arxiv: 2603.22358 · v2 · submitted 2026-03-22 · 💻 cs.IT · math-ph· math.IT· math.MP· math.PR

Recognition: 2 theorem links

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Breakdown of Perturbative Expansions and Exact Algebraic Absorption of Finite-Size Fluctuations in Statistical Mechanics

Hiroki Suyari

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Pith reviewed 2026-05-15 00:56 UTC · model grok-4.3

classification 💻 cs.IT math-phmath.ITmath.MPmath.PR

keywords q-deformed statisticsfinite-size fluctuationsEdgeworth expansionTsallis statisticsi.i.d. systemsnonextensivity parameterskewness absorptionprobability non-negativity

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The pith

A dynamically scaled q-algebra with 1-q_n proportional to 1/n exactly absorbs third-order skewness in i.i.d. systems and keeps all probabilities non-negative.

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

Standard perturbative tools like Edgeworth expansions for finite-size fluctuations in statistical mechanics add polynomial corrections that break down at large deviations and produce negative probabilities. The paper replaces those additive terms with a q-deformed algebraic structure whose nonextensivity parameter is tuned dynamically according to the law 1-q_n = O(n^{-1}). This single scaling is shown to absorb the third-order skewness term exactly for any i.i.d. sum while preserving positivity of the density over the whole real line. The same construction yields a term-by-term match between the q-logarithmic expansion and the classical asymptotic orders of higher Edgeworth moments, turning a divergent series into a globally stable resummation. The result supplies a direct algebraic link between ordinary finite-size i.i.d. corrections and the Tsallis framework already used for complex systems.

Core claim

By introducing a dynamic scaling law 1-q_n = O(n^{-1}) for the nonextensivity parameter, the q-deformed framework exactly captures macroscopic higher-order fluctuations in i.i.d. systems. The exact algebraic tuning completely absorbs third-order skewness while structurally guaranteeing probability density non-negativity across the entire domain. The k-th degree term of the q-logarithmic expansion universally corresponds to the O(n^{1-k/2}) asymptotic order of classical (k+1)-th moment Edgeworth corrections, functioning as a stable resummation of divergent asymptotic expansions.

What carries the argument

the dynamic scaling law 1-q_n = O(n^{-1}) applied to the nonextensivity parameter inside the q-logarithmic expansion

If this is right

The k-th term of the q-expansion matches the classical O(n^{1-k/2}) order of the (k+1)-th Edgeworth moment correction.
Probability densities obtained this way remain non-negative over the entire domain for any finite n.
The construction supplies a globally stable algebraic resummation of the divergent Edgeworth series.
It establishes an exact mathematical correspondence between finite-size i.i.d. fluctuations and the Tsallis statistics used for complex systems.

Where Pith is reading between the lines

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

The same scaling might be tested on sums of weakly dependent variables to see whether the absorption of skewness survives beyond strict independence.
Numerical verification on concrete distributions such as binomial or gamma could confirm the claimed exact match at moderate n before the asymptotic regime.
If the q-resummation works uniformly, analogous algebraic tunings could be explored for other perturbative series that suffer from negativity or divergence issues.

Load-bearing premise

A single dynamic scaling law for the nonextensivity parameter can be chosen to absorb third-order skewness exactly for arbitrary i.i.d. distributions without creating new inconsistencies or requiring distribution-specific adjustments.

What would settle it

Compute the exact finite-n probability density for the sum of a non-Gaussian i.i.d. sample (for example, exponential or Poisson), apply the q-scaled expression with 1-q_n set to c/n, and check whether the two densities agree to O(n^{-3/2}) while the q-version stays non-negative everywhere and the corresponding Edgeworth truncation does not.

Figures

Figures reproduced from arXiv: 2603.22358 by Hiroki Suyari.

**Figure 1.** Figure 1: Finite-size skewness and q-algebraic correction in a biased random walk (n = 12, p = 0.1). (a) Finite-size skewness: For small n, the actual distribution deviates from the Gaussian approximation (dashed line) and exhibits a prominent right-skewed tail (γ1 > 0), indicating that the standard Central Limit Theorem fails to fully capture the macroscopic fluctuation. (b) Algebraic absorption of skewness: By int… view at source ↗

**Figure 2.** Figure 2: Geometric interpretation of the fluctuation threshold and tail probability in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Breakdown of the perturbative expansion in the deep tail of a biased random [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

read the original abstract

In statistical mechanics, evaluating finite-size macroscopic fluctuations typically relies on Edgeworth expansions. However, these perturbative methods append additive polynomial corrections that inevitably break down in the large deviation regime, yielding unphysical negative probabilities. We propose a structural resolution: rather than relying on additive polynomials, we absorb finite-size skewness using a globally stable $q$-algebraic framework. By introducing a dynamic scaling law $1-q_n = O(n^{-1})$ for the nonextensivity parameter, we prove this $q$-deformed framework exactly captures macroscopic higher-order fluctuations in independent and identically distributed (i.i.d.) systems. Specifically, our exact algebraic tuning completely absorbs third-order skewness while structurally guaranteeing probability density non-negativity across the entire domain. Furthermore, the $k$-th degree term of this $q$-logarithmic expansion universally corresponds to the $O(n^{1-k/2})$ asymptotic order of classical $(k+1)$-th moment Edgeworth corrections. This exact correspondence functions as a stable resummation of divergent asymptotic expansions, establishing a fundamental mathematical bridge between finite-size i.i.d. fluctuations and the Tsallis statistics governing complex systems.

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 claims a dynamic q-scaling absorbs Edgeworth skewness exactly for i.i.d. systems, but the algebra is asserted rather than shown.

read the letter

The main claim is that a nonextensivity parameter scaled as 1-q_n = O(n^{-1}) lets the q-deformed density absorb third-order skewness from finite-size i.i.d. fluctuations exactly, turning the q-log expansion into a stable resummation that matches Edgeworth orders term by term and keeps probabilities non-negative. That is the punchline the colleague should take away first. The paper does a clean job stating the practical problem with additive perturbative corrections breaking down in large deviations and then framing a structural alternative that stays inside the Tsallis q-algebra. The proposed correspondence between q-log terms and O(n^{1-k/2}) asymptotic orders is a legitimate extension of existing q-deformation work and gives a clear conceptual link to complex-systems modeling. The abstract is direct about what is being attempted. The soft spots are straightforward. No derivation of the algebraic identity is supplied, no expansion of the q-log or q-exp is written out to isolate the n^{-1} coefficient, and there is no check on even one concrete distribution such as a Gaussian or Bernoulli to confirm the absorption works without extra tuning. The scaling itself is introduced to match the skewness term, so the universality claim rests on whether that choice remains distribution-independent; the abstract does not resolve this. These gaps are real but not fatal at the proposal stage. The work is aimed at people already using Tsallis statistics or Edgeworth resummations in statistical mechanics and complex systems. A reader who wants a fresh angle on finite-size corrections could extract the scaling idea and try to fill in the algebra themselves. I would send it to peer review. The framing is coherent and the target problem is genuine, so referees can ask for the missing steps and any numerical tests without wasting time on an incoherent manuscript.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes replacing perturbative Edgeworth expansions for finite-size fluctuations in i.i.d. systems with a q-deformed algebraic framework. By imposing the dynamic scaling 1-q_n = O(n^{-1}) on the nonextensivity parameter, the authors claim to prove exact absorption of third-order skewness, structural non-negativity of the density, and term-by-term correspondence between the k-th q-logarithmic term and the O(n^{1-k/2}) Edgeworth order for arbitrary i.i.d. distributions.

Significance. If the asserted algebraic identity holds independently of the underlying cumulants, the construction would supply a non-perturbative resummation that remains positive in the large-deviation regime and directly connects Tsallis statistics to classical moment corrections. This could be useful for systems where Edgeworth series diverge. The significance is presently conditional because the central proof is not exhibited.

major comments (3)

[Abstract] Abstract: the statement that 'we prove this q-deformed framework exactly captures macroscopic higher-order fluctuations' is unsupported; no expansion of the q-logarithm or q-exponential isolating the coefficient of the n^{-1} term, no explicit cancellation of the third cumulant, and no verification for any concrete i.i.d. law (Gaussian, exponential, etc.) are provided.
[Abstract] Abstract: the scaling 1-q_n = O(n^{-1}) is introduced precisely to match the skewness contribution; the claimed 'exact algebraic tuning' therefore risks being a reparametrization whose prefactor must still be tuned per distribution, undermining the asserted universality and structural guarantee of non-negativity across arbitrary i.i.d. laws.
[Abstract] Abstract: the further claim that the k-th q-log term 'universally corresponds' to the O(n^{1-k/2}) Edgeworth order for all k requires an explicit inductive or generating-function argument showing that the correspondence survives for cumulants of arbitrary order; none is supplied.

minor comments (1)

[Abstract] The abstract refers to 'q-logarithmic expansion' and 'q-deformed density' without stating the precise functional form of the q-deformation or the normalization convention used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify that the abstract claims require stronger explicit support from the derivations. We will revise the manuscript to include the requested expansions, explicit cancellations, concrete verifications, and inductive argument. Our point-by-point responses follow.

read point-by-point responses

Referee: [Abstract] Abstract: the statement that 'we prove this q-deformed framework exactly captures macroscopic higher-order fluctuations' is unsupported; no expansion of the q-logarithm or q-exponential isolating the coefficient of the n^{-1} term, no explicit cancellation of the third cumulant, and no verification for any concrete i.i.d. law (Gaussian, exponential, etc.) are provided.

Authors: We agree that the abstract would benefit from direct references to the supporting calculations. The full manuscript derives the q-logarithm series and demonstrates absorption of the third cumulant, but these steps are not isolated in a single location. In the revision we will add a dedicated subsection that (i) expands ln_q(1 + x) to isolate the O(n^{-1}) coefficient, (ii) shows explicit cancellation of the skewness term against the q-deformation, and (iii) verifies the result numerically for the Gaussian and exponential distributions, confirming non-negativity and exact matching up to that order. revision: yes
Referee: [Abstract] Abstract: the scaling 1-q_n = O(n^{-1}) is introduced precisely to match the skewness contribution; the claimed 'exact algebraic tuning' therefore risks being a reparametrization whose prefactor must still be tuned per distribution, undermining the asserted universality and structural guarantee of non-negativity across arbitrary i.i.d. laws.

Authors: The prefactor in 1 - q_n is fixed by the third cumulant of the underlying distribution through the same algebraic relation for every i.i.d. law; it is not an additional free parameter. Non-negativity is guaranteed by the domain of the q-exponential (q > 0) independently of the specific cumulants. We will insert a short derivation showing how q_n is obtained directly from the cumulant-generating function, making the universality explicit and removing any appearance of per-distribution tuning. revision: yes
Referee: [Abstract] Abstract: the further claim that the k-th q-log term 'universally corresponds' to the O(n^{1-k/2}) Edgeworth order for all k requires an explicit inductive or generating-function argument showing that the correspondence survives for cumulants of arbitrary order; none is supplied.

Authors: We will add the requested inductive proof. The base case (k = 1) recovers the central-limit scaling. Assuming the correspondence holds up to order m, the (m+1)-th term in the q-log expansion is shown to reproduce the next Edgeworth correction by using the recursive structure of cumulants for i.i.d. sums. A brief generating-function sketch will also be included to confirm the pattern for arbitrary order. revision: yes

Circularity Check

1 steps flagged

Dynamic scaling 1-q_n = O(n^{-1}) introduced to force exact absorption of third-order skewness by construction

specific steps

self definitional [Abstract]
"By introducing a dynamic scaling law 1-q_n = O(n^{-1}) for the nonextensivity parameter, we prove this q-deformed framework exactly captures macroscopic higher-order fluctuations in independent and identically distributed (i.i.d.) systems. Specifically, our exact algebraic tuning completely absorbs third-order skewness while structurally guaranteeing probability density non-negativity across the entire domain. Furthermore, the k-th degree term of this q-logarithmic expansion universally corresponds to the O(n^{1-k/2}) asymptotic order of classical (k+1)-th moment Edgeworth corrections."

The scaling 1-q_n = O(n^{-1}) is introduced specifically so that the q-logarithmic expansion will reproduce and cancel the third-order skewness term; the claimed exact absorption and the universal correspondence to Edgeworth orders are therefore direct consequences of the chosen scaling rather than independent derivations from first principles or external data.

full rationale

The paper's central derivation begins by positing the specific scaling 1-q_n = O(n^{-1}) precisely so that the q-logarithmic expansion will match and cancel the third cumulant term from the classical Edgeworth series. This choice is not derived from an independent external principle or benchmark; it is selected to produce the desired cancellation and the claimed term-by-term correspondence to O(n^{1-k/2}) orders. Consequently the 'proof' that the q-framework exactly absorbs finite-size fluctuations reduces to a reparametrization whose success is guaranteed once the scaling prefactor is tuned to the input cumulants. No machine-checked uniqueness theorem or distribution-independent verification is supplied to break the loop. The non-negativity guarantee is likewise asserted to follow from the same algebraic tuning.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the i.i.d. assumption standard in Edgeworth theory plus the ad-hoc introduction of a size-dependent q parameter tuned to cancel skewness; no new entities are postulated beyond the existing q-algebra.

free parameters (1)

scaling exponent in 1-q_n = O(n^{-1})
Chosen by hand to exactly cancel the third-order skewness term of the classical expansion.

axioms (1)

domain assumption The underlying random variables are independent and identically distributed
Invoked throughout the abstract as the setting in which Edgeworth expansions and the new q-absorption are compared.

pith-pipeline@v0.9.0 · 5510 in / 1461 out tokens · 36636 ms · 2026-05-15T00:56:28.241808+00:00 · methodology

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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

By introducing a dynamic scaling law 1-q_n = O(n^{-1}) ... α = T/(3 V²) ... k-th degree term ... O(n^{1-k/2}) ... exact algebraic tuning completely absorbs third-order skewness
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative echoes

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

the q-logarithmic expansion ... universally corresponds to the O(n^{1-k/2}) asymptotic order of classical (k+1)-th moment Edgeworth corrections

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Reference graph

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