\(xq\)\\mathcal\{N\}\_\{k\}\(x\_\{q\}\)be the selected source\-space neighborhood\. We always includeqqitself whenq∈𝒳−q\\in\\mathcal\{X\}^\{\-\}, replacing the farthest selected neighbor if necessary\. This matches the dense operator’s convention thatKq​q=1K\_\{qq\}=1\. The remaining non\-forced neighbors are chosen by increasing source\-space distance, with ties broken by index order\. The sparse weights are

w~q​i=Kq​i​1​\{i∈𝒩k​\(xq\)\}∑j∈𝒩k​\(xq\)Kq​j\.\\widetilde\{w\}\_\{qi\}=\\frac\{K\_\{qi\}\\,\\mathbf\{1\}\\\{i\\in\\mathcal\{N\}\_\{k\}\(x\_\{q\}\)\\\}\}\{\\sum\_\{j\\in\\mathcal\{N\}\_\{k\}\(x\_\{q\}\)\}K\_\{qj\}\}\.The corresponding row\-stochastic matrix has at mostkknonzero entries per row\.

#### Discontinuity ofkk\-NN selection\.

Thekk\-NN approximation is not Lipschitz in the source cloud\. We record this formally for the non\-forced part of the selected neighborhood\.

###### Proposition 2\(Discontinuity of hardkk\-NN selection\)\.

Consider thekk\-nearest\-neighbor selection rule on a finite source cloud, excluding any neighbor indices that are fixed by convention, such as a forced self\-index\. For1≤k<\|𝒳\|−11\\leq k<\|\\mathcal\{X\}\|\-1, the selected non\-forced neighbor set is discontinuous at any configuration where, for some queryxqx\_\{q\}, thekk\-th and\(k\+1\)\(k\+1\)\-st non\-forced neighbor distances are equal\. Consequently, any observable that nontrivially depends on the selected adjacency pattern can change by an order\-one amount under arbitrarily small perturbations\.

###### Proof\.

Suppose two candidate non\-forced neighborsxpx\_\{p\}andxrx\_\{r\}satisfy

‖xq−xp‖=‖xq−xr‖=R,\\\|x\_\{q\}\-x\_\{p\}\\\|=\\\|x\_\{q\}\-x\_\{r\}\\\|=R,with exactlyk−1k\-1other non\-forced candidates strictly closer toxqx\_\{q\}\. Suppose the deterministic tie\-breaking rule selectsxpx\_\{p\}and excludesxrx\_\{r\}\. For anyη\>0\\eta\>0, perturbxrx\_\{r\}by less thanη\\etain a direction that makes it strictly closer toxqx\_\{q\}, while leaving all other points fixed\. The selected neighbor changes fromxpx\_\{p\}toxrx\_\{r\}, so the corresponding adjacency entry changes by one under an arbitrarily small perturbation\. ∎

The dense Gaussian source\-smoothing operator does not exhibit this adjacency discontinuity at any fixed bandwidth \(Theorem[2](https://arxiv.org/html/2607.01746#Thmtheorem2)\)\. Thekk\-NN approximation is therefore appropriate for sensitivity studies and large\-scale computation, but it introduces an estimator nonsmoothness that the dense operator avoids\. The empirical comparison appears in Appendix[F\.6](https://arxiv.org/html/2607.01746#A6.SS6)\.

### A\.7Bandwidth selection

Unless otherwise stated, we use the median\-heuristic bandwidth

εmed=14​mediani<j⁡‖xi−xj‖2,\\varepsilon\_\{\\rm med\}=\\frac\{1\}\{4\}\\,\\operatorname\{median\}\_\{i<j\}\\\|x\_\{i\}\-x\_\{j\}\\\|^\{2\},computed on the source coordinates, with subsampling to at most20002000points whenn\>2000n\>2000\. The factor of44matches the kernel denominator\. Under this choice the median pairwise affinity ise−1e^\{\-1\}, giving a reproducible default scale for the dense operator\.

If the median pairwise squared distance is numerically zero, we replaceεmed\\varepsilon\_\{\\rm med\}by a small positive floor\. Stability statements with adaptive bandwidth should be read on subsets where the selected bandwidth is bounded away from zero\. Theorem[2](https://arxiv.org/html/2607.01746#Thmtheorem2)is stated for fixedε\>0\\varepsilon\>0\. Whenε\\varepsilonis selected from the data, the same fixed\-bandwidth bound applies conditionally on the selected scale, and additional dependence enters through the bandwidth\-selection rule\.

We sweep the bandwidth scale

ε/εmed∈\{0\.1,0\.2,0\.3,0\.5,1\.0,2\.0\}\\varepsilon/\\varepsilon\_\{\\rm med\}\\in\\\{0\.1,0\.2,0\.3,0\.5,1\.0,2\.0\\\}in the resolution\-dependence experiment reported below\. Wider bandwidths smooth more aggressively, attribute more trace mass to conditional spread, and increase the effective neighborhood size\. Narrower bandwidths resolve more local structure but reduce effective neighborhood size and worsen finite\-sample stability constants\.

## Appendix BProofs for the framework

This appendix proves the basic structural propositions used in Section[3](https://arxiv.org/html/2607.01746#S3)\. We use the notation of Appendix[A](https://arxiv.org/html/2607.01746#A1)\.

### B\.1Well\-posedness of the empirical operator

###### Proposition 3\(Well\-posed empirical operator, restated\)\.

SupposeDq=∑iKq​i\>0D\_\{q\}=\\sum\_\{i\}K\_\{qi\}\>0at every evaluated source nodexqx\_\{q\}\. ThenQ^Δ\(⋅∣xq\)\\widehat\{Q\}\_\{\\Delta\}\(\\cdot\\mid x\_\{q\}\)is a probability measure for everyqq,P^Δ\\widehat\{P\}\_\{\\Delta\}is a positive operator on bounded observables, and the finite matrix

\(P^Δ\)q​i=wq​i\(\\widehat\{P\}\_\{\\Delta\}\)\_\{qi\}=w\_\{qi\}is row\-stochastic\.

###### Proof\.

The Gaussian kernel is positive, soKq​i\>0K\_\{qi\}\>0for all finite source coordinatesxq,xix\_\{q\},x\_\{i\}\. IfDq\>0D\_\{q\}\>0, then

wq​i=Kq​iDq≥0,∑iwq​i=1\.w\_\{qi\}=\\frac\{K\_\{qi\}\}\{D\_\{q\}\}\\geq 0,\\qquad\\sum\_\{i\}w\_\{qi\}=1\.Thus

Q^Δ​\(d​y∣xq\)=∑iwq​i​δyi​\(d​y\)\\widehat\{Q\}\_\{\\Delta\}\(dy\\mid x\_\{q\}\)=\\sum\_\{i\}w\_\{qi\}\\delta\_\{y\_\{i\}\}\(dy\)is a probability measure and the rows ofP^Δ\\widehat\{P\}\_\{\\Delta\}sum to one\. For any bounded observableffon hidden\-state coordinates,

\(P^Δ​f\)​\(xq\)=∑iwq​i​f​\(yi\)\.\(\\widehat\{P\}\_\{\\Delta\}f\)\(x\_\{q\}\)=\\sum\_\{i\}w\_\{qi\}f\(y\_\{i\}\)\.Iff≥0f\\geq 0, thenP^Δ​f≥0\\widehat\{P\}\_\{\\Delta\}f\\geq 0, establishing positivity\. ∎

In the operating regime of this paper,Dq\>0D\_\{q\}\>0holds automatically\. Forq∈𝒳−q\\in\\mathcal\{X\}^\{\-\}, the self\-kernelKq​q=1K\_\{qq\}=1is included in the sum\.

### B\.2Coordinate form and positive semidefiniteness

Motivated by the carré\-du\-champ covariance formula, define the finite\-lag quadratic form on observablesf,gf,gby

ΓΔ​\(f,g\)​\(x\)=12​τ​∫\(f​\(y\)−f​\(x\)\)​\(g​\(y\)−g​\(x\)\)​QΔ​\(d​y∣x\)\.\\Gamma\_\{\\Delta\}\(f,g\)\(x\)=\\frac\{1\}\{2\\tau\}\\int\\bigl\(f\(y\)\-f\(x\)\\bigr\)\\bigl\(g\(y\)\-g\(x\)\\bigr\)\\,Q\_\{\\Delta\}\(dy\\mid x\)\.
###### Proposition 4\(Coordinate form, restated\)\.

For coordinate functionsza​\(u\)=uaz\_\{a\}\(u\)=u\_\{a\},

ΓΔ​\(za,zb\)​\(x\)=\(GΔ​\(x\)\)a​b\.\\Gamma\_\{\\Delta\}\(z\_\{a\},z\_\{b\}\)\(x\)=\\bigl\(G\_\{\\Delta\}\(x\)\\bigr\)\_\{ab\}\.Moreover,GΔ​\(x\)⪰0G\_\{\\Delta\}\(x\)\\succeq 0for everyxx\.

###### Proof\.

Substitutingza​\(y\)−za​\(x\)=ya−xaz\_\{a\}\(y\)\-z\_\{a\}\(x\)=y\_\{a\}\-x\_\{a\}gives

ΓΔ​\(za,zb\)​\(x\)=12​τ​∫\(ya−xa\)​\(yb−xb\)​QΔ​\(d​y∣x\)=\(GΔ​\(x\)\)a​b\.\\Gamma\_\{\\Delta\}\(z\_\{a\},z\_\{b\}\)\(x\)=\\frac\{1\}\{2\\tau\}\\int\(y\_\{a\}\-x\_\{a\}\)\(y\_\{b\}\-x\_\{b\}\)\\,Q\_\{\\Delta\}\(dy\\mid x\)=\\bigl\(G\_\{\\Delta\}\(x\)\\bigr\)\_\{ab\}\.For positive semidefiniteness, letv∈ℝdv\\in\\mathbb\{R\}^\{d\}\. Then

v⊤​GΔ​\(x\)​v=12​τ​∫\(v⊤​\(y−x\)\)2​QΔ​\(d​y∣x\)≥0\.v^\{\\top\}G\_\{\\Delta\}\(x\)v=\\frac\{1\}\{2\\tau\}\\int\\bigl\(v^\{\\top\}\(y\-x\)\\bigr\)^\{2\}\\,Q\_\{\\Delta\}\(dy\\mid x\)\\geq 0\.∎

For the empirical operator,

\(G^Δ​\(xq\)\)a​b=12​τ​∑iwq​i​\(yi,a−xq,a\)​\(yi,b−xq,b\),\\bigl\(\\widehat\{G\}\_\{\\Delta\}\(x\_\{q\}\)\\bigr\)\_\{ab\}=\\frac\{1\}\{2\\tau\}\\sum\_\{i\}w\_\{qi\}\(y\_\{i,a\}\-x\_\{q,a\}\)\(y\_\{i,b\}\-x\_\{q,b\}\),and the same quadratic\-form argument gives

G^Δ​\(xq\)⪰0\\widehat\{G\}\_\{\\Delta\}\(x\_\{q\}\)\\succeq 0at every evaluated source node\.

### B\.3Source\-centered decomposition

###### Proposition 5\(Source\-centered decomposition, restated\)\.

Whenever the conditional second moment ofY−XY\-Xexists,

2​GΔ​\(x\)=CΔ​\(x\)\+τ​mΔ​\(x\)​mΔ​\(x\)⊤\.2G\_\{\\Delta\}\(x\)=C\_\{\\Delta\}\(x\)\+\\tau\\,m\_\{\\Delta\}\(x\)m\_\{\\Delta\}\(x\)^\{\\top\}\.

###### Proof\.

Condition onX=xX=xand writeD=Y−XD=Y\-X\. The second\-moment identity gives

𝔼​\[D​D⊤∣X=x\]=Cov⁡\(D∣X=x\)\+𝔼​\[D∣X=x\]​𝔼​\[D∣X=x\]⊤\.\\mathbb\{E\}\[DD^\{\\top\}\\mid X=x\]=\\operatorname\{Cov\}\(D\\mid X=x\)\+\\mathbb\{E\}\[D\\mid X=x\]\\mathbb\{E\}\[D\\mid X=x\]^\{\\top\}\.By definition,

2​GΔ​\(x\)=τ−1​𝔼​\[D​D⊤∣X=x\],2G\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\mathbb\{E\}\[DD^\{\\top\}\\mid X=x\],CΔ​\(x\)=τ−1​Cov⁡\(D∣X=x\),mΔ​\(x\)=τ−1​𝔼​\[D∣X=x\]\.C\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\operatorname\{Cov\}\(D\\mid X=x\),\\qquad m\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\mathbb\{E\}\[D\\mid X=x\]\.Therefore

2​GΔ​\(x\)=CΔ​\(x\)\+τ−1​𝔼​\[D∣X=x\]​𝔼​\[D∣X=x\]⊤\.2G\_\{\\Delta\}\(x\)=C\_\{\\Delta\}\(x\)\+\\tau^\{\-1\}\\mathbb\{E\}\[D\\mid X=x\]\\mathbb\{E\}\[D\\mid X=x\]^\{\\top\}\.Since𝔼​\[D∣X=x\]=τ​mΔ​\(x\)\\mathbb\{E\}\[D\\mid X=x\]=\\tau m\_\{\\Delta\}\(x\), the second term is

τ​mΔ​\(x\)​mΔ​\(x\)⊤\.\\tau\\,m\_\{\\Delta\}\(x\)m\_\{\\Delta\}\(x\)^\{\\top\}\.∎

The empirical identity

2​G^Δ​\(xq\)=C^Δ​\(xq\)\+τ​m^Δ​\(xq\)​m^Δ​\(xq\)⊤2\\widehat\{G\}\_\{\\Delta\}\(x\_\{q\}\)=\\widehat\{C\}\_\{\\Delta\}\(x\_\{q\}\)\+\\tau\\,\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)^\{\\top\}holds pointwise by the same argument with finite sums\. Taking traces and averaging againstρ\\rhogives Equation \([2](https://arxiv.org/html/2607.01746#S3.E2)\)\.

### B\.4Deterministic transport corollary

###### Corollary 1\(Deterministic transport\)\.

IfY=TΔ​\(x\)Y=T\_\{\\Delta\}\(x\)deterministically givenX=xX=x, then

GΔ​\(x\)=12​τ​\(TΔ​\(x\)−x\)​\(TΔ​\(x\)−x\)⊤\.G\_\{\\Delta\}\(x\)=\\frac\{1\}\{2\\tau\}\\bigl\(T\_\{\\Delta\}\(x\)\-x\\bigr\)\\bigl\(T\_\{\\Delta\}\(x\)\-x\\bigr\)^\{\\top\}\.In particular,GΔ​\(x\)≠0G\_\{\\Delta\}\(x\)\\neq 0wheneverTΔ​\(x\)≠xT\_\{\\Delta\}\(x\)\\neq x\.

###### Proof\.

Under deterministic transport,

Cov⁡\(Y−X∣X=x\)=0,𝔼​\[Y−X∣X=x\]=TΔ​\(x\)−x\.\\operatorname\{Cov\}\(Y\-X\\mid X=x\)=0,\\qquad\\mathbb\{E\}\[Y\-X\\mid X=x\]=T\_\{\\Delta\}\(x\)\-x\.Substituting into Definition[2](https://arxiv.org/html/2607.01746#Thmdefinition2)gives the stated formula\. The rank\-one matrix is zero if and only ifTΔ​\(x\)=xT\_\{\\Delta\}\(x\)=x\. ∎

This is a population statement for the exact conditional law\. At finite bandwidth, the dense empirical estimator can have nonzeroC^Δ​\(xq\)\\widehat\{C\}\_\{\\Delta\}\(x\_\{q\}\)even for deterministic underlying dynamics, becauseQ^Δ\(⋅∣xq\)\\widehat\{Q\}\_\{\\Delta\}\(\\cdot\\mid x\_\{q\}\)averages successors attached to nearby source points\. This is the finite\-resolution spread of the empirical operator\.

### B\.5Abstract finite\-state current and detailed balance

This subsection records standard current identities for a row\-stochastic Markov matrix on a common finite state space\. These identities apply directly whenPΔP\_\{\\Delta\}is interpreted as a transition matrix whose row and column indices refer to the same state nodes\. For the empirical source\-smoothing matrixP^Δ\\widehat\{P\}\_\{\\Delta\}, the same algebra defines a source\-neighborhood imbalance diagnostic\.

###### Proposition 6\(Skew\-symmetry ofJρJ^\{\\rho\}\)\.

LetPΔP\_\{\\Delta\}be a row\-stochastic matrix on a common finite state space\. For any distributionρ\\rhoon that state space, define

Jp​qρ=ρp​\(PΔ\)p​q−ρq​\(PΔ\)q​p\.J^\{\\rho\}\_\{pq\}=\\rho\_\{p\}\(P\_\{\\Delta\}\)\_\{pq\}\-\\rho\_\{q\}\(P\_\{\\Delta\}\)\_\{qp\}\.Then

\(Jρ\)⊤=−Jρ\.\(J^\{\\rho\}\)^\{\\top\}=\-J^\{\\rho\}\.

###### Proof\.

By definition,

Jq​pρ=ρq​\(PΔ\)q​p−ρp​\(PΔ\)p​q=−Jp​qρ\.J^\{\\rho\}\_\{qp\}=\\rho\_\{q\}\(P\_\{\\Delta\}\)\_\{qp\}\-\\rho\_\{p\}\(P\_\{\\Delta\}\)\_\{pq\}=\-J^\{\\rho\}\_\{pq\}\.∎

###### Proposition 7\(Detailed balance characterization\)\.

LetPΔP\_\{\\Delta\}be a row\-stochastic matrix on a common finite state space, and letπ\\pibe stationary:

π⊤​PΔ=π⊤\.\\pi^\{\\top\}P\_\{\\Delta\}=\\pi^\{\\top\}\.ThenJπ=0J^\{\\pi\}=0if and only ifPΔP\_\{\\Delta\}satisfies detailed balance with respect toπ\\pi:

πp​\(PΔ\)p​q=πq​\(PΔ\)q​pfor all​p,q\.\\pi\_\{p\}\(P\_\{\\Delta\}\)\_\{pq\}=\\pi\_\{q\}\(P\_\{\\Delta\}\)\_\{qp\}\\qquad\\text\{for all \}p,q\.

###### Proof\.

The conditionJp​qπ=0J^\{\\pi\}\_\{pq\}=0for allp,qp,qis exactly

πp​\(PΔ\)p​q=πq​\(PΔ\)q​pfor all​p,q,\\pi\_\{p\}\(P\_\{\\Delta\}\)\_\{pq\}=\\pi\_\{q\}\(P\_\{\\Delta\}\)\_\{qp\}\\qquad\\text\{for all \}p,q,which is detailed balance\. ∎

#### Current divergence\.

For any distributionρ\\rho, define

divρ⁡\(p\)=∑qJp​qρ\.\\operatorname\{div\}\_\{\\rho\}\(p\)=\\sum\_\{q\}J^\{\\rho\}\_\{pq\}\.Using row\-stochasticity,

divρ⁡\(p\)=∑q\[ρp​\(PΔ\)p​q−ρq​\(PΔ\)q​p\]=ρp−\(ρ⊤​PΔ\)p\.\\operatorname\{div\}\_\{\\rho\}\(p\)=\\sum\_\{q\}\\left\[\\rho\_\{p\}\(P\_\{\\Delta\}\)\_\{pq\}\-\\rho\_\{q\}\(P\_\{\\Delta\}\)\_\{qp\}\\right\]=\\rho\_\{p\}\-\(\\rho^\{\\top\}P\_\{\\Delta\}\)\_\{p\}\.Ifρ=π\\rho=\\piis stationary, then

divρ⁡\(p\)=0for all​p\.\\operatorname\{div\}\_\{\\rho\}\(p\)=0\\qquad\\text\{for all \}p\.For nonstationaryρ\\rho, this divergence is the nodewise imbalance betweenρ\\rhoand its one\-step pushforwardρ⊤​PΔ\\rho^\{\\top\}P\_\{\\Delta\}\.

### B\.6Skew\-symmetry of coordinate circulation

###### Proposition 8\(Skew\-symmetry of𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\)\.

The coordinate circulation

𝒲Δρ=τ−1​\(𝔼ρ​\[X~​Y~⊤\]−𝔼ρ​\[Y~​X~⊤\]\)\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}=\\tau^\{\-1\}\\left\(\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{X\}\\,\\widetilde\{Y\}^\{\\top\}\]\-\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{Y\}\\,\\widetilde\{X\}^\{\\top\}\]\\right\)is real and skew\-symmetric for any source measureρ\\rho\. Hence its eigenvalues are purely imaginary and occur in conjugate pairs\. IfQΔQ\_\{\\Delta\}is a Markov transition law on a common state space with stationary distributionπ\\pi, and if detailed balance holds with respect toπ\\pi, then

𝒲Δπ=0\.\\mathcal\{W\}\_\{\\Delta\}^\{\\pi\}=0\.

###### Proof\.

Taking the transpose gives

\(𝒲Δρ\)⊤=τ−1​\(𝔼ρ​\[Y~​X~⊤\]−𝔼ρ​\[X~​Y~⊤\]\)=−𝒲Δρ\.\(\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\)^\{\\top\}=\\tau^\{\-1\}\\left\(\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{Y\}\\,\\widetilde\{X\}^\{\\top\}\]\-\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{X\}\\,\\widetilde\{Y\}^\{\\top\}\]\\right\)=\-\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\.Thus𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}is real and skew\-symmetric\. A real skew\-symmetric matrix is normal and has purely imaginary eigenvalues, occurring in conjugate pairs\.

For the reversible case, detailed balance gives exchangeability of the stationary two\-time law:

π​\(d​x\)​QΔ​\(d​y∣x\)=π​\(d​y\)​QΔ​\(d​x∣y\)\.\\pi\(dx\)Q\_\{\\Delta\}\(dy\\mid x\)=\\pi\(dy\)Q\_\{\\Delta\}\(dx\\mid y\)\.Therefore

𝔼π​\[X~​Y~⊤\]=𝔼π​\[Y~​X~⊤\],\\mathbb\{E\}\_\{\\pi\}\[\\widetilde\{X\}\\,\\widetilde\{Y\}^\{\\top\}\]=\\mathbb\{E\}\_\{\\pi\}\[\\widetilde\{Y\}\\,\\widetilde\{X\}^\{\\top\}\],and hence

𝒲Δπ=0\.\\mathcal\{W\}\_\{\\Delta\}^\{\\pi\}=0\.∎

The Frobenius norm

‖𝒲Δρ‖F,\\\|\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\},the maximum imaginary eigenvalue magnitude

ωmax=maxλ⁡\|Im⁡λ​\(𝒲Δρ\)\|,\\omega\_\{\\max\}=\\max\_\{\\lambda\}\|\\operatorname\{Im\}\\lambda\(\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\)\|,and the relative circulation

rcirc=‖𝒲Δρ‖Ftr⁡\(G¯Δρ\)r\_\{\\rm circ\}=\\frac\{\\\|\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\}\}\{\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}^\{\\rho\}\)\}are invariant under orthogonal coordinate changes\. Under the center\-RMS normalization convention of Appendix[A\.5](https://arxiv.org/html/2607.01746#A1.SS5), the relative circulation is also invariant to global scalar rescaling whenever the denominator is nonzero\.

### B\.7Source current and coordinate circulation

The source currentJρJ^\{\\rho\}and the coordinate circulation𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}are distinct finite\-lag diagnostics\.

The currentJρJ^\{\\rho\}is a node\-level object for a finite\-state Markov matrix\. It records pairwise imbalance

ρp​\(PΔ\)p​q−ρq​\(PΔ\)q​p\\rho\_\{p\}\(P\_\{\\Delta\}\)\_\{pq\}\-\\rho\_\{q\}\(P\_\{\\Delta\}\)\_\{qp\}and gives the detailed\-balance characterization above\. When applied to the empirical source\-smoothing matrix, it measures imbalance between source neighborhood weights\.

The coordinate circulation𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}is a coordinate\-level antisymmetric moment between sources and successors\. It is the object used for representation\-geometric reporting\. In the linear\-Gaussian model of Theorem[4](https://arxiv.org/html/2607.01746#Thmtheorem4), it has the closed form

𝒲Δ=τ−1​\(Σ​AΔ⊤−AΔ​Σ\)\.\\mathcal\{W\}\_\{\\Delta\}=\\tau^\{\-1\}\\left\(\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigma\\right\)\.
For Markov chains on a unified finite state space, coordinate\-level circulations can be obtained by contracting node currents with coordinate observables\. In the empirical source–successor construction used here, source indices and successor observations are paired but are not identified as the same finite state nodes by the source smoother\. We therefore keepJρJ^\{\\rho\}as a finite\-state imbalance object and𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}as the coordinate\-level circulation statistic\.

## Appendix CProofs of Structural Properties of Finite\-Lag Geometry

This appendix proves the structural theorems of Section[4](https://arxiv.org/html/2607.01746#S4)\. Notation follows Appendix[A](https://arxiv.org/html/2607.01746#A1)\. We write

𝒵=\{\(xi,yi\)\}i=1n\\mathcal\{Z\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}for a paired source–successor cloud and

‖𝒵−𝒵~‖∞=maxi⁡max⁡\{‖xi−x~i‖,‖yi−y~i‖\}\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}=\\max\_\{i\}\\max\\\{\\\|x\_\{i\}\-\\widetilde\{x\}\_\{i\}\\\|,\\\|y\_\{i\}\-\\widetilde\{y\}\_\{i\}\\\|\\\}for the perturbation size\.

### C\.1Affine covariance

###### Theorem 5\(Affine covariance and metric dependence, restated\)\.

Letϕ​\(x\)=A​x\+b\\phi\(x\)=Ax\+b, withA∈ℝd×dA\\in\\mathbb\{R\}^\{d\\times d\}invertible, and letQΔ′Q^\{\\prime\}\_\{\\Delta\}be the pushed\-forward conditional law

QΔ′​\(B∣x′\)=QΔ​\(ϕ−1​\(B\)∣ϕ−1​\(x′\)\)\.Q^\{\\prime\}\_\{\\Delta\}\(B\\mid x^\{\\prime\}\)=Q\_\{\\Delta\}\(\\phi^\{\-1\}\(B\)\\mid\\phi^\{\-1\}\(x^\{\\prime\}\)\)\.Letρ′=ϕ\#​ρ\\rho^\{\\prime\}=\\phi\_\{\\\#\}\\rhobe the pushed\-forward source measure\. The following covariance statements are for the unnormalized tensorial quantities\. For𝒲Δρ\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}, centered coordinates are used before any RMS renormalization\.

Then, forx′=ϕ​\(x\)x^\{\\prime\}=\\phi\(x\),

mΔ′​\(x′\)=A​mΔ​\(x\),CΔ′​\(x′\)=A​CΔ​\(x\)​A⊤,m^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=A\\,m\_\{\\Delta\}\(x\),\\qquad C^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=A\\,C\_\{\\Delta\}\(x\)\\,A^\{\\top\},GΔ′​\(x′\)=A​GΔ​\(x\)​A⊤,\(𝒲Δ′\)ρ′=A​𝒲Δρ​A⊤\.G^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=A\\,G\_\{\\Delta\}\(x\)\\,A^\{\\top\},\\qquad\(\\mathcal\{W\}^\{\\prime\}\_\{\\Delta\}\)^\{\\rho^\{\\prime\}\}=A\\,\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\,A^\{\\top\}\.The decomposition identity is coordinate\-covariant:

2​GΔ′​\(x′\)=CΔ′​\(x′\)\+τ​mΔ′​\(x′\)​mΔ′​\(x′\)⊤\.2G^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=C^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)\+\\tau\\,m^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)m^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)^\{\\top\}\.
ForM≻0M\\succ 0, define the metric\-weighted spread and coherent displacement summaries by

SCM=∫tr⁡\(M​CΔ​\(x\)\)​ρ​\(d​x\),SmM=τ​∫mΔ​\(x\)⊤​M​mΔ​\(x\)​ρ​\(d​x\),S\_\{C\}^\{M\}=\\int\\operatorname\{tr\}\\\!\\bigl\(MC\_\{\\Delta\}\(x\)\\bigr\)\\,\\rho\(dx\),\\qquad S\_\{m\}^\{M\}=\\tau\\int m\_\{\\Delta\}\(x\)^\{\\top\}Mm\_\{\\Delta\}\(x\)\\,\\rho\(dx\),and

FM=SmMSCM\+SmMF^\{M\}=\\frac\{S\_\{m\}^\{M\}\}\{S\_\{C\}^\{M\}\+S\_\{m\}^\{M\}\}whenever the denominator is nonzero\. Under the metric transformation

M′=A−⊤​M​A−1,M^\{\\prime\}=A^\{\-\\top\}MA^\{\-1\},one has

SC′M′=SCM,Sm′M′=SmM,FM′=FM\.S\_\{C^\{\\prime\}\}^\{M^\{\\prime\}\}=S\_\{C\}^\{M\},\\qquad S\_\{m^\{\\prime\}\}^\{M^\{\\prime\}\}=S\_\{m\}^\{M\},\\qquad F^\{M^\{\\prime\}\}=F^\{M\}\.

###### Proof\.

LetD=Y−XD=Y\-X\. Under the affine change of coordinates,

D′=Y′−X′=A​\(Y−X\)=A​D\.D^\{\\prime\}=Y^\{\\prime\}\-X^\{\\prime\}=A\(Y\-X\)=AD\.Therefore

𝔼​\[D′∣X′=x′\]=A​𝔼​\[D∣X=x\],\\mathbb\{E\}\[D^\{\\prime\}\\mid X^\{\\prime\}=x^\{\\prime\}\]=A\\,\\mathbb\{E\}\[D\\mid X=x\],and

Cov⁡\(D′∣X′=x′\)=A​Cov⁡\(D∣X=x\)​A⊤\.\\operatorname\{Cov\}\(D^\{\\prime\}\\mid X^\{\\prime\}=x^\{\\prime\}\)=A\\,\\operatorname\{Cov\}\(D\\mid X=x\)\\,A^\{\\top\}\.Dividing byτ\\taugives

mΔ′​\(x′\)=A​mΔ​\(x\),CΔ′​\(x′\)=A​CΔ​\(x\)​A⊤\.m^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=A\\,m\_\{\\Delta\}\(x\),\\qquad C^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=A\\,C\_\{\\Delta\}\(x\)\\,A^\{\\top\}\.Similarly,

GΔ′​\(x′\)=12​τ​𝔼​\[D′​D′⁣⊤∣X′=x′\]=12​τ​A​𝔼​\[D​D⊤∣X=x\]​A⊤=A​GΔ​\(x\)​A⊤\.G^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=\\frac\{1\}\{2\\tau\}\\mathbb\{E\}\[D^\{\\prime\}D^\{\\prime\\top\}\\mid X^\{\\prime\}=x^\{\\prime\}\]=\\frac\{1\}\{2\\tau\}A\\,\\mathbb\{E\}\[DD^\{\\top\}\\mid X=x\]\\,A^\{\\top\}=A\\,G\_\{\\Delta\}\(x\)\\,A^\{\\top\}\.Substituting these identities into the source\-centered decomposition gives

2​GΔ′​\(x′\)=CΔ′​\(x′\)\+τ​mΔ′​\(x′\)​mΔ′​\(x′\)⊤\.2G^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=C^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)\+\\tau\\,m^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)m^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)^\{\\top\}\.
For circulation, letX~,Y~\\widetilde\{X\},\\widetilde\{Y\}denote centered coordinates in the original system andX~′,Y~′\\widetilde\{X\}^\{\\prime\},\\widetilde\{Y\}^\{\\prime\}centered coordinates in the transformed system, before RMS renormalization\. Centering removes the translationbb, so

X~′=A​X~,Y~′=A​Y~\.\\widetilde\{X\}^\{\\prime\}=A\\widetilde\{X\},\\qquad\\widetilde\{Y\}^\{\\prime\}=A\\widetilde\{Y\}\.Thus

\(𝒲Δ′\)ρ′=τ−1​\(𝔼ρ′​\[X~′​Y~′⁣⊤\]−𝔼ρ′​\[Y~′​X~′⁣⊤\]\)\(\\mathcal\{W\}^\{\\prime\}\_\{\\Delta\}\)^\{\\rho^\{\\prime\}\}=\\tau^\{\-1\}\\left\(\\mathbb\{E\}\_\{\\rho^\{\\prime\}\}\[\\widetilde\{X\}^\{\\prime\}\\widetilde\{Y\}^\{\\prime\\top\}\]\-\\mathbb\{E\}\_\{\\rho^\{\\prime\}\}\[\\widetilde\{Y\}^\{\\prime\}\\widetilde\{X\}^\{\\prime\\top\}\]\\right\)=τ−1​\(A​𝔼ρ​\[X~​Y~⊤\]​A⊤−A​𝔼ρ​\[Y~​X~⊤\]​A⊤\)=A​𝒲Δρ​A⊤\.=\\tau^\{\-1\}\\left\(A\\,\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{X\}\\widetilde\{Y\}^\{\\top\}\]\\,A^\{\\top\}\-A\\,\\mathbb\{E\}\_\{\\rho\}\[\\widetilde\{Y\}\\widetilde\{X\}^\{\\top\}\]\\,A^\{\\top\}\\right\)=A\\,\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\,A^\{\\top\}\.
For the metric\-weighted summaries, cyclicity of trace gives

tr⁡\(M′​CΔ′​\(x′\)\)=tr⁡\(A−⊤​M​A−1​A​CΔ​\(x\)​A⊤\)=tr⁡\(M​CΔ​\(x\)\)\.\\operatorname\{tr\}\\\!\\bigl\(M^\{\\prime\}C^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)\\bigr\)=\\operatorname\{tr\}\\\!\\bigl\(A^\{\-\\top\}MA^\{\-1\}AC\_\{\\Delta\}\(x\)A^\{\\top\}\\bigr\)=\\operatorname\{tr\}\\\!\\bigl\(MC\_\{\\Delta\}\(x\)\\bigr\)\.Likewise,

mΔ′​\(x′\)⊤​M′​mΔ′​\(x′\)=\(A​mΔ​\(x\)\)⊤​A−⊤​M​A−1​\(A​mΔ​\(x\)\)=mΔ​\(x\)⊤​M​mΔ​\(x\)\.m^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)^\{\\top\}M^\{\\prime\}m^\{\\prime\}\_\{\\Delta\}\(x^\{\\prime\}\)=\(Am\_\{\\Delta\}\(x\)\)^\{\\top\}A^\{\-\\top\}MA^\{\-1\}\(Am\_\{\\Delta\}\(x\)\)=m\_\{\\Delta\}\(x\)^\{\\top\}Mm\_\{\\Delta\}\(x\)\.Integrating against the pushed\-forward source measure gives

SC′M′=SCM,Sm′M′=SmM\.S\_\{C^\{\\prime\}\}^\{M^\{\\prime\}\}=S\_\{C\}^\{M\},\\qquad S\_\{m^\{\\prime\}\}^\{M^\{\\prime\}\}=S\_\{m\}^\{M\}\.The fractionFM=SmM/\(SCM\+SmM\)F^\{M\}=S\_\{m\}^\{M\}/\(S\_\{C\}^\{M\}\+S\_\{m\}^\{M\}\)is invariant because both numerator and denominator are invariant\. ∎

The Euclidean trace conventionM=IM=Iis invariant under translations and orthogonal transformations\. Under global scalar rescaling, center\-RMS normalization removes the scalar factor before Euclidean traces are reported\. Anisotropic reparameterizations require the metric correction above\.

### C\.2Stability of the dense empirical operator

The proof of Theorem[2](https://arxiv.org/html/2607.01746#Thmtheorem2)uses three elementary bounds, one for squared distances, one for Gaussian kernel entries, and one for row\-normalization\.

###### Lemma 1\(Squared\-distance perturbation\)\.

Let𝒵,𝒵~\\mathcal\{Z\},\\widetilde\{\\mathcal\{Z\}\}satisfy

maxi⁡\{‖xi‖,‖yi‖,‖x~i‖,‖y~i‖\}≤R\.\\max\_\{i\}\\\{\\\|x\_\{i\}\\\|,\\\|y\_\{i\}\\\|,\\\|\\widetilde\{x\}\_\{i\}\\\|,\\\|\\widetilde\{y\}\_\{i\}\\\|\\\}\\leq R\.Then, for alli,ji,j,

\|‖xi−xj‖2−‖x~i−x~j‖2\|≤8​R​‖𝒵−𝒵~‖∞\.\\left\|\\\|x\_\{i\}\-x\_\{j\}\\\|^\{2\}\-\\\|\\widetilde\{x\}\_\{i\}\-\\widetilde\{x\}\_\{j\}\\\|^\{2\}\\right\|\\leq 8R\\,\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}\.The same bound holds for squared distances formed from any corresponding source or successor coordinates\.

###### Proof\.

Let

a=xi−xj,b=x~i−x~j\.a=x\_\{i\}\-x\_\{j\},\\qquad b=\\widetilde\{x\}\_\{i\}\-\\widetilde\{x\}\_\{j\}\.Then

‖a‖2−‖b‖2=⟨a\+b,a−b⟩,\\\|a\\\|^\{2\}\-\\\|b\\\|^\{2\}=\\langle a\+b,a\-b\\rangle,so

\|‖a‖2−‖b‖2\|≤‖a\+b‖​‖a−b‖\.\\left\|\\\|a\\\|^\{2\}\-\\\|b\\\|^\{2\}\\right\|\\leq\\\|a\+b\\\|\\,\\\|a\-b\\\|\.The boundedness assumption gives

‖a‖≤2​R,‖b‖≤2​R,‖a\+b‖≤4​R\.\\\|a\\\|\\leq 2R,\\qquad\\\|b\\\|\\leq 2R,\\qquad\\\|a\+b\\\|\\leq 4R\.Also,

‖a−b‖=‖\(xi−x~i\)−\(xj−x~j\)‖≤2​‖𝒵−𝒵~‖∞\.\\\|a\-b\\\|=\\\|\(x\_\{i\}\-\\widetilde\{x\}\_\{i\}\)\-\(x\_\{j\}\-\\widetilde\{x\}\_\{j\}\)\\\|\\leq 2\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}\.Multiplying the two bounds gives the claim\. ∎

###### Lemma 2\(Kernel perturbation\)\.

For the Gaussian kernel

Kq​i=kε​\(xq,xi\)=exp⁡\(−‖xq−xi‖24​ε\),K\_\{qi\}=k\_\{\\varepsilon\}\(x\_\{q\},x\_\{i\}\)=\\exp\\\!\\left\(\-\\frac\{\\\|x\_\{q\}\-x\_\{i\}\\\|^\{2\}\}\{4\\varepsilon\}\\right\),one has

\|Kq​i​\(𝒵\)−Kq​i​\(𝒵~\)\|≤2​Rε​‖𝒵−𝒵~‖∞\.\|K\_\{qi\}\(\\mathcal\{Z\}\)\-K\_\{qi\}\(\\widetilde\{\\mathcal\{Z\}\}\)\|\\leq\\frac\{2R\}\{\\varepsilon\}\\,\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}\.

###### Proof\.

The function

u↦e−u/\(4​ε\)u\\mapsto e^\{\-u/\(4\\varepsilon\)\}has derivative

−14​ε​e−u/\(4​ε\),\-\\frac\{1\}\{4\\varepsilon\}e^\{\-u/\(4\\varepsilon\)\},whose absolute value is at most1/\(4​ε\)1/\(4\\varepsilon\)foru≥0u\\geq 0\. By Lemma[1](https://arxiv.org/html/2607.01746#Thmlemma1),

\|Kq​i​\(𝒵\)−Kq​i​\(𝒵~\)\|≤14​ε​\|‖xq−xi‖2−‖x~q−x~i‖2\|≤2​Rε​‖𝒵−𝒵~‖∞\.\|K\_\{qi\}\(\\mathcal\{Z\}\)\-K\_\{qi\}\(\\widetilde\{\\mathcal\{Z\}\}\)\|\\leq\\frac\{1\}\{4\\varepsilon\}\\left\|\\\|x\_\{q\}\-x\_\{i\}\\\|^\{2\}\-\\\|\\widetilde\{x\}\_\{q\}\-\\widetilde\{x\}\_\{i\}\\\|^\{2\}\\right\|\\leq\\frac\{2R\}\{\\varepsilon\}\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}\.∎

###### Lemma 3\(Row\-normalization Lipschitz bound\)\.

LetW,W~∈ℝ≥0n×nW,\\widetilde\{W\}\\in\\mathbb\{R\}\_\{\\geq 0\}^\{n\\times n\}have row sums

Dq=∑jWq​j≥n​β,D~q=∑jW~q​j≥n​βD\_\{q\}=\\sum\_\{j\}W\_\{qj\}\\geq n\\beta,\\qquad\\widetilde\{D\}\_\{q\}=\\sum\_\{j\}\\widetilde\{W\}\_\{qj\}\\geq n\\betafor someβ\>0\\beta\>0\. Let

wq​i=Wq​iDq,w~q​i=W~q​iD~q\.w\_\{qi\}=\\frac\{W\_\{qi\}\}\{D\_\{q\}\},\\qquad\\widetilde\{w\}\_\{qi\}=\\frac\{\\widetilde\{W\}\_\{qi\}\}\{\\widetilde\{D\}\_\{q\}\}\.If

maxq,i⁡\|Wq​i−W~q​i\|≤δ,\\max\_\{q,i\}\|W\_\{qi\}\-\\widetilde\{W\}\_\{qi\}\|\\leq\\delta,then, for everyqq,

∑i\|wq​i−w~q​i\|≤2​δβ\.\\sum\_\{i\}\|w\_\{qi\}\-\\widetilde\{w\}\_\{qi\}\|\\leq\\frac\{2\\delta\}\{\\beta\}\.

###### Proof\.

Fixqqand write

Wi=Wq​i,W~i=W~q​i,D=Dq,D~=D~q\.W\_\{i\}=W\_\{qi\},\\qquad\\widetilde\{W\}\_\{i\}=\\widetilde\{W\}\_\{qi\},\\qquad D=D\_\{q\},\\qquad\\widetilde\{D\}=\\widetilde\{D\}\_\{q\}\.Then

∑i\|Wi−W~i\|≤n​δ,\|D−D~\|≤n​δ\.\\sum\_\{i\}\|W\_\{i\}\-\\widetilde\{W\}\_\{i\}\|\\leq n\\delta,\\qquad\|D\-\\widetilde\{D\}\|\\leq n\\delta\.For eachii,

\|WiD−W~iD~\|≤\|Wi−W~i\|D\+W~i​\|1D−1D~\|\\left\|\\frac\{W\_\{i\}\}\{D\}\-\\frac\{\\widetilde\{W\}\_\{i\}\}\{\\widetilde\{D\}\}\\right\|\\leq\\frac\{\|W\_\{i\}\-\\widetilde\{W\}\_\{i\}\|\}\{D\}\+\\widetilde\{W\}\_\{i\}\\left\|\\frac\{1\}\{D\}\-\\frac\{1\}\{\\widetilde\{D\}\}\\right\|=\|Wi−W~i\|D\+W~i​\|D−D~\|D​D~\.=\\frac\{\|W\_\{i\}\-\\widetilde\{W\}\_\{i\}\|\}\{D\}\+\\widetilde\{W\}\_\{i\}\\frac\{\|D\-\\widetilde\{D\}\|\}\{D\\widetilde\{D\}\}\.Summing overiigives

∑i\|wq​i−w~q​i\|≤n​δD\+\|D−D~\|D​D~​∑iW~i\.\\sum\_\{i\}\|w\_\{qi\}\-\\widetilde\{w\}\_\{qi\}\|\\leq\\frac\{n\\delta\}\{D\}\+\\frac\{\|D\-\\widetilde\{D\}\|\}\{D\\widetilde\{D\}\}\\sum\_\{i\}\\widetilde\{W\}\_\{i\}\.Since∑iW~i=D~\\sum\_\{i\}\\widetilde\{W\}\_\{i\}=\\widetilde\{D\},

∑i\|wq​i−w~q​i\|≤n​δD\+\|D−D~\|D≤2​n​δD\.\\sum\_\{i\}\|w\_\{qi\}\-\\widetilde\{w\}\_\{qi\}\|\\leq\\frac\{n\\delta\}\{D\}\+\\frac\{\|D\-\\widetilde\{D\}\|\}\{D\}\\leq\\frac\{2n\\delta\}\{D\}\.UsingD≥n​βD\\geq n\\betayields

∑i\|wq​i−w~q​i\|≤2​δβ\.\\sum\_\{i\}\|w\_\{qi\}\-\\widetilde\{w\}\_\{qi\}\|\\leq\\frac\{2\\delta\}\{\\beta\}\.∎

###### Theorem 6\(Dense stability, restated\)\.

Fix bandwidthε\>0\\varepsilon\>0and radiusR\>0R\>0\. On the set of paired clouds satisfying

maxi⁡\{‖xi‖,‖yi‖\}≤R,\\max\_\{i\}\\\{\\\|x\_\{i\}\\\|,\\\|y\_\{i\}\\\|\\\}\\leq R,the dense source\-smoothing operator satisfies

‖P^Δ​\(𝒵\)−P^Δ​\(𝒵~\)‖∞→∞≤4​Rε​exp⁡\(R2ε\)​‖𝒵−𝒵~‖∞\\left\\\|\\widehat\{P\}\_\{\\Delta\}\(\\mathcal\{Z\}\)\-\\widehat\{P\}\_\{\\Delta\}\(\\widetilde\{\\mathcal\{Z\}\}\)\\right\\\|\_\{\\infty\\to\\infty\}\\leq\\frac\{4R\}\{\\varepsilon\}\\exp\\\!\\left\(\\frac\{R^\{2\}\}\{\\varepsilon\}\\right\)\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}whenever both clouds lie in this bounded set\.

###### Proof\.

By Lemma[2](https://arxiv.org/html/2607.01746#Thmlemma2),

\|Kq​i​\(𝒵\)−Kq​i​\(𝒵~\)\|≤δ:=2​Rε​‖𝒵−𝒵~‖∞\.\|K\_\{qi\}\(\\mathcal\{Z\}\)\-K\_\{qi\}\(\\widetilde\{\\mathcal\{Z\}\}\)\|\\leq\\delta:=\\frac\{2R\}\{\\varepsilon\}\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}\.On the radius\-RRset,

‖xq−xi‖≤2​R,\\\|x\_\{q\}\-x\_\{i\}\\\|\\leq 2R,so each Gaussian kernel entry is bounded below by

Kq​i\(𝒵\)≥exp\(−\(2​R\)24​ε\)=exp\(−R2ε\)=:β\.K\_\{qi\}\(\\mathcal\{Z\}\)\\geq\\exp\\\!\\left\(\-\\frac\{\(2R\)^\{2\}\}\{4\\varepsilon\}\\right\)=\\exp\\\!\\left\(\-\\frac\{R^\{2\}\}\{\\varepsilon\}\\right\)=:\\beta\.The same lower bound holds forK~q​i\\widetilde\{K\}\_\{qi\}\. Thus every row sum is at leastn​βn\\beta\. Applying Lemma[3](https://arxiv.org/html/2607.01746#Thmlemma3)gives, for each rowqq,

∑i\|wq​i​\(𝒵\)−wq​i​\(𝒵~\)\|≤2​δβ=4​Rε​exp⁡\(R2ε\)​‖𝒵−𝒵~‖∞\.\\sum\_\{i\}\|w\_\{qi\}\(\\mathcal\{Z\}\)\-w\_\{qi\}\(\\widetilde\{\\mathcal\{Z\}\}\)\|\\leq\\frac\{2\\delta\}\{\\beta\}=\\frac\{4R\}\{\\varepsilon\}\\exp\\\!\\left\(\\frac\{R^\{2\}\}\{\\varepsilon\}\\right\)\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}\.Taking the maximum row sum gives the∞→∞\\infty\\to\\inftyoperator norm bound\. ∎

#### Stability of derived observables\.

The empirical observables

m^Δ,G^Δ,C^Δ,𝒲^Δρ,\\widehat\{m\}\_\{\\Delta\},\\quad\\widehat\{G\}\_\{\\Delta\},\\quad\\widehat\{C\}\_\{\\Delta\},\\quad\\widehat\{\\mathcal\{W\}\}\_\{\\Delta\}^\{\\rho\},and their scalar trace summaries inherit Lipschitz dependence on𝒵\\mathcal\{Z\}on bounded sets\.

For example,

m^Δ​\(xq\)=τ−1​\[\(P^Δ​Y\)q−xq\],\\widehat\{m\}\_\{\\Delta\}\(x\_\{q\}\)=\\tau^\{\-1\}\\left\[\(\\widehat\{P\}\_\{\\Delta\}Y\)\_\{q\}\-x\_\{q\}\\right\],whereYYis the matrix of successor coordinates\. The term\(P^Δ​Y\)q\(\\widehat\{P\}\_\{\\Delta\}Y\)\_\{q\}is Lipschitz because bothP^Δ\\widehat\{P\}\_\{\\Delta\}andYYare Lipschitz in the paired cloud on the bounded set, and the term−xq\-x\_\{q\}is directly Lipschitz\.

Similarly,

G^Δ​\(xq\)=12​τ​∑iwq​i​\(yi−xq\)​\(yi−xq\)⊤\.\\widehat\{G\}\_\{\\Delta\}\(x\_\{q\}\)=\\frac\{1\}\{2\\tau\}\\sum\_\{i\}w\_\{qi\}\(y\_\{i\}\-x\_\{q\}\)\(y\_\{i\}\-x\_\{q\}\)^\{\\top\}\.Each entry is a row\-weighted sum of bounded quadratic functions of the coordinates\. Perturbations enter through the weights and through the coordinate factors, both linearly in‖𝒵−𝒵~‖∞\\\|\\mathcal\{Z\}\-\\widetilde\{\\mathcal\{Z\}\}\\\|\_\{\\infty\}on bounded sets\. The same argument applies toC^Δ\\widehat\{C\}\_\{\\Delta\}, using either its covariance form or the decomposition

C^Δ=2​G^Δ−τ​m^Δ​m^Δ⊤\.\\widehat\{C\}\_\{\\Delta\}=2\\widehat\{G\}\_\{\\Delta\}\-\\tau\\,\\widehat\{m\}\_\{\\Delta\}\\widehat\{m\}\_\{\\Delta\}^\{\\top\}\.The smoothed empirical circulation

𝒲^Δρ=τ−1​∑qρq​∑iwq​i​\(x~q​y~i⊤−y~i​x~q⊤\)\\widehat\{\\mathcal\{W\}\}\_\{\\Delta\}^\{\\rho\}=\\tau^\{\-1\}\\sum\_\{q\}\\rho\_\{q\}\\sum\_\{i\}w\_\{qi\}\\left\(\\widetilde\{x\}\_\{q\}\\widetilde\{y\}\_\{i\}^\{\\top\}\-\\widetilde\{y\}\_\{i\}\\widetilde\{x\}\_\{q\}^\{\\top\}\\right\)is a weighted bilinear coordinate moment and is Lipschitz by the same bounded\-product argument\.

Thus the derived observables are Lipschitz on bounded sets at fixedε\>0\\varepsilon\>0\. Their constants are polynomial inRRandτ−1\\tau^\{\-1\}, with the dense\-operator dependence entering through the factor

1ε​exp⁡\(R2ε\)\.\\frac\{1\}\{\\varepsilon\}\\exp\\\!\\left\(\\frac\{R^\{2\}\}\{\\varepsilon\}\\right\)\.

#### Stability after center\-RMS normalization\.

The main experiments compute observables after center\-RMS normalization \(Appendix[A\.5](https://arxiv.org/html/2607.01746#A1.SS5)\)\. The center\-RMS map is smooth away from collapsed clouds and ill\-conditioned nearσRMS=0\\sigma\_\{\\rm RMS\}=0\. Hence the same stability conclusions extend to normalized observables on subsets where

σRMS≥σ0\>0\.\\sigma\_\{\\rm RMS\}\\geq\\sigma\_\{0\}\>0\.On such subsets, the resulting constants are polynomial inRR,1/σ01/\\sigma\_\{0\}, andτ−1\\tau^\{\-1\}, together with the fixed\-bandwidth dense\-operator constant above\. Near collapsed clouds, whereσRMS→0\\sigma\_\{\\rm RMS\}\\to 0, the normalization itself is ill\-conditioned\.

### C\.3Finite\-lag separation

###### Theorem 7\(Finite\-lag detection of deterministic motion, restated\)\.

LetT:ℝd→ℝdT:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}be measurable and fixΔ≥1\\Delta\\geq 1\. Assume

∫‖TΔ​\(x\)−x‖2​ρ​\(d​x\)<∞\\int\\\|T^\{\\Delta\}\(x\)\-x\\\|^\{2\}\\,\\rho\(dx\)<\\inftyand supposeTΔ​\(x\)≠xT^\{\\Delta\}\(x\)\\neq xon aρ\\rho\-positive set\. If

Xt\+Δ=TΔ​\(Xt\)X\_\{t\+\\Delta\}=T^\{\\Delta\}\(X\_\{t\}\)deterministically, then

G¯Δρ=12​τ​∫\(TΔ​\(x\)−x\)​\(TΔ​\(x\)−x\)⊤​ρ​\(d​x\),\\bar\{G\}\_\{\\Delta\}^\{\\rho\}=\\frac\{1\}\{2\\tau\}\\int\\bigl\(T^\{\\Delta\}\(x\)\-x\\bigr\)\\bigl\(T^\{\\Delta\}\(x\)\-x\\bigr\)^\{\\top\}\\,\\rho\(dx\),and

tr⁡\(G¯Δρ\)\>0\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}^\{\\rho\}\)\>0at the chosen lagτ\\tau\.

For an oriented cyclic shiftTΔ=ΠT^\{\\Delta\}=\\Piin centered coordinates with

Cov⁡\(X\)=σx2​I,\\operatorname\{Cov\}\(X\)=\\sigma\_\{x\}^\{2\}I,one has

‖𝒲Δρ‖F=σx2τ​‖Π⊤−Π‖F\>0\\\|\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\}=\\frac\{\\sigma\_\{x\}^\{2\}\}\{\\tau\}\\\|\\Pi^\{\\top\}\-\\Pi\\\|\_\{F\}\>0wheneverΠ\\Piis not symmetric\.

By contrast, if a deterministic continuous\-time flow

X˙t=b​\(Xt\)\\dot\{X\}\_\{t\}=b\(X\_\{t\}\)is represented by the first\-order generatorL=b⋅∇L=b\\cdot\\nabla, then its infinitesimal carré du champ satisfies

###### Proof\.

Under deterministic dynamics, Corollary[1](https://arxiv.org/html/2607.01746#Thmcorollary1)gives

GΔ​\(x\)=12​τ​\(TΔ​\(x\)−x\)​\(TΔ​\(x\)−x\)⊤\.G\_\{\\Delta\}\(x\)=\\frac\{1\}\{2\\tau\}\\bigl\(T^\{\\Delta\}\(x\)\-x\\bigr\)\\bigl\(T^\{\\Delta\}\(x\)\-x\\bigr\)^\{\\top\}\.Averaging againstρ\\rhogives the displayed formula forG¯Δρ\\bar\{G\}\_\{\\Delta\}^\{\\rho\}\. Taking traces gives

tr⁡\(G¯Δρ\)=12​τ​∫‖TΔ​\(x\)−x‖2​ρ​\(d​x\)\.\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}^\{\\rho\}\)=\\frac\{1\}\{2\\tau\}\\int\\\|T^\{\\Delta\}\(x\)\-x\\\|^\{2\}\\,\\rho\(dx\)\.The integrand is nonnegative and strictly positive on aρ\\rho\-positive set, so the integral is strictly positive\.

For the cyclic shift, letY=Π​XY=\\Pi X\. In centered coordinates with𝔼​\[X​X⊤\]=σx2​I\\mathbb\{E\}\[XX^\{\\top\}\]=\\sigma\_\{x\}^\{2\}I,

𝔼​\[X​Y⊤\]=𝔼​\[X​X⊤​Π⊤\]=σx2​Π⊤,\\mathbb\{E\}\[XY^\{\\top\}\]=\\mathbb\{E\}\[XX^\{\\top\}\\Pi^\{\\top\}\]=\\sigma\_\{x\}^\{2\}\\Pi^\{\\top\},and

𝔼​\[Y​X⊤\]=Π​𝔼​\[X​X⊤\]=σx2​Π\.\\mathbb\{E\}\[YX^\{\\top\}\]=\\Pi\\mathbb\{E\}\[XX^\{\\top\}\]=\\sigma\_\{x\}^\{2\}\\Pi\.Therefore

𝒲Δρ=τ−1​\(σx2​Π⊤−σx2​Π\)=σx2τ​\(Π⊤−Π\),\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}=\\tau^\{\-1\}\\left\(\\sigma\_\{x\}^\{2\}\\Pi^\{\\top\}\-\\sigma\_\{x\}^\{2\}\\Pi\\right\)=\\frac\{\\sigma\_\{x\}^\{2\}\}\{\\tau\}\(\\Pi^\{\\top\}\-\\Pi\),so

‖𝒲Δρ‖F=σx2τ​‖Π⊤−Π‖F\.\\\|\\mathcal\{W\}\_\{\\Delta\}^\{\\rho\}\\\|\_\{F\}=\\frac\{\\sigma\_\{x\}^\{2\}\}\{\\tau\}\\\|\\Pi^\{\\top\}\-\\Pi\\\|\_\{F\}\.This is positive wheneverΠ⊤≠Π\\Pi^\{\\top\}\\neq\\Pi\.

For the infinitesimal contrast, letL=b⋅∇L=b\\cdot\\nabla\. SinceLLis first order, it satisfies the Leibniz rule:

L​\(f​g\)=f​L​g\+g​L​f\.L\(fg\)=fLg\+gLf\.Thus

ΓL​\(f,g\)=12​\(L​\(f​g\)−f​L​g−g​L​f\)=0\\Gamma\_\{L\}\(f,g\)=\\frac\{1\}\{2\}\\left\(L\(fg\)\-fLg\-gLf\\right\)=0for all smoothf,gf,g\. HenceΓL≡0\\Gamma\_\{L\}\\equiv 0\. ∎

### C\.4Linear\-Gaussian closed form

###### Theorem 8\(Linear\-Gaussian closed form, restated\)\.

Let

Xt\+Δ=AΔ​Xt\+ξt,X\_\{t\+\\Delta\}=A\_\{\\Delta\}X\_\{t\}\+\\xi\_\{t\},whereXtX\_\{t\}has mean zero and covarianceΣ\\Sigma, and whereξt\\xi\_\{t\}is independent ofXtX\_\{t\}, mean zero, with covarianceΣξ\\Sigma\_\{\\xi\}\. Then

mΔ​\(x\)=τ−1​\(AΔ−I\)​x,CΔ​\(x\)=τ−1​Σξ,m\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\(A\_\{\\Delta\}\-I\)x,\\qquad C\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\Sigma\_\{\\xi\},G¯Δ=12​τ​\[Σξ\+\(AΔ−I\)​Σ​\(AΔ−I\)⊤\],\\bar\{G\}\_\{\\Delta\}=\\frac\{1\}\{2\\tau\}\\left\[\\Sigma\_\{\\xi\}\+\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\}\\right\],and

𝒲Δ=τ−1​\(Σ​AΔ⊤−AΔ​Σ\)\.\\mathcal\{W\}\_\{\\Delta\}=\\tau^\{\-1\}\\left\(\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigma\\right\)\.

###### Proof\.

GivenXt=xX\_\{t\}=x, the finite\-lag displacement is

D=Xt\+Δ−Xt=\(AΔ−I\)​x\+ξt\.D=X\_\{t\+\\Delta\}\-X\_\{t\}=\(A\_\{\\Delta\}\-I\)x\+\\xi\_\{t\}\.Sinceξt\\xi\_\{t\}is independent ofXtX\_\{t\}and has mean zero,

𝔼​\[D∣Xt=x\]=\(AΔ−I\)​x,\\mathbb\{E\}\[D\\mid X\_\{t\}=x\]=\(A\_\{\\Delta\}\-I\)x,and

Cov⁡\(D∣Xt=x\)=Σξ\.\\operatorname\{Cov\}\(D\\mid X\_\{t\}=x\)=\\Sigma\_\{\\xi\}\.Dividing byτ\\taugives

mΔ​\(x\)=τ−1​\(AΔ−I\)​x,CΔ​\(x\)=τ−1​Σξ\.m\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\(A\_\{\\Delta\}\-I\)x,\\qquad C\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\Sigma\_\{\\xi\}\.Using the source\-centered decomposition,

2​GΔ​\(x\)=τ−1​Σξ\+τ−1​\(AΔ−I\)​x​x⊤​\(AΔ−I\)⊤\.2G\_\{\\Delta\}\(x\)=\\tau^\{\-1\}\\Sigma\_\{\\xi\}\+\\tau^\{\-1\}\(A\_\{\\Delta\}\-I\)xx^\{\\top\}\(A\_\{\\Delta\}\-I\)^\{\\top\}\.Averaging overXtX\_\{t\}, with

𝔼​\[Xt​Xt⊤\]=Σ,\\mathbb\{E\}\[X\_\{t\}X\_\{t\}^\{\\top\}\]=\\Sigma,gives

G¯Δ=12​τ​\[Σξ\+\(AΔ−I\)​Σ​\(AΔ−I\)⊤\]\.\\bar\{G\}\_\{\\Delta\}=\\frac\{1\}\{2\\tau\}\\left\[\\Sigma\_\{\\xi\}\+\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\}\\right\]\.
For circulation,

𝔼​\[Xt​Xt\+Δ⊤\]=𝔼​\[Xt​\(AΔ​Xt\+ξt\)⊤\]=Σ​AΔ⊤,\\mathbb\{E\}\[X\_\{t\}X\_\{t\+\\Delta\}^\{\\top\}\]=\\mathbb\{E\}\\\!\\left\[X\_\{t\}\(A\_\{\\Delta\}X\_\{t\}\+\\xi\_\{t\}\)^\{\\top\}\\right\]=\\Sigma A\_\{\\Delta\}^\{\\top\},because𝔼​\[Xt​ξt⊤\]=0\\mathbb\{E\}\[X\_\{t\}\\xi\_\{t\}^\{\\top\}\]=0\. Similarly,

𝔼​\[Xt\+Δ​Xt⊤\]=𝔼​\[\(AΔ​Xt\+ξt\)​Xt⊤\]=AΔ​Σ\.\\mathbb\{E\}\[X\_\{t\+\\Delta\}X\_\{t\}^\{\\top\}\]=\\mathbb\{E\}\\\!\\left\[\(A\_\{\\Delta\}X\_\{t\}\+\\xi\_\{t\}\)X\_\{t\}^\{\\top\}\\right\]=A\_\{\\Delta\}\\Sigma\.Therefore

𝒲Δ=τ−1​\(Σ​AΔ⊤−AΔ​Σ\)\.\\mathcal\{W\}\_\{\\Delta\}=\\tau^\{\-1\}\\left\(\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigma\\right\)\.∎

The trace decomposition becomes

2​tr⁡\(G¯Δ\)=τ−1​tr⁡\(Σξ\)\+τ−1​tr⁡\(\(AΔ−I\)​Σ​\(AΔ−I\)⊤\)\.2\\,\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\)=\\tau^\{\-1\}\\operatorname\{tr\}\(\\Sigma\_\{\\xi\}\)\+\\tau^\{\-1\}\\operatorname\{tr\}\\left\(\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\}\\right\)\.Thus the conditional spread trace is

τ−1​tr⁡\(Σξ\),\\tau^\{\-1\}\\operatorname\{tr\}\(\\Sigma\_\{\\xi\}\),and the coherent displacement trace is

τ−1​tr⁡\(\(AΔ−I\)​Σ​\(AΔ−I\)⊤\)\.\\tau^\{\-1\}\\operatorname\{tr\}\\left\(\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\}\\right\)\.These formulas are stated in centered coordinates before empirical normalization\. After applying center\-RMS normalization, the same identities hold for the normalized variables and their transformed covariance matrices\. They give the parametric calibration used in Section[4\.4](https://arxiv.org/html/2607.01746#S4.SS4)and the mechanism signatures in Appendix[D](https://arxiv.org/html/2607.01746#A4)\.

## Appendix DLinear\-Gaussian mechanism signatures

This appendix specializes the linear\-Gaussian closed form \(Theorem[8](https://arxiv.org/html/2607.01746#Thmtheorem8)\) to several canonical recurrent mechanisms\. These examples are analytical calibrations, showing how the framework’s observables respond to simple update structures, and providing reference patterns for interpreting trained\-network measurements\. We then make explicit the structural parallel between this recurrent closed form and the Gaussian bridge in static feedforward operator geometry\.

### D\.1Zero\-circulation relaxation and isotropic contraction

Suppose

AΔ​Σ=Σ​AΔ⊤\.A\_\{\\Delta\}\\Sigma=\\Sigma A\_\{\\Delta\}^\{\\top\}\.Then by Theorem[8](https://arxiv.org/html/2607.01746#Thmtheorem8),

𝒲Δ=τ−1​\(Σ​AΔ⊤−AΔ​Σ\)=0\.\\mathcal\{W\}\_\{\\Delta\}=\\tau^\{\-1\}\\left\(\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigma\\right\)=0\.Thus a linear\-Gaussian system can have large finite\-lag transport scale while having no antisymmetric coordinate circulation\. The condition above is the zero\-circulation condition for the lagged cross\-moment\.

For isotropic contraction,

AΔ=α​I,α∈ℝ,A\_\{\\Delta\}=\\alpha I,\\qquad\\alpha\\in\\mathbb\{R\},we obtain

G¯Δ=12​τ​\[Σξ\+\(α−1\)2​Σ\],𝒲Δ=0\.\\bar\{G\}\_\{\\Delta\}=\\frac\{1\}\{2\\tau\}\\left\[\\Sigma\_\{\\xi\}\+\(\\alpha\-1\)^\{2\}\\Sigma\\right\],\\qquad\\mathcal\{W\}\_\{\\Delta\}=0\.The conditional spread trace is

τ−1​tr⁡\(Σξ\),\\tau^\{\-1\}\\operatorname\{tr\}\(\\Sigma\_\{\\xi\}\),and the coherent displacement trace is

τ−1​\(α−1\)2​tr⁡\(Σ\)\.\\tau^\{\-1\}\(\\alpha\-1\)^\{2\}\\operatorname\{tr\}\(\\Sigma\)\.Hence the coherent displacement fraction is

FΔ=\(α−1\)2​tr⁡\(Σ\)tr⁡\(Σξ\)\+\(α−1\)2​tr⁡\(Σ\)\.F\_\{\\Delta\}=\\frac\{\(\\alpha\-1\)^\{2\}\\operatorname\{tr\}\(\\Sigma\)\}\{\\operatorname\{tr\}\(\\Sigma\_\{\\xi\}\)\+\(\\alpha\-1\)^\{2\}\\operatorname\{tr\}\(\\Sigma\)\}\.At fixed innovation covariance,FΔ=0F\_\{\\Delta\}=0whenα=1\\alpha=1, andFΔ→1F\_\{\\Delta\}\\to 1as\|α−1\|→∞\|\\alpha\-1\|\\to\\infty\. Isotropic contraction therefore increases coherent transport scale without creating coordinate circulation\.

### D\.2Two\-dimensional rotation

Letd=2d=2, let

Σ=σx2​I,\\Sigma=\\sigma\_\{x\}^\{2\}I,and let

AΔ=r​Rθ,Rθ=\(cos⁡θ−sin⁡θsin⁡θcos⁡θ\)\.A\_\{\\Delta\}=rR\_\{\\theta\},\\qquad R\_\{\\theta\}=\\begin\{pmatrix\}\\cos\\theta&\-\\sin\\theta\\\\ \\sin\\theta&\\cos\\theta\\end\{pmatrix\}\.Then

Σ​AΔ⊤−AΔ​Σ=r​σx2​\(Rθ⊤−Rθ\)\.\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigma=r\\sigma\_\{x\}^\{2\}\\left\(R\_\{\\theta\}^\{\\top\}\-R\_\{\\theta\}\\right\)\.Since

Rθ⊤−Rθ=2​sin⁡θ​\(01−10\),R\_\{\\theta\}^\{\\top\}\-R\_\{\\theta\}=2\\sin\\theta\\begin\{pmatrix\}0&1\\\\ \-1&0\\end\{pmatrix\},the circulation matrix is

𝒲Δ=2​r​σx2​sin⁡θτ​\(01−10\)\.\\mathcal\{W\}\_\{\\Delta\}=\\frac\{2r\\sigma\_\{x\}^\{2\}\\sin\\theta\}\{\\tau\}\\begin\{pmatrix\}0&1\\\\ \-1&0\\end\{pmatrix\}\.Its Frobenius norm is

‖𝒲Δ‖F=2​2​r​σx2​\|sin⁡θ\|τ\.\\\|\\mathcal\{W\}\_\{\\Delta\}\\\|\_\{F\}=\\frac\{2\\sqrt\{2\}\\,r\\sigma\_\{x\}^\{2\}\|\\sin\\theta\|\}\{\\tau\}\.Thus the antisymmetric statistic grows linearly with radial gainrr, source varianceσx2\\sigma\_\{x\}^\{2\}, and oriented angle magnitude\|sin⁡θ\|\|\\sin\\theta\|, and vanishes exactly whensin⁡θ=0\\sin\\theta=0\.

For pure rotation,r=1r=1andΣξ=0\\Sigma\_\{\\xi\}=0\. In that case,

G¯Δ=12​τ​\(Rθ−I\)​Σ​\(Rθ−I\)⊤\.\\bar\{G\}\_\{\\Delta\}=\\frac\{1\}\{2\\tau\}\(R\_\{\\theta\}\-I\)\\Sigma\(R\_\{\\theta\}\-I\)^\{\\top\}\.Using

\(Rθ−I\)​\(Rθ−I\)⊤=2​\(1−cos⁡θ\)​I,\(R\_\{\\theta\}\-I\)\(R\_\{\\theta\}\-I\)^\{\\top\}=2\(1\-\\cos\\theta\)I,we get

G¯Δ=σx2​\(1−cos⁡θ\)τ​I\.\\bar\{G\}\_\{\\Delta\}=\\frac\{\\sigma\_\{x\}^\{2\}\(1\-\\cos\\theta\)\}\{\\tau\}I\.Pure rotation therefore produces symmetric transport scale through1−cos⁡θ1\-\\cos\\thetaand antisymmetric circulation throughsin⁡θ\\sin\\theta\. The two components distinguish displacement magnitude from oriented finite\-lag flow\.

### D\.3Permutation and shift transport

LetAΔ=ΠA\_\{\\Delta\}=\\Pibe a permutation matrix and let

Σ=σx2​I\.\\Sigma=\\sigma\_\{x\}^\{2\}I\.Then

G¯Δ=12​τ​\[Σξ\+σx2​\(Π−I\)​\(Π−I\)⊤\],\\bar\{G\}\_\{\\Delta\}=\\frac\{1\}\{2\\tau\}\\left\[\\Sigma\_\{\\xi\}\+\\sigma\_\{x\}^\{2\}\(\\Pi\-I\)\(\\Pi\-I\)^\{\\top\}\\right\],and

𝒲Δ=σx2τ​\(Π⊤−Π\)\.\\mathcal\{W\}\_\{\\Delta\}=\\frac\{\\sigma\_\{x\}^\{2\}\}\{\\tau\}\(\\Pi^\{\\top\}\-\\Pi\)\.
#### Symmetric permutations\.

If

then

𝒲Δ=0\.\\mathcal\{W\}\_\{\\Delta\}=0\.This includes the identity and nontrivial involutions, such as the two\-state swap

Π=\(0110\)\.\\Pi=\\begin\{pmatrix\}0&1\\\\ 1&0\\end\{pmatrix\}\.Such updates can have nonzero transport scale through

\(Π−I\)​\(Π−I\)⊤=2​I−Π−Π⊤,\(\\Pi\-I\)\(\\Pi\-I\)^\{\\top\}=2I\-\\Pi\-\\Pi^\{\\top\},while having zero coordinate circulation\.

#### Oriented cyclic shifts\.

For a cyclic shift of lengthL≥3L\\geq 3, the supports ofΠ\\PiandΠ⊤\\Pi^\{\\top\}are disjoint\. Therefore

‖Π⊤−Π‖F=2​L,\\\|\\Pi^\{\\top\}\-\\Pi\\\|\_\{F\}=\\sqrt\{2L\},and

‖𝒲Δ‖F=σx2τ​2​L\.\\\|\\mathcal\{W\}\_\{\\Delta\}\\\|\_\{F\}=\\frac\{\\sigma\_\{x\}^\{2\}\}\{\\tau\}\\sqrt\{2L\}\.An oriented cyclic shift therefore has both nonzero transport scale and nonzero circulation\.

#### Open shift matrices\.

An open delay line requires a boundary rule and is therefore not represented by a unique permutation matrix\. For example, the nilpotent shiftSSwith

Si\+1,i=1,i=1,…,L−1,S\_\{i\+1,i\}=1,\\qquad i=1,\\dots,L\-1,gives

𝒲Δ=σx2τ​\(S⊤−S\),\\mathcal\{W\}\_\{\\Delta\}=\\frac\{\\sigma\_\{x\}^\{2\}\}\{\\tau\}\(S^\{\\top\}\-S\),whereas reset, leakage, or input\-injection rules at the boundary change bothG¯Δ\\bar\{G\}\_\{\\Delta\}and𝒲Δ\\mathcal\{W\}\_\{\\Delta\}\. Thus open delay\-line behavior is boundary\-sensitive rather than a single canonical linear\-Gaussian signature\. The framework still records the resulting finite\-lag transport, but its signature depends on the chosen boundary dynamics\.

### D\.4Identity plus innovation noise

For identity dynamics with additive innovation,

AΔ=I,Xt\+Δ=Xt\+ξt,A\_\{\\Delta\}=I,\\qquad X\_\{t\+\\Delta\}=X\_\{t\}\+\\xi\_\{t\},the closed form gives

G¯Δ=Σξ2​τ,mΔ​\(x\)≡0,𝒲Δ=0\.\\bar\{G\}\_\{\\Delta\}=\\frac\{\\Sigma\_\{\\xi\}\}\{2\\tau\},\\qquad m\_\{\\Delta\}\(x\)\\equiv 0,\\qquad\\mathcal\{W\}\_\{\\Delta\}=0\.The conditional spread trace is

τ−1​tr⁡\(Σξ\),\\tau^\{\-1\}\\operatorname\{tr\}\(\\Sigma\_\{\\xi\}\),and the coherent displacement trace is zero\. Hence

This is the recurrent counterpart of pure diffusion at the chosen lag: transport is entirely conditional spread, with no coherent displacement and no directed circulation\.

### D\.5Structural parallel with the static feedforward Gaussian bridge

Static feedforward operator geometry admits a closed\-form Gaussian bridge under a balanced shared\-covariance class\-conditional model\[[36](https://arxiv.org/html/2607.01746#bib.bib1)\]\. If

z∣y=a∼𝒩​\(μa,Σ\),a∈\[K\],z\\mid y=a\\sim\\mathcal\{N\}\(\\mu\_\{a\},\\Sigma\),\\qquad a\\in\[K\],then the coarse class geometry of the Gaussian\-kernel diffusion operator is controlled by the regularized Mahalanobis separations

cε\(a,b\)=14​\(μa−μb\)⊤​\(ε​I\+Σ\)−1​\(μa−μb\),cε\(a,a\)=0\.c\_\{\\varepsilon\}^\{\(a,b\)\}=\\frac\{1\}\{4\}\(\\mu\_\{a\}\-\\mu\_\{b\}\)^\{\\top\}\(\\varepsilon I\+\\Sigma\)^\{\-1\}\(\\mu\_\{a\}\-\\mu\_\{b\}\),\\qquad c\_\{\\varepsilon\}^\{\(a,a\)\}=0\.The class\-affinity matrix has entries

αa​b=α0​e−cε\(a,b\),α0=det\(I\+Σ/ε\)−1/2\.\\alpha\_\{ab\}=\\alpha\_\{0\}e^\{\-c\_\{\\varepsilon\}^\{\(a,b\)\}\},\\qquad\\alpha\_\{0\}=\\det\(I\+\\Sigma/\\varepsilon\)^\{\-1/2\}\.Thus coarse class transport, leakage, and spectral quantities reduce to functions of the pairwise scalarscε\(a,b\)c\_\{\\varepsilon\}^\{\(a,b\)\}\. The static closed form converts class\-mean offsets into operator observables through a bandwidth\-regularized inverse\-covariance metric\.

The recurrent linear\-Gaussian bridge has the same role for finite\-lag transport\. If

Xt\+Δ=AΔ​Xt\+ξt,X\_\{t\+\\Delta\}=A\_\{\\Delta\}X\_\{t\}\+\\xi\_\{t\},then

G¯Δ=12​τ​Σξ\+12​τ​\(AΔ−I\)​Σ​\(AΔ−I\)⊤,\\bar\{G\}\_\{\\Delta\}=\\frac\{1\}\{2\\tau\}\\Sigma\_\{\\xi\}\+\\frac\{1\}\{2\\tau\}\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\},and

𝒲Δ=τ−1​\(Σ​AΔ⊤−AΔ​Σ\)\.\\mathcal\{W\}\_\{\\Delta\}=\\tau^\{\-1\}\(\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigma\)\.The coherent displacement trace is

τ−1​tr⁡\(\(AΔ−I\)​Σ​\(AΔ−I\)⊤\),\\tau^\{\-1\}\\operatorname\{tr\}\\left\(\(A\_\{\\Delta\}\-I\)\\Sigma\(A\_\{\\Delta\}\-I\)^\{\\top\}\\right\),a covariance\-weighted quadratic expression in the deterministic update offsetAΔ−IA\_\{\\Delta\}\-I\. The circulation is the antisymmetric part of the lagged cross\-covariance mismatch,

Σ​AΔ⊤−AΔ​Σ\.\\Sigma A\_\{\\Delta\}^\{\\top\}\-A\_\{\\Delta\}\\Sigma\.
The structural parallel is that both closed forms reduce operator\-level observables to quadratic expressions in a displacement parameter, modulated by the data covariance\. The displacement parameter is the class\-mean offset

in the static case and the update offset

in the recurrent case\. The covariance enters differently\. In the static Gaussian bridge,

\(ε​I\+Σ\)−1\(\\varepsilon I\+\\Sigma\)^\{\-1\}defines a bandwidth\-regularized inverse\-covariance metric on class\-mean offsets\. In the recurrent linear\-Gaussian bridge,Σ\\Sigmaweights the average squared finite\-lag displacement induced byAΔ−IA\_\{\\Delta\}\-I\. The population recurrent closed form contains no additive bandwidth regularizer, bandwidth enters through the finite\-sample source\-smoothing estimator\.

The recurrent setting also has an antisymmetric directed component,𝒲Δ\\mathcal\{W\}\_\{\\Delta\}, which has no counterpart in the symmetric reversible diffusion operator of the static feedforward construction\. This is the structural distinction isolated by Theorem[3](https://arxiv.org/html/2607.01746#Thmtheorem3), that deterministic finite\-lag motion contributes directly to the source\-centered transport tensor and, when the lagged cross\-covariance is antisymmetric, to coordinate circulation\.

In both settings, the parametric closed form is calibration rather than identification\. The Gaussian models are tractable cases in which the operator\-level observables can be written explicitly\. Neither the static feedforward paper nor the present recurrent paper assumes that learned representations are universally described by these parametric models\.

The recurrent paper is therefore the trajectory\-directed counterpart of the static feedforward construction\. The same operator\-first philosophy and closed\-form calibration strategy are adapted from static diffusion geometry to directed finite\-lag transport geometry\.

## Appendix EFinite lag versus infinitesimal carré du champ

This appendix makes the relationship between finite\-lag transport geometry and the infinitesimal carré\-du\-champ limit precise\. Theorem[3](https://arxiv.org/html/2607.01746#Thmtheorem3)formalizes the structural distinction by exhibiting deterministic dynamics for which the finite\-lag tensor is positive at the chosen lag while the infinitesimalΓL\\Gamma\_\{L\}vanishes\. Here we give the broader picture, including the small\-lag consistency that holds when an exact diffusion description is available\.

### E\.1Three regimes

It is useful to distinguish three settings that differ in what mathematical object is taken as primary\.

#### Exact Markov diffusions with infinitesimal generator\.

LetXtX\_\{t\}be an Itô diffusion satisfying

d​Xt=b​\(Xt\)​d​t\+σ​\(Xt\)​d​WtdX\_\{t\}=b\(X\_\{t\}\)\\,dt\+\\sigma\(X\_\{t\}\)\\,dW\_\{t\}onℝd\\mathbb\{R\}^\{d\}\. The backward generator is

L​f=b⋅∇f\+12​tr⁡\(σ​σ⊤​∇2f\),Lf=b\\cdot\\nabla f\+\\frac\{1\}\{2\}\\operatorname\{tr\}\\\!\\left\(\\sigma\\sigma^\{\\top\}\\nabla^\{2\}f\\right\),and the carré du champ is

ΓL​\(f,g\)=12​\(L​\(f​g\)−f​L​g−g​L​f\)\.\\Gamma\_\{L\}\(f,g\)=\\frac\{1\}\{2\}\\left\(L\(fg\)\-fLg\-gLf\\right\)\.In this setting the infinitesimal generator is the natural object\. On coordinate functions,ΓL\\Gamma\_\{L\}recovers the second\-order diffusion tensor, while the first\-order drift cancels in the carré\-du\-champ expression\.

#### Augmented or non\-Markov effective dynamics\.

For input\-driven or finitely observed systems, the hidden statehth\_\{t\}alone is generally not Markov\. A Markov representation may exist on an augmented state space that includes input or memory variables, but an infinitesimal generator onhth\_\{t\}alone is not then an intrinsic object\. In this regime, “the carré du champ onhth\_\{t\}” is not well\-defined without a Markov closure assumption\.

#### Finite\-lag empirical transfer operators\.

The empirical transfer operatorPΔP\_\{\\Delta\}is constructed from observed pairs

\(ht,ht\+Δ\)\(h\_\{t\},h\_\{t\+\\Delta\}\)regardless of whetherhth\_\{t\}is Markov and regardless of whether a small\-lag diffusion description exists\. The conditional lawQΔQ\_\{\\Delta\}is the law of the successor given the source under the joint sequence and input distribution\. Its symmetric and antisymmetric moments are well defined at every chosen lagΔ\\Delta, provided the corresponding conditional second moments are finite\.

This paper works in regime three\. The framework is finite\-lag by construction\. It does not require regime one and remains well\-defined under regime two\.

### E\.2First\-order drift cancellation

We restate the infinitesimal identity used in Theorem[3](https://arxiv.org/html/2607.01746#Thmtheorem3)\.

###### Proposition 9\(First\-order drift cancellation\)\.

Let

be a first\-order differential operator on smooth functions\. ThenLLsatisfies the Leibniz rule

L​\(f​g\)=f​L​g\+g​L​f,L\(fg\)=fLg\+gLf,and consequently

ΓL​\(f,g\)=0\\Gamma\_\{L\}\(f,g\)=0for all smoothf,gf,g\.

###### Proof\.

For smoothf,gf,g,

L​\(f​g\)=b⋅∇\(f​g\)=b⋅\(g​∇f\+f​∇g\)=g​\(b⋅∇f\)\+f​\(b⋅∇g\)=g​L​f\+f​L​g\.L\(fg\)=b\\cdot\\nabla\(fg\)=b\\cdot\(g\\nabla f\+f\\nabla g\)=g\(b\\cdot\\nabla f\)\+f\(b\\cdot\\nabla g\)=gLf\+fLg\.Substituting into the carré\-du\-champ definition gives

ΓL​\(f,g\)=12​\(L​\(f​g\)−f​L​g−g​L​f\)=12​\(g​L​f\+f​L​g−f​L​g−g​L​f\)=0\.\\Gamma\_\{L\}\(f,g\)=\\frac\{1\}\{2\}\\left\(L\(fg\)\-fLg\-gLf\\right\)=\\frac\{1\}\{2\}\\left\(gLf\+fLg\-fLg\-gLf\\right\)=0\.∎

Thus the infinitesimal carré du champ records the second\-order part of the generator and cancels the first\-order drift contribution\. For an Itô diffusion,

L​f=b⋅∇f\+12​tr⁡\(σ​σ⊤​∇2f\),Lf=b\\cdot\\nabla f\+\\frac\{1\}\{2\}\\operatorname\{tr\}\\left\(\\sigma\\sigma^\{\\top\}\\nabla^\{2\}f\\right\),one obtains

ΓL​\(f,g\)=12​∇f⊤​\(σ​σ⊤\)​∇g\.\\Gamma\_\{L\}\(f,g\)=\\frac\{1\}\{2\}\\nabla f^\{\\top\}\(\\sigma\\sigma^\{\\top\}\)\\nabla g\.For deterministic continuous\-time dynamics,σ≡0\\sigma\\equiv 0, and therefore

### E\.3Small\-lag diffusion consistency

When an exact small\-lag diffusion description exists, the finite\-lag construction is consistent with the infinitesimal carré du champ in the small\-lag limit\.

Suppose

Pτ=eτ​LP\_\{\\tau\}=e^\{\\tau L\}is the Markov semigroup generated byLLin regime one\. Define the finite\-lag generator and finite\-lag carré du champ by

Lτ=Pτ−Iτ,Γτ​\(f,g\)=12​\(Lτ​\(f​g\)−f​Lτ​g−g​Lτ​f\)\.L\_\{\\tau\}=\\frac\{P\_\{\\tau\}\-I\}\{\\tau\},\\qquad\\Gamma\_\{\\tau\}\(f,g\)=\\frac\{1\}\{2\}\\left\(L\_\{\\tau\}\(fg\)\-fL\_\{\\tau\}g\-gL\_\{\\tau\}f\\right\)\.Equivalently,

Γτ\(f,g\)\(x\)=12​τ𝔼\[\(f\(Xt\+τ\)−f\(x\)\)\(g\(Xt\+τ\)−g\(x\)\)\|Xt=x\]\.\\Gamma\_\{\\tau\}\(f,g\)\(x\)=\\frac\{1\}\{2\\tau\}\\mathbb\{E\}\\left\[\\bigl\(f\(X\_\{t\+\\tau\}\)\-f\(x\)\\bigr\)\\bigl\(g\(X\_\{t\+\\tau\}\)\-g\(x\)\\bigr\)\\,\\middle\|\\,X\_\{t\}=x\\right\]\.Thus the finite\-lag quadratic form used in the paper is exactly the finite\-time carré\-du\-champ expression associated withPτP\_\{\\tau\}whenever an exact semigroup exists\.

###### Proposition 10\(Small\-lag consistency\)\.

Assumeff,gg, andf​gfglie in the domain ofL2L^\{2\}, or equivalently take smooth compactly supported test functions under smooth bounded diffusion coefficients\. Then, uniformly on compact sets,

Lτ​f=L​f\+O​\(τ\),Γτ​\(f,g\)=ΓL​\(f,g\)\+O​\(τ\)L\_\{\\tau\}f=Lf\+O\(\\tau\),\\qquad\\Gamma\_\{\\tau\}\(f,g\)=\\Gamma\_\{L\}\(f,g\)\+O\(\\tau\)asτ→0\\tau\\to 0\.

###### Proof\.

The semigroup expansion gives

Pτ​f=f\+τ​L​f\+τ22​L2​f\+O​\(τ3\)P\_\{\\tau\}f=f\+\\tau Lf\+\\frac\{\\tau^\{2\}\}\{2\}L^\{2\}f\+O\(\\tau^\{3\}\)under the stated regularity\. Hence

Lτ​f=Pτ​f−fτ=L​f\+O​\(τ\)\.L\_\{\\tau\}f=\\frac\{P\_\{\\tau\}f\-f\}\{\\tau\}=Lf\+O\(\\tau\)\.Applying the same expansion tof​gfg,ff, andgg, and substituting into

Γτ​\(f,g\)=12​\(Lτ​\(f​g\)−f​Lτ​g−g​Lτ​f\),\\Gamma\_\{\\tau\}\(f,g\)=\\frac\{1\}\{2\}\\left\(L\_\{\\tau\}\(fg\)\-fL\_\{\\tau\}g\-gL\_\{\\tau\}f\\right\),gives

Γτ​\(f,g\)=12​\(L​\(f​g\)−f​L​g−g​L​f\)\+O​\(τ\)=ΓL​\(f,g\)\+O​\(τ\)\.\\Gamma\_\{\\tau\}\(f,g\)=\\frac\{1\}\{2\}\\left\(L\(fg\)\-fLg\-gLf\\right\)\+O\(\\tau\)=\\Gamma\_\{L\}\(f,g\)\+O\(\\tau\)\.∎

This proposition is the consistency check for regime one\. It does not change the design choice in this paper, the lag is part of the object being measured\. We do not attempt to reconstruct an underlying continuous\-time generator from discrete recurrent trajectories\.

### E\.4Why finite lag retains recurrent motion

The previous subsection describes the limiting relation in regime one\. The structural content of finite\-lag transport geometry is what happens at a chosenτ\>0\\tau\>0\.

Let

D=Xt\+τ−XtD=X\_\{t\+\\tau\}\-X\_\{t\}and assume the conditional second moment ofDDexists\. The finite\-lag transport tensor is

Gτ​\(x\)=12​τ​𝔼​\[D​D⊤∣Xt=x\]\.G\_\{\\tau\}\(x\)=\\frac\{1\}\{2\\tau\}\\mathbb\{E\}\[DD^\{\\top\}\\mid X\_\{t\}=x\]\.By the second\-moment identity,

Gτ​\(x\)=12​τ​Cov⁡\(D∣Xt=x\)\+12​τ​𝔼​\[D∣Xt=x\]​𝔼​\[D∣Xt=x\]⊤\.G\_\{\\tau\}\(x\)=\\frac\{1\}\{2\\tau\}\\operatorname\{Cov\}\(D\\mid X\_\{t\}=x\)\+\\frac\{1\}\{2\\tau\}\\mathbb\{E\}\[D\\mid X\_\{t\}=x\]\\mathbb\{E\}\[D\\mid X\_\{t\}=x\]^\{\\top\}\.The first term is the conditional spread contribution\. The second term is the coherent finite\-lag displacement contribution\.

For a smooth Itô diffusion,

𝔼​\[D∣Xt=x\]=τ​b​\(x\)\+O​\(τ2\),\\mathbb\{E\}\[D\\mid X\_\{t\}=x\]=\\tau b\(x\)\+O\(\\tau^\{2\}\),and

Cov⁡\(D∣Xt=x\)=τ​σ​σ⊤​\(x\)\+O​\(τ2\)\.\\operatorname\{Cov\}\(D\\mid X\_\{t\}=x\)=\\tau\\sigma\\sigma^\{\\top\}\(x\)\+O\(\\tau^\{2\}\)\.Therefore

Gτ​\(x\)=12​σ​σ⊤​\(x\)\+τ2​b​\(x\)​b​\(x\)⊤\+O​\(τ\)\.G\_\{\\tau\}\(x\)=\\frac\{1\}\{2\}\\sigma\\sigma^\{\\top\}\(x\)\+\\frac\{\\tau\}\{2\}b\(x\)b\(x\)^\{\\top\}\+O\(\\tau\)\.The leading drift\-induced correction is the rank\-one term

τ2​b​\(x\)​b​\(x\)⊤,\\frac\{\\tau\}\{2\}b\(x\)b\(x\)^\{\\top\},and all finite\-lag corrections vanish asτ→0\\tau\\to 0\. Thus

Gτ​\(x\)→12​σ​σ⊤​\(x\),G\_\{\\tau\}\(x\)\\to\\frac\{1\}\{2\}\\sigma\\sigma^\{\\top\}\(x\),which is the coordinate form ofΓL\\Gamma\_\{L\}\.

For deterministic continuous\-time dynamics,σ≡0\\sigma\\equiv 0, so at finite lag

Gτ​\(x\)=12​τ​\(Tτ​\(x\)−x\)​\(Tτ​\(x\)−x\)⊤\.G\_\{\\tau\}\(x\)=\\frac\{1\}\{2\\tau\}\(T\_\{\\tau\}\(x\)\-x\)\(T\_\{\\tau\}\(x\)\-x\)^\{\\top\}\.If

Tτ​\(x\)−x=τ​b​\(x\)\+o​\(τ\),T\_\{\\tau\}\(x\)\-x=\\tau b\(x\)\+o\(\\tau\),then

Gτ​\(x\)=τ2​b​\(x\)​b​\(x\)⊤\+o​\(τ\),G\_\{\\tau\}\(x\)=\\frac\{\\tau\}\{2\}b\(x\)b\(x\)^\{\\top\}\+o\(\\tau\),and the tensor vanishes in the infinitesimal limit, consistent withΓL≡0\\Gamma\_\{L\}\\equiv 0\.

For discrete\-time recurrent systems, however,τ\\tauis not taken to zero\. The measured object is the finite\-step displacement\. If the update is a mapTTobserved at integer lagΔ\\Delta, then

GΔ​\(x\)=12​τ​\(TΔ​\(x\)−x\)​\(TΔ​\(x\)−x\)⊤,G\_\{\\Delta\}\(x\)=\\frac\{1\}\{2\\tau\}\(T^\{\\Delta\}\(x\)\-x\)\(T^\{\\Delta\}\(x\)\-x\)^\{\\top\},which is positive whenever the finite\-step displacement is nonzero\. There need not be an underlying small\-time diffusion limit on hidden state\. In that case, the infinitesimal carré\-du\-champ comparison is a structural contrast, not a limiting approximation\.

The finite\-lag construction therefore makes recurrent motion measurable in two complementary regimes\. When a smooth diffusion description exists, finite\-lagGτG\_\{\\tau\}retains drift\-induced corrections at the chosen lag even though they disappear in the infinitesimal limit\. When no such description is available, as for general trained recurrent networks operating on integer time steps, the finite\-lag operator is the well\-defined object\.

### E\.5Implications for the framework’s use

Finite\-lag transport geometry is not the same object as infinitesimal carré\-du\-champ geometry\. The difference is the coherent displacement contribution at finite lag\. Three implications guide the use of the framework in the rest of the paper\.

First,GΔG\_\{\\Delta\}is reported at the operating lagΔ\\Deltaused in the experiments, not as an estimate of a limiting object\. The lag is part of the operator specification\.

Second, the framework does not estimate a continuous\-time generator from discrete recurrent trajectories\. The operatorP^Δ\\widehat\{P\}\_\{\\Delta\}is the empirical finite\-lag transfer operator at the chosen lag, not an approximation of an underlyingLL\. This avoids imposing a generator model that may not exist for the observed hidden state alone\.

Third, the distinction in Theorem[3](https://arxiv.org/html/2607.01746#Thmtheorem3)is structural\. Discrete cyclic shifts, shift\-like memory transport, and other coherent finite\-step hidden\-state motions are precisely the cases where finite\-lag source\-centered transport differs from an infinitesimal second\-order geometry\. The paper’s operator choice is matched to the dynamics it aims to describe\.

## Appendix FExperimental details and additional results

This appendix gives experimental details and additional empirical results for Section[5](https://arxiv.org/html/2607.01746#S5)\. Unless otherwise stated, empirical operators use the dense Gaussian source kernel of Equation \([1](https://arxiv.org/html/2607.01746#S3.E1)\), center\-RMS normalization, lagΔ=1\\Delta=1, and the median\-heuristic bandwidthεmed\\varepsilon\_\{\\rm med\}\. The exception is the linear\-Gaussian moment calibration, where spread and coherent displacement are computed from known conditional moments rather than from the kernel smoother\.

### F\.1Controlled decomposition and circulation

The decomposition calibration uses linear\-Gaussian dynamics

Y=A​X\+ξ,X∼𝒩​\(0,I\),ξ∼𝒩​\(0,σ2​I\),A=I\+β​B,Y=AX\+\\xi,\\qquad X\\sim\\mathcal\{N\}\(0,I\),\\qquad\\xi\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\),\\qquad A=I\+\\beta B,whereBBis a fixed Frobenius\-normalized perturbation\. We sweep

β∈\{0,0\.25,0\.5,0\.75,1\.0\},σ∈\{0,0\.1,0\.25,0\.5\},\\beta\\in\\\{0,0\.25,0\.5,0\.75,1\.0\\\},\\qquad\\sigma\\in\\\{0,0\.1,0\.25,0\.5\\\},atd=16d=16\. Since this experiment validates the population linear\-Gaussian closed form, the empirical conditional spread and coherent displacement traces are computed from the known conditional moments:

D=Y−X,𝔼​\[D∣X\]=\(A−I\)​X,Cov⁡\(D∣X\)=σ2​I\.D=Y\-X,\\qquad\\mathbb\{E\}\[D\\mid X\]=\(A\-I\)X,\\qquad\\operatorname\{Cov\}\(D\\mid X\)=\\sigma^\{2\}I\.Thus

2​tr⁡\(G¯Δ\)=τ−1​𝔼​‖D‖2,2\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\)=\\tau^\{\-1\}\\mathbb\{E\}\\\|D\\\|^\{2\},coherent displacement trace=τ−1​𝔼​‖\(A−I\)​X‖2,conditional spread trace=τ−1​tr⁡\(σ2​I\)\.\\text\{coherent displacement trace\}=\\tau^\{\-1\}\\mathbb\{E\}\\\|\(A\-I\)X\\\|^\{2\},\\qquad\\text\{conditional spread trace\}=\\tau^\{\-1\}\\operatorname\{tr\}\(\\sigma^\{2\}I\)\.BecauseBBis Frobenius\-normalized andΣ=I\\Sigma=I, the true coherent trace isβ2\\beta^\{2\}, while the true spread trace isd​σ2d\\sigma^\{2\}\. Across all twenty configurations, the maximum relative error in2​tr⁡\(G¯Δ\)2\\operatorname\{tr\}\(\\bar\{G\}\_\{\\Delta\}\)is0\.65%0\.65\\%, and the maximum relative error in the coherent trace, excluding zero\-denominator cases, is0\.70%0\.70\\%\. The identity case\(β,σ\)=\(0,0\)\(\\beta,\\sigma\)=\(0,0\)gives zero transport\.

This closed\-form evaluation also avoids a finite\-bandwidth smoothing artifact, as whenA=IA=Iandσ=0\\sigma=0, the true finite\-lag transport is zero, while a source\-neighborhood smoother can still report nonzero local displacement by averaging successors attached to nearby but distinct source states\.

For the circulation calibration, we use a two\-dimensional system

A=α​I\+γ​J,J=\(0−110\),A=\\alpha I\+\\gamma J,\\qquad J=\\begin\{pmatrix\}0&\-1\\\\ 1&0\\end\{pmatrix\},with fixedα=0\.85\\alpha=0\.85andσ=0\.05\\sigma=0\.05, sweepingγ∈\{0,0\.125,0\.25,…,1\.0\}\\gamma\\in\\\{0,0\.125,0\.25,\\ldots,1\.0\\\}\. The coordinate circulation‖𝒲^Δ‖F\\\|\\widehat\{\\mathcal\{W\}\}\_\{\\Delta\}\\\|\_\{F\}grows approximately linearly withγ\\gamma, from0\.00020\.0002atγ=0\\gamma=0to1\.1911\.191atγ=1\\gamma=1, matching Theorem[4](https://arxiv.org/html/2607.01746#Thmtheorem4)\.

![Refer to caption](https://arxiv.org/html/2607.01746v1/x3.png)Figure 3:Controlled circulation sweep\. The Frobenius norm of coordinate circulation increases with the oriented rotation parameterγ\\gamma\.
### F\.2Affine covariance experiment

The affine covariance experiment uses a length\-66cyclic\-shift trajectory cloud ind=12d=12, with3232trials and1010time steps per trial\. The empirical observables are computed at the dense median\-heuristic bandwidth\.

We run two versions\. In the push\-forward version, the dense source\-smoothing weights are computed on the base cloud and then held fixed while the paired coordinates are transformed as

\(X,Y\)↦\(A​X\+b,A​Y\+b\)\.\(X,Y\)\\mapsto\(AX\+b,AY\+b\)\.This is the empirical analogue of pushing forward the conditional law in Theorem[1](https://arxiv.org/html/2607.01746#Thmtheorem1)\. In the re\-kernelized version, the Euclidean Gaussian kernel is rebuilt after transforming the source coordinates\. Under anisotropic transformations this changes the empirical conditional law, so the resulting values test estimator sensitivity rather than tensorial covariance\.

Table 3:Affine covariance and metric dependence\. Trace columns are normalized by the base value\. Push\-forward rows hold the conditional weights fixed and test Theorem[1](https://arxiv.org/html/2607.01746#Thmtheorem1); metric correction restores the base scalar summaries\. Re\-kernelized anisotropic rows rebuild the Euclidean Gaussian kernel after transformation and therefore test estimator sensitivity\.Orthogonal transformations preserve Euclidean trace summaries\. Scalar dilation rescales traces by323^\{2\}, and the transformed metric restores the base values\. In the anisotropic push\-forward case, metric\-corrected traces match the base values to numerical precision\. In the re\-kernelized anisotropic case, the corrected values remain close but not identical because the Gaussian source\-smoothing operator has changed\.

### F\.3Dense stability experiment

The dense stability experiment perturbs a fixed normalized cyclic\-shift trajectory cloud by additive Gaussian noise of scale

σp∈\{0,10−4,3⋅10−4,10−3,3⋅10−3,10−2,3⋅10−2\}\.\\sigma\_\{p\}\\in\\\{0,10^\{\-4\},3\\cdot 10^\{\-4\},10^\{\-3\},3\\cdot 10^\{\-3\},10^\{\-2\},3\\cdot 10^\{\-2\}\\\}\.The bandwidth is fixed to the base\-cloud median\-heuristic value\. For each scale we average over eight perturbations and measure absolute changes in transport scale, coherent displacement trace, circulation norm, and dominant imaginary eigenvalue\. Log\-log slopes over the nonzero perturbation scales are1\.161\.16,1\.131\.13,1\.071\.07, and1\.011\.01, respectively, consistent with the Lipschitz behavior of Theorem[2](https://arxiv.org/html/2607.01746#Thmtheorem2)\.

Table 4:Dense stability perturbation test\. Entries are mean absolute changes over eight perturbations\. The near\-linear scaling matches the fixed\-bandwidth stability theorem\.
### F\.4Repeat\-copy task and training protocol

All recurrent experiments use repeat\-copy with feature dimensionF=4F=4\. Each sequence has a pattern phase, a delimiter, an optional blank delay, and a recall phase\. For copy lengthLL, a pattern

P∈\{−1,\+1\}L×FP\\in\\\{\-1,\+1\\\}^\{L\\times F\}is sampled uniformly\. Inputs present the pattern forLLsteps, then a delimiter cue, then blank inputs\. Targets are zero until the recall phase and then reproduce the pattern\.

The five seed performance matched run and the dense sensitivity sweeps use the standard repeat\-copy configuration withL=10L=10\. The capacity\-control and memory\-horizon experiments useL=8L=8, as specified below\. In all cases the geometry is computed on validation hidden trajectories after restoring the best validation\-recall checkpoint\.

We use three architectures\. Elman is a tanh recurrent network \(nn\.RNN\) with a linear readout\. GRU is a standardnn\.GRUwith a linear readout\. LSTM is a one\-layer LSTM unrolled explicitly so that both hidden statehth\_\{t\}and cell statectc\_\{t\}are available as trajectories\. For LSTM we reporthth\_\{t\},ctc\_\{t\}, and the concatenation\(ht,ct\)\(h\_\{t\},c\_\{t\}\)in the appendix\.

Training uses a recall\-window\-weighted mean\-squared loss:

ℒ=1N​T​F​∑n,t,fwt​\(y^n,t,f−yn,t,f\)2,wt=\{λrecall,t∈recall window,1,otherwise,\\mathcal\{L\}=\\frac\{1\}\{NTF\}\\sum\_\{n,t,f\}w\_\{t\}\\bigl\(\\hat\{y\}\_\{n,t,f\}\-y\_\{n,t,f\}\\bigr\)^\{2\},\\qquad w\_\{t\}=\\begin\{cases\}\\lambda\_\{\\rm recall\},&t\\in\\text\{recall window\},\\\\ 1,&\\text\{otherwise\},\\end\{cases\}withλrecall=5\.0\\lambda\_\{\\rm recall\}=5\.0\. Models are optimized with Adam\. The five\-seed performance\-matched run uses hidden dimension6464, dense mode, center\-RMS normalization, and architecture\-specific optimization settings\. Elman uses learning rate10−310^\{\-3\}and120120requested epochs\. GRU and LSTM use learning rate3⋅10−33\\cdot 10^\{\-3\}and240240requested epochs\.

The capacity\-control experiment uses20482048training sequences,256256validation sequences, batch size6464, patience4040, minimum8080epochs, and copy lengthL=8L=8\. Elman uses120120requested epochs, while GRU and LSTM use240240\. The memory\-horizon and phase\-profile experiments use20482048training sequences,192192validation sequences, patience6060, minimum100100epochs, copy lengthL=8L=8, and requested epoch budgets180180for Elman and320320for GRU/LSTM\. The phase\-profile experiment fixes delayD=4D=4and computes separate phase\-local operators on write, cue, delay, and recall source–successor pairs\.

### F\.5Operator estimation and baselines

For trained recurrent networks, validation hidden trajectories are pooled across sequences to form source–successor pairs\. In the five\-seed performance\-matched and sensitivity runs, the pooled source set contains48644864valid source nodes per run\. The densen×nn\\times nGaussian kernel matrix is therefore manageable in memory\. Local moments are computed in chunks to avoid materializing the full\(chunk,n,d\)\(\\text\{chunk\},n,d\)displacement tensor\.

The default bandwidth is

εmed=14​mediani<j⁡‖xi−xj‖2,\\varepsilon\_\{\\rm med\}=\\frac\{1\}\{4\}\\operatorname\{median\}\_\{i<j\}\\\|x\_\{i\}\-x\_\{j\}\\\|^\{2\},estimated from at most20002000source states when the full source cloud is larger\. Thekk\-NN approximation of Appendix[A\.6](https://arxiv.org/html/2607.01746#A1.SS6)is reported only in sensitivity studies\.

For comparison, we compute several standard summaries on the same normalized hidden trajectories\. Static effective rank is

exp⁡\(−∑ipi​log⁡pi\),pi=λi/∑jλj,\\exp\\\!\\left\(\-\\sum\_\{i\}p\_\{i\}\\log p\_\{i\}\\right\),\\qquad p\_\{i\}=\\lambda\_\{i\}/\\sum\_\{j\}\\lambda\_\{j\},whereλi\\lambda\_\{i\}are covariance eigenvalues\. Dynamic mode decomposition fitsY≈X​A⊤Y\\approx XA^\{\\top\}by ridge regression with regularization10−410^\{\-4\}and reports pooled validation\-pair predictionR2R^\{2\}\. TICA reports eigenvalues of the symmetrized lagged covariance operator, and VAMP reports singular values of the whitened lagged cross\-covariance, both with ridge regularization10−410^\{\-4\}\. We also report relative lagged\-covariance skew,

‖C0​t−C0​t⊤‖F/‖C0​t‖F\.\\\|C\_\{0t\}\-C\_\{0t\}^\{\\top\}\\\|\_\{F\}/\\\|C\_\{0t\}\\\|\_\{F\}\.

### F\.6Resolution andkk\-NN sensitivity

The dense bandwidth sweep covers

ε/εmed∈\{0\.1,0\.2,0\.3,0\.5,1\.0,2\.0\}\\varepsilon/\\varepsilon\_\{\\rm med\}\\in\\\{0\.1,0\.2,0\.3,0\.5,1\.0,2\.0\\\}on hidden states from the performance\-matched repeat\-copy case study\. Table[5](https://arxiv.org/html/2607.01746#A6.T5)reports coherent displacement fraction atΔ=1\\Delta=1, center\-RMS normalization, averaged over three seeds for the dense matched sensitivity run\.

Table 5:Coherent displacement fraction versus dense bandwidth scaleε/εmed\\varepsilon/\\varepsilon\_\{\\rm med\},Δ=1\\Delta=1, center\-RMS normalization\. The fraction is generally larger at narrower bandwidths, showing that it is a resolution\-dependent scalar summary\.The effective\-neighborhood diagnostics confirm that the smallest dense bandwidth is not a self\-loop artifact\. Atε/εmed=0\.1\\varepsilon/\\varepsilon\_\{\\rm med\}=0\.1, average self\-mass overhh\-state runs is about0\.090\.09and the entropy\-effective neighborhood size is about8\.3×1028\.3\\times 10^\{2\}\. At the median bandwidth, self\-mass is near zero and the effective neighborhood size is about4\.6×1034\.6\\times 10^\{3\}\.

Whitening changes absolute trace scale and shifts coherent fractions\. This is consistent with Theorem[1](https://arxiv.org/html/2607.01746#Thmtheorem1)\. Scalar trace summaries depend on the metric used to contract the transport tensor\. We report center\-RMS Euclidean summaries in the main text and use whitening as a sensitivity analysis\.

Atk=16k=16, the sparse approximation increasesFΔρF\_\{\\Delta\}^\{\\rho\}from roughly0\.460\.46–0\.500\.50for the dense estimator to roughly0\.680\.68–0\.700\.70across states, and increases circulation norms by about an order of magnitude\. This confirms that hard sparse conditioning changes the local resolution, so we therefore use it only as a sensitivity analysis\.

### F\.7Performance matched repeat\-copy details

Table 6:Full state comparison for the performance\-matched repeat\-copy experiment\. Values are means over five seeds\. LSTM cell states have lower coherent displacement fraction than LSTM hidden outputs under this metric\.The main text reportshth\_\{t\}for architectural comparison\. The additional LSTM rows show thathth\_\{t\},ctc\_\{t\}, and\(ht,ct\)\(h\_\{t\},c\_\{t\}\)have distinct finite\-lag geometry under the same estimator\.

### F\.8Capacity controls

The capacity\-control experiment uses copy lengthL=8L=8and sweeps

Elman\-64,Elman\-128,GRU\-36,GRU\-64,LSTM\-32,LSTM\-64\.\\text\{Elman\-64\},\\ \\text\{Elman\-128\},\\ \\text\{GRU\-36\},\\ \\text\{GRU\-64\},\\ \\text\{LSTM\-32\},\\ \\text\{LSTM\-64\}\.GRU\-36 and LSTM\-32 are approximate parameter\-count controls for Elman\-64\. All configurations achieve near\-perfect recall sign accuracy, although the smaller gated models have slightly larger recall MSE\.

Table 7:Capacity controls, three seeds\. Within this grid, the transport\-scale gap between Elman and gated networks is not explained solely by hidden width or parameter count\.The coherent displacement fraction is not a robust family separator in this sweep, as LSTM\-32 has a fraction close to Elman despite lower transport scale\. The more stable finite\-lag distinctions are total transport scale and coherent displacement trace\.

### F\.9Phase\-resolved repeat\-copy profile

The phase\-profile experiment localizes where finite\-lag transport differences arise within a solved recurrent computation\. We use repeat\-copy with delayD=4D=4, copy lengthL=8L=8, hidden dimension6464, three seeds, dense mode, center\-RMS normalization, lagΔ=1\\Delta=1, and median\-heuristic bandwidth\. Allhth\_\{t\}runs solve the task with recall sign accuracy1\.01\.0\.

We split source–successor pairs into task phases\. The write phase contains source times receiving pattern inputs, the cue phase contains the delimiter transition, the delay phase contains blank\-memory transitions, and the recall phase contains output transitions\. We apply one global center\-RMS normalization to the validation trajectory cloud, then build a separate dense phase\-local operator for each phase\. The cue phase has only one source time, so it is included as a transition diagnostic rather than a primary architecture\-level conclusion\.

Table 8:Phase\-resolved finite\-lag geometry for repeat\-copy at delayD=4D=4, statehth\_\{t\}, three seeds\. Values are means over solved runs\. Separate phase\-local dense operators are built after global center\-RMS normalization\.The global transport\-scale differences are concentrated in write and recall phases\. Elman has substantially larger transport and coherent displacement during write and recall, while GRU and LSTM maintain lower transport through write, cue, and delay and increase at recall\. Static rank follows a different pattern: for example, Elman has high static rank during delay, while its largest finite\-lag transport occurs during recall\. This illustrates the distinction between snapshot geometry and finite\-lag transport geometry\.

### F\.10Memory\-horizon experiment

The memory\-horizon experiment inserts a blank delay

D∈\{0,2,4,6,8\}D\\in\\\{0,2,4,6,8\\\}between delimiter and recall\. We use copy lengthL=8L=8, hidden dimension6464, three seeds, dense mode, center\-RMS normalization, lagΔ=1\\Delta=1, and median\-heuristic bandwidth\. Geometry is interpreted only on solved rows, defined by validation recall sign accuracy at least0\.90\.9\. GRU and LSTM solve all delays across all three seeds\. Elman solves all seeds throughD=6D=6and two of three seeds atD=8D=8\.

Table 9:Recall sign accuracy in the small\-delay memory\-horizon experiment, averaged over three seeds\. Elman has one unsolved seed atD=8D=8, geometry is interpreted on solved rows\.Table 10:Memory\-horizon geometry forhth\_\{t\}on solved rows\. GRU and LSTM show decreasing transport scale and coherent displacement trace as delay increases\. Elman remains higher\-transport at short delays but is less consistently solved atD=8D=8\.
### F\.11Compute and reproducibility

The dense estimator scales asO​\(n2​d\)O\(n^\{2\}d\)onnnpooled source nodes\. For the repeat\-copy experiments, dense kernel matrices contain roughly5×1035\\times 10^\{3\}source nodes and are feasible on a single workstation\. Synthetic experiments run in seconds to minutes\. The main repeat\-copy and sensitivity experiments use five and three seeds, respectively, while capacity controls, memory\-horizon experiments, and phase\-profile experiments use three seeds\.