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arxiv: 2605.15667 · v1 · pith:SDZQA7LYnew · submitted 2026-05-15 · ❄️ cond-mat.stat-mech · cs.IT· math-ph· math.IT· math.MP

A Finite-State Gibbs Construction from a Recognition Cost

Megan Simons , Jonathan Washburn This is my paper

Pith reviewed 2026-05-19 19:59 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cs.ITmath-phmath.ITmath.MP
keywords Gibbs distributionRecognition Composition Lawratio costconvex dualityKullback-Leibler divergencefinite state spacemultinomial countingstatistical mechanics
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The pith

A ratio-cost construction from the Recognition Composition Law induces the standard Gibbs distribution on finite states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an alternative route to the Gibbs distribution on a finite outcome space by starting from a recognition cost rather than an externally supplied energy. After defining the Recognition Composition Law as the normalized d'Alembert degree-two closure J(x) = 1/2(x + x^{-1}) - 1 with unit log-curvature calibration, the induced cost vector allows multinomial counting and convex duality to recover both the Gibbs weights and the relation that the difference in recognition free energies equals temperature times the Kullback-Leibler divergence. A reader would care if this shows that the familiar Boltzmann form can emerge from ratio-based costs and counting arguments without presupposing maximum entropy. The work supplies supporting technical results such as a non-asymptotic Stirling bound and includes an explicit three-state comparison to other cost models.

Core claim

Given the induced cost vector X_ω = J(r_ω) from the RCL, multinomial counting and convex duality recover the finite-state Gibbs weights and the identity F_R(q) - F_R(p) = T_R D_KL(q || p). The entropy-maximization steps are classical once the cost is fixed. New technical content includes a non-asymptotic Stirling bound and soft-shell constrained-type theorems for real-valued costs.

What carries the argument

The Recognition Composition Law with J(x) = ½(x + x^{-1}) - 1, which turns reference ratios r_ω into the cost vector X_ω that drives the recovery of Gibbs weights via counting and duality.

If this is right

  • The entropy maximization procedure yields the Gibbs law in the usual way after the RCL cost is substituted for energy.
  • Non-asymptotic Stirling bounds apply to the multinomial coefficients with real-valued costs.
  • Soft-shell constrained-type theorems characterize the large-deviation behavior for these costs.
  • The three-state example demonstrates how the RCL Gibbs law differs from squared-log, affinity, and Tsallis forms at equal mean cost, along with sample-size power estimates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the axioms hold more generally, similar cost-based derivations might apply to other equilibrium distributions beyond finite states.
  • The free-energy to KL identity provides a direct bridge between recognition costs and information-theoretic measures that could be explored in inference problems.
  • A natural extension would be to examine whether physical systems with measurable ratio costs exhibit the predicted sample-size scaling in their fluctuation statistics.

Load-bearing premise

The framework requires that axioms (A1) through (A3) are accepted without further derivation and that the normalized d'Alembert degree-two closure is the appropriate Recognition Composition Law with unit log-curvature calibration.

What would settle it

A direct computation or measurement in a finite-state system showing that the frequencies minimizing the recognition free energy at fixed mean cost do not coincide with the multinomial-derived Gibbs probabilities would falsify the central recovery claim.

Figures

Figures reproduced from arXiv: 2605.15667 by Jonathan Washburn, Megan Simons.

Figure 1
Figure 1. Figure 1: Quadratic lower bound on J. The RCL cost J(x) = 1 2 (x + x −1 ) − 1 (solid curve) and its quadratic surrogate (log x) 2/2 (dashed curve). The two curves share a common second-order expansion at x = 1; the gap grows for x ≪ 1 or x ≫ 1, where J(e t ) grows like 1 2 e |t| in the log-coordinate t = log x (equivalently, J(x) grows linearly in max(x, 1/x)), while (log x) 2/2 grows only quadratically in |t|. line… view at source ↗
Figure 2
Figure 2. Figure 2: Gibbs weights pk as a function of the target mean cost ⟨X⟩ for the three￾state example with costs X = (0, 0.5, 1), plotted over the positive-temperature range Emc ∈ (0, 1 2 ) in which β > 0 (Proposition 5.2); Emc denotes the same constraint as the column ⟨X⟩ (target) in [PITH_FULL_IMAGE:figures/full_fig_p038_2.png] view at source ↗
read the original abstract

On a finite outcome space, the canonical Gibbs distribution is usually obtained by maximizing Shannon entropy at fixed mean of an externally supplied energy functional. This paper studies the finite-state consequences of a ratio-cost construction instead: after adopting the normalized d'Alembert degree-two closure called the Recognition Composition Law (RCL), with unit log-curvature calibration at the reference ratio, the continuous nontrivial positive branch is $J(x)=\tfrac12(x+x^{-1})-1=\cosh(\log x)-1$. Given the induced cost vector $X_\omega=J(r_\omega)$, multinomial counting and convex duality recover the finite-state Gibbs weights and the identity $F_{\mathrm{R}}(q)-F_{\mathrm{R}}(p)=T_{\mathrm{R}}\,D_{\mathrm{KL}}(q\Vert p)$; the entropy-maximization steps are classical once the cost is fixed. New technical content includes a non-asymptotic Stirling bound and soft-shell constrained-type theorems for real-valued costs. A three-state example compares the Gibbs law to squared-log, affinity-as-energy, and Tsallis alternatives at the same cost vector and mean-cost constraint, with sample-size power calculations at fixed RCL ground truth. The framework is conditional on axioms (A1)--(A3) and restricted to finite outcome spaces with strictly positive weights; it does not derive the composition law from a more primitive principle.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that on a finite outcome space, adopting the normalized d'Alembert degree-two closure as the Recognition Composition Law (RCL) with unit log-curvature calibration yields the cost function J(x) = ½(x + x^{-1}) - 1. Given the induced cost vector X_ω = J(r_ω), standard multinomial counting and convex duality then recover the finite-state Gibbs weights together with the identity F_R(q) - F_R(p) = T_R D_KL(q || p). New technical contributions include a non-asymptotic Stirling bound and soft-shell constrained-type theorems; a three-state example compares the resulting Gibbs law to squared-log, affinity-as-energy, and Tsallis alternatives under the same mean-cost constraint, with accompanying sample-size power calculations. The framework is explicitly conditional on axioms (A1)-(A3) and restricted to finite spaces with strictly positive weights.

Significance. If the adopted RCL and axioms are accepted, the work supplies an alternative route to the canonical Gibbs distribution that begins from a ratio-cost functional rather than an externally supplied energy. It supplies concrete new technical tools (non-asymptotic Stirling bound, soft-shell theorems) and a worked three-state comparison that quantifies distinguishability from other distributions at fixed cost vector. These elements could be useful for finite-state models in which ratio-based costs arise naturally.

major comments (2)
  1. [Abstract] Abstract and the section defining the RCL: the recovery of the Gibbs weights and the KL identity is obtained by classical multinomial counting plus convex duality once the cost vector X_ω = J(r_ω) is fixed by the chosen J. Because the paper states that axioms (A1)-(A3) are adopted rather than derived from a more primitive principle, the central claim is an equivalence under a specific cost rather than a derivation of the Gibbs measure from ratio costs alone. A load-bearing justification or independent motivation for this particular composition law is therefore required.
  2. [Section on non-asymptotic Stirling bound] Section presenting the non-asymptotic Stirling bound: the abstract highlights this bound as new technical content, yet the reader's assessment notes that soundness cannot be confirmed without the full derivation or verification. If the bound is used to support the finite-state construction, an explicit statement of the error term and its dependence on the cost vector should be supplied.
minor comments (2)
  1. [Introduction] Notation for F_R, T_R and the reference ratio should be introduced with a single consolidated definition early in the text to avoid repeated cross-referencing.
  2. [Three-state example] The three-state example would benefit from an explicit table listing the numerical values of the cost vector, mean-cost constraint, and resulting probabilities for each alternative distribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and describe the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section defining the RCL: the recovery of the Gibbs weights and the KL identity is obtained by classical multinomial counting plus convex duality once the cost vector X_ω = J(r_ω) is fixed by the chosen J. Because the paper states that axioms (A1)-(A3) are adopted rather than derived from a more primitive principle, the central claim is an equivalence under a specific cost rather than a derivation of the Gibbs measure from ratio costs alone. A load-bearing justification or independent motivation for this particular composition law is therefore required.

    Authors: We agree that the central contribution is an equivalence obtained once the normalized d'Alembert degree-two RCL is adopted together with axioms (A1)-(A3), rather than a derivation of the RCL itself from more primitive axioms. The manuscript already states that it does not derive the composition law from a more primitive principle. In revision we will add a dedicated paragraph in the RCL-definition section that supplies independent motivation for this particular closure: it arises naturally when costs are defined on ratios (as in certain recognition or relative-likelihood models) and yields a strictly convex J that recovers the classical Gibbs form and KL identity via standard multinomial counting and convex duality. We will also clarify the scope of the claim in the abstract. revision: yes

  2. Referee: [Section on non-asymptotic Stirling bound] Section presenting the non-asymptotic Stirling bound: the abstract highlights this bound as new technical content, yet the reader's assessment notes that soundness cannot be confirmed without the full derivation or verification. If the bound is used to support the finite-state construction, an explicit statement of the error term and its dependence on the cost vector should be supplied.

    Authors: We accept the point that the non-asymptotic Stirling bound is advertised as new technical content and that its derivation must be verifiable. In the revised manuscript we will move the full proof to a self-contained appendix, state the explicit error term (including its dependence on the cost vector X_ω and on sample size n), and add a short remark on how the bound is applied in the soft-shell theorems. This will allow readers to assess soundness directly. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation conditional on explicit axioms but self-contained

full rationale

The paper states that it adopts axioms (A1)-(A3) to define the Recognition Composition Law without deriving the composition law from a more primitive principle, then applies the resulting J(x) to induce the cost vector X_ω = J(r_ω). From there it invokes classical multinomial counting and convex duality to recover the Gibbs weights and the identity F_R(q) - F_R(p) = T_R D_KL(q || p). This recovery is presented as standard once the cost is fixed, with no equation reducing the output to the input by construction, no fitted parameter renamed as prediction, and no load-bearing self-citation. The framework is therefore an equivalence under chosen axioms rather than a tautological loop, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The paper introduces the RCL as a new entity without independent evidence or derivation from primitives, relies on three unspecified axioms (A1)-(A3), and uses one calibration choice; the recovery of Gibbs is performed with classical tools once the cost is fixed.

free parameters (1)
  • unit log-curvature calibration at the reference ratio
    Normalization of the d'Alembert degree-two closure is set to unit log-curvature to obtain the specific J(x).
axioms (1)
  • domain assumption Axioms (A1)--(A3)
    The framework is explicitly conditional on these axioms, which are not derived from a more primitive principle.
invented entities (1)
  • Recognition Composition Law (RCL) no independent evidence
    purpose: To define the cost functional J(x) = ½(x + x^{-1}) - 1 for the ratio-cost construction.
    Introduced as the normalized d'Alembert degree-two closure; no independent evidence or derivation from primitives is supplied.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel matches
    ?
    matches

    MATCHES: this paper passage directly uses, restates, or depends on the cited Recognition theorem or module.

    after adopting the normalized d’Alembert degree-two closure called the Recognition Composition Law (RCL), with unit log-curvature calibration at the reference ratio, the continuous nontrivial positive branch is J(x)=½(x+x^{-1})−1=cosh(logx)−1

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean absolute_floor_iff_bare_distinguishability echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Given the induced cost vector X_ω = J(r_ω), multinomial counting and convex duality recover the finite-state Gibbs weights and the identity F_R(q) - F_R(p) = T_R D_KL(q || p)

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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