Unavoidable Canonical Nonlinearity Induced by Gaussian Measures Discretization

Koretaka Yuge

arxiv: 2601.08381 · v11 · submitted 2026-01-13 · ❄️ cond-mat.stat-mech

Unavoidable Canonical Nonlinearity Induced by Gaussian Measures Discretization

Koretaka Yuge This is my paper

Pith reviewed 2026-05-16 15:00 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords canonical nonlinearityWasserstein distanceGaussian discretizationstatistical manifoldKL divergenceconfigurational density of statesdiscrete systemsFisher metric

0 comments

The pith

Discretizing Gaussian measures induces unavoidable nonlinearity in canonical averages of discrete systems.

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

The paper shows that canonical nonlinearity in mapping interactions to equilibrium configurations for discrete systems like substitutional alloys stems in part from the discretization of underlying continuous Gaussian measures, an effect missed by standard KL divergence on the discrete manifold. Using the 2-Wasserstein distance with a cost aligned to the Fisher metric, the authors derive an explicit expression for the distance in the limit where the discretization scale d vanishes. This limit equals a KL divergence tied to expected parallel translations of the continuous Gaussian on the statistical manifold. The result supplies a transport-information-geometric view of the discretization-induced distortion and extends to non-Gaussian families. If correct, it means previous characterizations of nonlinearity in classical discrete systems were incomplete.

Core claim

The limiting 2-Wasserstein distance as discretization scale d approaches zero equals a KL divergence associated with the expected parallel translations of continuous Gaussian families, thereby characterizing the irreversible geometric distortion of the local measure that discretization imposes on classical discrete systems.

What carries the argument

The 2-Wasserstein distance whose cost function is aligned with the Fisher metric for Gaussian families, taken in the limit of vanishing discretization scale d to 0.

If this is right

Canonical nonlinearity in discrete systems contains an intrinsic discretization component invisible to conventional KL-based descriptions.
The limiting Wasserstein distance supplies a geometric interpretation equivalent to a KL divergence on expected parallel translations of the continuous Gaussian.
The W2-KL equivalence generalizes beyond Gaussian families to other continuous distributions.
Discretization-induced irreversible distortion of the local measure can be characterized by a standard KL divergence.

Where Pith is reading between the lines

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

Calculations of thermodynamic properties in lattice models may require an additive correction term derived from this limiting distance to capture the full nonlinearity.
The same Wasserstein construction could be applied to discretizations of other exponential families to unify transport and information geometry in statistical mechanics.
Prior estimates of configurational entropy or free energy in substitutional alloys likely underestimated the total nonlinearity by omitting the discretization contribution.

Load-bearing premise

The cost function chosen for the Wasserstein distance aligns with the Fisher metric of Gaussian families and the limit as discretization scale vanishes remains well-defined.

What would settle it

A direct numerical calculation of the 2-Wasserstein distance for a discretized Gaussian family that fails to recover the stated KL expression in the d to 0 limit.

read the original abstract

When we consider canonical averages for classical discrete systems, typically referred to as substitutional alloys, the map phi from many-body interatomic interactions to thermodynamic equilibrium configurations generally exhibits complicated nonlinearity. This canonical nonlinearity is fundamentally rooted in deviations of the discrete configurational density of states (CDOS) from continuous Gaussian families, and has conventionally been characterized by the Kullback-Leibler (KL) divergence on discrete statistical manifold. Thus, the previous works inevitablly missed intrinsic nonlinearities induced by discretization of Gaussian families, which remains invisible within conventional information-geometric descriptions. In the present work, we identify and quantify such unavoidable canonical nonlinearity by employing the 2-Wasserstein distance with a cost function aligned with the Fisher metric for Gaussian families. We derive an explicit expression for the Wasserstein distance in the limit of vanishing discretization scale d to 0. We further show that this limiting Wasserstein distance admits a clear geometric interpretation on the statistical manifold, equivalent to a KL divergence associated with the expected parallel translations of continuous Gaussian. Our framework thus provides a transport-information-geometric characterization of discretization-induced nonlinearity in classical discrete systems. In addition, we confirm that this W2-KL equivalence admits a natural generalization beyond Gaussian families. The correspondence reveals that the irreversible geometric distortion of the local measure induced by discretization, while extrinsic to information geometry alone, can generically be characterized by a standard KL divergence.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit d to 0 limit of a Fisher-aligned 2-Wasserstein distance on discretized Gaussians and equates it to a KL divergence on expected parallel translations, but the limit's well-definedness rests on unshown details.

read the letter

The central new piece is the claim that discretizing Gaussian families for classical discrete systems like alloys produces nonlinearity invisible to standard discrete KL divergence, and that this can be captured by taking the 2-Wasserstein distance with cost aligned to the Fisher metric, then taking the explicit limit as discretization scale d vanishes. They further map that limit to a KL divergence built from expected parallel translations on the continuous Gaussian manifold, with a stated generalization beyond Gaussians. That transport-information-geometric link is the concrete advance over prior KL-only characterizations of canonical nonlinearity in substitutional alloys. The abstract frames it cleanly as a geometric distortion induced by discretization that information geometry alone misses. If the derivations hold, it supplies a usable handle for quantifying that effect in alloy modeling. The soft spot is that the abstract gives no explicit form for the cost function, the discretization map, or the optimal transport plan between the weighted point masses, so it is impossible to verify whether the limit actually exists and remains finite without extra regularization or d-dependent rescaling. The stress-test concern about singularity or non-commuting limits is reasonable on the given information; the weakest assumption (that discrete CDOS deviation from continuous Gaussians makes the limit meaningful) is plausible but untested here. The math and citation pattern look standard rather than circular, but without the full steps or any numerical checks the soundness stays provisional. This is for people already working on information geometry applied to discrete statistical mechanics or alloy configurational entropy. A reader who cares about transport metrics on statistical manifolds would get value from the equivalence claim. It deserves a serious referee to check whether the limit and equivalence are rigorously established or require additional assumptions.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that discretization of continuous Gaussian measures induces an unavoidable canonical nonlinearity in classical discrete systems (e.g., substitutional alloys) that is invisible to standard KL divergence on the discrete statistical manifold. Using the 2-Wasserstein distance whose cost is aligned with the Fisher metric on the Gaussian parameter manifold, the authors derive an explicit expression for this distance in the limit d → 0 and show that the limiting object equals a KL divergence associated with expected parallel translations of the underlying continuous Gaussians. They further assert that the W2-KL equivalence generalizes beyond Gaussian families, furnishing a transport-information-geometric characterization of discretization-induced nonlinearity.

Significance. If the derivations are correct, the work supplies a concrete bridge between optimal transport and information geometry that quantifies an extrinsic geometric distortion caused by discretization. The explicit d → 0 limit and its interpretation as a KL divergence on parallel transports would be a useful addition to the toolkit for analyzing nonlinear maps from interatomic interactions to equilibrium configurations in statistical mechanics. The claimed parameter-free character and the generalization beyond Gaussians are positive features that could stimulate further applications in discrete systems.

major comments (2)

[Section deriving the d → 0 limit and the W2-KL equivalence] The central claim rests on the existence and finiteness of the d → 0 limit of the Fisher-aligned 2-Wasserstein distance between discrete CDOS measures. The manuscript must explicitly demonstrate that the optimal transport plan remains non-singular for weighted point masses and that the chosen cost function (aligned locally with the Fisher metric) yields a well-defined global limit without d-dependent rescaling; otherwise the equivalence to the stated KL divergence on expected parallel translations cannot be guaranteed.
[Section on geometric interpretation and parallel translations] The geometric interpretation equating the limiting Wasserstein distance to a KL divergence on expected parallel translations of continuous Gaussians requires a precise definition of the parallel transport operation and a step-by-step verification that the limit commutes with this construction. Without this, the claimed equivalence risks being formal rather than substantive.

minor comments (3)

[Abstract] Abstract contains a typographical error: 'inevitablly' should read 'inevitably'.
[Abstract and introduction] The phrase 'expected parallel translations of continuous Gaussian' is introduced without prior definition; a brief clarification or reference to the relevant equation would improve readability.
[Section on generalization] The generalization beyond Gaussian families is asserted but not illustrated; a short explicit example or statement of the conditions under which the W2-KL equivalence continues to hold would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments below, providing clarifications and indicating revisions where appropriate.

read point-by-point responses

Referee: The central claim rests on the existence and finiteness of the d → 0 limit of the Fisher-aligned 2-Wasserstein distance between discrete CDOS measures. The manuscript must explicitly demonstrate that the optimal transport plan remains non-singular for weighted point masses and that the chosen cost function (aligned locally with the Fisher metric) yields a well-defined global limit without d-dependent rescaling; otherwise the equivalence to the stated KL divergence on expected parallel translations cannot be guaranteed.

Authors: In Section 3 of the manuscript, we derive the explicit limit of the 2-Wasserstein distance as d approaches 0, showing that it converges to the KL divergence associated with expected parallel translations without any d-dependent rescaling, because the cost function is the infinitesimal Fisher metric which becomes d-independent in the continuum limit. For the optimal transport plan between the discrete measures (weighted point masses approximating the Gaussian), the plan is non-singular due to the strict convexity of the quadratic cost and the finite support; uniqueness follows from the properties of the Wasserstein space. We acknowledge that making this demonstration more explicit would benefit the reader. In the revised manuscript, we will add a paragraph or subsection explicitly proving the non-singularity of the transport plan and confirming the well-definedness of the global limit. revision: yes
Referee: The geometric interpretation equating the limiting Wasserstein distance to a KL divergence on expected parallel translations of continuous Gaussians requires a precise definition of the parallel transport operation and a step-by-step verification that the limit commutes with this construction. Without this, the claimed equivalence risks being formal rather than substantive.

Authors: The parallel transport is defined using the affine connection induced by the Fisher metric on the manifold of Gaussian distributions. The expected parallel translation refers to integrating the transported tangent vectors with respect to the Gaussian measure. In the manuscript, the equivalence is established by direct computation of the limit, which implicitly shows the commutation. However, to strengthen the geometric interpretation, we will provide a precise definition of the parallel transport operator and include a step-by-step verification that the d → 0 limit commutes with the parallel transport construction in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard Wasserstein and Fisher properties

full rationale

The paper derives the d→0 limit of the 2-Wasserstein distance with Fisher-aligned cost on discretized Gaussians and equates it to a KL divergence on expected parallel transports. This follows from explicit limiting expressions and geometric properties of optimal transport on statistical manifolds, without reducing to fitted parameters, self-referential definitions, or load-bearing self-citations. The abstract and description indicate the result is obtained from first-principles alignment of the cost function with the Fisher metric, independent of the target equivalence. No quoted equations show a prediction that is forced by construction or an ansatz smuggled via prior work by the same author. The framework remains self-contained against external benchmarks of Wasserstein geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Framework rests on standard properties of Wasserstein distance for Gaussians and the Fisher metric; no free parameters or invented entities are introduced in the abstract. The key domain assumption is the alignment of the transport cost with the Fisher metric.

axioms (2)

standard math 2-Wasserstein distance admits an explicit closed-form expression for Gaussian measures
Invoked to derive the limit expression as d approaches 0
domain assumption Cost function can be aligned with the Fisher metric on the Gaussian statistical manifold
Required for the geometric interpretation and equivalence to KL divergence

pith-pipeline@v0.9.0 · 5544 in / 1349 out tokens · 40560 ms · 2026-05-16T15:00:43.435612+00:00 · methodology

discussion (0)

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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

W_{2}^{2}(Pc,Pd) = d^{2}/12 Tr(Γ^{-1}) ... = 2·E_ρ[D(Pc(µ+δµ,Γ):Pc(µ,Γ))]
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

cost function aligned with the Fisher metric for Gaussian families

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.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Orthogonal Decomposition of Discretization-Induced Transport-Information Cost under Rank-Deficient Parametrizations
cond-mat.stat-mech 2026-05 unverdicted novelty 6.0

An orthogonal decomposition separates discretization costs into observable and unobservable parts under rank-deficient parametrizations, providing a geometric view of parametrization-dependent information loss in tran...
Path-Integral Formulation of Unavoidable Canonical Nonlinearity: Dynamic Discretization Cost over Variable Supports
cond-mat.stat-mech 2026-04 unverdicted novelty 6.0

A path-integral formulation of unavoidable canonical nonlinearity (PUCN) is introduced to measure geometric costs between arbitrary states including those with different supports.