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arxiv: 2604.08784 · v1 · submitted 2026-04-09 · ❄️ cond-mat.stat-mech

Path-Integral Formulation of Unavoidable Canonical Nonlinearity: Dynamic Discretization Cost over Variable Supports

Koretaka Yuge This is my paper

Pith reviewed 2026-05-10 16:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords canonical nonlinearitypath integralinformation geometrydiscretization coststatistical thermodynamicsFisher metrice-mixtureexponential family
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The pith

A path-integral formulation extends the measure of unavoidable canonical nonlinearity to arbitrary discrete distributions including those with different supports.

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

The paper develops a path-integral version of unavoidable canonical nonlinearity to capture the cumulative information-geometric cost of discretization when moving between distinct classical discrete distributions. Standard Kullback-Leibler divergence misses the intrinsic nonlinearity that arises from discretizing continuous Gaussian families, and prior work on unavoidable canonical nonlinearity handled only single distributions. By constructing a path that keeps the canonical distribution as an exponential family through the e-mixture and enforces covariant cell changes through the harmonic mixture of the second-moment matrix, the new measure works for states whose supports differ. This supplies an explicit way to quantify canonical nonlinearity between different systems and to separate the total effect into the unavoidable discretization part plus a residual contribution.

Core claim

The central claim is that defining the path via retention of the canonical distribution as an exponential family through the e-mixture, which produces an arithmetic mixture of the Fisher metric, together with covariant discretization enforced by the harmonic mixture of the second-moment matrix M, yields a path-integral unavoidable canonical nonlinearity that quantifies the geometric cost between arbitrary states, including those with essentially different supports, and permits a natural decomposition of the overall canonical nonlinearity into the unavoidable component and a residual term.

What carries the argument

The Path-Integral Unavoidable Canonical Nonlinearity (PUCN) constructed by combining the e-mixture path for the exponential family with the harmonic mixture of the discretization second-moment matrix.

If this is right

  • Explicit quantification of canonical nonlinearity becomes possible between different CDOS systems.
  • The total canonical nonlinearity decomposes naturally into the unavoidable discretization contribution and a residual part.
  • Geometric cost can be measured between arbitrary states, including those whose probability supports differ substantially.
  • The measure extends previous unavoidable canonical nonlinearity results, which were restricted to single continuous distributions.

Where Pith is reading between the lines

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

  • The same path construction could be tested on simple lattice models to see whether the decomposed terms align with known equilibrium properties.
  • If the decomposition holds, it may clarify how microscopic interaction parameters translate into observed nonlinearities in thermodynamic observables.
  • The approach supplies a concrete starting point for comparing information-geometric costs across systems whose variable ranges change during a process.

Load-bearing premise

The chosen path that retains the canonical distribution via the e-mixture and handles discretization changes via the harmonic mixture of the second-moment matrix correctly captures the information-geometric cost and uncertainty in parameter variations.

What would settle it

Direct numerical evaluation of total canonical nonlinearity between two discrete distributions with different supports, followed by checking whether that value equals the sum of the computed PUCN along the defined path and the remaining residual term.

read the original abstract

In the statistical thermodynamics of classical discrete systems, the map from microscopic interactions to thermodynamic equilibrium configurations generally exhibits complex nonlinearity, known as "canonical nonlinearity" (CN). While conventionally characterized by the Kullback-Leibler (KL) divergence, this approach inevitably misses intrinsic nonlinearities arising from the discretization of continuous Gaussian families themselves. This intrinsic effect of unavoidable CN (UCN), has recently been quantified within a transport-information-geometric framework. However, the UCN is fundamentally limited to evaluating the discretization-induced cost for a single continuous distribution. It therefore does not capture the information-geometric cost between a continuous Gaussian reference and an actual discrete distribution, nor between states with fundamentally different supports, making it conceptually unclear how to decompose the overall CN. To address this limitation, we propose the Path-Integral UCN (PUCN) quantifying the cumulative information-geometric cost between distinct distributions. The PUCN adopts a path by (i) retaining the canonical distribution as an exponential family via the $e$-mixture (geometric mean) of the base measure, leading to an arithmetic mixture of the Fisher metric as the CN standard, and (ii) enforcing covariant changes in the discretization cell through the harmonic mixture of its second-moment matrix $M$, reflecting the uncertainty in parameter variations on the statistical manifold. The resulting PUCN provides a flexible measure of the geometric cost between arbitrary states, including those with essentially different supports. This formulation enables an explicit quantification of CN between different CDOS systems and a natural decomposition of the total CN into the UCN and a residual contribution, which has not been clearly separated in existing approaches.

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

3 major / 2 minor

Summary. The paper proposes the Path-Integral Unavoidable Canonical Nonlinearity (PUCN) to extend the prior single-distribution UCN framework to arbitrary pairs of discrete distributions (including variable supports). It defines a path on the statistical manifold by retaining the canonical distribution as an exponential family via the e-mixture of the base measure (yielding an arithmetic mixture of the Fisher metric) and enforcing covariant discretization via the harmonic mixture of the second-moment matrix M; the resulting line integral is claimed to quantify the cumulative geometric cost and to decompose total canonical nonlinearity (CN) into UCN plus a residual term.

Significance. If the specific mixture choices and path construction can be shown to be canonical and path-independent, the PUCN would supply a missing decomposition of CN that standard KL-based measures do not provide and would allow direct comparison of CDOS systems with differing supports. This would be a useful technical advance within the information-geometric treatment of discretization costs in classical statistical mechanics.

major comments (3)
  1. [formulation of the path] The central construction (abstract and formulation section) asserts without derivation that the e-mixture of the base measure correctly preserves the exponential-family structure while producing the arithmetic mixture of the Fisher metric as the appropriate CN standard. No proof is given that this choice is unique or that alternative mixtures (e.g., m-mixture) produce inconsistent or non-geometric results.
  2. [covariant discretization] The harmonic mixture of the second-moment matrix M is introduced to enforce covariant discretization under parameter variation, yet the manuscript provides no explicit demonstration that this rule recovers the single-distribution UCN when supports coincide, nor that the resulting PUCN is path-independent.
  3. [decomposition claim] The claimed natural decomposition total CN = UCN + residual is stated as a consequence of the path integral, but no reduction formula or limiting case is derived showing that the residual vanishes or matches the prior UCN when the two distributions share the same support.
minor comments (2)
  1. [notation] Notation for the second-moment matrix M and the parameter-variation uncertainty is introduced without a self-contained definition or reference to its origin in the earlier UCN paper.
  2. [abstract] The abstract is dense and would benefit from a short schematic diagram illustrating the e-mixture path versus a direct KL comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below, agreeing that explicit derivations and limiting cases are required to support the claims. Revisions will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: [formulation of the path] The central construction (abstract and formulation section) asserts without derivation that the e-mixture of the base measure correctly preserves the exponential-family structure while producing the arithmetic mixture of the Fisher metric as the appropriate CN standard. No proof is given that this choice is unique or that alternative mixtures (e.g., m-mixture) produce inconsistent or non-geometric results.

    Authors: We agree a derivation is absent and will add a dedicated subsection in revision. The e-mixture is selected to preserve the exponential-family form of the canonical distribution, which is required for consistency with the statistical mechanics definition of CN. The arithmetic mixture of the Fisher metric then follows from the e-connection in information geometry. We will also include an argument for uniqueness by demonstrating that an m-mixture breaks the exponential structure, yielding a non-canonical and geometrically inconsistent measure. revision: yes

  2. Referee: [covariant discretization] The harmonic mixture of the second-moment matrix M is introduced to enforce covariant discretization under parameter variation, yet the manuscript provides no explicit demonstration that this rule recovers the single-distribution UCN when supports coincide, nor that the resulting PUCN is path-independent.

    Authors: We acknowledge the missing demonstrations. The revised manuscript will contain an explicit limiting-case calculation showing recovery of the original UCN when supports coincide, with the harmonic mixture reducing to the standard second-moment matrix. For path-independence, the construction follows a geodesic on the mixed metric; we will add either a proof under the chosen mixtures or a clear statement of the conditions under which independence holds, together with numerical checks. revision: partial

  3. Referee: [decomposition claim] The claimed natural decomposition total CN = UCN + residual is stated as a consequence of the path integral, but no reduction formula or limiting case is derived showing that the residual vanishes or matches the prior UCN when the two distributions share the same support.

    Authors: We agree the reduction must be derived. We will insert an explicit reduction formula and limiting-case analysis demonstrating that, when supports coincide, the residual term vanishes identically and the path integral reduces to the prior UCN, thereby justifying the decomposition total CN = UCN + residual both analytically and via the path construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity in PUCN path-integral extension

full rationale

The paper starts from the established single-distribution UCN (cited as recently quantified in a transport-information-geometric framework) and explicitly addresses its limitation for variable-support cases by proposing a new path-integral construction. The e-mixture and harmonic mixture of M are introduced as the adopted path that retains the exponential-family form and enforces covariance, with the resulting PUCN claimed to enable decomposition into UCN plus residual. These steps are presented as a proposal rather than a reduction of the output to prior definitions or fitted inputs by construction; no equations equate PUCN directly to re-expressed UCN quantities, and the central claims introduce new content for inter-distribution costs without self-definitional collapse or load-bearing reliance on unverified self-citations for uniqueness.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on extending the transport-information-geometric framework with two specific mixture operations for path definition; no free parameters or invented physical entities are introduced in the abstract.

axioms (2)
  • domain assumption Canonical distribution retained as exponential family via e-mixture of the base measure, leading to arithmetic mixture of the Fisher metric
    Invoked to define the CN standard along the path.
  • domain assumption Covariant changes in discretization cell enforced through harmonic mixture of its second-moment matrix M
    Used to reflect uncertainty in parameter variations on the statistical manifold.
invented entities (1)
  • Path-Integral UCN (PUCN) no independent evidence
    purpose: Quantifying cumulative information-geometric cost between distinct distributions with different supports
    Newly proposed formulation to overcome single-distribution limitation of prior UCN.

pith-pipeline@v0.9.0 · 5593 in / 1416 out tokens · 47884 ms · 2026-05-10T16:39:45.585958+00:00 · methodology

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Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

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    J.M. Sanchez, F. Ducastelle, and D. Gratias, Physica A 128, 334 (1984)

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    S. Kullback and R. A. Leibler, Ann. Math. Stat. 22, 79 (1951)

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    K. Yuge, J. Phys. Soc. Jpn. 85, 024802 (2016)

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    Unavoidable Canonical Nonlinearity Induced by Gaussian Measures Discretization

    K. Yuge, arXiv:2601.08381 [cond-mat.stat-mech] (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025