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Every aggregate counting distribution admits a poset of positive factorizations into independent latent processes, and the maximum latent entropy (factorization entropy) is attained at maximal atomizations.

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2026-08-02 02:10 UTC pith:NKAYKP6S

load-bearing objection A clean, original math-ph paper introducing the positive factorization poset and factorization entropy; the core theorems hold up and it deserves a real referee.

arxiv 2607.14409 v1 pith:NKAYKP6S submitted 2026-07-15 math-ph cs.ITmath.ITmath.MPmath.PRphysics.data-an

On Factorizing Aggregate Counting Distributions into Independent Latent Processes

Israel Klich This is my paper
classification math-ph cs.ITmath.ITmath.MPmath.PRphysics.data-an
keywords entropyfactorizationlatentpositivecountingaggregatefactorizationsindependent
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

In counting experiments, often only the total number of events is measured—for example, the number of photons hitting a detector or the number of electrons transmitted through a wire. The probability of each total count is summarized by a probability-generating function, a polynomial whose coefficients are the probabilities. If the counted events come from several independent hidden sources, this polynomial factors into a product of smaller polynomials, one per source. This paper studies when such factorization is possible and what it reveals.

The central new idea is the 'positive factorization poset': the family of all ways to split a given counting distribution into independent parts, ordered by how finely they split. On this family, the paper defines 'factorization entropy'—the largest total Shannon entropy that the hidden sources can have while still producing the observed totals. A sharp inequality shows that splitting a source can only increase total entropy, so the maximum is always reached at the finest possible splitting. Equality holds exactly when the sums of outcomes from the different sources identify the individual outcomes uniquely.

The paper also gives exact criteria for small cases: for counts taking values 0,1,2, a factorization exists exactly when the coefficients satisfy b² ≥ 4ac; for degree three, the condition is p₁p₂ ≥ p₀p₃. It computes the volume of factorable polynomials inside the probability simplex for low degrees and provides numerical estimates for higher degrees, showing that factorability depends strongly on the mean and variance of the observed count.

Core claim

For any finite-support probability distribution, the maximum latent entropy compatible with the observed aggregate distribution equals Sfac(P) = max over positive factorizations of Σ_j H(P_j), and this maximum is always attained at a maximal positive atomization (Theorem 3.2). Moreover, Sfac(P) has the exact forms Σ_j h(ν_j) for real-rooted laws (Theorem 4.3) and via degree-≤2 factors for Hurwitz-stable laws (Corollary 4.4), and the factorable regions in degrees two and three are exactly characterized by simple coefficient inequalities with volumes µ2=1/3 and µ3=1/2.

Load-bearing premise

The observed counting distribution is exactly a probability polynomial with finite support, so that the positive factorization poset F+(P) is finite and factorization entropy is attained. The entire framework (Definitions 2.2–2.3, Theorem 3.2, Remark 2.6) depends on polynomiality; for infinite-support distributions, such as the full counting statistics of systems with unbounded charge transfer, the poset could be infinite and Sfac may not be attained. The paper explicitly defers this to Open Problem 2 ('Extend the theory to analytic probability-generating functions of infinite-support distributions'), confirming that finite support is a load-bearing domain restriction rather than a technical convenience.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper develops a theory of positive factorizations of probability-generating polynomials with nonnegative coefficients. It introduces the positive factorization poset F_+(P), defines the factorization entropy Sfac(P) as the maximum total Shannon entropy over all positive factorizations, and proves that this maximum is attained on a maximal positive atomization. The main results include an entropy dominance inequality with an injectivity equality condition, monotonicity under refinement, examples of incomparable maximal atomizations with different entropies, support-based obstructions, characterizations of the real-rooted and Hurwitz-stable sectors, a local stability theorem for coprime factorizations, exact factorable-region volumes in degrees two and three, an exact quartic Hurwitz volume ν4 = 1/6, and numerical estimates of factorable volume in higher degree. The central derivations are elementary and were checked; the finite-support restriction is explicitly stated and deferred to Open Problem 2.

Significance. If the result holds, the paper supplies a natural information-theoretic invariant for the inverse problem of inferring independent latent counting processes from an aggregate counting distribution. The exact values of μ2, μ3, and ν4, together with the entropy inequality, give concrete, falsifiable anchors for the theory. The paper is careful to distinguish exact results from exploratory numerics, and the finite-support domain is a declared scope limitation rather than a hidden assumption. The work connects binding-polynomial theory, stable polynomials, and counting statistics in a way that is likely to be useful to later researchers. A particular strength is that the main theorems are proved by explicit algebra that can be independently verified; no fitted parameters enter the exact statements.

minor comments (4)
  1. [Proposition 2.5 / Example 2.4] The proof that 1+z^2+z^4 is a positive atom only treats factors of the special form (a+bz^2)(c+dz^2). A general real quadratic factor would be (a+bz+cz^2), and the vanishing z-coefficient together with nonnegativity of the coefficients forces the linear terms to vanish before the AM-GM argument applies. The conclusion is correct, but this step should be stated explicitly.
  2. [Theorem 4.5] The proof of local stability is compressed. In the differential equation, the passage from Σ_i dotP_i ∏_{j≠i}P_j = 0 to the factorization dotP_i ∏_{j≠i}P_j = P_i M should be written out, including the sign and the transfer of terms. It would also help to state explicitly why the domain and target have the same dimension, namely Σ_i d_i = N.
  3. [Section 7] The Monte Carlo sections do not provide code or a data repository, and the description of the hit-and-run algorithm contains an incomplete sentence ('fixed thinning intervals'). For reproducibility, please specify the numerical root-finding method, coefficient-positivity tolerance, and how exact arithmetic was handled for near-boundary cases. These issues do not affect the exact mathematical results.
  4. [Throughout] Minor typographical and presentation issues: 'Section 8 is gives a brief list' should read 'Section 8 gives a brief list'; the phrases 'factorizable' and 'factorable' are used interchangeably; Fig. 1's label 'Hurwitz polynomials' should be 'Hurwitz-stable polynomials'; and the discussion in Corollary 4.4 would be more self-contained if the degree-≤2 property of positive atoms were derived directly from the factor splitting rather than cited only to Briggs.

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained, and the author's self-citations are non-load-bearing.

full rationale

I checked the paper's derivation chain and found no circular reduction. Sfac is defined as a maximum over the finite positive-factorization poset F+(P), whose finiteness follows from unique factorization over R[z] and normalization (Remark 2.6). The central entropy inequality (Theorem 3.2) is a direct application of the standard chain rule H(X1,...,Xm)=H(X)+H(X1,...,Xm|X), and monotonicity under refinement plus the atomization principle follow from that inequality without any fitted parameter or renamed input. The finite-support assumption is explicitly stated in Definitions 2.2–2.3 and deferred for infinite supports in Open Problem 2, so it is a declared domain restriction, not a concealed identification of output with input. Theorem 4.3 uses unique factorization into real linear factors together with Theorem 3.2, and the exact low-degree and Hurwitz volumes are obtained by coefficient inequalities and explicit integration, not by fitting. The Monte Carlo estimates are explicitly exploratory and are not used to define or prove any exact result. The author's self-citations, references [5,6,7,8], appear only in the physical-motivation passages and are not load-bearing for any theorem; the load-bearing citations are to Briggs, Routh–Hurwitz, and independent probability literature. Thus I find no circularity and no prediction that reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 2 invented entities

The paper introduces no fitted parameters and no ad hoc physical entities. The frameworks rely on standard mathematical facts (UFD, chain rule, inverse function theorem, Routh-Hurwitz, Dirichlet representation) and one cited theorem from Briggs. The new mathematical objects (poset, factorization entropy) are definitions that constitute the contribution.

axioms (7)
  • standard math The polynomial ring over R is a unique factorization domain; every positive factorization of P corresponds to a grouping of its irreducible real factors.
    Used in Remark 2.6 (finiteness of F+(P)), Prop 3.5 (maximality of the two factorizations), and the enumeration algorithm in Remark 7.1.
  • domain assumption The probability-generating function of a sum of independent nonnegative integer variables is the product of their PGFs.
    Used in Equation (2) and throughout to equate polynomial factorization with latent independence.
  • standard math Shannon entropy chain rule: H(X,Y)=H(X)+H(Y|X), and additivity for independent variables.
    Used in the proof of Theorem 3.2.
  • standard math Inverse function theorem for smooth maps between manifolds of equal dimension.
    Used in Theorem 4.5 (local stability).
  • standard math Routh–Hurwitz stability criterion for quartic polynomials (two inequalities imply Hurwitz stability for positive coefficients).
    Used in Proposition 5.3 to characterize H4.
  • standard math Dirichlet(1,...,1) distribution is the law of normalized independent unit-rate exponential variables.
    Used in Proposition 5.3 to compute ν4.
  • domain assumption Briggs's theorem: for positive Hurwitz-stable polynomials, the factorization into positive linear and p-irreducible quadratic factors is unique.
    Invoked in Corollary 4.4 to assert Sfac(P) = Sfac^(≤2)(P).
invented entities (2)
  • positive factorization poset F+(P) no independent evidence
    purpose: Orders all positive factorizations of a probability polynomial P by refinement; the maximal elements are atomizations relevant to entropy optimization.
    New mathematical object defined in Section 3; no falsifiable handle outside the paper, it is a definition.
  • factorization entropy Sfac(P) no independent evidence
    purpose: Measures the maximum sum of Shannon entropies over all positive factorizations; operationalizes the maximal latent randomness compatible with the observed distribution.
    Defined via a max over the poset; a mathematical functional rather than a physical entity. Its value can be computed for any distribution, but no independent empirical handle is introduced.

pith-pipeline@v1.3.0-alltime-deepseek · 12086 in / 26193 out tokens · 222308 ms · 2026-08-02T02:10:09.562998+00:00 · methodology

0 comments
read the original abstract

Given only the probability distribution of an aggregate counting variable, what independent latent counting processes are compatible with the observation? Equivalently, when does a probability-generating function admit a factorization into normalized polynomials with nonnegative coefficients? We develop a mathematical theory of such positive factorizations. We introduce the positive factorization poset, whose elements are all positive factorizations ordered by refinement, and define the factorization entropy, measuring the maximal latent Shannon entropy compatible with the observed distribution. We prove a sharp entropy inequality, characterize the equality case by injectivity of the latent addition map, show that entropy optimization may be restricted to maximal atomizations, and exhibit examples where distinct maximal atomizations have different entropy. We further establish support-based obstructions to positive factorization, characterize the real-rooted and Hurwitz-stable sectors, prove a local stability theorem for coprime factorizations, determine exactly the factorable regions in degrees two and three, obtain an exact quartic Hurwitz volume, and investigate the geometry of the factorable region inside the probability simplex through exact calculations and Monte Carlo experiments. These results identify the positive factorization poset as a natural algebraic object associated with probability-generating functions and provide a framework for studying latent independent structure in aggregate counting statistics.

Figures

Figures reproduced from arXiv: 2607.14409 by Israel Klich.

Figure 1
Figure 1. Figure 1: Monte Carlo estimates of µN and νN the fraction of factorizable probability polynomials and the subset of Hurwitz polynomials on the probability simplex. The cross markers show the exact values. For N = 4 the exact result ν4 = 1/6 is described in proposition 5.3 We concentrate on distributions centered in the support, m = N 2 , and compare two width scalings: v = 1 and v = N 2 . The first ensemble remains … view at source ↗
Figure 2
Figure 2. Figure 2: Estimated fractions of positively reducible polynomials in the centered moment [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

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