arXiv:2606.28295v1

Equilibrium as a Limit: The Competitive Canon Nested in an Adaptive, Information-Theoretic Economy

Avishek Bhandari

econ.TH91B5091B5294A1737A3591A22

Paper metadata

Original submission: June 26, 2026
Claimed MSC: 91B50, 91B52, 94A17, 37A35, 91A22
arXiv categories: econ.TH
Paper license: CC BY-NC-ND 4.0 ↗
Comments: EL classification: C62, D58, D80, E10

Abstract

The competitive equilibrium of general equilibrium theory exists as a fixed point and is, by the theorys own results on aggregate excess demand, in general silent on whether that fixed point is unique, stable, or attained. This paper takes the economy to be not a configuration to be solved for but a process to be recovered, an asymptotically mean stationary information source carrying a partially identified operator of statistical dependence, populated by agents that are finite-capacity information channels. Within this adaptive order the competitive, rational expectations equilibrium is recovered exactly, as a joint limit taken along an explicit scaling path. Three parameter limits and two fixed-point conditions deliver it, the entropy rate falls to zero, agent channel capacity diverges, selection intensity grows infinitely sharp, adaptive learning reaches its expectationally stable rest point, and the recovered structure ceases to coevolve. At that corner the limiting object satisfies the axioms of the canon and its rest state is a Walrasian equilibrium, away from it the adaptive economy is a strict generalisation, carrying a positive entropy rate and a recovered dependence structure that the equilibrium primitive cannot express. We give the nesting as a theorem, establish the result by result correspondence with existence, with the Sonnenschein Mantel Debreu indeterminacy, and with the regular economies recovery, and characterise exactly what the equilibrium limit erases.

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Audited against arXiv v1

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.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsContains wrong statements

Several formal claims used by the proposed nesting theorem are false or not mathematically specified. In particular, arbitrary random preference relations are not equivalent to random utilities, and the claimed universal eigenpair-recovery threshold is not a valid spiked-covariance theorem.

Definition 6Incorrect

Random relations are not equivalent to random-utility fields

Page 8 · Definition 6 · arXiv:2606.28295v1

A scalar utility function $u:X\to\mathbb R$ induces the complete transitive relation $a\succeq b$ exactly when $u(a)\geq u(b)$ . A probability law on such functions can therefore represent only stochastic choices arising from mixtures of complete transitive rankings and obeying the associated random-utility constraints. Definition 6 allows arbitrary binary relations, explicitly including intransitive cycles and context-dependent reversals, and then calls a random-utility representation 'equivalent.' For three alternatives, the deterministic cycle $a\succ b\succ c\succ a$ is an immediate counterexample: no real-valued $u$ can satisfy all three strict inequalities.

Full paper, version 1 ↗

Proposition 5(iv)Incorrect

The eigenpair threshold and consistency claim are not universal

Pages 7–8 · Proposition 5(iv) and proof sketch · arXiv:2606.28295v1

The paper asserts recovery whenever $\lambda_{\max}>\nu\sqrt{d/N}$ without specifying a covariance model, normalization, aspect-ratio limit, or bulk spectrum. Even in the standard rank-one spiked covariance model with $d/N\to c>0$ , the BBP transition is expressed relative to the bulk covariance, and above threshold the empirical eigenvector has a limiting overlap strictly below one for a fixed spike; it is not a consistent estimator of the population eigenvector. Thus neither the threshold nor the stated conclusion follows from the cited phase transition.

Theorem 11Not able to verify

The hypotheses do not imply a Walrasian equilibrium

Pages 11–13 · Theorem 11 and Lemma 12 · arXiv:2606.28295v1

Conditions (A1)–(A4) concern a Lyapunov function, selection, learning, and structural fixity. No commodity space, endowments, budget correspondence, excess-demand function, or market-clearing condition is defined in the theorem. The proof nevertheless adds 'satisfying the market-clearing conditions' only at the conjunction step. That conclusion is not a formal consequence of the listed assumptions. Lemma 12 likewise invokes uniform convergence in the other parameters, but Definition 10 and (A1)–(A4) contain no such uniformity hypothesis.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The central arguments are proof sketches that assume the conclusions they need: a covariance is identified with an unspecified operator, market clearing appears without a model that defines it, and path independence is deduced from uniform convergence that was never assumed or established.

Proof of Proposition 5Incorrect or incomplete

Second-order comovement does not identify the asserted operator without a forward model

Pages 7–8 · proof sketch of Proposition 5 · arXiv:2606.28295v1

A covariance form is a positive self-adjoint operator, but the paper's directed dependence operator $\widehat T$ is introduced independently. No equation maps $\widehat T$ to the observed covariance. Consequently, the sentence that covariance 'determines, and is determined by' $(\widehat T+\widehat T^\top)/2$ is an additional identification assumption, not a proof. The later resolvent and finite-sample claims therefore have no defined statistical experiment from which they can follow.

Proof of Theorem 11 and Lemma 12Incorrect or incomplete

The limiting conclusion is inserted rather than derived

Pages 28–29 · proofs of Theorem 11 and Lemma 12 · arXiv:2606.28295v1

Laplace concentration of a finite Gibbs law proves only choice concentration at a unique maximizer; it does not construct prices or prove feasibility and market clearing. The final paragraph simply declares those missing properties. The Moore–Osgood interchange in Lemma 12 requires uniform convergence or joint continuity at the corner, but the proof asserts both without deriving them from the standing assumptions.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.28295v1
Authors listed: Avishek Bhandari
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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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

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P(delta)

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requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

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Equilibrium as a Limit: The Competitive Canon Nested in an Adaptive, Information-Theoretic Economy

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Random relations are not equivalent to random-utility fields

The eigenpair threshold and consistency claim are not universal

The hypotheses do not imply a Walrasian equilibrium

Second-order comovement does not identify the asserted operator without a forward model

The limiting conclusion is inserted rather than derived

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