arXiv:2606.25826v2

Weight geometry governs functional memory in complex systems

Elkaïoum M. Moutuou, Habib Benali

cond-mat.dis-nncs.SImath-phq-bio.NC37A6046L5505C5060J10

Paper metadata

Original submission: June 24, 2026
Latest revision: July 22, 2026 · version 2
Claimed MSC: 37A60, 46L55, 05C50, 60J10
arXiv categories: cond-mat.dis-nn, cs.SI, math-ph, q-bio.NC
Paper license: CC BY 4.0 ↗
Comments: GFM comparison with random models added; few changes in the abstract, 51 pages

Abstract

Complex systems, from gene regulatory networks to neural circuits and ecological food webs, exhibit rich functional behaviour that topology alone does not capture. Yet functional complexity remains difficult to quantify independently of structural organisation. Here we introduce a thermodynamic framework in which functional complexity is characterised through the hierarchical organisation of functional memory, quantifying how the influence of past interactions is distributed and progressively compressed across scales. Across thirty-four empirical networks spanning biological, ecological, social, technological, and biophysical systems and several orders of magnitude in size and density, real interaction strengths organise functional memory at greater hierarchical depth than random weight assignment on the same topology in every domain studied. The framework further reveals that functional memory occupies a remarkably low-dimensional space, collapsing onto four recurrent dynamical organisations. Comparisons with null models that selectively perturb weighted transport geometry, mesoscale wiring, and directionality show that these structural ingredients play distinct roles: weighted transport geometry systematically governs memory depth, whereas mesoscale wiring organises memory across scales and directionality modulates the response of the cascade to structural perturbation. The same comparisons provide an operational criterion for determining whether network weights encode functionally meaningful structure beyond topology. These results establish weighted transport geometry as a primary organiser of functional memory and provide a quantitative framework for studying functional complexity in directed weighted networks.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsContains unsupported statements

Several central interpretations of the proposed memory signature are not supported by the definitions or controls: protocol independence is stronger than the proved scale cancellation, the unsigned Wasserstein distance does not encode direction, and the geometric null model retains the original edgewise expected weight pattern.

Protocol-independence claimNot able to verify

Only a scale factor cancels; arbitrary monotone protocols can change the signature

Pages 5–7 · Section S4 and the protocol-independence claim · arXiv:2606.25826v2

Section S4 shows cancellation of the single scale parameter $\kappa$ . The remaining exponent $\alpha$ and the distribution of protocol points enter the ratios and normalized masses. A general monotone reparameterization preserves ordering but not those ratios, so the broader claim that the signature is independent of the specific protocol is not established.

Directional interpretation of W1Incorrect

A positive Wasserstein distance does not show movement toward deeper scales

Pages 4–6 · interpretation of Figure 2 · arXiv:2606.25826v2

The first Wasserstein distance is nonnegative and symmetric. Its positivity shows that two distributions differ, but contains no sign or orientation from which redistribution toward deeper rather than shallower scales can be inferred. A directional statistic or an additional stochastic-order comparison would be needed.

Geometric-free modelNot able to verify

Independent edge resampling does not erase the original expected placement pattern

Pages 7–9 · definition and interpretation of the GFM control · arXiv:2606.25826v2

Each edge is sampled independently with mean equal to that same edge's original weight. Consequently, for a path with distinct edges, the expected product equals the original product, so the ensemble retains the original spatially indexed expected weight pattern. The control therefore does not by itself isolate the effect of weight geometry in the manner claimed.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The algebraic argument proves only a narrow scale invariance, and the statistical controls do not justify the directional and causal interpretations assigned to them.

Section S4Incomplete as written

The invariance argument is extended beyond its proved parameter

Pages 5–7 · Section S4 · arXiv:2606.25826v2

The calculation removes $\kappa$ but supplies no invariance under changes of $\alpha$ or under arbitrary monotone protocol maps. Those transformations alter the normalized scale spacing used in the signature. No additional lemma establishes the stronger conclusion.

Null-model inferenceIncorrect as written

The control does not randomize the edgewise mean field

Pages 7–9 · GFM analysis · arXiv:2606.25826v2

Independence removes cross-edge sampling correlations, but preserving each individual edge mean preserves the full original expected matrix. Thus the comparison cannot be interpreted as removing weight placement while holding only the marginal distribution fixed. A verified repair would require a genuinely exchangeable or explicitly permuted null model and a renewed analysis; none is supplied.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.25826v2
Authors listed: Elkaïoum M. Moutuou, Habib Benali
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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P(delta)

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Weight geometry governs functional memory in complex systems

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Only a scale factor cancels; arbitrary monotone protocols can change the signature

A positive Wasserstein distance does not show movement toward deeper scales

Independent edge resampling does not erase the original expected placement pattern

The invariance argument is extended beyond its proved parameter

The control does not randomize the edgewise mean field

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