Spectral Indistinguishability and Learning-Curve Identifiability in Quadratic Networks

# Spectral Indistinguishability and Learning-Curve Identifiability in Quadratic Networks
Source: [https://arxiv.org/html/2610.20678](https://arxiv.org/html/2610.20678)
###### Abstract

We study what spectral summaries and scalar learning curves reveal about centered quadratic networks\. We construct analytic positive input densities with exactly Gaussian one\- and two\-coordinate marginals\. Their complete fourth\-moment operators are intertwined by an orthogonal map fixing the entire diagonal teacher algebra, so the corresponding anchored spectral measures agree\. For selected common teachers with a nonzero cycle invariant, their factorized loss curves differ because the map does not preserve matrix multiplication\. Cycle lengthrr, which can be arbitrarily large, givesO⁡\(εr\)O\(\\varepsilon^\{r\}\)contrast uniformly in training time for each fixed construction\. For a supplied pair, noisy scalar loss queries require orderε−2​r\\varepsilon^\{\-2r\}observations, while bounded initialization interventions with scalar gradient readouts require orderε−2\\varepsilon^\{\-2\}\. A shared\-input score, equal to a paired\-gradient difference, attains the optimal input\-sample exponent without knowing the teacher\. An explicit family of diverging auxiliary teacher scales instead admits a nonzero finite limiting contrast at coupling1/log⁡L1/\\log L\. Conversely, for Gaussian inputs and a rationally independent teacher spectrum, one exact scalar loss curve identifies the initial predictor up to sign conjugation\. Within this rationally independent class, with nonzero diagonal inverse\-state error in every coordinate and calibrated time and loss scales, it also identifies the unknown teacher spectrum and active dimension, modulo simultaneous rotations\. For fixed dimension and distinct unordered teacher sums, dimension\-minimal exact observations give generic joint local recovery\. Finite exact global bounds hold for a known teacher and, on the active class, under a blind dimension cap\. Exact certificates and full\-weight replay verify the construction; numerical inverse checks reveal severe ill\-conditioning\.

Keywords:quadratic neural networks; spectral alignment; learning dynamics; nonlinear observability; signed cycles\.

## 1Introduction

Spectral descriptions explain how a learner distributes sensitivity across modes\. Their explanatory value depends on what information is retained with those modes\. Eigenvalues alone omit their orientation; task alignment restores some of that information\. In nonlinear learning, even a complete family of task\-anchored spectral measures can leave an additional operation unspecified: how matrix\-valued modes multiply during training\.

We study this issue in centered quadratic neural networks\. Their population dynamics close on the positive predictor matrixM=W​W⊤M=WW^\{\\top\}, allowing exact comparisons of actual input distributions and full weight initializations\. Our negative result matches more than the operator spectrum\. An explicit isometry fixes every diagonal matrix and intertwines the complete moment operators of two input laws\. Consequently all spectral responses anchored on the diagonal teacher algebra agree\. The two learners nevertheless have different loss histories\. The isometry preserves inner products but does not preserve the multiplication used in the factorized gradient flow\.

The converse question gives the second part of the paper\. Under Gaussian inputs and a known nonresonant teacher, an exact scalar loss history is a complete invariant of a positive initial predictor, modulo the unavoidable diagonal sign symmetry\. Its exponential expansion encodes entry magnitudes and products around hidden cycles\. Within the rationally independent, diagonally active class, the expansion’s observed rate set also determines the unknown teacher spectrum and active dimension\. For fixed dimension and distinct unordered teacher sums, finitely many exact scalar observations give generic joint local recovery\. Larger sufficient counts give exact global identification for a known teacher or for the active blind class with a dimension cap\. These are different observation problems: initial spectral information need not predict a nonlinear trajectory, whereas the trajectory itself can reveal information absent from those summaries\.

We establish four linked statements\. First, complete spectral equivalence can fix the whole teacher algebra and still fail to transport the learning vector field \(Theorem[3\.1](https://arxiv.org/html/2610.20678#S3.Thmtheorem1)and Proposition[3\.2](https://arxiv.org/html/2610.20678#S3.Thmtheorem2)\)\. Second, under the nonzero cycle certificate, the length of a hidden cycle controls the first nonzero perturbative loss response throughout the entire trajectory for each fixed construction, with an exact polynomial counterpart for finitely many weight updates \(Theorems[4\.1](https://arxiv.org/html/2610.20678#S4.Thmtheorem1)and[4\.3](https://arxiv.org/html/2610.20678#S4.Thmtheorem3), Corollary[4\.2](https://arxiv.org/html/2610.20678#S4.Thmtheorem2)\)\. A bounded initialization intervention reduces the sharp noisy scalar query exponent from2​r2rto22for the same supplied pair \(Theorem[4\.4](https://arxiv.org/html/2610.20678#S4.Thmtheorem4)\); paired gradients give a finite\-input implementation without the teacher \(Corollary[4\.5](https://arxiv.org/html/2610.20678#S4.Thmtheorem5)\)\. Third, an explicitly unbounded teacher family has a logarithmic amplification threshold: vanishing moment perturbations create a finite effective state during a vanishing initial layer \(Theorem[5\.1](https://arxiv.org/html/2610.20678#S5.Thmtheorem1)\)\. Finally, the scalar Gaussian loss curve identifies that state and, under an explicit excitation condition, the unknown teacher itself, while a sharp two\-point bound limits what noisy loss measurements can resolve \(Theorems[6\.1](https://arxiv.org/html/2610.20678#S6.Thmtheorem1)–[6\.7](https://arxiv.org/html/2610.20678#S6.Thmtheorem7)\)\.

#### Relation to existing theory\.

Kernel generalization depends on task\-model alignment as well as eigenvalues\([Canatar et al\., 2021](https://arxiv.org/html/2610.20678#bib.bib5)\)\. Our construction matches the full diagonal\-anchored population spectral measures\. Pointwise raw\-input kernel values and empirical Gram matrices remain different observation objects\.

Predictor closure, solvable mode dynamics and factorization\-dependent learning are established\([Saxe et al\., 2014](https://arxiv.org/html/2610.20678#bib.bib14);[Li et al\., 2018](https://arxiv.org/html/2610.20678#bib.bib10);[Arora et al\., 2018](https://arxiv.org/html/2610.20678#bib.bib1);[Liu Ziyin et al\., 2026](https://arxiv.org/html/2610.20678#bib.bib11)\)\. In particular, Neural Quadratic Forms includes both commuting reductions and a noncommuting isotropic matrix\-Riccati solution\([Liu Ziyin et al\., 2026](https://arxiv.org/html/2610.20678#bib.bib11), Theorem 7\)\. We build on that solution to analyze spectral equivalence and scalar observability\. Extensive\-width Gaussian\-sensing theory\([Martin et al\., 2026](https://arxiv.org/html/2610.20678#bib.bib12)\)concerns a different asymptotic regime\. Kernel/rich transitions can depend on initialization\([Woodworth et al\., 2020](https://arxiv.org/html/2610.20678#bib.bib20)\); our amplification result instead keeps the initial factor fixed and varies teacher scale and the input moment perturbation\.

Signed\-graph switching and closed walks are classical\([Harary, 1953](https://arxiv.org/html/2610.20678#bib.bib6);[Stanić, 2023](https://arxiv.org/html/2610.20678#bib.bib17)\)\. Recovery from magnitudes and cycle products has a particularly close precedent in principal\-minor and determinantal point process learning\([Urschel et al\., 2017](https://arxiv.org/html/2610.20678#bib.bib19);[Brunel, 2018](https://arxiv.org/html/2610.20678#bib.bib3);[Brunel and Urschel, 2024](https://arxiv.org/html/2610.20678#bib.bib4)\)\. The inverse contribution here is the proof that a scalar learning\-loss curve supplies these invariants, the recovery of the unknown teacher spectrum from its rate set, and the observation count for joint generic local recovery\. General nonlinear observability\([Hermann and Krener, 1977](https://arxiv.org/html/2610.20678#bib.bib7)\)and matrix factorization geometry\([Bhatia et al\., 2019](https://arxiv.org/html/2610.20678#bib.bib2)\)supply background\. The testing argument uses standard Gaussian divergence inequalities; the dynamic ingredient is a cycle\-dependent signal bound uniform over all observation times\. General finite\-experiment identifiability and exponential\-sum reconstruction are established\([Sontag, 2002](https://arxiv.org/html/2610.20678#bib.bib16);[Potts and Tasche, 2010](https://arxiv.org/html/2610.20678#bib.bib13)\)\. Our blind recovery argument specifies the equivalence class and excitation conditions associated with this neural loss\.

Gradient\-query models also distinguish observation channels in sparse function learning\([Joshi et al\., 2024](https://arxiv.org/html/2610.20678#bib.bib8)\)\. Our comparison concerns identification of a supplied input\-law pair from its neural responses\. The loss and gradient experiments each return one scalar with its own fixed Gaussian measurement variance\. This makes the exponent comparison precise without identifying either channel with iid training data or assigning them equal computational cost\. Separately, a shared\-input gradient difference cancels the label and realizes an ordinary fourth\-moment test with an optimal input\-sample exponent for the supplied alternatives\.

The principal\-minor problem has quadratic query complexity and approximate recovery guarantees\([Brunel and Urschel, 2024](https://arxiv.org/html/2610.20678#bib.bib4)\)\. Our fixed\-time observation count has the same dimensional scale and applies to the scalar neural\-loss map\. Table[1](https://arxiv.org/html/2610.20678#S1.T1)distinguishes the established reconstruction tools from the dynamic statements\.

Table 1:The contribution is in the learning dynamics and observation channel\. The listed algebraic and statistical tools are prior results\.
#### Scope\.

The dynamical results concern exact population dynamics of a specified quadratic learner and constructed laws\. The finite\-input test is restricted to a supplied pair\. The finite limiting contrast requires diverging teacher norm and initial loss\. The inverse results use exact observations and nonresonance\. Blind recovery additionally requires nonzero diagonal inverse\-state error; an unexcited coordinate can remain invisible\. The minimal observation counts are local, and the bounds valid for every global observation design are exponential in dimension\. Practical reconstruction on natural data and transfer to ordinary deep networks remain outside these results\.

## 2Model and an admissible paired construction

For a mean\-zero inputx∈ℝdx\\in\\mathbb\{R\}^\{d\}with covarianceII, define

A⁡\(x\)=x​x⊤−I2,fW​\(x\)=⟨W​W⊤,A⁡\(x\)⟩,y⁡\(x\)=⟨T,A⁡\(x\)⟩,T≻0\.A\(x\)=\\frac\{xx^\{\\top\}\-I\}\{\\sqrt\{2\}\},\\qquad f\_\{W\}\(x\)=\\langle WW^\{\\top\},A\(x\)\\rangle,\\qquad y\(x\)=\\langle T,A\(x\)\\rangle,\\quad T\\succ 0\.The columns ofWWare quadratic neural units\. With squared population loss, the complete moment operator and training equations are

H⁡\(E\)\\displaystyle H\(E\)=𝔼⁡\[A⁡\(x\)​⟨A⁡\(x\),E⟩\],\\displaystyle=\\mathbb\{E\}\[A\(x\)\\langle A\(x\),E\\rangle\],G⁡\(M\)\\displaystyle G\(M\)=H⁡\(M−T\),\\displaystyle=H\(M\-T\),W˙\\displaystyle\\dot\{W\}=−2​G​\(W​W⊤\)​W,\\displaystyle=\-2G\(WW^\{\\top\}\)W,M˙\\displaystyle\\dot\{M\}=−2​\{G⁡\(M\),M\},M=W​W⊤\.\\displaystyle=\-2\\\{G\(M\),M\\\},\\quad M=WW^\{\\top\}\.\(1\)We take the same square full factorW0=c​IW\_\{0\}=\\sqrt\{c\}Iin both arms\. The exact predictor update induced by weight gradient descent is

Mk\+1=Mk−2​h​\(Gk​Mk\+Mk​Gk\)\+4​h2​Gk​Mk​Gk\.M\_\{k\+1\}=M\_\{k\}\-2h\(G\_\{k\}M\_\{k\}\+M\_\{k\}G\_\{k\}\)\+4h^\{2\}G\_\{k\}M\_\{k\}G\_\{k\}\.Dropping the last term would change the optimizer\.

Table 2:Notation across the construction and inverse problem\. The coordinate setCCand inverse\-state matrixCCoccur in separate sections\. Arguments and subscripts distinguishA⁡\(x\)A\(x\)andA±A\_\{\\pm\}from the coordinate setAA\.Fixq,r≥3q,r\\geq 3and splitd=q\+2​r−1d=q\+2r\-1coordinates into an auxiliary setAAand interaction setCC\. OnCC, two chordlessrr\-cycles share one hub\. Corresponding edges have the same positive weightsw1,…,wrw\_\{1\},\\ldots,w\_\{r\}\. InB\+B\_\{\+\}the two cycle products have signs\(\+,−\)\(\+,\-\); inB−B\_\{\-\}they have signs\(−,\+\)\(\-,\+\)\. Choose the changed edge in each branch correspondingly\. The branch\-exchange involutionPPsatisfiesB−=P​B\+​P⊤B\_\{\-\}=PB\_\{\+\}P^\{\\top\}, while\|B−\|=\|B\+\|\|B\_\{\-\}\|=\|B\_\{\+\}\|\.

LetG0=diag⁡\(g\)G\_\{0\}=\\operatorname\{diag\}\(g\)have distinct negative entries and letT=c​I−G0T=cI\-G\_\{0\}\. For curve separation we additionally chooseggso that

Jr=tr⁡\[diag⁡\(gC\)​\(B\+r−B−r\)\]≠0\.J\_\{r\}=\\operatorname\{tr\}\\\!\\left\[\\operatorname\{diag\}\(g\_\{C\}\)\(B\_\{\+\}^\{r\}\-B\_\{\-\}^\{r\}\)\\right\]\\neq 0\.By \([28](https://arxiv.org/html/2610.20678#A1.E28)\), this is equivalent to unequal sums of the teacher errors on the two branches excluding the hub\. Distinct negative entries alone do not ensure it\. Choosing all errors on one branch closer to zero than those on the other gives an admissible noncancelling choice\. The auxiliary barycentric vector

ℓi=∏j∈Aj≠i\(gi−gj\)−1\\ell\_\{i\}=\\prod\_\{\\begin\{subarray\}\{c\}j\\in A\\\\ j\\neq i\\end\{subarray\}\}\(g\_\{i\}\-g\_\{j\}\)^\{\-1\}satisfiesℓ⊤​gAk=0\\ell^\{\\top\}g\_\{A\}^\{k\}=0fork<q−1k<q\-1andℓ⊤​gAq−1=1\\ell^\{\\top\}g\_\{A\}^\{q\-1\}=1\. In particularℓ⊤​𝟏=ℓ⊤​TA=0\\ell^\{\\top\}\\mathbf\{1\}=\\ell^\{\\top\}T\_\{A\}=0\. For physical rescalings below, only the specified normalization changes; the moment\-operator formula remains explicit\.

#### Positive input laws\.

Letp0p\_\{0\}be the product five\-node Gaussian quadrature law for a standard normal input\. In one coordinate its nodes are0,±5−10,±5\+100,\\pm\\sqrt\{5\-\\sqrt\{10\}\},\\pm\\sqrt\{5\+\\sqrt\{10\}\}; the mass at zero is8/158/15, at each inner node\(7\+2​10\)/60\(7\+2\\sqrt\{10\}\)/60, and at each outer node\(7−2​10\)/60\(7\-2\\sqrt\{10\}\)/60\. Its moments through degree nine match the standard normal moments\. Set

hs​\(x\)=12​\(∑i∈Aℓi​xi2\)​∑e∈E⁡\(C\)\(Bs\)e​xe1​xe2,ps​\(x\)=p0​\(x\)​\(1\+ε​hs​\(x\)\)\.h\_\{s\}\(x\)=\\frac\{1\}\{2\}\\left\(\\sum\_\{i\\in A\}\\ell\_\{i\}x\_\{i\}^\{2\}\\right\)\\sum\_\{e\\in E\(C\)\}\(B\_\{s\}\)\_\{e\}x\_\{e\_\{1\}\}x\_\{e\_\{2\}\},\\qquad p\_\{s\}\(x\)=p\_\{0\}\(x\)\(1\+\\varepsilon h\_\{s\}\(x\)\)\.\(2\)If\|ε\|​R5<1\|\\varepsilon\|R\_\{5\}<1, where

R5=\(5\+10\)24​‖ℓ‖1​∑e∈E⁡\(C\)we,R\_\{5\}=\\frac\{\(5\+\\sqrt\{10\}\)^\{2\}\}\{4\}\\\|\\ell\\\|\_\{1\}\\sum\_\{e\\in E\(C\)\}w\_\{e\},every mass is positive\. The bound uses∑ℓi=0\\sum\\ell\_\{i\}=0and the largest squared node\. Oddness at interaction endpoints and∑ℓi=0\\sum\\ell\_\{i\}=0show that the tilted law has mass one, zero mean and covarianceII\. Fourth\-moment contractions against the tilt have coordinate degrees at most eight, so Gaussian quadrature is exact\. They give the complete operator in \([6](https://arxiv.org/html/2610.20678#S3.E6)\), including the auxiliary–interaction directions\. Thus the operator is realized by actual input laws, not merely a proposed positive linear map\.

###### Proposition 2\.1\(Exact realization by analytic Gaussian perturbations\)\.

For the fixed construction there are bounded analytic functionsψs\\psi\_\{s\}such that, for all sufficiently small realε\\varepsilon,

ps,ε​\(x\)=ϕd​\(x\)​\(1\+ε​ψs​\(x\)\)p\_\{s,\\varepsilon\}\(x\)=\\phi\_\{d\}\(x\)\(1\+\\varepsilon\\psi\_\{s\}\(x\)\)is an everywhere positive density with complete moment operatorId\+ε​Ks\\operatorname\{Id\}\+\\varepsilon K\_\{s\}\. Every one\- or two\-coordinate marginal is exactly the corresponding standard Gaussian marginal\. For an explicit constantRψ<∞R\_\{\\psi\}<\\infty,

\(1−\|ε\|​Rψ\)​ϕd≤ps,ε≤\(1\+\|ε\|​Rψ\)​ϕd,‖Ps,ε−N⁡\(0,I\)‖TV≤\|ε\|​Rψ/2\.\(1\-\|\\varepsilon\|R\_\{\\psi\}\)\\phi\_\{d\}\\leq p\_\{s,\\varepsilon\}\\leq\(1\+\|\\varepsilon\|R\_\{\\psi\}\)\\phi\_\{d\},\\qquad\\\|P\_\{s,\\varepsilon\}\-N\(0,I\)\\\|\_\{\\rm TV\}\\leq\|\\varepsilon\|R\_\{\\psi\}/2\.In particular, all subsequent exact dynamics and spectral identities hold under these smooth laws without an approximation of the moments\.

###### Proof\.

LetX∼N⁡\(0,1\)X\\sim N\(0,1\)and define the bounded analytic functions

u⁡\(x\)=1−e​cos⁡\(2​x\)2,v⁡\(x\)=e​sin⁡x\.u\(x\)=\\frac\{1\-e\\cos\(\\sqrt\{2\}x\)\}\{2\},\\qquad v\(x\)=\\sqrt\{e\}\\sin x\.The Gaussian characteristic function and its first two derivatives give

𝔼​u​\(X\)=0,𝔼⁡\[X​u​\(X\)\]=0,𝔼⁡\[X2​u​\(X\)\]=1,𝔼​v​\(X\)=0,𝔼⁡\[X​v​\(X\)\]=1\.\\mathbb\{E\}u\(X\)=0,\\quad\\mathbb\{E\}\[Xu\(X\)\]=0,\\quad\\mathbb\{E\}\[X^\{2\}u\(X\)\]=1,\\qquad\\mathbb\{E\}v\(X\)=0,\\quad\\mathbb\{E\}\[Xv\(X\)\]=1\.Set

ψs​\(x\)=\(∑i∈Aℓi​u​\(xi\)\)​∑\(j,k\)∈E⁡\(C\)\(Bs\)j​k​v​\(xj\)​v​\(xk\)\.\\psi\_\{s\}\(x\)=\\left\(\\sum\_\{i\\in A\}\\ell\_\{i\}u\(x\_\{i\}\)\\right\)\\sum\_\{\(j,k\)\\in E\(C\)\}\(B\_\{s\}\)\_\{jk\}v\(x\_\{j\}\)v\(x\_\{k\}\)\.Each perturbation term is a product of centered functions on three distinct coordinates\. Integrating out all but any one or two coordinates therefore removes every term\. This proves the Gaussian marginal statement and, in particular, zero mean and covarianceII\. For a product monomial to have a nonzero expectation against one term of this perturbation, its auxiliary coordinate must occur at least twice and both interaction endpoints at least once\. All moments of degree at most three therefore remain Gaussian\. At degree four the only possibilities are

∫ϕd​\(x\)​ψs​\(x\)​xi2​xj​xk​𝑑x=ℓi​\(Bs\)j​k,i∈A,\(j,k\)∈E⁡\(C\)\.\\int\\phi\_\{d\}\(x\)\\psi\_\{s\}\(x\)x\_\{i\}^\{2\}x\_\{j\}x\_\{k\}\\,dx=\\ell\_\{i\}\(B\_\{s\}\)\_\{jk\},\\qquad i\\in A,\\ \(j,k\)\\in E\(C\)\.These are exactly the fourth moments generating \([6](https://arxiv.org/html/2610.20678#S3.E6)\)\. All other lower or fourth moments agree with the standard Gaussian\.

Since∑iℓi=0\\sum\_\{i\}\\ell\_\{i\}=0, the constant part ofuucancels in its weighted sum\. Consequently the explicit bound is

‖ψs‖∞≤Rψ:=e22​‖ℓ‖1​∑e∈E⁡\(C\)we\.\\\|\\psi\_\{s\}\\\|\_\{\\infty\}\\leq R\_\{\\psi\}:=\\frac\{e^\{2\}\}\{2\}\\\|\\ell\\\|\_\{1\}\\sum\_\{e\\in E\(C\)\}w\_\{e\}\.For\|ε\|​Rψ<1\|\\varepsilon\|R\_\{\\psi\}<1this proves positivity, normalization, the density\-ratio bounds and the total\-variation bound\. ∎

Products of centered bounded functions are an established way to construct distributions with prescribed marginals, as in the Sarmanov family and its extensions\([Lee, 1996](https://arxiv.org/html/2610.20678#bib.bib9)\)\. Likewise, pairwise independence without joint independence and three\-variable interaction tests are established\([Sejdinovic et al\., 2013](https://arxiv.org/html/2610.20678#bib.bib15)\)\. The role of this elementary analytic construction is to realize the specific complete neural moment operator exactly\. For the logarithmic amplification family below, these functions andRψR\_\{\\psi\}are independent ofLL\. Consequently those laws converge even in uniform density ratio to the actual Gaussian, while the teacher norm diverges\. The finite contrast is therefore not a consequence of atoms or vanishing support probabilities\. The construction gives no dimension\-uniform lower bound on the admissible coupling\.

###### Proposition 2\.2\(The marginal\-order boundary for diagonal training\)\.

In a fixed coordinate basis, suppose every marginal on at most three coordinates of an input law is standard Gaussian\. For any diagonal teacherTTand positive diagonal initial predictor, its factorized population flow and exact weight\-gradient\-descent predictor sequence coincide with those under the full standard Gaussian law, for the same initialization and step sizes\. Both original\-law and Gaussian\-evaluation losses coincide as well\. Requiring only the one\- and two\-coordinate marginals to be Gaussian does not have this implication\.

###### Proof\.

For every diagonalDD,

H​\(D\)i​j=12​∑kDk​k​𝔼​\[\(xi​xj−δi​j\)​\(xk2−1\)\]\.H\(D\)\_\{ij\}=\\frac\{1\}\{2\}\\sum\_\{k\}D\_\{kk\}\\,\\mathbb\{E\}\[\(x\_\{i\}x\_\{j\}\-\\delta\_\{ij\}\)\(x\_\{k\}^\{2\}\-1\)\]\.Each expectation depends on at most three coordinates, so it equals its Gaussian value\. ThusH⁡\(D\)=DH\(D\)=D\. The diagonal predictor subspace is invariant under both the flow and the exact predictor update in Section[2](https://arxiv.org/html/2610.20678#S2); on this subspace their equations are the Gaussian equations\. Uniqueness of the flow and induction over updates prove equality\. For diagonal errorEE, the loss is⟨E,H⁡\(E\)⟩/2=‖E‖F2/2\\langle E,H\(E\)\\rangle/2=\\\|E\\\|\_\{F\}^\{2\}/2\. For the final assertion, Proposition[2\.1](https://arxiv.org/html/2610.20678#S2.Thmtheorem1)gives exactly Gaussian two\-coordinate marginals, whereas the nonzero cycle certificate below gives different learning curves for admissible diagonal teachers and a common diagonal initialization\. ∎

This boundary concerns the specified diagonal training protocol and coordinate basis\. It does not state that three\-coordinate marginals determine the complete moment operator or the dynamics from arbitrary nondiagonal states\.

The diagonal auxiliary/interaction block space is invariant\. WritingMs=diag⁡\(as\)⊕NsM\_\{s\}=\\operatorname\{diag\}\(a\_\{s\}\)\\oplus N\_\{s\},Ss=ℓ⊤​asS\_\{s\}=\\ell^\{\\top\}a\_\{s\},Qs=tr⁡\(Bs​Ns\)Q\_\{s\}=\\operatorname\{tr\}\(B\_\{s\}N\_\{s\}\)gives

a˙s,i\\displaystyle\\dot\{a\}\_\{s,i\}=−4​as,i​\(as,i−Ti\)−2​ε​ℓi​as,i​Qs,\\displaystyle=\-4a\_\{s,i\}\(a\_\{s,i\}\-T\_\{i\}\)\-2\\varepsilon\\ell\_\{i\}a\_\{s,i\}Q\_\{s\},N˙s\\displaystyle\\dot\{N\}\_\{s\}=−4​Ns2\+2​\{TC,Ns\}−ε​Ss​\{Bs,Ns\}\.\\displaystyle=\-4N\_\{s\}^\{2\}\+2\\\{T\_\{C\},N\_\{s\}\\\}\-\\varepsilon S\_\{s\}\\\{B\_\{s\},N\_\{s\}\\\}\.\(3\)The restriction ofKs=\(Hs−Id\)/εK\_\{s\}=\(H\_\{s\}\-\\operatorname\{Id\}\)/\\varepsilonis

Ks​\(diag⁡\(eA\)⊕EC\)=12​diag⁡\(ℓ​tr⁡\(Bs​EC\)\)⊕12​\(ℓ⊤​eA\)​Bs\.K\_\{s\}\(\\operatorname\{diag\}\(e\_\{A\}\)\\oplus E\_\{C\}\)=\\tfrac\{1\}\{2\}\\operatorname\{diag\}\(\\ell\\,\\operatorname\{tr\}\(B\_\{s\}E\_\{C\}\)\)\\oplus\\tfrac\{1\}\{2\}\(\\ell^\{\\top\}e\_\{A\}\)B\_\{s\}\.\(4\)It givesHs​\(I\)=IH\_\{s\}\(I\)=I,Hs​\(T\)=TH\_\{s\}\(T\)=TandHs​\(G0\)=G0H\_\{s\}\(G\_\{0\}\)=G\_\{0\}\. We distinguish original\-law training loss and evaluation of both predictors under one common standard Gaussian law:

LsG\\displaystyle L\_\{s\}^\{G\}=12​∑i∈A\(as,i−Ti\)2\+12​‖Ns−TC‖F2,\\displaystyle=\\tfrac\{1\}\{2\}\\sum\_\{i\\in A\}\(a\_\{s,i\}\-T\_\{i\}\)^\{2\}\+\\tfrac\{1\}\{2\}\\\|N\_\{s\}\-T\_\{C\}\\\|\_\{F\}^\{2\},LsP\\displaystyle L\_\{s\}^\{P\}=LsG\+ε2​Ss​Qs\.\\displaystyle=L\_\{s\}^\{G\}\+\\tfrac\{\\varepsilon\}\{2\}S\_\{s\}Q\_\{s\}\.\(5\)Their initial values agree between arms\.

## 3Spectral equivalence with the entire teacher algebra fixed

We use the complete moment operators from the two\-cycle construction,Hs=Id\+ε​KsH\_\{s\}=\\operatorname\{Id\}\+\\varepsilon K\_\{s\},s∈\{\+,−\}s\\in\\\{\+,\-\\\}\. Write a symmetric matrix as

E=\(EAXX⊤Y\),a=diag⁡\(EA\)\.E=\\begin\{pmatrix\}E\_\{A\}&X\\\\ X^\{\\top\}&Y\\end\{pmatrix\},\\qquad a=\\operatorname\{diag\}\(E\_\{A\}\)\.The complete perturbation is

\(Ks​E\)A,i​i=12​ℓi​tr⁡\(Bs​Y\),\(Ks​E\)C=12​\(ℓ⊤​a\)​Bs,\(Ks​E\)A,C=diag⁡\(ℓ\)​X​Bs,\(K\_\{s\}E\)\_\{A,ii\}=\\frac\{1\}\{2\}\\ell\_\{i\}\\operatorname\{tr\}\(B\_\{s\}Y\),\\quad\(K\_\{s\}E\)\_\{C\}=\\frac\{1\}\{2\}\(\\ell^\{\\top\}a\)B\_\{s\},\\quad\(K\_\{s\}E\)\_\{A,C\}=\\operatorname\{diag\}\(\\ell\)XB\_\{s\},\(6\)with zero remaining auxiliary components\. The two weightedrr\-cycles share their hub and have opposite sign products\. A permutationPPexchanges the branches, soB−=P​B\+​P⊤B\_\{\-\}=PB\_\{\+\}P^\{\\top\}\. All teachers in this section are diagonal in these fixed coordinates\.

###### Theorem 3\.1\(Complete equivalence of diagonal\-anchored spectral data\)\.

There is a Frobenius\-orthogonal involutionUUonSym⁡\(d\)\\operatorname\{Sym\}\(d\)such that

UI=I,UE=Efor every diagonalE,H−=UH\+U∗\.UI=I,\\qquad UE=E\\quad\\text\{for every diagonal \}E,\\qquad H\_\{\-\}=UH\_\{\+\}U^\{\*\}\.Consequently, for any diagonalE,FE,Fand any real functionsf,gf,gdefined on the finite spectrum,

⟨f⁡\(H\+\)​E,g⁡\(H\+\)​F⟩=⟨f⁡\(H−\)​E,g⁡\(H−\)​F⟩\.\\langle f\(H\_\{\+\}\)E,g\(H\_\{\+\}\)F\\rangle=\\langle f\(H\_\{\-\}\)E,g\(H\_\{\-\}\)F\\rangle\.\(7\)This includes all orders of the operator Krylov response Grams, the matrix\-valued spectral measure anchored on the complete diagonal algebra, and its resolvents\. It includes the teacher and initial\-error spectral weights of the population initial tangent kernel, whose nonzero feature\-space operator is4​c​Hs4cH\_\{s\}atW0=c​IW\_\{0\}=\\sqrt\{c\}I\.

###### Proof\.

For supported interaction edges putσi​j=\(B−\)i​j/\(B\+\)i​j∈\{−1,1\}\\sigma\_\{ij\}=\(B\_\{\-\}\)\_\{ij\}/\(B\_\{\+\}\)\_\{ij\}\\in\\\{\-1,1\\\}, and putσi​j=1\\sigma\_\{ij\}=1on the diagonal and unsupported entries\. LetV⁡\(Y\)=\(σi​j​Yi​j\)V\(Y\)=\(\\sigma\_\{ij\}Y\_\{ij\}\)\. Define

U​\(EAXX⊤Y\)=\(EAX​P⊤P​X⊤V⁡\(Y\)\)\.U\\begin\{pmatrix\}E\_\{A\}&X\\\\ X^\{\\top\}&Y\\end\{pmatrix\}=\\begin\{pmatrix\}E\_\{A\}&XP^\{\\top\}\\\\ PX^\{\\top\}&V\(Y\)\\end\{pmatrix\}\.Both components are orthogonal involutions,V​B\+=B−VB\_\{\+\}=B\_\{\-\}, andUUfixes every diagonal matrix\. The first two components of \([6](https://arxiv.org/html/2610.20678#S3.E6)\) intertwine because⟨B−,V​Y⟩=⟨B\+,Y⟩\\langle B\_\{\-\},VY\\rangle=\\langle B\_\{\+\},Y\\rangle\. The cross component intertwines becauseX​P⊤​B−=X​B\+​P⊤XP^\{\\top\}B\_\{\-\}=XB\_\{\+\}P^\{\\top\}\. ThusU​K\+=K−​UUK\_\{\+\}=K\_\{\-\}U\. Functional calculus and orthogonality give \([7](https://arxiv.org/html/2610.20678#S3.E7)\)\. Finally,∇WfW​\(x\)=2​A​\(x\)​W\\nabla\_\{W\}f\_\{W\}\(x\)=2A\(x\)Wgives the initial kernel4​c​⟨A⁡\(x\),A⁡\(x′\)⟩4c\\langle A\(x\),A\(x^\{\\prime\}\)\\rangle; its nonzero eigenvalues are those of4​c​Hs4cH\_\{s\}\. The teacher and initial error are diagonal anchors\. ∎

This theorem concerns population spectral information\. It does not say that the two kernels agree pointwise on raw inputs, that empirical Gram matrices coincide, or that a coordinate\-resolved queryH⁡\(E\)H\(E\)is hidden\. In fact a suitable diagonal query returns the different matricesBsB\_\{s\}\. Nor does it identify the full initial parameter Hessian spectra\.

###### Proposition 3\.2\(The multiplication condition for universal transport\)\.

LetU:Sym⁡\(d\)→Sym⁡\(d\)U:\\operatorname\{Sym\}\(d\)\\to\\operatorname\{Sym\}\(d\)be a unital Frobenius isometry, and writeE∘F=\(E​F\+F​E\)/2E\\circ F=\(EF\+FE\)/2\. The following conditions are equivalent:

1. 1\.U⁡\(E∘F\)=U​E∘U​FU\(E\\circ F\)=UE\\circ UFfor allE,FE,F\.
2. 2\.UUtransports the factorized vector fieldFH,b\(M\)=−4M∘\(HM−b\)F\_\{H,b\}\(M\)=\-4M\\circ\(HM\-b\)toFU​H​U∗,U​b​\(U​M\)F\_\{UHU^\{\*\},Ub\}\(UM\)for every self\-adjointHH, everybb, and everyMM\.
3. 3\.Such transport holds forH=IdH=\\operatorname\{Id\},b=τ​Ib=\\tau I, and everyMMin an open neighborhood of a positive scalar matrix\.

Under these conditions it also transports the exact predictor update\(I−2​h​G\)​M​\(I−2​h​G\)\(I\-2hG\)M\(I\-2hG\)of weight gradient descent\.

###### Proof\.

The first condition directly implies the second; the second implies the third\. Under the third, cancellation of the linear term inFId,τ​I​\(M\)=−4​M2\+4​τ​MF\_\{\\operatorname\{Id\},\\tau I\}\(M\)=\-4M^\{2\}\+4\\tau MgivesU⁡\(M2\)=\(U​M\)2U\(M^\{2\}\)=\(UM\)^\{2\}on an open set\. This polynomial identity holds everywhere\. Polarization proves the first condition\. The exact update follows fromG​M​G=2​G∘\(G∘M\)−G2∘MGMG=2G\\circ\(G\\circ M\)\-G^\{2\}\\circ M\. ∎

This proposition applies a classical Jordan\-preservation condition to the learning vector field; it is not a new classification of Jordan automorphisms or Bures isometries\. On real symmetric matrices these automorphisms are orthogonal conjugations\. In the diagonal\-fixed case one can see the obstruction directly: an automorphism fixing all diagonal projectors must sendEi​j\+Ej​iE\_\{ij\}\+E\_\{ji\}toϵi​ϵj​\(Ei​j\+Ej​i\)\\epsilon\_\{i\}\\epsilon\_\{j\}\(E\_\{ij\}\+E\_\{ji\}\), and therefore preserve every named cycle product\. TheUUin Theorem[3\.1](https://arxiv.org/html/2610.20678#S3.Thmtheorem1)changes these products and is not such an automorphism\.

For thisUU, on the invariant training spaceℬ=diag⁡\(Sym⁡\(q\)\)⊕Sym⁡\(2​r−1\)\\mathcal\{B\}=\\operatorname\{diag\}\(\\operatorname\{Sym\}\(q\)\)\\oplus\\operatorname\{Sym\}\(2r\-1\), diagonal Jordan multiplication commutes withUU\. Thus the affine linearizations about any common diagonal reference path are intertwined\. Quadratic multiplication of two nondiagonal state components is the additional operation they omit\. This assertion is restricted toℬ\\mathcal\{B\}\. On the auxiliary–interaction directions the complete Jacobian contains the different pencilsDC\+2​c​ε​ℓi​BsD\_\{C\}\+2c\\varepsilon\\ell\_\{i\}B\_\{s\}; their spectral moments can distinguish the arms\.

## 4Cycle order throughout training and under interventions

Fix a teacher, initialization and graph size\. LetLsG​\(t,ε\)=12​‖Ms​\(t,ε\)−T‖F2L\_\{s\}^\{G\}\(t,\\varepsilon\)=\\frac\{1\}\{2\}\\\|M\_\{s\}\(t,\\varepsilon\)\-T\\\|\_\{F\}^\{2\}andLsP​\(t,ε\)=12​⟨Ms−T,Hs​\(Ms−T\)⟩L\_\{s\}^\{P\}\(t,\\varepsilon\)=\\frac\{1\}\{2\}\\langle M\_\{s\}\-T,H\_\{s\}\(M\_\{s\}\-T\)\\rangle\. The fixed teacher must satisfyℓ⊤​TA=0\\ell^\{\\top\}T\_\{A\}=0\.

###### Theorem 4\.1\(Signed\-cycle order of the entire weak\-coupling response\)\.

For every finiteT0\>0T\_\{0\}\>0there isε0\>0\\varepsilon\_\{0\}\>0for which, uniformly on0≤t≤T00\\leq t\\leq T\_\{0\},

L\+ζ​\(t,ε\)−L−ζ​\(t,ε\)=εr​Φrζ​\(t\)\+OT0​\(εr\+2\),ζ∈\{G,P\}\.L\_\{\+\}^\{\\zeta\}\(t,\\varepsilon\)\-L\_\{\-\}^\{\\zeta\}\(t,\\varepsilon\)=\\varepsilon^\{r\}\\Phi\_\{r\}^\{\\zeta\}\(t\)\+O\_\{T\_\{0\}\}\(\\varepsilon^\{r\+2\}\),\\qquad\\zeta\\in\\\{G,P\\\}\.\(8\)Every lowerε\\varepsiloncoefficient vanishes identically in time\. Only powers congruent torrmodulo two can occur in the difference\. For the normalized auxiliary barycentric construction withJr=tr⁡\[DC​\(B\+r−B−r\)\]≠0J\_\{r\}=\\operatorname\{tr\}\[D\_\{C\}\(B\_\{\+\}^\{r\}\-B\_\{\-\}^\{r\}\)\]\\neq 0,

Φrζ​\(t\)=C¯q,rζ​tr​q\+O⁡\(tr​q\+1\),C¯q,rζ=Cq,rζ/εr≠0,\\Phi\_\{r\}^\{\\zeta\}\(t\)=\\overline\{C\}\_\{q,r\}^\{\\zeta\}t^\{rq\}\+O\(t^\{rq\+1\}\),\\qquad\\overline\{C\}\_\{q,r\}^\{\\zeta\}=C\_\{q,r\}^\{\\zeta\}/\\varepsilon^\{r\}\\neq 0,whereCq,rζC\_\{q,r\}^\{\\zeta\}is the exact initial\-time coefficient\. In particular, on a sufficiently short compact interval\[t0,t1\]⊂\(0,∞\)\[t\_\{0\},t\_\{1\}\]\\subset\(0,\\infty\), all sufficiently small positive couplings give a loss difference with a fixed nonzero sign\.

###### Proof\.

At zero coupling the positive diagonal logistic solution exists on every finite interval\. Analytic dependence of a polynomial ODE on parameters, continued over a finite cover of that interval, gives a common complex parameter neighborhood and uniformly convergent coupling expansions\. This also gives a uniform remainder bound\.

Assign an independent variable to each signed edge\. Each coefficient of degreekkin the coupling is a homogeneous polynomial of total degreekkin these edge variables\. Changing signs at interaction vertices conjugates the state by the associated diagonal sign matrix and leaves both scalar losses invariant\. Consequently every nonzero edge monomial has even total incident degree at each vertex\. Its odd\-multiplicity edges form an even subgraph\. On the two\-cycle graph the possibilities are the empty set, either individual cycle, or their union\.

The empty set and the union give the same sign for cycle products\(\+,−\)\(\+,\-\)and\(−,\+\)\(\-,\+\)\. A differing monomial must therefore have exactly one whole cycle as its odd edge set\. Its degree isr\+2​mr\+2m,m≥0m\\geq 0\. This proves the vanishing and parity assertions, including ther\+2r\+2remainder\. The initial\-time coefficient identifies the first nonzero Taylor coefficient ofΦrζ\\Phi\_\{r\}^\{\\zeta\}\. Analyticity and compactness then give the final fixed\-sign assertion\. ∎

The fixed teacher is essential to the uniformity in Theorem[4\.1](https://arxiv.org/html/2610.20678#S4.Thmtheorem1)\. The next result determines an explicit family in which this uniformity fails\.

###### Corollary 4\.2\(Exact divisibility for a finite number of weight updates\)\.

For every fixed finite number of actual weight\-gradient steps and every fixed step size, the between\-arm difference of either loss is a polynomial of the form

L\+,Kζ​\(ε\)−L−,Kζ​\(ε\)=εr​PK,hζ​\(ε2\)\.L\_\{\+,K\}^\{\\zeta\}\(\\varepsilon\)\-L\_\{\-,K\}^\{\\zeta\}\(\\varepsilon\)=\\varepsilon^\{r\}P\_\{K,h\}^\{\\zeta\}\(\\varepsilon^\{2\}\)\.The polynomial is permitted to be zero, for example before the hidden interaction has been reached\.

###### Proof\.

Each updateW↦W−2​h​Hs​\(W​W⊤−T\)​WW\\mapsto W\-2hH\_\{s\}\(WW^\{\\top\}\-T\)Wis polynomial in the weights and edge variables\. A finite number of updates and either quadratic predictor loss are therefore polynomials\. Diagonal sign conjugation of the interaction coordinates preserves the initialization, teacher, update and scalar loss\. The even\-incidence and two\-cycle parity argument in the preceding proof applies to every monomial of this polynomial, and gives the stated divisibility without an asymptotic remainder\. ∎

### 4\.1An infinite\-horizon bound and the value of an intervention

For a fixed bounded teacher, late observations do not evade the cycle\-order obstruction\. The next theorem strengthens the compact\-time statement to the entire training trajectory\.

###### Theorem 4\.3\(Uniform cycle order throughout training\)\.

Fix the construction,T≻0T\\succ 0,c\>0c\>0and a positive\-law realization\. There areε0,C,β\>0\\varepsilon\_\{0\},C,\\beta\>0such that, for both losses and all\|ε\|≤ε0\|\\varepsilon\|\\leq\\varepsilon\_\{0\},t≥0t\\geq 0,

L\+ζ​\(t,ε\)−L−ζ​\(t,ε\)=εr​Φrζ​\(t\)\+εr\+2​Rζ​\(t,ε\),\|Φrζ​\(t\)\|\+\|Rζ​\(t,ε\)\|≤C​e−β​t\.L\_\{\+\}^\{\\zeta\}\(t,\\varepsilon\)\-L\_\{\-\}^\{\\zeta\}\(t,\\varepsilon\)=\\varepsilon^\{r\}\\Phi\_\{r\}^\{\\zeta\}\(t\)\+\\varepsilon^\{r\+2\}R\_\{\\zeta\}\(t,\\varepsilon\),\\qquad\|\\Phi\_\{r\}^\{\\zeta\}\(t\)\|\+\|R\_\{\\zeta\}\(t,\\varepsilon\)\|\\leq Ce^\{\-\\beta t\}\.The constants depend on the fixed dimension, graph, teacher and initialization; they are not uniform asrrgrows\. If the initial\-time certificate hasJr≠0J\_\{r\}\\neq 0, there are a fixed timet†\>0t\_\{\\dagger\}\>0and constantsa,A\>0a,A\>0such that, for sufficiently small positiveε\\varepsilon,

a​εr≤\|L\+ζ​\(t†,ε\)−L−ζ​\(t†,ε\)\|≤supt≥0\|L\+ζ−L−ζ\|≤A​εr\.a\\varepsilon^\{r\}\\leq\|L\_\{\+\}^\{\\zeta\}\(t\_\{\\dagger\},\\varepsilon\)\-L\_\{\-\}^\{\\zeta\}\(t\_\{\\dagger\},\\varepsilon\)\|\\leq\\sup\_\{t\\geq 0\}\|L\_\{\+\}^\{\\zeta\}\-L\_\{\-\}^\{\\zeta\}\|\\leq A\\varepsilon^\{r\}\.

###### Proof\.

Puta−=min⁡TAa\_\{\-\}=\\min T\_\{A\},a\+=max⁡TAa\_\{\+\}=\\max T\_\{A\},t−=λmin​\(TC\)t\_\{\-\}=\\lambda\_\{\\min\}\(T\_\{C\}\),t\+=λmax​\(TC\)t\_\{\+\}=\\lambda\_\{\\max\}\(T\_\{C\}\),A∗=2​max⁡\(c,a\+\)A\_\{\*\}=2\\max\(c,a\_\{\+\}\)andN∗=2​max⁡\(c,t\+\)N\_\{\*\}=2\\max\(c,t\_\{\+\}\)\. Letb∗=maxs⁡‖Bs‖∗b\_\{\*\}=\\max\_\{s\}\\\|B\_\{s\}\\\|\_\{\*\}andbo=maxs⁡‖Bs‖opb\_\{o\}=\\max\_\{s\}\\\|B\_\{s\}\\\|\_\{\\rm op\}\. The norm of the complete perturbation is

k=maxs⁡\{12​‖ℓ‖2​‖Bs‖F,‖ℓ‖∞​‖Bs‖op\}\.k=\\max\_\{s\}\\left\\\{\\tfrac\{1\}\{2\}\\\|\\ell\\\|\_\{2\}\\\|B\_\{s\}\\\|\_\{F\},\\,\\\|\\ell\\\|\_\{\\infty\}\\\|B\_\{s\}\\\|\_\{\\rm op\}\\right\\\}\.Chooseε¯\>0\\bar\{\\varepsilon\}\>0strictly below the law\-positivity radius and

min⁡\{k−1,a−‖ℓ‖∞​b∗​N∗,t−‖ℓ‖1​A∗​bo\}\.\\min\\left\\\{k^\{\-1\},\\ \\frac\{a\_\{\-\}\}\{\\\|\\ell\\\|\_\{\\infty\}b\_\{\*\}N\_\{\*\}\},\\ \\frac\{t\_\{\-\}\}\{\\\|\\ell\\\|\_\{1\}A\_\{\*\}b\_\{o\}\}\\right\\\}\.A zero denominator is interpreted as\+∞\+\\infty\. The effective teachers in \([3](https://arxiv.org/html/2610.20678#S2.E3)\) areTA,i−\(ε/2\)​ℓi​QsT\_\{A,i\}\-\(\\varepsilon/2\)\\ell\_\{i\}Q\_\{s\}andTC−\(ε/2\)​Ss​BsT\_\{C\}\-\(\\varepsilon/2\)S\_\{s\}B\_\{s\}\. On the boxai≤A∗a\_\{i\}\\leq A\_\{\*\},Ns⪯N∗​IN\_\{s\}\\preceq N\_\{\*\}I, their deviations from the original teachers are bounded bya−/2a\_\{\-\}/2andt−/2t\_\{\-\}/2, respectively\. The scalar logistic equation points inward atmin⁡\(c,a−/2\)\\min\(c,a\_\{\-\}/2\)andA∗A\_\{\*\}\. Ifvvis an eigenvector ofNNwith eigenvalueλ\\lambda, then

v⊤​N˙​v=4​λ​\{v⊤​\[TC−\(ε/2\)​S​Bs\]​v−λ\}\.v^\{\\top\}\\dot\{N\}v=4\\lambda\\\{v^\{\\top\}\[T\_\{C\}\-\(\\varepsilon/2\)SB\_\{s\}\]v\-\\lambda\\\}\.Extremal\-eigenvalue Dini derivatives therefore point inward atmin⁡\(c,t−/2\)​I\\min\(c,t\_\{\-\}/2\)IandN∗​IN\_\{\*\}I\. A first\-exit argument proves invariance of this compact positive box for allt≥0t\\geq 0and\|ε\|≤ε¯\|\\varepsilon\|\\leq\\bar\{\\varepsilon\}\. In particularMs⪰m​IM\_\{s\}\\succeq mIfor a fixedm\>0m\>0\.

WriteEs=Ms−TE\_\{s\}=M\_\{s\}\-T,μ=1−ε¯​k\\mu=1\-\\bar\{\\varepsilon\}k,Λ=1\+ε¯​k\\Lambda=1\+\\bar\{\\varepsilon\}kandVs=⟨Es,Hs​Es⟩/2V\_\{s\}=\\langle E\_\{s\},H\_\{s\}E\_\{s\}\\rangle/2\. WithGs=Hs​EsG\_\{s\}=H\_\{s\}E\_\{s\},

V˙s=−4​tr⁡\(Ms​Gs2\)≤−ω​Vs,ω=8​m​μ2Λ\.\\dot\{V\}\_\{s\}=\-4\\operatorname\{tr\}\(M\_\{s\}G\_\{s\}^\{2\}\)\\leq\-\\omega V\_\{s\},\\qquad\\omega=\\frac\{8m\\mu^\{2\}\}\{\\Lambda\}\.Consequently∥Es\(t\)∥F≤\(Λ/μ\)1/2∥cI−T∥Fe−ωt/2\\\|E\_\{s\}\(t\)\\\|\_\{F\}\\leq\(\\Lambda/\\mu\)^\{1/2\}\\\|cI\-T\\\|\_\{F\}e^\{\-\\omega t/2\}, uniformly in the coupling\.

It remains to justify a uniform Taylor remainder on an unbounded time interval\. The state variational equation has

Js​\(t\)​Z=−2​\{Hs​Z,Ms\}−2​\{Gs,Z\}\.J\_\{s\}\(t\)Z=\-2\\\{H\_\{s\}Z,M\_\{s\}\\\}\-2\\\{G\_\{s\},Z\\\}\.ForWZ=⟨Z,Hs​Z⟩/2W\_\{Z\}=\\langle Z,H\_\{s\}Z\\rangle/2,

W˙Z≤\[−ω\+8​Λμ​‖Gs​\(t\)‖F\]​WZ\.\\dot\{W\}\_\{Z\}\\leq\\left\[\-\\omega\+\\frac\{8\\Lambda\}\{\\mu\}\\\|G\_\{s\}\(t\)\\\|\_\{F\}\\right\]W\_\{Z\}\.The second coefficient is integrable uniformly inε\\varepsilon\. Thus the variational propagator satisfies∥Us\(t,u\)∥≤CUe−ω\(t−u\)/2\\\|U\_\{s\}\(t,u\)\\\|\\leq C\_\{U\}e^\{\-\\omega\(t\-u\)/2\}for allt≥u≥0t\\geq u\\geq 0\.

The parameter\-derivative induction can be written explicitly\. Suppress the arm subscript and define

𝒜H​Y=−2​\{H​Y,T\},ℬH​\(Y,Z\)=−2​\{H​Y,Z\}\.\\mathcal\{A\}\_\{H\}Y=\-2\\\{HY,T\\\},\\qquad\\mathcal\{B\}\_\{H\}\(Y,Z\)=\-2\\\{HY,Z\\\}\.ThenE˙=𝒜H​E\+ℬH​\(E,E\)\\dot\{E\}=\\mathcal\{A\}\_\{H\}E\+\\mathcal\{B\}\_\{H\}\(E,E\)and∂εH=K\\partial\_\{\\varepsilon\}H=K\. PutX0=EX\_\{0\}=EandXj=∂εjEX\_\{j\}=\\partial\_\{\\varepsilon\}^\{j\}Eforj≥1j\\geq 1\. Leibniz’s rule givesX˙j=Js​Xj\+Rj\\dot\{X\}\_\{j\}=J\_\{s\}X\_\{j\}\+R\_\{j\},Xj​\(0\)=0X\_\{j\}\(0\)=0, with

Rj=\\displaystyle R\_\{j\}=\{\}∑a=1j−1\(ja\)​ℬH​\(Xa,Xj−a\)\+j​𝒜K​Xj−1\\displaystyle\\sum\_\{a=1\}^\{j\-1\}\\binom\{j\}\{a\}\\mathcal\{B\}\_\{H\}\(X\_\{a\},X\_\{j\-a\}\)\+j\\mathcal\{A\}\_\{K\}X\_\{j\-1\}\+j∑a=0j−1\(j−1a\)ℬK\(Xa,Xj−1−a\)\.\\displaystyle\+j\\sum\_\{a=0\}^\{j\-1\}\\binom\{j\-1\}\{a\}\\mathcal\{B\}\_\{K\}\(X\_\{a\},X\_\{j\-1\-a\}\)\.All factors on the right areEEor lower\-order derivatives\. The linear and bilinear operator norms are uniformly bounded on the fixed coupling interval\. Forj=1j=1, the exponential bound onEEgives∥R1\(t\)∥≤C1e−ωt/2\\\|R\_\{1\}\(t\)\\\|\\leq C\_\{1\}e^\{\-\\omega t/2\}\. If the lower\-order derivatives decay ase−β​te^\{\-\\beta t\}, whereβ=ω/4\\beta=\\omega/4, then‖Rj​\(t\)‖≤Cj​e−β​t\\\|R\_\{j\}\(t\)\\\|\\leq C\_\{j\}e^\{\-\\beta t\}for2≤j≤r\+22\\leq j\\leq r\+2\. Using the faster propagator above,

∥Xj\(t\)∥≤CU∫0te−ω\(t−u\)/2∥Rj\(u\)∥du\.\\\|X\_\{j\}\(t\)\\\|\\leq C\_\{U\}\\int\_\{0\}^\{t\}e^\{\-\\omega\(t\-u\)/2\}\\\|R\_\{j\}\(u\)\\\|\\,du\.Forj=1j=1this is bounded byCte−ωt/2Cte^\{\-\\omega t/2\}; forj≥2j\\geq 2it is bounded byCj​e−β​t/\(ω/2−β\)C\_\{j\}e^\{\-\\beta t\}/\(\\omega/2\-\\beta\)\. Induction therefore proves

sup\|ε\|≤ε¯/2,t≥0eβ​t​‖∂εjEs​\(t,ε\)‖F<∞\.\\sup\_\{\|\\varepsilon\|\\leq\\bar\{\\varepsilon\}/2,\\ t\\geq 0\}e^\{\\beta t\}\\\|\\partial\_\{\\varepsilon\}^\{j\}E\_\{s\}\(t,\\varepsilon\)\\\|\_\{F\}<\\infty\.The constants are uniform in time and coupling for every0≤j≤r\+20\\leq j\\leq r\+2at the fixedrr\. Both losses are quadratic inEE, andHHis affine in the coupling\. A further application of Leibniz’s rule gives

sup\|ε\|≤ε¯/2,t≥0eβ​t​\|∂εjLsζ​\(t,ε\)\|<∞,0≤j≤r\+2,ζ∈\{G,P\}\.\\sup\_\{\|\\varepsilon\|\\leq\\bar\{\\varepsilon\}/2,\\ t\\geq 0\}e^\{\\beta t\}\|\\partial\_\{\\varepsilon\}^\{j\}L\_\{s\}^\{\\zeta\}\(t,\\varepsilon\)\|<\\infty,\\qquad 0\\leq j\\leq r\+2,\\quad\\zeta\\in\\\{G,P\\\}\.
The signed\-cycle proof of Theorem[4\.1](https://arxiv.org/html/2610.20678#S4.Thmtheorem1)makes all between\-arm derivatives below orderrr, and the derivative of orderr\+1r\+1, vanish at zero coupling for eachtt\. Taylor’s formula with the preceding uniform derivative bound proves the claimed expansion and exponential envelope\. FinallyΦrζ​\(t\)=C¯q,rζ​tr​q\+O⁡\(tr​q\+1\)\\Phi\_\{r\}^\{\\zeta\}\(t\)=\\overline\{C\}\_\{q,r\}^\{\\zeta\}t^\{rq\}\+O\(t^\{rq\+1\}\)withC¯q,rζ≠0\\overline\{C\}\_\{q,r\}^\{\\zeta\}\\neq 0gives a nonzero coefficient at one sufficiently small fixedt†t\_\{\\dagger\}\. Shrinkingε0\\varepsilon\_\{0\}makes the remainder less than half that signal\. This proves the two\-sided order\. ∎

###### Theorem 4\.4\(A scalar passive–intervention separation\)\.

Fix a supplied pair satisfying the nonzero\-certificate condition in Theorem[4\.3](https://arxiv.org/html/2610.20678#S4.Thmtheorem3)\. The unknown hypothesis is which of its two input laws is used\. The teacher and alternatives are known\. In both experiments the query complexity is the minimal fixed deterministic budgetnn, with adaptive query choices within that budget\.

In the passive experiment, each query returnsLsζ​\(tj,ε\)\+σL​ZjL\_\{s\}^\{\\zeta\}\(t\_\{j\},\\varepsilon\)\+\\sigma\_\{L\}Z\_\{j\}, with independent standard GaussianZjZ\_\{j\}, fixedσL\>0\\sigma\_\{L\}\>0, and an adaptively chosentj≥0t\_\{j\}\\geq 0on the original trajectory fromW0=c​IW\_\{0\}=\\sqrt\{c\}I\. For error probability at mostη\\etaunder each alternative, where0<η≤1/80<\\eta\\leq 1/8, the optimal number of scalar queries has order

Θ⁡\(σL2​ε−2​r​log⁡\(1/η\)\)\(ε↓0\)\.\\Theta\\\!\\left\(\\sigma\_\{L\}^\{2\}\\varepsilon^\{\-2r\}\\log\(1/\\eta\)\\right\)\\qquad\(\\varepsilon\\downarrow 0\)\.\(9\)
In the intervention experiment, choose a fixed0<ρ<c0<\\rho<c\. A query may initialize a fresh factor with

M⁡\(v\)=diag⁡\(c​𝟏\+v\)⊕c​I,‖v‖2≤ρ,W⁡\(v\)=M​\(v\)1/2,M\(v\)=\\operatorname\{diag\}\(c\\mathbf\{1\}\+v\)\\oplus cI,\\qquad\\\|v\\\|\_\{2\}\\leq\\rho,\\qquad W\(v\)=M\(v\)^\{1/2\},and return the single noisy scalar

Y=⟨V,Gs​\(M⁡\(v\)\)C⟩\+σG​Z,Y=\\langle V,G\_\{s\}\(M\(v\)\)\_\{C\}\\rangle\+\\sigma\_\{G\}Z,whereV∈Sym⁡\(2​r−1\)V\\in\\operatorname\{Sym\}\(2r\-1\),‖V‖F≤1\\\|V\\\|\_\{F\}\\leq 1andσG\>0\\sigma\_\{G\}\>0\. This is a normalized directional measurement of the initial weight gradient:

Gs​\(M⁡\(v\)\)C=\(∇WLsP​\(W⁡\(v\)\)\)C​C/\(2​c\)\.G\_\{s\}\(M\(v\)\)\_\{C\}=\(\\nabla\_\{W\}L\_\{s\}^\{P\}\(W\(v\)\)\)\_\{CC\}/\(2\\sqrt\{c\}\)\.The choices ofv,Vv,Vmay be adaptive\. In this bounded intervention class the optimal number of scalar queries instead has order

Θ⁡\(σG2​ε−2​log⁡\(1/η\)\)\.\\Theta\\\!\\left\(\\sigma\_\{G\}^\{2\}\\varepsilon^\{\-2\}\\log\(1/\\eta\)\\right\)\.\(10\)Both exponents concern the same two laws and the same teacher; each observation is scalar\. They do not count input samples, the cost of training, or the cost of producing a population\-gradient measurement\.

###### Proof\.

For a pair of scalar response functions whose maximum difference isDD, the Gaussian chain rule gives, for any adaptive query policy,

DKL\(P\+\(n\)∥P−\(n\)\)≤n​D22​σ2\.D\_\{\\rm KL\}\(P\_\{\+\}^\{\(n\)\}\\\|P\_\{\-\}^\{\(n\)\}\)\\leq\\frac\{nD^\{2\}\}\{2\\sigma^\{2\}\}\.The policy has the same conditional law under each hypothesis, so adaptivity adds no extra divergence\. The two\-point inequalityP\+\(error\)\+P−\(error\)≥12e−DKL\(P\+∥P−\)P\_\{\+\}\(\\mathrm\{error\}\)\+P\_\{\-\}\(\\mathrm\{error\}\)\\geq\\tfrac\{1\}\{2\}e^\{\-D\_\{\\rm KL\}\(P\_\{\+\}\\\|P\_\{\-\}\)\}\([Tsybakov, 2009](https://arxiv.org/html/2610.20678#bib.bib18)\)impliesn≥2​σ2​D−2​log⁡\(1/\(4​η\)\)n\\geq 2\\sigma^\{2\}D^\{\-2\}\\log\(1/\(4\\eta\)\)\. At a fixed query with signald\>0d\>0, repeated measurements followed by a midpoint test have each error at mostexp\[−nd2/\(8σ2\)\]\\exp\[\-nd^\{2\}/\(8\\sigma^\{2\}\)\]\. Forη≤1/8\\eta\\leq 1/8the two logarithmic factors are comparable\. The uniform upper signal and one fixed\-time lower signal in Theorem[4\.3](https://arxiv.org/html/2610.20678#S4.Thmtheorem3)establish \([9](https://arxiv.org/html/2610.20678#S4.E9)\)\.

For the intervention,Hs​\(c​I−T\)=c​I−TH\_\{s\}\(cI\-T\)=cI\-Tand the complete block formula give

G\+​\(M⁡\(v\)\)C−G−​\(M⁡\(v\)\)C=ε2​\(ℓ⊤​v\)​\(B\+−B−\)\.G\_\{\+\}\(M\(v\)\)\_\{C\}\-G\_\{\-\}\(M\(v\)\)\_\{C\}=\\frac\{\\varepsilon\}\{2\}\(\\ell^\{\\top\}v\)\(B\_\{\+\}\-B\_\{\-\}\)\.Thus every allowed scalar query has signal at mostDG=ε​ρ​‖ℓ‖2​‖B\+−B−‖F/2D\_\{G\}=\\varepsilon\\rho\\\|\\ell\\\|\_\{2\}\\\|B\_\{\+\}\-B\_\{\-\}\\\|\_\{F\}/2\. Equality is attained byv=ρ​ℓ/‖ℓ‖2v=\\rho\\ell/\\\|\\ell\\\|\_\{2\}andV=\(B\+−B−\)/‖B\+−B−‖FV=\(B\_\{\+\}\-B\_\{\-\}\)/\\\|B\_\{\+\}\-B\_\{\-\}\\\|\_\{F\}\. The factor is positive becausec\+vi≥c−ρ\>0c\+v\_\{i\}\\geq c\-\\rho\>0\. Applying the same divergence bound and midpoint test proves \([10](https://arxiv.org/html/2610.20678#S4.E10)\)\. ∎

The intervention queries a gradient at a new initialization, while the passive channel observes the original trajectory\. Both experiments test the same supplied pair with the same teacher and scalar readouts\. The initialization change removes the cycle penalty in this testing problem\. Query\-dependent statistical difficulty is well established, including gradient\-query models\([Joshi et al\., 2024](https://arxiv.org/html/2610.20678#bib.bib8)\)\. Here the separation is realized within one positive\-input\-law neural construction with exactly matched diagonal\-anchored spectra\.

###### Corollary 4\.5\(An intervention implemented with finitely many inputs\)\.

Use the smooth laws in Proposition[2\.1](https://arxiv.org/html/2610.20678#S2.Thmtheorem1), with their fixed perturbationsψ±\\psi\_\{\\pm\}, and suppose0<ε​Rψ≤1/20<\\varepsilon R\_\{\\psi\}\\leq 1/2\. For a supplied pair let

BΔ=B\+−B−,v∗=ρ​ℓ/‖ℓ‖2,V∗=BΔ/‖BΔ‖F,D∗=diag⁡\(v∗\)⊕0,0<ρ<c\.B\_\{\\Delta\}=B\_\{\+\}\-B\_\{\-\},\\quad v\_\{\*\}=\\rho\\ell/\\\|\\ell\\\|\_\{2\},\\quad V\_\{\*\}=B\_\{\\Delta\}/\\\|B\_\{\\Delta\}\\\|\_\{F\},\\quad D\_\{\*\}=\\operatorname\{diag\}\(v\_\{\*\}\)\\oplus 0,\\qquad 0<\\rho<c\.On each independent input form the scalar

Z⁡\(x\)=⟨V∗,A​\(x\)C⟩​⟨D∗,A⁡\(x\)⟩\.Z\(x\)=\\langle V\_\{\*\},A\(x\)\_\{C\}\\rangle\\langle D\_\{\*\},A\(x\)\\rangle\.\(11\)This score is also the projection of the difference of the two per\-example weight gradients atW⁡\(v∗\)W\(v\_\{\*\}\)andW⁡\(0\)W\(0\), divided by2​c2\\sqrt\{c\}, when the same input and label are used in both gradients\. The reduced scoreZ⁡\(x\)Z\(x\)can be evaluated from the input alone, without knowing the teacher or a label\. An implementation through two ordinary per\-example gradients uses the same available label in both evaluations; that label cancels from their difference\. Its means under the two alternatives are

𝔼±Z=±dε/2,dε=ε​ρ​‖ℓ‖2​‖BΔ‖F2,Var±Z≤32ρ2\.\\mathbb\{E\}\_\{\\pm\}Z=\\pm d\_\{\\varepsilon\}/2,\\qquad d\_\{\\varepsilon\}=\\frac\{\\varepsilon\\rho\\\|\\ell\\\|\_\{2\}\\\|B\_\{\\Delta\}\\\|\_\{F\}\}\{2\},\\qquad\\operatorname\{Var\}\_\{\\pm\}Z\\leq\\frac\{3\}\{2\}\\rho^\{2\}\.A median\-of\-means estimate followed by a zero\-threshold test therefore has error at mostη\\etaunder each alternative using

n=O⁡\(log⁡\(1/η\)ε2​‖ℓ‖22​‖BΔ‖F2\),0<η≤1/8\.n=O\\\!\\left\(\\frac\{\\log\(1/\\eta\)\}\{\\varepsilon^\{2\}\\\|\\ell\\\|\_\{2\}^\{2\}\\\|B\_\{\\Delta\}\\\|\_\{F\}^\{2\}\}\\right\),\\qquad 0<\\eta\\leq 1/8\.\(12\)For the fixed pair this has the optimalε\\varepsilonandη\\etaorders among all tests based on independent raw inputs\. This is a finite\-input implementation of pair testing, not a procedure for recovering an arbitrary unknown graph\.

###### Proof\.

Write the per\-example loss asℒ⁡\(W,x,y\)=12​\(fW​\(x\)−y\)2\\mathcal\{L\}\(W;x,y\)=\\tfrac\{1\}\{2\}\(f\_\{W\}\(x\)\-y\)^\{2\}\. Since∇Wℒ=2​\(fW−y\)​A​\(x\)​W\\nabla\_\{W\}\\mathcal\{L\}=2\(f\_\{W\}\-y\)A\(x\)Wand both initializations haveWC​C=c​IW\_\{CC\}=\\sqrt\{c\}Iand zero off\-diagonal blocks,

\[∇Wℒ​\(W⁡\(v∗\),x,y\)−∇Wℒ​\(W⁡\(0\),x,y\)\]C​C2​c=A​\(x\)C​⟨D∗,A⁡\(x\)⟩\.\\frac\{\[\\nabla\_\{W\}\\mathcal\{L\}\(W\(v\_\{\*\}\);x,y\)\-\\nabla\_\{W\}\\mathcal\{L\}\(W\(0\);x,y\)\]\_\{CC\}\}\{2\\sqrt\{c\}\}=A\(x\)\_\{C\}\\langle D\_\{\*\},A\(x\)\\rangle\.Projection ontoV∗V\_\{\*\}proves the paired\-gradient identity\. The label cancels before averaging, including any shared label noise\. The moment operator gives

𝔼s​Z=⟨V∗,Hs​\(D∗\)C⟩=ε2​\(ℓ⊤​v∗\)​⟨V∗,Bs⟩\.\\mathbb\{E\}\_\{s\}Z=\\langle V\_\{\*\},H\_\{s\}\(D\_\{\*\}\)\_\{C\}\\rangle=\\frac\{\\varepsilon\}\{2\}\(\\ell^\{\\top\}v\_\{\*\}\)\\langle V\_\{\*\},B\_\{s\}\\rangle\.The twoBsB\_\{s\}have the same Frobenius norm; consequently⟨V∗,B±⟩=±∥BΔ∥F/2\\langle V\_\{\*\},B\_\{\\pm\}\\rangle=\\pm\\\|B\_\{\\Delta\}\\\|\_\{F\}/2\. Under the product Gaussian, the two factors in \([11](https://arxiv.org/html/2610.20678#S4.E11)\) depend on disjoint coordinate blocks and have second moments11andρ2\\rho^\{2\}, respectively\. Thus𝔼ϕd​Z2=ρ2\\mathbb\{E\}\_\{\\phi\_\{d\}\}Z^\{2\}=\\rho^\{2\}\. The density\-ratio upper boundps,ε≤\(3/2\)​ϕdp\_\{s,\\varepsilon\}\\leq\(3/2\)\\phi\_\{d\}implies the stated variance bound\.

For completeness, divide the inputs into an odd numberbbof independent blocks of sizemm\. Ifm≥64​\(3​ρ2/2\)/dε2m\\geq 64\(3\\rho^\{2\}/2\)/d\_\{\\varepsilon\}^\{2\}, Chebyshev’s inequality puts each block mean withindε/4d\_\{\\varepsilon\}/4of its true mean with probability at least3/43/4\. The probability that half the blocks fail is at moste−b/8e^\{\-b/8\}\. Choosing the smallest oddb≥8​log⁡\(1/η\)b\\geq 8\\log\(1/\\eta\)proves \([12](https://arxiv.org/html/2610.20678#S4.E12)\), with integer rounding absorbed in the asymptotic bound\.

For the lower bound putDψ=𝔼ϕd​\(ψ\+−ψ−\)2\>0D\_\{\\psi\}=\\mathbb\{E\}\_\{\\phi\_\{d\}\}\(\\psi\_\{\+\}\-\\psi\_\{\-\}\)^\{2\}\>0\. The two perturbations differ because their fourth moments differ\. The one\-input divergence satisfies

DKL\(P\+,ε∥P−,ε\)≤χ2\(P\+,ε∥P−,ε\)=ε2𝔼ϕd\(ψ\+−ψ−\)21\+ε​ψ−≤ε2​Dψ1−ε​Rψ\.D\_\{\\rm KL\}\(P\_\{\+,\\varepsilon\}\\\|P\_\{\-,\\varepsilon\}\)\\leq\\chi^\{2\}\(P\_\{\+,\\varepsilon\}\\\|P\_\{\-,\\varepsilon\}\)=\\varepsilon^\{2\}\\mathbb\{E\}\_\{\\phi\_\{d\}\}\\frac\{\(\\psi\_\{\+\}\-\\psi\_\{\-\}\)^\{2\}\}\{1\+\\varepsilon\\psi\_\{\-\}\}\\leq\\frac\{\\varepsilon^\{2\}D\_\{\\psi\}\}\{1\-\\varepsilon R\_\{\\psi\}\}\.The two\-point inequality used above then requires every fixed input budget with both errors at mostη\\etato satisfy

n≥1−ε​Rψε2​Dψ​log⁡14​η\.n\\geq\\frac\{1\-\\varepsilon R\_\{\\psi\}\}\{\\varepsilon^\{2\}D\_\{\\psi\}\}\\log\\frac\{1\}\{4\\eta\}\.Labels generated by the same conditional law given the input under both hypotheses do not change this divergence\. For a fixed pair andη≤1/8\\eta\\leq 1/8, the lower and upper orders agree\. ∎

The shared inputs are essential to the exact cancellation in this implementation\. No independent additive measurement noise is introduced after the two gradients are computed\. The passive exponent in Theorem[4\.4](https://arxiv.org/html/2610.20678#S4.Thmtheorem4)belongs to its explicitly defined noisy\-loss oracle; it is not a lower bound for a learner that can inspect raw inputs\. The score above is an ordinary fourth\-moment statistic, which also can be evaluated directly without computing gradients\.

###### Proposition 4\.6\(The bounded\-teacher continuity boundary\)\.

Fix the dimension, a finite time interval,c\>0c\>0, and bounded sets of positive teachers and perturbationsKsK\_\{s\}\. ForHs=Id\+ε​KsH\_\{s\}=\\operatorname\{Id\}\+\\varepsilon K\_\{s\}, the two predictors starting atc​IcIand their common Gaussian losses differ byO⁡\(ε\)O\(\\varepsilon\), uniformly over those sets for sufficiently smallε\\varepsilon\.

###### Proof\.

The Gaussian reference solutions stay in a common bounded set: their largest eigenvalues cannot cross the larger ofccand the teacher’s largest eigenvalue\. On a slightly larger matrix ball of radiusRR, the perturbed polynomial vector fields have a common Lipschitz constant and their differences are bounded by4​R​\(R\+‖T‖F\)​‖H\+−H−‖op4R\(R\+\\\|T\\\|\_\{F\}\)\\\|H\_\{\+\}\-H\_\{\-\}\\\|\_\{\\rm op\}\. Gronwall’s inequality gives anO⁡\(ε\)O\(\\varepsilon\)difference from the reference and closes the ball bootstrap on the fixed time interval\. The common loss is Lipschitz on this ball\. This proves the assertion\. ∎

## 5A logarithmic amplification threshold

Fixq,r≥3q,r\\geq 3,c\>0c\>0,a\>ca\>c, and a positive diagonal interaction teacherTCT\_\{C\}with distinct entries, none equal toaa\. Put

ℓi=\(−1\)ii\!​\(q−1−i\)\!,i=0,…,q−1,TA,i\(L\)=a\+iL,TL=diag\(TA\(L\)\)⊕TC\.\\ell\_\{i\}=\\frac\{\(\-1\)^\{i\}\}\{i\!\(q\-1\-i\)\!\},\\quad i=0,\\ldots,q\-1,\\qquad T\_\{A,i\}^\{\(L\)\}=a\+iL,\\qquad T\_\{L\}=\\operatorname\{diag\}\(T\_\{A\}^\{\(L\)\}\)\\oplus T\_\{C\}\.Thusℓ⊤​𝟏=ℓ⊤​TA\(L\)=0\\ell^\{\\top\}\\mathbf\{1\}=\\ell^\{\\top\}T\_\{A\}^\{\(L\)\}=0andℓ0=1/\(q−1\)\!\\ell\_\{0\}=1/\(q\-1\)\!\. For all sufficiently largeLLthe teacher is simple\. Use the fixed moment perturbations \([6](https://arxiv.org/html/2610.20678#S3.E6)\), with physical couplingεL≥0\\varepsilon\_\{L\}\\geq 0\. These are legal positive GH5 laws for all sufficiently smallεL\\varepsilon\_\{L\}; their support and positivity constant are independent ofLL\. The canonical normalized barycentric vector would beℓ/Lq−1\\ell/L^\{q\-1\}, with compensating couplingεL​Lq−1\\varepsilon\_\{L\}L^\{q\-1\}\. We keep the physical operator normalization explicit throughout\.

###### Theorem 5\.1\(Weak moment perturbations with a finite nonlinear limit\)\.

Suppose

εL⟶0,εL​log⁡L⟶κ∈\[0,∞\)\.\\varepsilon\_\{L\}\\longrightarrow 0,\\qquad\\varepsilon\_\{L\}\\log L\\longrightarrow\\kappa\\in\[0,\\infty\)\.Both learners start at the same full factorW0=c​IW\_\{0\}=\\sqrt\{c\}Iand train on the same teacherTLT\_\{L\}\. Their complete operators converge in norm toId\\operatorname\{Id\}and satisfy Theorem[3\.1](https://arxiv.org/html/2610.20678#S3.Thmtheorem1)for eachLL\. Nevertheless, for every compact interval\[t0,T0\]⊂\(0,∞\)\[t\_\{0\},T\_\{0\}\]\\subset\(0,\\infty\)their interaction predictors converge uniformly to

Nsκ​\(t\)\\displaystyle N\_\{s\}^\{\\kappa\}\(t\)=\[c−1​e−2​TC​t​exp⁡\(κ​ℓ02​Bs\)​e−2​TC​t\+TC−1​\(I−e−4​TC​t\)\]−1,\\displaystyle=\\left\[c^\{\-1\}e^\{\-2T\_\{C\}t\}\\exp\\\!\\left\(\\frac\{\\kappa\\ell\_\{0\}\}\{2\}B\_\{s\}\\right\)e^\{\-2T\_\{C\}t\}\+T\_\{C\}^\{\-1\}\(I\-e^\{\-4T\_\{C\}t\}\)\\right\]^\{\-1\},\(13\)Nsκ​\(0\)\\displaystyle N\_\{s\}^\{\\kappa\}\(0\)=c​exp⁡\(−κ​ℓ02​Bs\)\.\\displaystyle=c\\exp\\\!\\left\(\-\\frac\{\\kappa\\ell\_\{0\}\}\{2\}B\_\{s\}\\right\)\.\(14\)The two own\-loss and common\-loss differences have the same limit,

Δκ​\(t\)=12​‖N\+κ​\(t\)−TC‖F2−12​‖N−κ​\(t\)−TC‖F2\.\\Delta\_\{\\kappa\}\(t\)=\\frac\{1\}\{2\}\\\|N\_\{\+\}^\{\\kappa\}\(t\)\-T\_\{C\}\\\|\_\{F\}^\{2\}\-\\frac\{1\}\{2\}\\\|N\_\{\-\}^\{\\kappa\}\(t\)\-T\_\{C\}\\\|\_\{F\}^\{2\}\.\(15\)Forκ=0\\kappa=0this limit vanishes\. IfDC=c​I−TCD\_\{C\}=cI\-T\_\{C\}andJr=tr⁡\[DC​\(B\+r−B−r\)\]≠0J\_\{r\}=\\operatorname\{tr\}\[D\_\{C\}\(B\_\{\+\}^\{r\}\-B\_\{\-\}^\{r\}\)\]\\neq 0, then

Δκ​\(0\)=cr\!​\(−κ​ℓ02\)r​Jr\+O⁡\(κr\+2\)\.\\Delta\_\{\\kappa\}\(0\)=\\frac\{c\}\{r\!\}\\left\(\-\\frac\{\\kappa\\ell\_\{0\}\}\{2\}\\right\)^\{r\}J\_\{r\}\+O\(\\kappa^\{r\+2\}\)\.\(16\)Thus every sufficiently small fixedκ\>0\\kappa\>0gives nonzero limiting separation on some positive time interval\. More quantitatively, set

τL=\(log⁡L\)2L,bL=L2​e−2​\(log⁡L\)2,ηL=\|εL​log⁡L−κ\|\+εL\+τL\+bL\.\\tau\_\{L\}=\\frac\{\(\\log L\)^\{2\}\}\{L\},\\qquad b\_\{L\}=L^\{2\}e^\{\-2\(\\log L\)^\{2\}\},\\qquad\\eta\_\{L\}=\|\\varepsilon\_\{L\}\\log L\-\\kappa\|\+\\varepsilon\_\{L\}\+\\tau\_\{L\}\+b\_\{L\}\.For every fixedT0\>0T\_\{0\}\>0there is a constantCC, independent ofLLand of the arm, such that for all sufficiently largeLL,

supτL≤t≤T0‖Ns​\(t\)−Nsκ​\(t\)‖op\\displaystyle\\sup\_\{\\tau\_\{L\}\\leq t\\leq T\_\{0\}\}\\\|N\_\{s\}\(t\)\-N\_\{s\}^\{\\kappa\}\(t\)\\\|\_\{\\rm op\}≤C​ηL,\\displaystyle\\leq C\\eta\_\{L\},\(17\)supτL≤t≤T0\|\(L\+ζ−L−ζ\)​\(t\)−Δκ​\(t\)\|\\displaystyle\\sup\_\{\\tau\_\{L\}\\leq t\\leq T\_\{0\}\}\|\(L\_\{\+\}^\{\\zeta\}\-L\_\{\-\}^\{\\zeta\}\)\(t\)\-\\Delta\_\{\\kappa\}\(t\)\|≤C​ηL,ζ∈\{G,P\}\.\\displaystyle\\leq C\\eta\_\{L\},\\quad\\zeta\\in\\\{G,P\\\}\.\(18\)In particular,εL=κ/log⁡L\\varepsilon\_\{L\}=\\kappa/\\log Lgives anO⁡\(1/log⁡L\+\(log⁡L\)2/L\)O\(1/\\log L\+\(\\log L\)^\{2\}/L\)error bound\. The constants may depend on the fixed teacher, graph, initialization,κ\\kappa, andT0T\_\{0\}\.

###### Lemma 5\.2\(Uniform initial\-layer estimates\)\.

Under the assumptions of Theorem[5\.1](https://arxiv.org/html/2610.20678#S5.Thmtheorem1), both arms have, uniformly on\[0,τL\]\[0,\\tau\_\{L\}\], boundedNsN\_\{s\}and bounded exposureus=εL​∫0tSs​\(v\)​𝑑vu\_\{s\}=\\varepsilon\_\{L\}\\int\_\{0\}^\{t\}S\_\{s\}\(v\)\\,dv\. At the end of this interval,

\|a0​\(τL\)−c\|\\displaystyle\|a\_\{0\}\(\\tau\_\{L\}\)\-c\|≤C​τL,\\displaystyle\\leq C\\tau\_\{L\},\|ai​\(τL\)−\(a\+i​L\)\|\\displaystyle\|a\_\{i\}\(\\tau\_\{L\}\)\-\(a\+iL\)\|≤C⁡\(εL\+bL\),i≥1,\\displaystyle\\leq C\(\\varepsilon\_\{L\}\+b\_\{L\}\),\\quad i\\geq 1,\|us​\(τL\)−κ​ℓ0/4\|\\displaystyle\|u\_\{s\}\(\\tau\_\{L\}\)\-\\kappa\\ell\_\{0\}/4\|≤C⁡\(\|εL​log⁡L−κ\|\+εL\),\\displaystyle\\leq C\(\|\\varepsilon\_\{L\}\\log L\-\\kappa\|\+\\varepsilon\_\{L\}\),∥Ns\(τL\)−ce−κℓ0Bs/2∥op\\displaystyle\\\|N\_\{s\}\(\\tau\_\{L\}\)\-ce^\{\-\\kappa\\ell\_\{0\}B\_\{s\}/2\}\\\|\_\{\\rm op\}≤C⁡\(\|εL​log⁡L−κ\|\+εL\+τL\)\.\\displaystyle\\leq C\(\|\\varepsilon\_\{L\}\\log L\-\\kappa\|\+\\varepsilon\_\{L\}\+\\tau\_\{L\}\)\.

###### Proof\.

Writeai​\(t\)a\_\{i\}\(t\)for auxiliary states,S=∑iℓi​aiS=\\sum\_\{i\}\\ell\_\{i\}a\_\{i\},Q=tr⁡\(Bs​N\)Q=\\operatorname\{tr\}\(B\_\{s\}N\)andu⁡\(t\)=εL​∫0tS⁡\(v\)​𝑑vu\(t\)=\\varepsilon\_\{L\}\\int\_\{0\}^\{t\}S\(v\)\\,dv\. The exact block equations are

a˙i=−4​ai​\(ai−TA,i\(L\)\)−2​εL​ℓi​ai​Q,N˙=−4​N2\+2​\{TC,N\}−εL​S​\{Bs,N\}\.\\dot\{a\}\_\{i\}=\-4a\_\{i\}\(a\_\{i\}\-T\_\{A,i\}^\{\(L\)\}\)\-2\\varepsilon\_\{L\}\\ell\_\{i\}a\_\{i\}Q,\\qquad\\dot\{N\}=\-4N^\{2\}\+2\\\{T\_\{C\},N\\\}\-\\varepsilon\_\{L\}S\\\{B\_\{s\},N\\\}\.Positivity permits integration of the logarithmic auxiliary equations\. Usingℓ⊤​TA\(L\)=0\\ell^\{\\top\}T\_\{A\}^\{\(L\)\}=0gives the exact identity

u\(t\)=−εL4∑iℓilogai​\(t\)c−εL22∥ℓ∥22∫0tQ\(v\)dv\.u\(t\)=\-\\frac\{\\varepsilon\_\{L\}\}\{4\}\\sum\_\{i\}\\ell\_\{i\}\\log\\frac\{a\_\{i\}\(t\)\}\{c\}\-\\frac\{\\varepsilon\_\{L\}^\{2\}\}\{2\}\\\|\\ell\\\|\_\{2\}^\{2\}\\int\_\{0\}^\{t\}Q\(v\)\\,dv\.\(19\)
Stop the solution when‖N‖op=K\\\|N\\\|\_\{\\rm op\}=K, for a fixedKKchosen below\. On this interval,\|Q\|≤‖Bs‖∗​K\|Q\|\\leq\\\|B\_\{s\}\\\|\_\{\*\}K\. Seth=\(εL/2\)​‖ℓ‖∞​maxs​‖Bs‖∗​Kh=\(\\varepsilon\_\{L\}/2\)\\\|\\ell\\\|\_\{\\infty\}\\max\_\{s\}\\\|B\_\{s\}\\\|\_\{\*\}K\. Scalar comparison places eachaia\_\{i\}between

bR\(t\)=R1\+\(R/c−1\)​e−4​R​t,R=a\+iL−handR=a\+iL\+h\.b\_\{R\}\(t\)=\\frac\{R\}\{1\+\(R/c\-1\)e^\{\-4Rt\}\},\\qquad R=a\+iL\-h\\quad\\hbox\{and\}\\quad R=a\+iL\+h\.For fixedKKand sufficiently largeLL, all these targets exceedcc\. They also satisfyR≤C′​LR\\leq C^\{\\prime\}L, withC′C^\{\\prime\}independent ofKKonceLLis large enough\. Thus

0≤log⁡\(ai​\(t\)/c\)≤log⁡\(C′​L/c\),\|u⁡\(t\)\|≤εL4​‖ℓ‖1​log⁡\(C′​L/c\)\+C′′​εL2​K​τL\.0\\leq\\log\(a\_\{i\}\(t\)/c\)\\leq\\log\(C^\{\\prime\}L/c\),\\qquad\|u\(t\)\|\\leq\\frac\{\\varepsilon\_\{L\}\}\{4\}\\\|\\ell\\\|\_\{1\}\\log\(C^\{\\prime\}L/c\)\+C^\{\\prime\\prime\}\\varepsilon\_\{L\}^\{2\}K\\tau\_\{L\}\.Chooseu¯\>1\+‖ℓ‖1​\(κ\+1\)/4\\bar\{u\}\>1\+\\\|\\ell\\\|\_\{1\}\(\\kappa\+1\)/4and thenK\>2​c​exp⁡\(2​u¯​maxs​‖Bs‖op\)K\>2c\\exp\(2\\bar\{u\}\\max\_\{s\}\\\|B\_\{s\}\\\|\_\{\\rm op\}\)\. For this fixedKK, the preceding inequality gives\|u\|≤u¯\|u\|\\leq\\bar\{u\}whenLLis sufficiently large\. This order of choices avoids a circular bootstrap constant\.

The change of variablesZ=eu​Bs​N​eu​BsZ=e^\{uB\_\{s\}\}Ne^\{uB\_\{s\}\}cancels the entireεL​S\\varepsilon\_\{L\}Sterm\. On the stopped interval its derivative iseu​Bs​\[−4​N2\+2​\{TC,N\}\]​eu​Bse^\{uB\_\{s\}\}\[\-4N^\{2\}\+2\\\{T\_\{C\},N\\\}\]e^\{uB\_\{s\}\}, with norm at most

e2​u¯​‖Bs‖op​\(4​K2\+4​‖TC‖op​K\)\.e^\{2\\bar\{u\}\\\|B\_\{s\}\\\|\_\{\\rm op\}\}\(4K^\{2\}\+4\\\|T\_\{C\}\\\|\_\{\\rm op\}K\)\.Hence‖Z⁡\(t\)−c​I‖op≤CK​τL\\\|Z\(t\)\-cI\\\|\_\{\\rm op\}\\leq C\_\{K\}\\tau\_\{L\}\. For largeLL, the bound‖N⁡\(t\)‖op≤e2​u¯​‖Bs‖op​\(c\+CK​τL\)<K\\\|N\(t\)\\\|\_\{\\rm op\}\\leq e^\{2\\bar\{u\}\\\|B\_\{s\}\\\|\_\{\\rm op\}\}\(c\+C\_\{K\}\\tau\_\{L\}\)<Kprevents the stopping event\.

Fori≥1i\\geq 1, the explicit logistic formula gives, for allt≥τLt\\geq\\tau\_\{L\}on any interval where the same bound onQQholds,

\|ai​\(t\)−\(a\+i​L\)\|≤h\+C​L2​e−2​L​t≤C⁡\(εL\+bL\)\.\|a\_\{i\}\(t\)\-\(a\+iL\)\|\\leq h\+CL^\{2\}e^\{\-2Lt\}\\leq C\(\\varepsilon\_\{L\}\+b\_\{L\}\)\.Fori=0i=0, boundedness of the two logistic targets gives\|a0​\(t\)−c\|≤C​t\|a\_\{0\}\(t\)\-c\|\\leq Ctin the initial layer\. Since∑i≥1ℓi=−ℓ0\\sum\_\{i\\geq 1\}\\ell\_\{i\}=\-\\ell\_\{0\},

∑iℓi​log⁡\(ai​\(τL\)/c\)=−ℓ0​log⁡L\+∑i≥1ℓi​log⁡\(i/c\)\+O⁡\(τL\+L−1​\(1\+εL\+bL\)\)\.\\sum\_\{i\}\\ell\_\{i\}\\log\(a\_\{i\}\(\\tau\_\{L\}\)/c\)=\-\\ell\_\{0\}\\log L\+\\sum\_\{i\\geq 1\}\\ell\_\{i\}\\log\(i/c\)\+O\\\!\\left\(\\tau\_\{L\}\+L^\{\-1\}\(1\+\\varepsilon\_\{L\}\+b\_\{L\}\)\\right\)\.The integral term in \([19](https://arxiv.org/html/2610.20678#S5.E19)\) has magnitude at mostC​εL2​τLC\\varepsilon\_\{L\}^\{2\}\\tau\_\{L\}\. This proves the exposure bound\. The matrix exponential is Lipschitz on the fixed bounded interval of exponents\. Combining it with‖Z⁡\(τL\)−c​I‖≤C​τL\\\|Z\(\\tau\_\{L\}\)\-cI\\\|\\leq C\\tau\_\{L\}proves the final predictor estimate\. ∎

###### Lemma 5\.3\(Uniform continuation and loss estimates\)\.

For every fixedT0\>0T\_\{0\}\>0, the constants in the preceding lemma can be enlarged so thatNsN\_\{s\}remains uniformly bounded on\[0,T0\]\[0,T\_\{0\}\]and

supτL≤t≤T0\|εLSs\(t\)\|≤CεL,supτL≤t≤T0\|as,i\(t\)−\(a\+iL\)\|≤C\(εL\+bL\)\(i≥1\)\.\\sup\_\{\\tau\_\{L\}\\leq t\\leq T\_\{0\}\}\|\\varepsilon\_\{L\}S\_\{s\}\(t\)\|\\leq C\\varepsilon\_\{L\},\\qquad\\sup\_\{\\tau\_\{L\}\\leq t\\leq T\_\{0\}\}\|a\_\{s,i\}\(t\)\-\(a\+iL\)\|\\leq C\(\\varepsilon\_\{L\}\+b\_\{L\}\)\\quad\(i\\geq 1\)\.Moreover \([17](https://arxiv.org/html/2610.20678#S5.E17)\) and \([18](https://arxiv.org/html/2610.20678#S5.E18)\) hold\.

###### Proof\.

The Gaussian equationN˙=−4​N2\+2​\{TC,N\}\\dot\{N\}=\-4N^\{2\}\+2\\\{T\_\{C\},N\\\}with initial statece−κℓ0Bs/2ce^\{\-\\kappa\\ell\_\{0\}B\_\{s\}/2\}has its largest eigenvalue bounded by the larger of the initial largest eigenvalue and‖TC‖op\\\|T\_\{C\}\\\|\_\{\\rm op\}\. Call a common boundK0K\_\{0\}\. Stop the actual solution afterτL\\tau\_\{L\}if‖Ns‖op\\\|N\_\{s\}\\\|\_\{\\rm op\}reaches a fixedK1\>K0\+1K\_\{1\}\>K\_\{0\}\+1, enlargingK1K\_\{1\}to cover the entire uniform initial\-layer bound if needed\. On this stopped interval, the preceding scalar comparisons apply withK1K\_\{1\}in place ofKK\. Their explicit formula proves the displayed fast\-coordinate estimate\. The slow coordinate remains bounded above and below by logistic solutions with targetsa±C​εLa\\pm C\\varepsilon\_\{L\}\. Sinceℓ⊤​TA\(L\)=0\\ell^\{\\top\}T\_\{A\}^\{\(L\)\}=0,

Ss=ℓ0​\(as,0−a\)\+∑i≥1ℓi​\(as,i−\(a\+i​L\)\)=O⁡\(1\)\+O⁡\(εL\+bL\)\.S\_\{s\}=\\ell\_\{0\}\(a\_\{s,0\}\-a\)\+\\sum\_\{i\\geq 1\}\\ell\_\{i\}\(a\_\{s,i\}\-\(a\+iL\)\)=O\(1\)\+O\(\\varepsilon\_\{L\}\+b\_\{L\}\)\.Thus\|εL​Ss\|≤C​εL\|\\varepsilon\_\{L\}S\_\{s\}\|\\leq C\\varepsilon\_\{L\}for largeLL\.

On the fixed matrix ball the Gaussian vector field has a Lipschitz constantC1C\_\{1\}\. Its perturbing term has norm at most2​εL​\|Ss\|​‖Bs‖op​K1≤C2​εL2\\varepsilon\_\{L\}\|S\_\{s\}\|\\\|B\_\{s\}\\\|\_\{\\rm op\}K\_\{1\}\\leq C\_\{2\}\\varepsilon\_\{L\}\. The initial mismatch atτL\\tau\_\{L\}, including theO⁡\(τL\)O\(\\tau\_\{L\}\)advance of the limiting trajectory, is at mostC3​ηLC\_\{3\}\\eta\_\{L\}\. Gronwall therefore gives

supτL≤t≤T0‖Ns​\(t\)−Nsκ​\(t\)‖op≤eC1​T0​\(C3​ηL\+C2​T0​εL\)\.\\sup\_\{\\tau\_\{L\}\\leq t\\leq T\_\{0\}\}\\\|N\_\{s\}\(t\)\-N\_\{s\}^\{\\kappa\}\(t\)\\\|\_\{\\rm op\}\\leq e^\{C\_\{1\}T\_\{0\}\}\(C\_\{3\}\\eta\_\{L\}\+C\_\{2\}T\_\{0\}\\varepsilon\_\{L\}\)\.This tends to zero and prevents the stopping event\. Together with the initial\-layer bound it also boundsQsQ\_\{s\}on\[0,T0\]\[0,T\_\{0\}\]\.

Leta¯\\bar\{a\}solvea¯˙=−4​a¯​\(a¯−a\)\\dot\{\\bar\{a\}\}=\-4\\bar\{a\}\(\\bar\{a\}\-a\),a¯​\(0\)=c\\bar\{a\}\(0\)=c\. The slow auxiliary forcing is bounded byC​εLC\\varepsilon\_\{L\}on\[0,T0\]\[0,T\_\{0\}\], so scalar Gronwall givessup\|as,0−a¯\|≤C​εL\\sup\|a\_\{s,0\}\-\\bar\{a\}\|\\leq C\\varepsilon\_\{L\}\. The fast\-coordinate squared\-error sum is bounded byC​\(εL\+bL\)2C\(\\varepsilon\_\{L\}\+b\_\{L\}\)^\{2\}on\[τL,T0\]\[\\tau\_\{L\},T\_\{0\}\]\. The interaction squared loss is Lipschitz on the fixed ball\. Consequently

supτL≤t≤T0\|LsG​\(t\)−\{12​\(a¯​\(t\)−a\)2\+12​‖Nsκ​\(t\)−TC‖F2\}\|≤C​ηL\.\\sup\_\{\\tau\_\{L\}\\leq t\\leq T\_\{0\}\}\\left\|L\_\{s\}^\{G\}\(t\)\-\\left\\\{\\tfrac\{1\}\{2\}\(\\bar\{a\}\(t\)\-a\)^\{2\}\+\\tfrac\{1\}\{2\}\\\|N\_\{s\}^\{\\kappa\}\(t\)\-T\_\{C\}\\\|\_\{F\}^\{2\}\\right\\\}\\right\|\\leq C\\eta\_\{L\}\.Finally\|LsP−LsG\|=\|εL​Ss​Qs\|/2≤C​εL\|L\_\{s\}^\{P\}\-L\_\{s\}^\{G\}\|=\|\\varepsilon\_\{L\}S\_\{s\}Q\_\{s\}\|/2\\leq C\\varepsilon\_\{L\}\. Subtraction between arms cancels the common slow\-coordinate limit and proves \([18](https://arxiv.org/html/2610.20678#S5.E18)\)\. ∎

###### Proof of Theorem[5\.1](https://arxiv.org/html/2610.20678#S5.Thmtheorem1)\.

The two lemmas prove the effective seed, both uniform convergences, and the quantitative bounds\. The inverse of the limiting predictor satisfiesR˙=4​I−2​\{TC,R\}\\dot\{R\}=4I\-2\\\{T\_\{C\},R\\\}, which integrates to \([13](https://arxiv.org/html/2610.20678#S5.E13)\)\. Complete spectral equivalence holds at eachLLby Theorem[3\.1](https://arxiv.org/html/2610.20678#S3.Thmtheorem1); the fixed operatorsKsK\_\{s\}are bounded, soHs→IdH\_\{s\}\\to\\operatorname\{Id\}\. At the effective initial state, conjugacy ofB\+B\_\{\+\}andB−B\_\{\-\}makestr⁡\(N\+2\)=tr⁡\(N−2\)\\operatorname\{tr\}\(N\_\{\+\}^\{2\}\)=\\operatorname\{tr\}\(N\_\{\-\}^\{2\}\)andtr⁡\(N\+\)=tr⁡\(N−\)\\operatorname\{tr\}\(N\_\{\+\}\)=\\operatorname\{tr\}\(N\_\{\-\}\), leaving

Δκ\(0\)=ctr\[DC\(e−κℓ0B\+/2−e−κℓ0B−/2\)\]\.\\Delta\_\{\\kappa\}\(0\)=c\\,\\operatorname\{tr\}\\\!\\left\[D\_\{C\}\\left\(e^\{\-\\kappa\\ell\_\{0\}B\_\{\+\}/2\}\-e^\{\-\\kappa\\ell\_\{0\}B\_\{\-\}/2\}\\right\)\\right\]\.Shorter signed closed walks agree\. The first differing term is the length\-rrterm, and parity excludes lengthr\+1r\+1, proving \([16](https://arxiv.org/html/2610.20678#S5.E16)\)\. Continuity in positive time finishes the proof\. ∎

#### Scope of the threshold\.

The thresholdεL​log⁡L\\varepsilon\_\{L\}\\log Lis exact for this specified teacher family, not a universal condition\-number law for neural networks\. Teacher norm and initial loss diverge withLL, so the result does not contradict continuous dependence for bounded teachers\. Initialization is fixed and full rank; the singularity is in teacher scale and moment coupling\. The effective initial state in \([14](https://arxiv.org/html/2610.20678#S5.E14)\) is produced by the original dynamics in a vanishing initial time interval, not imposed as a reset or substituted training objective\. For every finiteLL, the true initial state remainsc​IcI\. Whenκ\>0\\kappa\>0andBs≠0B\_\{s\}\\neq 0, the limiting state has a discontinuity at zero relative to this true initial condition\. Uniform convergence is asserted only after the shrinking layer\[0,τL\]\[0,\\tau\_\{L\}\]\.

The direct predictor flowM˙=−4​Hs​\(M−TL\)\\dot\{M\}=\-4H\_\{s\}\(M\-T\_\{L\}\)is a useful control\. HereHs​\(c​I−TL\)=c​I−TLH\_\{s\}\(cI\-T\_\{L\}\)=cI\-T\_\{L\}, so it equalsTL\+e−4​t​\(c​I−TL\)T\_\{L\}\+e^\{\-4t\}\(cI\-T\_\{L\}\)under both laws for everyLL\. Its between\-arm loss gap is identically zero\.

## 6What a scalar learning curve identifies

The preceding results compare spectral information available before training\. We now ask a different question: what information is present in the scalar history after nonlinear learning has occurred? Consider the Gaussian factorized flow

N˙=−4​N2\+2​\{T,N\},N⁡\(0\)=M0≻0,T=diag⁡\(t1,…,td\)≻0\.\\dot\{N\}=\-4N^\{2\}\+2\\\{T,N\\\},\\qquad N\(0\)=M\_\{0\}\\succ 0,\\qquad T=\\operatorname\{diag\}\(t\_\{1\},\\ldots,t\_\{d\}\)\\succ 0\.We first take the teacher and time scale as known; Section[6\.1](https://arxiv.org/html/2610.20678#S6.SS1)removes the teacher assumption\. PutC=M0−1−T−1C=M\_\{0\}^\{\-1\}\-T^\{\-1\}andDs=e−2​T​sD\_\{s\}=e^\{\-2Ts\}\. The established inverse\-matrix solution is

NC​\(s\)=\[T−1\+Ds​C​Ds\]−1,LC​\(s\)=12​‖NC​\(s\)−T‖F2\.N\_\{C\}\(s\)=\[T^\{\-1\}\+D\_\{s\}CD\_\{s\}\]^\{\-1\},\\qquad L\_\{C\}\(s\)=\\tfrac\{1\}\{2\}\\\|N\_\{C\}\(s\)\-T\\\|\_\{F\}^\{2\}\.\(20\)The question is not how to solve this Riccati equation, but whether its scalar loss identifies the initial matrix\.

###### Theorem 6\.1\(The exact scalar tail is a complete invariant\)\.

Supposet1,…,tdt\_\{1\},\\ldots,t\_\{d\}are linearly independent overℚ\\mathbb\{Q\}\. For anyC,C~C,\\widetilde\{C\}such thatT−1\+C≻0T^\{\-1\}\+C\\succ 0andT−1\+C~≻0T^\{\-1\}\+\\widetilde\{C\}\\succ 0, the following are equivalent:

1. 1\.LC​\(s\)=LC~​\(s\)L\_\{C\}\(s\)=L\_\{\\widetilde\{C\}\}\(s\)for all sufficiently largess;
2. 2\.C~=S​C​S\\widetilde\{C\}=SCSfor a diagonal matrixS=diag⁡\(ϵ1,…,ϵd\)S=\\operatorname\{diag\}\(\\epsilon\_\{1\},\\ldots,\\epsilon\_\{d\}\),ϵi∈\{−1,1\}\\epsilon\_\{i\}\\in\\\{\-1,1\\\};
3. 3\.M~0=S​M0​S\\widetilde\{M\}\_\{0\}=SM\_\{0\}Sfor such a matrixSS\.

There is no density or connectedness assumption on the off\-diagonal support ofCC\.

###### Proof\.

The inverse state satisfiesR˙=4​I−2​\{T,R\}\\dot\{R\}=4I\-2\\\{T,R\\\}, giving \([20](https://arxiv.org/html/2610.20678#S6.E20)\)\. DefineZ=T1/2​Ds​C​Ds​T1/2Z=T^\{1/2\}D\_\{s\}CD\_\{s\}T^\{1/2\}\. At sufficiently late times the entrywise absolute matrix has‖\|Z\|‖∞<1\\\|\|Z\|\\\|\_\{\\infty\}<1\. The Neumann expansion therefore converges absolutely even after expanding each trace into paths:

LC​\(s\)=12​∑m,n≥1\(−1\)m\+n​tr⁡\(T​Zm​T​Zn\)\.L\_\{C\}\(s\)=\\frac\{1\}\{2\}\\sum\_\{m,n\\geq 1\}\(\-1\)^\{m\+n\}\\operatorname\{tr\}\(TZ^\{m\}TZ^\{n\}\)\.\(21\)Every closed path has a vertex occupation vectorν∈ℕd\\nu\\in\\mathbb\{N\}^\{d\},\|ν\|≥2\|\\nu\|\\geq 2, and exponential rate4​ν⋅t4\\nu\\cdot t\. Absolute convergence permits regrouping to obtain

LC​\(s\)=∑\|ν\|≥2aν​\(C\)​e−4​\(ν⋅t\)​s\.L\_\{C\}\(s\)=\\sum\_\{\|\\nu\|\\geq 2\}a\_\{\\nu\}\(C\)e^\{\-4\(\\nu\\cdot t\)s\}\.The positive rates form a locally finite set; rational independence makes different occupation vectors have different rates\. These coefficients are unique\. Indeed, in a difference of two such series, take the least rate with a nonzero coefficient, multiply by its inverse exponential and sendssto infinity\. Absolute convergence at one fixed late time supplies a summable dominating sequence, so every larger\-rate term vanishes\. The nonzero leading coefficient is a contradiction\.

Expanding the degree\-two and degree\-three terms gives

a2​ei\\displaystyle a\_\{2e\_\{i\}\}=12​ti4​Ci​i2,\\displaystyle=\\tfrac\{1\}\{2\}t\_\{i\}^\{4\}C\_\{ii\}^\{2\},a3​ei\\displaystyle a\_\{3e\_\{i\}\}=−ti5​Ci​i3,\\displaystyle=\-t\_\{i\}^\{5\}C\_\{ii\}^\{3\},aei\+ej\\displaystyle a\_\{e\_\{i\}\+e\_\{j\}\}=ti2tj2Ci​j2\(i<j\)\.\\displaystyle=t\_\{i\}^\{2\}t\_\{j\}^\{2\}C\_\{ij\}^\{2\}\\quad\(i<j\)\.\(22\)Thus every diagonal entry, every off\-diagonal magnitude and the support graphGCG\_\{C\}are known\.

LetV⊂\[d\]V\\subset\[d\],\|V\|=k≥3\|V\|=k\\geq 3\. IfHam⁡\(GC​\[V\]\)\\operatorname\{Ham\}\(G\_\{C\}\[V\]\)denotes the undirected Hamiltonian cycles, counted as unoriented edge sets, then the coefficient of occupation vector𝟏V\\mathbf\{1\}\_\{V\}is

a𝟏V=\(−1\)k​\(∏v∈Vtv\)​\(∑a,b∈Va≠bta​tb\)⏟KV\>0​∑H∈Ham⁡\(GC​\[V\]\)∏\{i,j\}∈E⁡\(H\)Ci​j\.a\_\{\\mathbf\{1\}\_\{V\}\}=\(\-1\)^\{k\}\\underbrace\{\\left\(\\prod\_\{v\\in V\}t\_\{v\}\\right\)\\left\(\\sum\_\{\\begin\{subarray\}\{c\}a,b\\in V\\\\ a\\neq b\\end\{subarray\}\}t\_\{a\}t\_\{b\}\\right\)\}\_\{K\_\{V\}\>0\}\\sum\_\{H\\in\\operatorname\{Ham\}\(G\_\{C\}\[V\]\)\}\\prod\_\{\\\{i,j\\\}\\in E\(H\)\}C\_\{ij\}\.\(23\)To check the factor, each cyclic path uses every vertex once\. Its edge factors supply∏v∈Vtv\\prod\_\{v\\in V\}t\_\{v\}; the two marked positions of the externalTTfactors supplyta​tbt\_\{a\}t\_\{b\}\. Summing the initial vertex and the splitm=1,…,k−1m=1,\\ldots,k\-1supplies every ordered distinct pair\. The two path orientations cancel the prefactor1/21/2in \([21](https://arxiv.org/html/2610.20678#S6.E21)\)\. No diagonal stay or repeated vertex has this occupation vector, and no other vector shares its rate\.

IfGC​\[V\]G\_\{C\}\[V\]is a chordless cycle, its cycle is the only Hamiltonian edge set\. Equation \([23](https://arxiv.org/html/2610.20678#S6.E23)\) determines its signed edge product\. A cycle containing a chord decomposes into two shorter cycles, so the chordless products determine every cycle\-sign invariant\. This last reconstruction step is classical signed\-graph and principal\-minor machinery\([Harary, 1953](https://arxiv.org/html/2610.20678#bib.bib6);[Urschel et al\., 2017](https://arxiv.org/html/2610.20678#bib.bib19);[Brunel, 2018](https://arxiv.org/html/2610.20678#bib.bib3)\)\. For completeness, the ratio of the two edge signs then has product\+1\+1around every cycle\. Choose a root in each connected component and propagate vertex signs along paths from that root\. The cycle condition makes these signs path\-independent and givesC~i​j=ϵi​ϵj​Ci​j\\widetilde\{C\}\_\{ij\}=\\epsilon\_\{i\}\\epsilon\_\{j\}C\_\{ij\}\. Diagonal entries already agree, proving the second condition\. It is equivalent to the third by inversion\. ConverselySScommutes withTTandDsD\_\{s\}, soNS​C​S​\(s\)=S​NC​\(s\)​SN\_\{SCS\}\(s\)=SN\_\{C\}\(s\)S, which leaves the loss unchanged\. ∎

The theorem uses an exact continuum of observations with unlimited precision\. Analyticity implies that exact equality on any nonempty open time interval also suffices\. This implication is not a stable analytic\-continuation algorithm\. Rational independence is a sufficient condition for the coefficient proof, not a claimed necessary condition for identifiability\.

###### Theorem 6\.2\(A minimal number of exact scalar observations for generic local recovery\)\.

Letp=d⁡\(d\+1\)/2p=d\(d\+1\)/2and suppose all unordered sumsti\+tjt\_\{i\}\+t\_\{j\},i≤ji\\leq j, are distinct\. Fix anyppdistinct nonnegative timess1,…,sps\_\{1\},\\ldots,s\_\{p\}\. On the domain𝒟=\{C:T−1\+C≻0\}\\mathcal\{D\}=\\\{C:T^\{\-1\}\+C\\succ 0\\\}, the map

F⁡\(C\)=\(LC​\(s1\),…,LC​\(sp\)\)F\(C\)=\(L\_\{C\}\(s\_\{1\}\),\\ldots,L\_\{C\}\(s\_\{p\}\)\)has an invertible derivative on an open dense subset\. At each such point it has a local analytic inverse\. Fewer thanppreal scalar observations cannot have a continuous local inverse on an open subset of the fullpp\-dimensional parameter space\.

###### Proof\.

The matrix inside the inverse in \([20](https://arxiv.org/html/2610.20678#S6.E20)\) is positive: it equalsDs​\(T−1\+C\)​Ds\+T−1​\(I−Ds2\)D\_\{s\}\(T^\{\-1\}\+C\)D\_\{s\}\+T^\{\-1\}\(I\-D\_\{s\}^\{2\}\)\. ThusFFis real analytic throughout the open convex set𝒟\\mathcal\{D\}\. NearC=0C=0,

LC​\(s\)=∑i≤jwi​j​Ci​j2​e−λi​j​s\+O⁡\(‖C‖3\),λi​j=4​\(ti\+tj\),wi​i=12​ti4,wi​j=ti2​tj2​\(i<j\)\.L\_\{C\}\(s\)=\\sum\_\{i\\leq j\}w\_\{ij\}C\_\{ij\}^\{2\}e^\{\-\\lambda\_\{ij\}s\}\+O\(\\\|C\\\|^\{3\}\),\\quad\\lambda\_\{ij\}=4\(t\_\{i\}\+t\_\{j\}\),\\quad w\_\{ii\}=\\tfrac\{1\}\{2\}t\_\{i\}^\{4\},\\quad w\_\{ij\}=t\_\{i\}^\{2\}t\_\{j\}^\{2\}\\ \(i<j\)\.Choose a symmetricC¯\\overline\{C\}whose independent entries are nonzero\. For sufficiently smallδ≠0\\delta\\neq 0,δ​C¯∈𝒟\\delta\\overline\{C\}\\in\\mathcal\{D\}and

detD​F​\(δ​C¯\)\\displaystyle\\det DF\(\\delta\\overline\{C\}\)=\(2​δ\)p​\(∏i≤jwi​j​C¯i​j\)​det\[e−λi​j​sk\]k,\(i,j\)\+O⁡\(δp\+1\)\.\\displaystyle=\(2\\delta\)^\{p\}\\left\(\\prod\_\{i\\leq j\}w\_\{ij\}\\overline\{C\}\_\{ij\}\\right\)\\det\[e^\{\-\\lambda\_\{ij\}s\_\{k\}\}\]\_\{k,\(i,j\)\}\+O\(\\delta^\{p\+1\}\)\.\(24\)The exponential evaluation determinant is nonzero\. One elementary proof is induction on the number of distinct rates: divide a linear combination by its slowest exponential and differentiate\. Rolle’s theorem and the induction hypothesis show that a nonzero combination ofppexponentials has at mostp−1p\-1distinct zeros\. ThereforedetD​F\\det DFis not identically zero\. A nonzero real\-analytic function on a connected domain has a zero set with empty interior, and continuity makes its nonzero set open\. The inverse function theorem proves local analytic recovery\. The lower bound follows from invariance of dimension: an open subset ofℝp\\mathbb\{R\}^\{p\}cannot be continuously embedded intoℝm\\mathbb\{R\}^\{m\}form<pm<p\. ∎

For an invertible observation Jacobian, the local noise amplification depends onσmin​\(D​F​\(C\)\)−1\\sigma\_\{\\min\}\(DF\(C\)\)^\{\-1\}\. Its exact derivative can be computed without differentiating a numerical ODE:

d​LC​\(s\)​\[E\]=−tr⁡\[Ds​NC​\(s\)​\(NC​\(s\)−T\)​NC​\(s\)​Ds​E\]\.dL\_\{C\}\(s\)\[E\]=\-\\operatorname\{tr\}\[D\_\{s\}N\_\{C\}\(s\)\(N\_\{C\}\(s\)\-T\)N\_\{C\}\(s\)D\_\{s\}E\]\.\(25\)Generic local identifiability gives no uniform lower bound on this singular value and no guarantee for a globally initialized solver\. In particularD​F​\(0\)=0DF\(0\)=0, and rates or observation times can approach collisions\. The local theorem and nonlinear output observability\([Hermann and Krener, 1977](https://arxiv.org/html/2610.20678#bib.bib7)\)address related questions; the present statement computes the observation threshold for this particular loss map\.

###### Corollary 6\.3\(Reading the interaction left by the initial layer\)\.

In Theorem[5\.1](https://arxiv.org/html/2610.20678#S5.Thmtheorem1), suppose the interaction\-teacher entries are rationally independent, and the known constants satisfyκ​ℓ0≠0\\kappa\\ell\_\{0\}\\neq 0\. After subtracting the common known slow auxiliary loss from the limiting curve, that curve identifiesBsB\_\{s\}up to diagonal sign conjugation\. Indeed

Bs=−2κ​ℓ0​log⁡\(Nsκ​\(0\)/c\)\.B\_\{s\}=\-\\frac\{2\}\{\\kappa\\ell\_\{0\}\}\\log\(N\_\{s\}^\{\\kappa\}\(0\)/c\)\.The finite\-observation local theorem also transfers to arbitrary symmetric interaction matrices through this analytic change of variables\.

###### Proof\.

The symmetric exponential and the principal logarithm are inverse analytic maps between symmetric and positive\-definite matrices\. They commute with orthogonal conjugation\. Apply Theorems[6\.1](https://arxiv.org/html/2610.20678#S6.Thmtheorem1)and[6\.2](https://arxiv.org/html/2610.20678#S6.Thmtheorem2)\. ∎

The known slow auxiliary loss in this corollary is necessary: the observation must be the interaction loss, or a full limiting loss from which that common term is known\. It is also a statement about the derived limit, not an exact recovery theorem for a finite\-LLcoupled trajectory\. The numerical teacher used for the amplification example has resonant entries and is not presented as a test of the global inverse theorem\.

### 6\.1Joint identification of the teacher and initial state

The known\-teacher assumption can be removed on an explicitly described set of initial states\. The time unit and the absolute scale of the loss are assumed known\. In a teacher eigenbasis putC=M0−1−T−1C=M\_\{0\}^\{\-1\}\-T^\{\-1\}as before\.

###### Theorem 6\.4\(An unknown teacher is identifiable from the scalar curve\)\.

Consider the class of pairs\(T,M0\)\(T,M\_\{0\}\)withT,M0≻0T,M\_\{0\}\\succ 0, rationally independent eigenvalues ofTT, andCi​i≠0C\_\{ii\}\\neq 0for everyiiin a teacher eigenbasis\. Two members of this class have the same exact loss curve on a nonempty open time interval if and only if their dimensions agree and they are simultaneously orthogonally conjugate\. Thus the curve identifies the dimension, the teacher eigenvalues and the initial state modulo this unavoidable symmetry\. For each fixed admissible teacher, the condition onCCis open dense and has full Lebesgue measure in𝒟\\mathcal\{D\}\. It imposes no condition on the off\-diagonal support\.

###### Proof\.

Analyticity extends equality on the interval to all positive times\. For either candidate the absolutely convergent late\-time expansion has rates

4​ν⋅t,ν∈ℕd,\|ν\|≥2\.4\\nu\\cdot t,\\qquad\\nu\\in\\mathbb\{N\}^\{d\},\\quad\|\\nu\|\\geq 2\.The union of the two candidates’ rate sets is locally finite\. The least\-rate argument in Theorem[6\.1](https://arxiv.org/html/2610.20678#S6.Thmtheorem1)therefore shows that the nonzero exponent support𝒮\\mathcal\{S\}of the scalar curve is unique even when the candidate teachers differ\.

Writebi=4​tib\_\{i\}=4t\_\{i\}and let𝒬\\mathcal\{Q\}be the additive monoid generated by𝒮\\mathcal\{S\}, including zero\. The two coefficients in \([22](https://arxiv.org/html/2610.20678#S6.E22)\) imply2​bi,3​bi∈𝒮2b\_\{i\},3b\_\{i\}\\in\\mathcal\{S\}, becauseCi​i≠0C\_\{ii\}\\neq 0and rational independence prevents collisions with other occupation vectors\. Consequently

𝒢:=𝒬−𝒬=⨁i=1dℤ​bi\.\\mathcal\{G\}:=\\mathcal\{Q\}\-\\mathcal\{Q\}=\\bigoplus\_\{i=1\}^\{d\}\\mathbb\{Z\}b\_\{i\}\.The following set is determined entirely by the observed support:

Sat𝒢⁡\(𝒬\)=\{g∈𝒢:n​g∈𝒬​for some integer​n≥1\}=⨁i=1dℕ​bi\.\\operatorname\{Sat\}\_\{\\mathcal\{G\}\}\(\\mathcal\{Q\}\)=\\\{g\\in\\mathcal\{G\}:ng\\in\\mathcal\{Q\}\\text\{ for some integer \}n\\geq 1\\\}=\\bigoplus\_\{i=1\}^\{d\}\\mathbb\{N\}b\_\{i\}\.For the forward inclusion, expressggin its unique integer coordinates in the rationally independent basisbib\_\{i\}\. Since every element of𝒬\\mathcal\{Q\}has nonnegative coordinates, so mustggwhenn​g∈𝒬ng\\in\\mathcal\{Q\}\. Conversely, for anyνi≥0\\nu\_\{i\}\\geq 0,2​∑iνi​bi=∑iνi​\(2​bi\)∈𝒬2\\sum\_\{i\}\\nu\_\{i\}b\_\{i\}=\\sum\_\{i\}\\nu\_\{i\}\(2b\_\{i\}\)\\in\\mathcal\{Q\}\. The nonzero elements of this saturated monoid that cannot be written as a sum of two nonzero elements are exactlyb1,…,bdb\_\{1\},\\ldots,b\_\{d\}\. The curve therefore determinesddand the unordered teacher spectrum\.

Order that spectrum, choose teacher eigenbases, and apply Theorem[6\.1](https://arxiv.org/html/2610.20678#S6.Thmtheorem1)\. The remaining freedom is diagonal sign conjugation in these bases, which is simultaneous orthogonal conjugation of the original pair\. Such conjugation preserves the dynamics and loss, proving the converse\. The excluded conditionsCi​i=0C\_\{ii\}=0are a finite collection of proper hyperplanes intersected with the open set𝒟\\mathcal\{D\}\. ∎

The monoid here is generated by the observed support; the support itself need not be closed under addition\. No assumption that every exponential coefficient is nonzero is needed\. There are real obstructions outside the stated class:C=0C=0gives the zero curve for every teacher, and appending an uncoupled coordinate withCi​i=0C\_\{ii\}=0leaves the loss unchanged\. The theorem fixes the time, loss and gradient\-flow normalizations; joint calibration from differently normalized measurements is a separate question\. Recovering all exact exponents is an identifiability argument, not a stable spectral estimation procedure\. Finite exponential\-sum estimation and its noise sensitivity are established topics in Prony methods\([Potts and Tasche, 2010](https://arxiv.org/html/2610.20678#bib.bib13)\)\.

###### Theorem 6\.5\(The minimal local observation count with an unknown teacher\)\.

Letp=d⁡\(d\+1\)/2p=d\(d\+1\)/2andm=p\+dm=p\+d\. Suppose all unordered sumsti\+tjt\_\{i\}\+t\_\{j\},i≤ji\\leq j, are distinct\. For anymmdistinct nonnegative times, the observation map

\(t,C\)⟼\(Lt,C​\(s1\),…,Lt,C​\(sm\)\)\(t,C\)\\longmapsto\(L\_\{t,C\}\(s\_\{1\}\),\\ldots,L\_\{t,C\}\(s\_\{m\}\)\)has an invertible joint derivative at an open dense, full\-measure set ofC∈𝒟C\\in\\mathcal\{D\}for each such fixedtt\. It has a local analytic inverse there, with bothttandCCallowed to vary\. Fewer thanmmreal scalar observations cannot have a continuous local inverse on an openmm\-dimensional parameter set\.

###### Proof\.

For fixed smallδ\>0\\delta\>0, use positive coordinates

Ci​j=δ​αi​j/wi​j​\(t\),αi​j\>0,C\_\{ij\}=\\delta\\sqrt\{\\alpha\_\{ij\}/w\_\{ij\}\(t\)\},\\qquad\\alpha\_\{ij\}\>0,wherewi​i=ti4/2w\_\{ii\}=t\_\{i\}^\{4\}/2andwi​j=ti2​tj2w\_\{ij\}=t\_\{i\}^\{2\}t\_\{j\}^\{2\}fori<ji<j\. This is a local analytic change of variables from\(α,t\)\(\\alpha,t\)to\(C,t\)\(C,t\)\. For fixed positiveα\\alphathe loss expansion and its first derivatives give

δ−2​Lt,C​\(s\)=∑i≤jαi​j​e−λi​j​s\+O⁡\(δ\),λi​j=4​\(ti\+tj\)\.\\delta^\{\-2\}L\_\{t,C\}\(s\)=\\sum\_\{i\\leq j\}\\alpha\_\{ij\}e^\{\-\\lambda\_\{ij\}s\}\+O\(\\delta\),\\qquad\\lambda\_\{ij\}=4\(t\_\{i\}\+t\_\{j\}\)\.The leading expression is linear inα\\alphaand extends to its boundary\. At the limiting choiceαi​i=1\\alpha\_\{ii\}=1andαi​j=0\\alpha\_\{ij\}=0fori<ji<j, its Jacobian has theppcolumnse−λi​j​se^\{\-\\lambda\_\{ij\}s\}and theddcolumns−8​s​e−8​ti​s\-8se^\{\-8t\_\{i\}s\}\. These functions form an extended exponential Chebyshev system: the real rate8​ti8t\_\{i\}has multiplicity two, and the other rates have multiplicity one\. Its evaluation matrix at anymmdistinct times is invertible\. The zero count follows by successively dividing by an exponential and differentiating; repeated rates produce polynomial factors and Rolle’s theorem gives at mostm−1m\-1distinct zeros for a nonzero combination\.

This boundary calculation concerns only the leading map, not the square\-root coordinate change\. By continuity, first fix strictly positive off\-diagonalαi​j\\alpha\_\{ij\}sufficiently close to zero that the leading determinant remains nonzero\. For this fixed interior choice, chooseδ\>0\\delta\>0sufficiently small that the true joint derivative is invertible andT−1\+C≻0T^\{\-1\}\+C\\succ 0\. The analytic coordinate change is applied only at that interior point\. For each fixedttthis gives one point where the determinant is nonzero\. That determinant is real analytic inCCon the connected domain𝒟\\mathcal\{D\}\. Its nonzero set is open dense and its zero set has Lebesgue measure zero\. The analytic inverse function theorem proves joint local recovery\. Invariance of dimension proves the lower bound\. The finite sign and permutation ambiguities do not alter the local dimension at these regular points\. ∎

This local count concerns the quotient by simultaneous rotations: after choosing a teacher eigenbasis the continuous parameters are itsddeigenvalues and theppentries ofCC\. It does not recover an externally fixed orientation from a scalar\. The smallest joint singular value, rather than the parameter count, controls local noise amplification\.

###### Proposition 6\.6\(A finite global observation bound\)\.

For a fixed rationally independent teacher in dimensiondd, any5d5^\{d\}distinct nonnegative exact loss observations identify every admissibleCCup to diagonal sign conjugation\. If the teacher and dimension are unknown but the dimension is at mostDD, any9D9^\{D\}such observations identify every member of the class in Theorem[6\.4](https://arxiv.org/html/2610.20678#S6.Thmtheorem4)up to simultaneous orthogonal conjugation\. These sufficient bounds are not asserted to be sharp\.

###### Proof\.

Setai=ti−1a\_\{i\}=t\_\{i\}^\{\-1\}andxi=e−4​ti​sx\_\{i\}=e^\{\-4t\_\{i\}s\}\. The determinant

P⁡\(a,x\)=det\(diag⁡\(a\)\+diag⁡\(x\)​C​diag⁡\(x\)\)=∑S⊆\[d\]detCS,S​∏i∈Sxi​∏i∉SaiP\(a,x\)=\\det\(\\operatorname\{diag\}\(a\)\+\\operatorname\{diag\}\(\\sqrt\{x\}\)C\\operatorname\{diag\}\(\\sqrt\{x\}\)\)=\\sum\_\{S\\subseteq\[d\]\}\\det C\_\{S,S\}\\prod\_\{i\\in S\}x\_\{i\}\\prod\_\{i\\notin S\}a\_\{i\}is multiaffine inxx, withdetC∅,∅=1\\det C\_\{\\varnothing,\\varnothing\}=1\. Let

U=∂zP⁡\(a\+z​𝟏,x\)\|z=0,V=∂z2P⁡\(a\+z​𝟏,x\)\|z=0,W=∑iti​∂aiP⁡\(a,x\),U=\\left\.\\partial\_\{z\}P\(a\+z\\mathbf\{1\},x\)\\right\|\_\{z=0\},\\qquad V=\\left\.\\partial\_\{z\}^\{2\}P\(a\+z\\mathbf\{1\},x\)\\right\|\_\{z=0\},\\qquad W=\\sum\_\{i\}t\_\{i\}\\,\\partial\_\{a\_\{i\}\}P\(a,x\),wherettis held fixed in the last differentiation andai=ti−1a\_\{i\}=t\_\{i\}^\{\-1\}is substituted afterwards\. The determinant derivative identities givetr⁡\(N2\)=\(U2−P​V\)/P2\\operatorname\{tr\}\(N^\{2\}\)=\(U^\{2\}\-PV\)/P^\{2\}andtr⁡\(N​T\)=W/P\\operatorname\{tr\}\(NT\)=W/P\. Hence

Lt,C​\(s\)=Q⁡\(x\)P​\(a,x\)2,Q=12​\{U2−P​V−2​P​W\+tr⁡\(T2\)​P2\},L\_\{t,C\}\(s\)=\\frac\{Q\(x\)\}\{P\(a,x\)^\{2\}\},\\qquad Q=\\tfrac\{1\}\{2\}\\\{U^\{2\}\-PV\-2PW\+\\operatorname\{tr\}\(T^\{2\}\)P^\{2\}\\\},whereQQhas degree at most two in each coordinate\. The denominator is positive throughouts≥0s\\geq 0\.

For two states with the same teacher, clearing denominators in their loss difference gives a polynomial of degree at most four in eachxix\_\{i\}\. On the trajectory it is therefore an exponential polynomial with at most5d5^\{d\}distinct real rates\. For two teachers of dimensionsd,d~≤Dd,\\widetilde\{d\}\\leq D, use separate variablesxi=e−4​ti​sx\_\{i\}=e^\{\-4t\_\{i\}s\}andx~j=e−4​t~j​s\\widetilde\{x\}\_\{j\}=e^\{\-4\\widetilde\{t\}\_\{j\}s\}\. The numerator has degree at most two in each variable, so it has at most3d\+d~≤9D3^\{d\+\\widetilde\{d\}\}\\leq 9^\{D\}rates\. Coincident rates only reduce this count\. The exponential zero bound used above implies that the asserted number of equal observations forces equality of the entire curves\. Apply Theorem[6\.1](https://arxiv.org/html/2610.20678#S6.Thmtheorem1)or Theorem[6\.4](https://arxiv.org/html/2610.20678#S6.Thmtheorem4), respectively\. ∎

General finite\-experiment identifiability is established:[Sontag \(2002\)](https://arxiv.org/html/2610.20678#bib.bib16)proves that2​k\+12k\+1generic experiments preserve the distinguishability of an analytickk\-parameter response\. Applied to this analytic loss map in fixed dimension, that result already gives smaller generic global observation counts,2​p\+12p\+1for a known teacher and2​\(p\+d\)\+12\(p\+d\)\+1for an unknown teacher\. The elementary bounds above concern*every*set of distinct times\. The model\-specific conclusions are the exact equivalence class, the blind spectrum recovery, and the dimension\-minimal local count for every distinct\-time design\. None of these results supplies a noise\-uniform inversion algorithm\.

### 6\.2Exact identifiability can have an arbitrarily weak statistical signal

We make the distinction between uniqueness and measurement explicit\. This experiment observes a known pair of Gaussian learning curves with additive measurement noise\. Its inputs are scalar losses, not samples from the input laws of Sections[3](https://arxiv.org/html/2610.20678#S3)–[5](https://arxiv.org/html/2610.20678#S5)\.

###### Theorem 6\.7\(Sharp cycle\-length exponent for noisy loss queries\)\.

Letd=r≥3d=r\\geq 3andT=diag⁡\(t1,…,tr\)≻0T=\\operatorname\{diag\}\(t\_\{1\},\\ldots,t\_\{r\}\)\\succ 0\. LetA±A\_\{\\pm\}have zero diagonal, the common support of one chordlessrr\-cycle, positive edge magnitudesaea\_\{e\}, and opposite cycle sign products\. PutC±=δ​A±C\_\{\\pm\}=\\delta A\_\{\\pm\}andP=∏eaeP=\\prod\_\{e\}a\_\{e\}\. Define

R=‖T1/2​\|A±\|​T1/2‖∞,ρ=δ0​R<1\.R=\\\|T^\{1/2\}\|A\_\{\\pm\}\|T^\{1/2\}\\\|\_\{\\infty\},\\quad\\rho=\\delta\_\{0\}R<1\.For all0<δ≤δ00<\\delta\\leq\\delta\_\{0\}, both initial states are admissible and

sups≥0\|L\+​\(s\)−L−​\(s\)\|≤Kup​δr,Kup=r​tmax2​Rr​\[r−11−ρ2\+2​ρ2\(1−ρ2\)2\]\.\\sup\_\{s\\geq 0\}\|L\_\{\+\}\(s\)\-L\_\{\-\}\(s\)\|\\leq K\_\{\\rm up\}\\delta^\{r\},\\qquad K\_\{\\rm up\}=r\\,t\_\{\\max\}^\{2\}R^\{r\}\\left\[\\frac\{r\-1\}\{1\-\\rho^\{2\}\}\+\\frac\{2\\rho^\{2\}\}\{\(1\-\\rho^\{2\}\)^\{2\}\}\\right\]\.\(26\)For all sufficiently smallδ\\deltathe gap at zero also satisfies\|L\+​\(0\)−L−​\(0\)\|≥c0​δr\|L\_\{\+\}\(0\)\-L\_\{\-\}\(0\)\|\\geq c\_\{0\}\\delta^\{r\}for a fixedc0\>0c\_\{0\}\>0\.

Suppose the supplied alternatives are observed throughYj=Lθ​\(sj\)\+σ​ZjY\_\{j\}=L\_\{\\theta\}\(s\_\{j\}\)\+\\sigma Z\_\{j\}, whereθ∈\{\+,−\}\\theta\\in\\\{\+,\-\\\},σ\>0\\sigma\>0, theZjZ\_\{j\}are independent standard Gaussians, and the nonnegative observation times may be chosen adaptively\. If both testing errors are at mostη<1/4\\eta<1/4, necessarily

n≥2​σ2Kup2​δ2​r​log⁡14​η\.n\\geq\\frac\{2\\sigma^\{2\}\}\{K\_\{\\rm up\}^\{2\}\\delta^\{2r\}\}\\log\\frac\{1\}\{4\\eta\}\.\(27\)Conversely, repeated measurements at zero and a midpoint test attain error at mostη\\etawithn=O⁡\(σ2​δ−2​r​log⁡\(1/η\)\)n=O\(\\sigma^\{2\}\\delta^\{\-2r\}\\log\(1/\\eta\)\)asδ→0\\delta\\to 0\. Thus adaptive selection of observation times cannot improve theδ−2​r\\delta^\{\-2r\}exponent for this supplied pair\.

###### Proof\.

The norm condition givesI\+δ​T1/2​A±​T1/2≻0I\+\\delta T^\{1/2\}A\_\{\\pm\}T^\{1/2\}\\succ 0by its spectral\-radius bound, and thereforeT−1\+δ​A±≻0T^\{\-1\}\+\\delta A\_\{\\pm\}\\succ 0\. For everys≥0s\\geq 0, the absolute Neumann matrix obeys‖\|Z±​\(s\)\|‖∞≤δ​R<1\\\|\|Z\_\{\\pm\}\(s\)\|\\\|\_\{\\infty\}\\leq\\delta R<1\. In a closed\-walk term of \([21](https://arxiv.org/html/2610.20678#S6.E21)\), changing one edge sign can matter only if that edge is used oddly\. The odd edge set of a closed walk is an even subgraph\. On a single chordless cycle it is empty or the whole cycle, so every differing term has degreer\+2​mr\+2m\. The zero diagonal excludes additional one\-step stays\. At degreekk, the sum hask−1k\-1choices of the two positive path lengths\. Bounding both traces in the difference gives

\|Δ​L​\(s\)\|≤r​tmax2​∑m≥0\(r\+2​m−1\)​\(δ​R\)r\+2​m\.\|\\Delta L\(s\)\|\\leq rt\_\{\\max\}^\{2\}\\sum\_\{m\\geq 0\}\(r\+2m\-1\)\(\\delta R\)^\{r\+2m\}\.The geometric sum is bounded by \([26](https://arxiv.org/html/2610.20678#S6.E26)\)\. At zero, the explicit cycle coefficient gives

Δ​L​\(0\)=2​\(−1\)r​\(∏i=1rti\)​\(∑a≠bta​tb\)​P​δr\+O⁡\(δr\+2\),\\Delta L\(0\)=2\(\-1\)^\{r\}\\left\(\\prod\_\{i=1\}^\{r\}t\_\{i\}\\right\)\\left\(\\sum\_\{a\\neq b\}t\_\{a\}t\_\{b\}\\right\)P\\,\\delta^\{r\}\+O\(\\delta^\{r\+2\}\),whose leading coefficient is nonzero\. This proves the lower gap bound\.

LetP±\(n\)P\_\{\\pm\}^\{\(n\)\}denote the adaptive transcript laws, including any policy randomness\. The policy uses the same conditional kernel under both alternatives\. The Gaussian conditional divergence and the KL chain rule therefore give

D\(P\+\(n\)∥P−\(n\)\)=∑j=1n𝔼\+\(L\+​\(sj\)−L−​\(sj\)\)22​σ2≤n​Kup2​δ2​r2​σ2\.D\(P\_\{\+\}^\{\(n\)\}\\\|P\_\{\-\}^\{\(n\)\}\)=\\sum\_\{j=1\}^\{n\}\\mathbb\{E\}\_\{\+\}\\frac\{\(L\_\{\+\}\(s\_\{j\}\)\-L\_\{\-\}\(s\_\{j\}\)\)^\{2\}\}\{2\\sigma^\{2\}\}\\leq\\frac\{nK\_\{\\rm up\}^\{2\}\\delta^\{2r\}\}\{2\\sigma^\{2\}\}\.The classical two\-point testing inequalityP\+\(error\)\+P−\(error\)≥12e−D\(P\+∥P−\)P\_\{\+\}\(\\mathrm\{error\}\)\+P\_\{\-\}\(\\mathrm\{error\}\)\\geq\\tfrac\{1\}\{2\}e^\{\-D\(P\_\{\+\}\\\|P\_\{\-\}\)\}\([Tsybakov, 2009](https://arxiv.org/html/2610.20678#bib.bib18)\)proves \([27](https://arxiv.org/html/2610.20678#S6.E27)\)\. For the upper bound, the sample mean of repeated measurements at zero has varianceσ2/n\\sigma^\{2\}/n\. Thresholding it halfway between the two known means has each error at mostexp\[−nc02δ2​r/\(8σ2\)\]\\exp\[\-nc\_\{0\}^\{2\}\\delta^\{2r\}/\(8\\sigma^\{2\}\)\]\. ∎

The exponent is sharp for this two\-point experiment, with constants depending on the fixed teacher and cycle\. The theorem gives neither arbitrary\-graph reconstruction nor uniform guarantees as dimension increases\. Rational independence is not needed for this two\-point test, although it can be imposed to place both candidates inside Theorem[6\.1](https://arxiv.org/html/2610.20678#S6.Thmtheorem1)\. Cycle\-dependent statistical difficulty has a close precedent in determinantal point process learning\([Urschel et al\., 2017](https://arxiv.org/html/2610.20678#bib.bib19)\)\. Here the additional statement is uniform over all nonnegative learning times and hence over adaptive noisy loss\-query policies\.

#### Failures and symmetries\.

IfT=I2T=I\_\{2\}, a45∘45^\{\\circ\}rotation ofM0=diag⁡\(2,3\)M\_\{0\}=\\operatorname\{diag\}\(2,3\)gives an initial matrix with diagonal entries2\.52\.5and off\-diagonal entries of magnitude0\.50\.5\. These states are not diagonal\-sign conjugate, but their entire loss curves agree by orthogonal invariance\. Distinct teacher entries alone do not justify the noncollision proof either: for\(1,2,3\)\(1,2,3\),4​\(t1\+t3\)=8​t24\(t\_\{1\}\+t\_\{3\}\)=8t\_\{2\}\. That resonance invalidates direct extraction of the two coefficients; it does not by itself prove nonidentifiability\. A chordless four\-cycle, in contrast, is fully covered by Theorem[6\.1](https://arxiv.org/html/2610.20678#S6.Thmtheorem1): its signs first appear at degree four, even when no triangle or nonzero star is available\.

## 7Exact certificates and numerical checks

The computations address admissibility, full\-weight realization, the singular limit, and the conditioning of the inverse problem\. They are deterministic examples\. Numerical integration discrepancies are tolerance diagnostics, not rigorous interval error bounds\. Exploratory choices and all declared cells are retained in the accompanying protocol and source data\.

### 7\.1A larger admissible perturbation on a fixed finite support

Forq=r=3q=r=3,c=1c=1,ℓ=\(1/2,−1,1/2\)\\ell=\(1/2,\-1,1/2\)and unit edge magnitudes, we solve the finite moment problem on\{0,±3\}8\\\{0,\\pm\\sqrt\{3\}\\\}^\{8\}\. The desired fourth tensor is Gaussian except

𝔼\[xa2xjxk\]=εℓa\(Bs\)j​k,a∈A,j,k∈C,j≠k\.\\mathbb\{E\}\[x\_\{a\}^\{2\}x\_\{j\}x\_\{k\}\]=\\varepsilon\\ell\_\{a\}\(B\_\{s\}\)\_\{jk\},\\quad a\\in A,\\quad j,k\\in C,\\ j\\neq k\.Auxiliary sign averaging and a common interaction sign flip reduce the problem to 976 orbit columns and 140 constraints\. A linear program findsε∗=9/26\\varepsilon\_\{\*\}=9/26\. An exact rational basic solution has 111 positive orbit weights\. Its mass and moment residuals are exactly zero\. A separate rational dual certificate has nonnegative slack on all 976 columns and value9/269/26, proving the optimum on this specified support and symmetry class\. We make no claim about the full moment cone\.

Mixing this law with1/51/5of the Gaussian\-moment three\-node product law yieldsε=18/65\\varepsilon=18/65and positive mass at all 6,561 grid points\. The second arm is the branch\-permutation image\. Independent expansion checks all 8 first, 36 second and 330 distinct fourth monomials in each arm with exact rational residual zero\. Contracting the raw centered features gives the complete3636\-dimensional operator; its maximum entrywise discrepancy from the analytic operator is3\.09×10−143\.09\\times 10^\{\-14\}, and the paired eigenvalue discrepancy is8\.88×10−168\.88\\times 10^\{\-16\}\. The explicit isometry fixes every diagonal projector and has a nonzero Jordan defect of norm2\\sqrt\{2\}in a two\-edge example\.

### 7\.2A finite\-time contrast in the original weight dynamics

The first tested teacher\(2,3,…,9\)\(2,3,\\ldots,9\)produces a small maximum common\-loss gap, approximately3\.99×10−73\.99\\times 10^\{\-7\}\. We retain it as evidence that a legal larger perturbation alone does not guarantee an appreciable contrast\. A subsequent declared 108\-cell teacher grid varies auxiliary offset and spacing, and interaction offset and spacing\. The displayed selected cell is

T=diag⁡\(1\.1,31\.1,61\.1,2,5,8,11,14\)\.T=\\operatorname\{diag\}\(1\.1,31\.1,61\.1,2,5,8,11,14\)\.It lies on the grid’s largest auxiliary\-spacing boundary and is exploratory\. No global optimization or representative\-task claim is made\.

The selected cell has initial loss2426\.5152426\.515\. The maximal recorded common\-loss gap is0\.1785280\.178528near time0\.0483980\.048398, or7\.36×10−57\.36\\times 10^\{\-5\}of initial loss\. Among recorded times where the mean current loss is at least10−410^\{\-4\}of initial loss, the largest relative common\-loss gap is4\.351%4\.351\\%near time0\.1962140\.196214, at mean loss0\.2563040\.256304\. These two normalizations answer different questions; the larger percentage does not describe the initial\-loss scale\.

Independent replay integrates the full8×88\\times 8weight equation using the raw finite\-law operator, rather than the generating block equation\. Tightening DOP853 tolerances changes the evaluated losses by at most2\.27×10−102\.27\\times 10^\{\-10\}, andW​W⊤WW^\{\\top\}agrees with a separately integrated complete predictor equation to6\.04×10−116\.04\\times 10^\{\-11\}\. At the fixed times0\.05,0\.1,0\.20\.05,0\.1,0\.2, the common\-loss gaps are respectively0\.176956,−0\.0922444,0\.01083010\.176956,\-0\.0922444,0\.0108301\. Their signs change: there is no uniform winner\. Actual weight gradient descent with step sizes10−410^\{\-4\}and5×10−55\\times 10^\{\-5\}reproduces the gaps, with maximal loss discrepancy from the flow decreasing from9\.85×10−39\.85\\times 10^\{\-3\}to4\.90×10−34\.90\\times 10^\{\-3\}\.

Figure 1:Spectral equivalence, finite training contrast and its scale dependence\.a, Paired common\-loss differences from complete weight flow and actual weight updates at the three fixed observation times\. Lines joining the flow observations only guide the eye\.b, Absolute errors between the common\-loss contrasts of the original coupled dynamics and the derived limiting contrasts at four fixed times, as the auxiliary teacher scale grows\. Some errors are nonmonotone\. All four declared scales are shown\. Both panels are deterministic checks without sampling error bars\. The teacher in panel a was selected by the disclosed 108\-cell exploration; panel b uses diverging teacher norms\.
### 7\.3Analytic laws and scalar interventions

For the trigonometric density in Proposition[2\.1](https://arxiv.org/html/2610.20678#S2.Thmtheorem1), independent one\-dimensional Gaussian quadrature followed by product expansion checks all 495 monomials of degree zero through four in each arm\. The largest target residual is6\.66×10−166\.66\\times 10^\{\-16\}\. Characteristic\-function derivatives provide an independent reference; the largest reported quadrature error is7\.75×10−127\.75\\times 10^\{\-12\}, which is retained as the numerical resolution diagnostic\. With the unit triangle pair,Rψ=6​e2R\_\{\\psi\}=6e^\{2\}andε=1/\(12​e2\)≈0\.011278\\varepsilon=1/\(12e^\{2\}\)\\approx 0\.011278certify a density ratio between1/21/2and3/23/2\. These analytic laws differ from the larger\-coupling discrete laws used for Figure[1](https://arxiv.org/html/2610.20678#S7.F1)a\.

The scalar intervention is separately checked by raw\-feature contraction of the finite positive GH3 laws\. We useρ=0\.2\\rho=0\.2, teacher\(2,3,…,9\)\(2,3,\\ldots,9\)and couplings\(18/65\)/16,\(18/65\)/8,\(18/65\)/4,\(18/65\)/2\(18/65\)/16,\(18/65\)/8,\(18/65\)/4,\(18/65\)/2\. Every coupling includes the maximizing direction, a zero\-intervention control and three seeded bounded directions\. The largest discrepancy from the complete scalar signal formula is7\.11×10−157\.11\\times 10^\{\-15\}\. The same\-input score in \([11](https://arxiv.org/html/2610.20678#S4.E11)\) is evaluated directly on every atom; its mean agrees with the normalized projected gradient increment to2\.78×10−172\.78\\times 10^\{\-17\}\. Its second moment under the GH3 product baseline is0\.04=ρ20\.04=\\rho^\{2\}\. These calculations check the score algebra and moments; the finite\-input error guarantee follows from the proof, not from a Monte Carlo power estimate\.

### 7\.4The logarithmic\-scale limit

SetTA=\(1\.1,1\.1\+L,1\.1\+2​L\)T\_\{A\}=\(1\.1,1\.1\+L,1\.1\+2L\),TC=diag⁡\(2,5,8,11,14\)T\_\{C\}=\\operatorname\{diag\}\(2,5,8,11,14\)andε=1/log⁡L\\varepsilon=1/\\log L\. The four declared scales are102,104,106,10810^\{2\},10^\{4\},10^\{6\},10^\{8\}\. All couplings lie below the certified finite\-support bound9/269/26\. Radau integration of the original coupled equations uses logarithmic auxiliary coordinates and an analytic Jacobian\. An initial run using a numerical Jacobian completed but emitted step\-adjustment overflow warnings at the largest scale; the reported rerun supplies the analytic Jacobian and treats runtime warnings as errors\. The exact exposure identity is satisfied to at worst3\.37×10−113\.37\\times 10^\{\-11\}across the checked states\.

At time0\.050\.05, the common\-loss gap rises from0\.1647360\.164736atL=102L=10^\{2\}to0\.2409290\.240929atL=108L=10^\{8\}, compared with limiting gap0\.2675550\.267555\. The convergence is slow, as the logarithmic scaling suggests\. The other declared times include sign reversals and nonmonotone errors \(Figure[1](https://arxiv.org/html/2610.20678#S7.F1)b\)\. These checks support the specified asymptotic formula\. The finite\-LLerror rate follows from the proof of Theorem[5\.1](https://arxiv.org/html/2610.20678#S5.Thmtheorem1), not from a fitted numerical slope; neither result asserts amplification for bounded teachers\. Directional finite\-difference checks of the analytic Jacobian are asserted only atL=102L=10^\{2\}\. Their larger\-LLvalues are retained as unasserted cancellation\-sensitive diagnostics\.

### 7\.5Finite inverse observations and their conditioning

The inverse check uses dimension three, teacher\(1,7,73\)\(1,\\sqrt\{7\},\\sqrt\{73\}\), six geometrically spaced observation times from0\.0030\.003to11, and

C=0\.02​\(10\.40\.30\.4−0\.50\.20\.30\.20\.7\)\.C=0\.02\\begin\{pmatrix\}1&0\.4&0\.3\\\\ 0\.4&\-0\.5&0\.2\\\\ 0\.3&0\.2&0\.7\\end\{pmatrix\}\.The exact observation Jacobian has smallest singular value2\.45×10−62\.45\\times 10^\{\-6\}and condition number2\.22×1072\.22\\times 10^\{7\}\. A directional finite\-difference check agrees with its analytic derivative to relative error1\.77×10−101\.77\\times 10^\{\-10\}\. Eleven exact symbolic checks cover the diagonal, magnitude, triangle and chordless\-square coefficients\. Sign\-gauge and repeated\-teacher rotation controls agree to numerical precision\.

All four locally initialized reconstruction attempts are reported in Figure[2](https://arxiv.org/html/2610.20678#S7.F2)\. Starting2%2\\%from the true parameter, the noise\-free fit reaches relative parameter error3\.62×10−123\.62\\times 10^\{\-12\}\. Absolute observation\-noise standard deviations10−10,10−8,10−610^\{\-10\},10^\{\-8\},10^\{\-6\}give errors approximately8\.96×10−4,0\.0846,0\.5538\.96\\times 10^\{\-4\},0\.0846,0\.553, using one fixed noise direction\. The last noisy fit has a nonzero residual despite optimizer termination\. No ensemble accuracy, global convergence or noise\-robust recovery claim follows from these four cases\. Their purpose is to expose the gap between an invertible observation map and a well\-conditioned one\.

For the blind extension we use the same teacher, replaceC22/0\.02C\_\{22\}/0\.02by0\.80\.8\(one\-based indexing\), and fix nine geometric times from0\.0030\.003to11\. The joint Jacobian contains six state and three teacher columns\. Its determinant at 90\-digit working precision is approximately−7\.24704×10−26\-7\.24704\\times 10^\{\-26\}; its smallest singular value is1\.33×10−71\.33\\times 10^\{\-7\}and condition number4\.81×1084\.81\\times 10^\{8\}\. Central differences agree with the analytic joint derivative to relative error3\.71×10−103\.71\\times 10^\{\-10\}, and independent high\-precision differentiation agrees to working precision\. An exact two\-dimensional symbolic check gives zero residual in the identityL=Q/P2L=Q/P^\{2\}used for the finite global bound\. Sign, permutation and simultaneous\-rotation controls preserve the loss; zero error and an uncoupled equilibrium coordinate demonstrate the stated failures\. These are fixed algebraic and rank checks, not a blind parameter\-fitting experiment\.

Figure 2:Finite identifiability does not ensure stable reconstruction\.a, All six singular values of the analytic known\-teacher observation Jacobian in the specified three\-dimensional example\.b, Relative parameter errors for every declared local inverse attempt, including the noise\-free control\. Each noise level uses the same seeded Gaussian direction; these are single deterministic realizations, not averaged performance estimates\. Initialization is within2%2\\%of the true parameter in every case\.

## 8Discussion

The results distinguish three information regimes\. Before training, even complete spectral data anchored on the whole diagonal teacher algebra can miss multiplication relations that control nonlinear evolution\. During training, these relations can remain weak to arbitrarily high perturbative order, or acquire a finite effect when a specified teacher scale diverges\. Bounded diagonal initialization interventions expose the same hidden distinction at first order in the coupling through one scalar gradient direction\. Shared\-input gradient differences implement pair testing without the teacher or population\-gradient access, with a sharp input\-sample exponent\. After training, an exact scalar history can encode the full hidden state modulo a natural symmetry\. The observation channel and parameter regime determine which description is sufficient\.

In the amplification limit, both laws approach the same base law as their moment perturbations vanish, but their teacher norms grow without bound\. On bounded teacher sets, ordinary continuous dependence forces the trajectory contrast to vanish\. The result identifies the nonuniform limit and its effective initial state for the specified diverging\-teacher family\. The numerical examples likewise retain their initial\-loss normalization and exploratory selection\.

The inverse results separate exact information from usable estimates\. With a known nonresonant teacher, identifiability survives arbitrary support sparsity\. On the stated active class the scalar curve also identifies an unknown teacher spectrum, by recovering the generators of its exponent monoid\. With distinct unordered teacher sums, the joint finite\-observation theorem gives a sharp generic local parameter count in fixed dimension\. For a known teacher or the active blind class with a dimension cap, the global bounds replace a continuum by a finite exact design\. Noise amplification remains uncontrolled by these counts\. Cycle information can be exponentially weak, and the joint numerical Jacobian is poorly conditioned\. Stable and computationally feasible recovery under finite\-sample loss estimation, with unknown teachers or constrained graph classes, remains open\.

For neural learning theory, the constructive lesson is that the linear operator and its task\-anchored spectral measures are not the whole dynamical object\. Matrix multiplication supplies additional relations among responses\. The same relations that obstruct a spectral forecast can become visible in a nonlinear trajectory\. Thus a nonlinear scalar trajectory can reveal multiplication\-dependent invariants absent from the matched spectral summaries\.

#### Data and code availability\.

Versioned code and synthetic data accompany this manuscript as arXiv ancillary files\. They include exact finite\-law weights, rational primal and dual certificates, every exploratory grid cell, full\-weight replay, the declared protocols, plotting scripts and a SHA\-256 file manifest\. Code is licensed under MIT and covered synthetic outputs under CC BY 4\.0\. No repository DOI has been assigned\.

#### Funding and competing interests\.

The author reports no funding and no competing interests\.

#### Use of AI assistance\.

AI tools assisted with theoretical exploration, drafting, coding, literature discovery and independent computational checks\. The mathematical claims, computational results and references remain the author’s responsibility\.

## Appendix AInitial\-time order and discrete\-step coefficients

For completeness, this appendix supplies the coefficient calculation used to show that the finite\-time response in Theorem[4\.1](https://arxiv.org/html/2610.20678#S4.Thmtheorem1)is nonzero\. Here the physical coupling is denoted byβ=ε\\beta=\\varepsilonand the auxiliary barycentric vector has the normalization in Section[2](https://arxiv.org/html/2610.20678#S2)\.

### A\.1The first teacher\-weighted closed walk

PutD=diag⁡\(gC\)D=\\operatorname\{diag\}\(g\_\{C\}\)and

Jr=tr⁡\{D⁡\(B\+r−B−r\)\}=4​Πw​\(∑i∈C1∖C2gi−∑i∈C2∖C1gi\),Πw=∏j=1rwj\.J\_\{r\}=\\operatorname\{tr\}\\\{D\(B\_\{\+\}^\{r\}\-B\_\{\-\}^\{r\}\)\\\}=4\\Pi\_\{w\}\\left\(\\sum\_\{i\\in C\_\{1\}\\setminus C\_\{2\}\}g\_\{i\}\-\\sum\_\{i\\in C\_\{2\}\\setminus C\_\{1\}\}g\_\{i\}\\right\),\\qquad\\Pi\_\{w\}=\\prod\_\{j=1\}^\{r\}w\_\{j\}\.\(28\)Chooseggso thatJr≠0J\_\{r\}\\neq 0; increasing distinct negative integer magnitudes along the two branches is one explicit choice\. All traces of pure powers ofBBagree, but this joint invariant of the teacher and interaction does not\. The shared hub’s two once\-around contributions cancel in the difference\. Every non\-hub cycle vertex gives\+4​Πw\+4\\Pi\_\{w\}or−4​Πw\-4\\Pi\_\{w\}\. Shorter closed walks have even parity on every edge and contribute no difference, including walks with arbitrary diagonal stays\.

###### Lemma A\.1\(Series degree and closed\-walk filtration\)\.

Let the two laws, teacher and initialization be as above, and putλ=\(r−1\)​q\+1\\lambda=\(r\-1\)q\+1\. Use the original continuous block equation or, for any fixed finite integer step count, the exact weight\-gradient update in Equation \([30](https://arxiv.org/html/2610.20678#A1.E30)\)\. In the corresponding formal variableu=tu=toru=hu=h,

valu⁡\(Q\+−Q−\)≥λ,valu⁡\(a\+−a−\)≥λ\+1,valu⁡\(L\+ζ−L−ζ\)≥r​q,ζ∈\{G,P\}\.\\operatorname\{val\}\_\{u\}\(Q\_\{\+\}\-Q\_\{\-\}\)\\geq\\lambda,\\qquad\\operatorname\{val\}\_\{u\}\(a\_\{\+\}\-a\_\{\-\}\)\\geq\\lambda\+1,\\qquad\\operatorname\{val\}\_\{u\}\(L\_\{\+\}^\{\\zeta\}\-L\_\{\-\}^\{\\zeta\}\)\\geq rq,\\quad\\zeta\\in\\\{G,P\\\}\.Here a zero series has infinite valuation\. At degreeλ\\lambdainQQ, a differing closed matrix\-spine walk must contain exactlyrredges and one interaction\-teacher insertionDD\. This assertion concerns the differing maximal\-spine coefficient: common scalar trace factors and common feedback may contribute toQQ, but do not change that coefficient\.

###### Proof\.

WriteS0=ℓ⊤​ρS\_\{0\}=\\ell^\{\\top\}\\rho, where each Gaussian reference coordinate solves

ρ˙i=−4​ρi​\(ρi−c\+gi\),ρi​\(0\)=c\.\\dot\{\\rho\}\_\{i\}=\-4\\rho\_\{i\}\(\\rho\_\{i\}\-c\+g\_\{i\}\),\\qquad\\rho\_\{i\}\(0\)=c\.Its coefficientρi,m\\rho\_\{i,m\}is a polynomial ingig\_\{i\}of degree at mostmm, with leading termc​\(−4\)m​gim/m\!c\(\-4\)^\{m\}g\_\{i\}^\{m\}/m\!\. Thus

S0=s0​tq−1\+O⁡\(tq\),s0=c​\(−4\)q−1\(q−1\)\!,γ=−2​β​c​s0q\.S\_\{0\}=s\_\{0\}t^\{q\-1\}\+O\(t^\{q\}\),\\qquad s\_\{0\}=\\frac\{c\(\-4\)^\{q\-1\}\}\{\(q\-1\)\!\},\\qquad\\gamma=\-\\frac\{2\\beta cs\_\{0\}\}\{q\}\.For the matrix filtration, first holdSSfixed as a common scalar series of valuation at leastq−1q\-1, mark every matrix\-spine occurrence ofBBbyzz, and mark every spine occurrence ofD=diag⁡\(gC\)D=\\operatorname\{diag\}\(g\_\{C\}\)byη\\eta\. The marked continuous recurrence is

N˙=−4​N2\+4​c​N−2​η​\{D,N\}−β​S​\{z​B,N\}\.\\dot\{N\}=\-4N^\{2\}\+4cN\-2\\eta\\\{D,N\\\}\-\\beta S\\\{zB,N\\\}\.IfN=∑n≥0N⁡\[n\]​tnN=\\sum\_\{n\\geq 0\}N\[n\]t^\{n\}, its coefficient recurrence is

\(n\+1\)N\[n\+1\]=−4∑a\+b=nN\[a\]N\[b\]\+4cN\[n\]−2η\{D,N\[n\]\}−β∑p\+b=nS\[p\]\{zB,N\[b\]\}\.\(n\+1\)N\[n\+1\]=\-4\\sum\_\{a\+b=n\}N\[a\]N\[b\]\+4cN\[n\]\-2\\eta\\\{D,N\[n\]\\\}\-\\beta\\sum\_\{p\+b=n\}S\[p\]\\\{zB,N\[b\]\\\}\.Induction innngives

valt⁡\[zj​ηd\]​N≥j​q\+d\.\\operatorname\{val\}\_\{t\}\[z^\{j\}\\eta^\{d\}\]N\\geq jq\+d\.At the first possible degree, thed=0d=0coefficient is the pure word

\[tj​q​zj\]​N=pj​Bj,pj=c​\(γ/c\)jj\!\.\[t^\{jq\}z^\{j\}\]N=p\_\{j\}B^\{j\},\\qquad p\_\{j\}=\\frac\{c\(\\gamma/c\)^\{j\}\}\{j\!\}\.Indeed, the leadingS0S\_\{0\}term supplies oneBBandqqtime degrees after integration; all quadratic terms and all higher coefficients ofSSoccur later\. At degreej​q\+1jq\+1, theDD\-free terms are still common scalar multiples ofBjB^\{j\}, while every non\-pure term at this degree has exactly oneDDinsertion\.

Put

Rj=\[tj​q\+1​zj​η\]​N\.R\_\{j\}=\[t^\{jq\+1\}z^\{j\}\\eta\]N\.Keeping exactly oneDDin the displayed recurrence yields

\(j​q\+1\)​Rj=−2​pj​\{D,Bj\}−β​s0​\{B,Rj−1\},R0=−4​c​D\.\(jq\+1\)R\_\{j\}=\-2p\_\{j\}\\\{D,B^\{j\}\\\}\-\\beta s\_\{0\}\\\{B,R\_\{j\-1\}\\\},\\qquad R\_\{0\}=\-4cD\.For everyk≥0k\\geq 0, cyclicity of trace and the relation definingγ\\gammagive

tr⁡\(Bk​Rj\)=fj​c​\(γ/c\)j​tr⁡\(D​Bj\+k\),f0=−4,fj=−4/j\!\+qfj−1j​q\+1=−4j\!\.\\operatorname\{tr\}\(B^\{k\}R\_\{j\}\)=f\_\{j\}c\(\\gamma/c\)^\{j\}\\operatorname\{tr\}\(DB^\{j\+k\}\),\\qquad f\_\{0\}=\-4,\\qquad f\_\{j\}=\\frac\{\-4/j\!\+qf\_\{j\-1\}\}\{jq\+1\}=\-\\frac\{4\}\{j\!\}\.This is the insertion calculation, rather than a word\-counting claim\. In particular, the only one\-DDmaximal\-spine contribution to the first differing scalar coefficient is obtained fromj=r−1j=r\-1,k=1k=1:

Q\+−Q−=V​tλ\+O⁡\(tλ\+1\),V=−4​γr−1​Jrcr−2​\(r−1\)\!\.Q\_\{\+\}\-Q\_\{\-\}=Vt^\{\\lambda\}\+O\(t^\{\\lambda\+1\}\),\\qquad V=\-\\frac\{4\\gamma^\{r\-1\}J\_\{r\}\}\{c^\{r\-2\}\(r\-1\)\!\}\.
We now close the scalar feedback induction\. If all earlier coefficients ofQ\+Q\_\{\+\}andQ−Q\_\{\-\}agree, the scalar auxiliary recurrence makes every coefficient ofS\+S\_\{\+\}andS−S\_\{\-\}needed for the current coefficient ofNNagree as well\. More explicitly, a differingQQcoefficient first entersaaafter one integration, and can first return throughSStoNN, hence toQQ, after a second integration\. Therefore a coefficient ofQQat degreemmcan use a returned differing scalar only from aQQcoefficient of degree at mostm−2m\-2\. Induction onmmconsequently makes the feedback common through degreeλ\\lambda, and givesa\+−a−=O⁡\(tλ\+1\)a\_\{\+\}\-a\_\{\-\}=O\(t^\{\\lambda\+1\}\)\.

This induction does not say that common feedback is zero\. For clarity, the first common terms are

\[tq\]​Q=γ​tr⁡\(B2\),\[tq\+1\]​\(S−S0\)=−2​β​c​γq\+1​‖ℓ‖2​tr⁡\(B2\),\[t^\{q\}\]Q=\\gamma\\operatorname\{tr\}\(B^\{2\}\),\\qquad\[t^\{q\+1\}\]\(S\-S\_\{0\}\)=\-\\frac\{2\\beta c\\gamma\}\{q\+1\}\\\|\\ell\\\|^\{2\}\\operatorname\{tr\}\(B^\{2\}\),and this correction induces the common contribution

\[tq\+2\]​Q\|S−S0=4​β2​c2​γ\(q\+1\)​\(q\+2\)​‖ℓ‖2​\(tr⁡\(B2\)\)2\.\[t^\{q\+2\}\]Q\\big\|\_\{S\-S\_\{0\}\}=\\frac\{4\\beta^\{2\}c^\{2\}\\gamma\}\{\(q\+1\)\(q\+2\)\}\\\|\\ell\\\|^\{2\}\\bigl\(\\operatorname\{tr\}\(B^\{2\}\)\\bigr\)^\{2\}\.It is generally nonzero, but it is identical for the two arms and hence cancels in their difference\. Thus common feedback is excluded only from the differing maximal\-spine coefficient above, not fromQQitself\.

It remains to identify which common coefficient can differ\. On the two\-cycle graph, the edges of a closed matrix\-spine word that are traversed an odd number of times form a cycle\-space element\. Fewer thanrrclosed spine edges are therefore switching\-even\. With exactlyrredges and noDD, the contribution is a pure trace and agrees becauseB−B\_\{\-\}is conjugate toB\+B\_\{\+\}\. The outerBBinQ=tr⁡\(B​N\)Q=\\operatorname\{tr\}\(BN\)is free, so the maximal spine at degreeλ=\(r−1\)​q\+1\\lambda=\(r\-1\)q\+1hasj=r−1j=r\-1state edges\. The preceding recurrence shows that its sole differing possibility is exactly oneDD, which gives the displayed formula forVV\. An already differingQQcannot alter this conclusion by the two\-degree return just proved\.

For the loss bounds, the interaction part of the common Gaussian risk has at degreer​qrqonly the pairing ofDDwith the pure\-statej=rj=rterm\. It equalsZ/rZ/r, where

Z=γr​Jrcr−1​\(r−1\)\!\.Z=\\frac\{\\gamma^\{r\}J\_\{r\}\}\{c^\{r\-1\}\(r\-1\)\!\}\.The auxiliary linear response aroundρ\\rhois

ΔLA=−2β∫0tΔQ\(u\)𝒦\(t,u\)du\+terms of later degree,𝒦\(t,u\)=∑i∈Aℓi\(ρi\(t\)−Ti\)Φi\(t,u\)ρi\(u\),\\Delta L\_\{A\}=\-2\\beta\\int\_\{0\}^\{t\}\\Delta Q\(u\)\\mathcal\{K\}\(t,u\)\\,du\+\\text\{terms of later degree\},\\qquad\\mathcal\{K\}\(t,u\)=\\sum\_\{i\\in A\}\\ell\_\{i\}\(\\rho\_\{i\}\(t\)\-T\_\{i\}\)\\Phi\_\{i\}\(t,u\)\\rho\_\{i\}\(u\),whereΦi\\Phi\_\{i\}is the scalar reference variational propagator\. To derive the relevant node\-polynomial degree, put

bi​\(v\)=1\+cTi​\(e4​Ti​v−1\),Ti=c−gi\.b\_\{i\}\(v\)=1\+\\frac\{c\}\{T\_\{i\}\}\(e^\{4T\_\{i\}v\}\-1\),\\qquad T\_\{i\}=c\-g\_\{i\}\.The exact scalar solution and propagator are

ρi​\(v\)=c​e4​Ti​vbi​\(v\),Φi​\(t,u\)=e4​Ti​\(t−u\)​bi​\(u\)2bi​\(t\)2\.\\rho\_\{i\}\(v\)=\\frac\{ce^\{4T\_\{i\}v\}\}\{b\_\{i\}\(v\)\},\\qquad\\Phi\_\{i\}\(t,u\)=e^\{4T\_\{i\}\(t\-u\)\}\\frac\{b\_\{i\}\(u\)^\{2\}\}\{b\_\{i\}\(t\)^\{2\}\}\.Sinceρi​\(t\)−Ti=gi/bi​\(t\)\\rho\_\{i\}\(t\)\-T\_\{i\}=g\_\{i\}/b\_\{i\}\(t\), the individual kernel summand is

\(ρi​\(t\)−Ti\)​Φi​\(t,u\)​ρi​\(u\)=c​gi​e4​\(c−gi\)​t​bi​\(u\)bi​\(t\)3\.\(\\rho\_\{i\}\(t\)\-T\_\{i\}\)\\Phi\_\{i\}\(t,u\)\\rho\_\{i\}\(u\)=cg\_\{i\}e^\{4\(c\-g\_\{i\}\)t\}\\frac\{b\_\{i\}\(u\)\}\{b\_\{i\}\(t\)^\{3\}\}\.The coefficient ofvnv^\{n\}inbi​\(v\)−1b\_\{i\}\(v\)\-1isc​4n​\(c−gi\)n−1/n\!c4^\{n\}\(c\-g\_\{i\}\)^\{n\-1\}/n\!forn≥1n\\geq 1, so its degree ingig\_\{i\}is at mostn−1n\-1\. At bivariate total degreemm, every term involving a nonconstant coefficient ofbi​\(u\)/bi​\(t\)3b\_\{i\}\(u\)/b\_\{i\}\(t\)^\{3\}has node degree at mostmm\. The sole node\-degree\-m\+1m\+1part is therefore

c​\(−4\)mm\!​gim\+1​tm\.\\frac\{c\(\-4\)^\{m\}\}\{m\!\}g\_\{i\}^\{m\+1\}t^\{m\}\.Thus the weighted sum vanishes form<q−2m<q\-2\. Atm=q−2m=q\-2,∑iℓi​giq−1=1\\sum\_\{i\}\\ell\_\{i\}g\_\{i\}^\{q\-1\}=1retains the displayed highest\-degree term and annihilates all the others, giving

𝒦⁡\(t,u\)=c​\(−4\)q−2\(q−2\)\!​tq−2\+Ototal​\(q−1\)\.\\mathcal\{K\}\(t,u\)=\\frac\{c\(\-4\)^\{q\-2\}\}\{\(q\-2\)\!\}t^\{q\-2\}\+O\_\{\\rm total\}\(q\-1\)\.Substitution of the displayed formula forVVgives the auxiliary contribution

Z​q⁡\(q−1\)\(r−1\)​q\+2​tr​q\+O⁡\(tr​q\+1\)\.Z\\,\\frac\{q\(q\-1\)\}\{\(r\-1\)q\+2\}\\,t^\{rq\}\+O\(t^\{rq\+1\}\)\.Common feedback changes this kernel only above the displayed degree, and nonlinear arm differences have valuation at least2​λ\+22\\lambda\+2\. Finally, the own\-law correction has

β2​Δ​\(S​Q\)=q​Z​tr​q\+O⁡\(tr​q\+1\);\\frac\{\\beta\}\{2\}\\Delta\(SQ\)=qZt^\{rq\}\+O\(t^\{rq\+1\}\);all its other difference terms start later\. These facts prove the continuous loss valuations in the statement\.

For completeness, the finite\-step bookkeeping is also coefficientwise\. At stepKK, set

GK=NK−c​I\+η​D\+β2​SK​z​B,NK\+1=NK−2​h​\(GK​NK\+NK​GK\)\+4​h2​GK​NK​GK\.G\_\{K\}=N\_\{K\}\-cI\+\\eta D\+\\frac\{\\beta\}\{2\}S\_\{K\}zB,\\qquad N\_\{K\+1\}=N\_\{K\}\-2h\(G\_\{K\}N\_\{K\}\+N\_\{K\}G\_\{K\}\)\+4h^\{2\}G\_\{K\}N\_\{K\}G\_\{K\}\.Thus\[hn​zj​ηd\]​NK\+1\[h^\{n\}z^\{j\}\\eta^\{d\}\]N\_\{K\+1\}is its previous value plus−2\-2times the two\-factor convolution athn−1​zj​ηdh^\{n\-1\}z^\{j\}\\eta^\{d\}, plus44times the three\-factor convolution athn−2​zj​ηdh^\{n\-2\}z^\{j\}\\eta^\{d\}\. The directDDterm hashh\-degree zero and weight one; theBBforcing hasSKS\_\{K\}\-degree at leastq−1q\-1and weightqq; the prefactorshhandh2h^\{2\}supply the missing one and two degrees\. Induction overnnin these explicit convolutions gives zero forn<j​q\+dn<jq\+d, a step\-count\-dependent scalar multiple ofBjB^\{j\}at the lowest degreej​qjq, and the first possible one\-DDterm atj​q\+1jq\+1\. These discrete coefficients need not equal the continuous coefficients at a fixed step count\. The scalar update

ai,K\+1=ai,K−4​h​ai,K​Ei,K\+4​h2​ai,K​Ei,K2,Ei,K=ai,K−Ti\+β2​ℓi​QK,a\_\{i,K\+1\}=a\_\{i,K\}\-4ha\_\{i,K\}E\_\{i,K\}\+4h^\{2\}a\_\{i,K\}E\_\{i,K\}^\{2\},\\qquad E\_\{i,K\}=a\_\{i,K\}\-T\_\{i\}\+\\tfrac\{\\beta\}\{2\}\\ell\_\{i\}Q\_\{K\},gives the same two\-step feedback induction\. Its linearized kernel has the same degree\-at\-most\-m\+1m\+1property ingig\_\{i\}, so barycentric annihilation and the preceding loss calculation give no discrete loss coefficient belowr​qrq\.

Write the exact map asMK\+1=MK\+h​F​\(MK\)\+h2​J​\(MK\)M\_\{K\+1\}=M\_\{K\}\+hF\(M\_\{K\}\)\+h^\{2\}J\(M\_\{K\}\), and letmn​\(K\)m\_\{n\}\(K\)denote any coefficient athnh^\{n\}\. Directly from the map,

mn​\(K\+1\)−mn​\(K\)=\[hn−1\]​F​\(MK\)\+\[hn−2\]​J​\(MK\)\.m\_\{n\}\(K\+1\)\-m\_\{n\}\(K\)=\[h^\{n\-1\}\]F\(M\_\{K\}\)\+\[h^\{n\-2\}\]J\(M\_\{K\}\)\.The first term has degree at mostn−1n\-1inKKand the second at mostn−2n\-2\. HencedegK⁡mn≤n\\deg\_\{K\}m\_\{n\}\\leq n; its highestKnK^\{n\}coefficient comes only fromFFand is the continuous Taylor coefficient\. After composition with either loss, the leadingKr​qK^\{rq\}coefficient is therefore the continuousCq,rζC^\{\\zeta\}\_\{q,r\}displayed below\. This is an algebraic coefficient induction for allq,r≥3q,r\\geq 3, not an inference from numerical cells\. ∎

### A\.2Loss coefficients and actual weight updates

Withγ=\(−4\)q​c2​β/\(2​q\!\)\\gamma=\(\-4\)^\{q\}c^\{2\}\\beta/\(2q\!\)andZ=γr​Jr/\[cr−1​\(r−1\)\!\]Z=\\gamma^\{r\}J\_\{r\}/\[c^\{r\-1\}\(r\-1\)\!\], the preceding calculation gives

L\+ζ​\(t\)−L−ζ​\(t\)\\displaystyle L\_\{\+\}^\{\\zeta\}\(t\)\-L\_\{\-\}^\{\\zeta\}\(t\)=Cq,rζ​tr​q\+O⁡\(tr​q\+1\),\\displaystyle=C^\{\\zeta\}\_\{q,r\}t^\{rq\}\+O\(t^\{rq\+1\}\),Cq,rG\\displaystyle C^\{G\}\_\{q,r\}=Z⁡\[1r\+q⁡\(q−1\)\(r−1\)​q\+2\],\\displaystyle=Z\\left\[\\frac\{1\}\{r\}\+\\frac\{q\(q\-1\)\}\{\(r\-1\)q\+2\}\\right\],Cq,rP\\displaystyle C^\{P\}\_\{q,r\}=Z⁡\[q\+1r\+q⁡\(q−1\)\(r−1\)​q\+2\]\.\\displaystyle=Z\\left\[q\+\\frac\{1\}\{r\}\+\\frac\{q\(q\-1\)\}\{\(r\-1\)q\+2\}\\right\]\.\(29\)Both coefficients are nonzero whenJr≠0J\_\{r\}\\neq 0\. Analyticity of the polynomial vector field justifies the local expansions\. The exact weight\-gradient map used in the proof is

Ms,k\+1=Ms,k−2​h​\{Gs​\(Ms,k\),Ms,k\}\+4​h2​Gs​\(Ms,k\)​Ms,k​Gs​\(Ms,k\)\.M\_\{s,k\+1\}=M\_\{s,k\}\-2h\\\{G\_\{s\}\(M\_\{s,k\}\),M\_\{s,k\}\\\}\+4h^\{2\}G\_\{s\}\(M\_\{s,k\}\)M\_\{s,k\}G\_\{s\}\(M\_\{s,k\}\)\.\(30\)For fixed integerkk, both loss differences have nohhcoefficient below degreer​qrq\. The degree\-r​qrqcoefficient is a polynomial inkkof degree exactlyr​qrq, with leading coefficientCq,rζC^\{\\zeta\}\_\{q,r\}\. It is therefore nonzero for all but finitely many positive integer step counts\. This assertion concerns small step size at fixed step count, not long\-time or stochastic\-training performance\.

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