pith. sign in

arxiv: 2604.25952 · v2 · pith:5AEEJNDVnew · submitted 2026-04-24 · 🧮 math.GM

Structural Results for 4 x n Chomp: Unique Extension, Bimodal Asymptotic Structure, and Period-112 Geometry

Arnav Garg This is my paper

Pith reviewed 2026-05-22 10:12 UTC · model grok-4.3

classification 🧮 math.GM
keywords ChompP-positionsunique extensionbimodalasymptoticscombinatorial game theoryshadow-array sieve
0
0 comments X

The pith

In 4-row Chomp, any three row lengths extend to at most one P-position, and the positions form two distinct asymptotic families.

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

The paper proves that in four-row Chomp, for any choice of lengths in the first three rows, there is at most one way to choose the fourth row length to make the position a second-player win. It further shows that these winning positions split into two separate groups, called HIGH and LOW, that stay apart as the board size increases, with their ratio of lengths approaching different values. This computational and structural analysis replaces earlier guesses about a single limiting ratio with a more precise picture of how the game positions behave for large boards. The new sieve method makes it feasible to check these properties on boards with thousands of columns.

Core claim

We prove the Unique Extension property: for any triple (a,b,c), there is at most one value of d such that (a,b,c,d) is a P-position. The P-positions exhibit a persistent bimodal decomposition into HIGH and LOW subfamilies separated by a clean gap in the per-a median of d/a. Within each family the two larger row-length ratios satisfy an exact quadratic relation at machine precision.

What carries the argument

The unique extension property proved by short contradiction using the move structure of Chomp

If this is right

  • The unique extension generalizes immediately to all k-row Chomp.
  • The global limit d/a -> 2/9 is a mixture artifact of the two families.
  • Numerical evidence suggests d/a -> 1/4 in the HIGH family and L3 ~ 0.183 in the LOW family.
  • The HIGH subfamily maintains a stable density of 56.2%.
  • The gap in median d/a grows monotonically from 0.040 at n=500 to 0.062 at n=3000.

Where Pith is reading between the lines

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

  • The exact quadratic relations may reflect an algebraic identity hidden in the definition of P-positions.
  • Further computation beyond n=3000 could distinguish between 1/4 and the fitted asymptote of 0.248 for the HIGH limit.
  • Similar bimodal structures might exist in other row-based impartial games like subtraction games.
  • The shadow-array sieve could be adapted to compute P-positions in higher-row variants more efficiently.

Load-bearing premise

The shadow-array sieve correctly computes all P-positions without omissions or errors for n up to 3000.

What would settle it

A single counterexample triple (a,b,c) with two distinct d values both yielding P-positions would falsify the unique extension property.

Figures

Figures reproduced from arXiv: 2604.25952 by Arnav Garg.

Figure 1
Figure 1. Figure 1: Convergence of row-length ratios b/a, c/a, d/a for P-positions with n ≤ 500. Solid lines show windowed medians; dashed lines show conjectured limit values L1 ≈ 0.762, L2 ≈ 0.499, L3 ≈ 0.224. 3.4 Linear Cone Geometry Conjecture 4 (Linear Cone). The set of extending triples (a, b, c) forms a linear cone in R 3 with asymptotic width width(c) ≈ 11 8 · c + f(c mod 112), where f is a bounded periodic function wi… view at source ↗
Figure 2
Figure 2. Figure 2: Autocorrelation of the d-value sequence for n ≤ 500. The sharp peak at lag 112 indicates the fundamental modular period. Subsidiary peaks appear at multiples of 112. on triples. That is a stronger global statement than anything proven or conjectured for 3×n Chomp. 4.2 Generalization to k × n If the Unique Extension property holds for 4 × n, a natural conjecture is that it generalizes: every k × n P-positio… view at source ↗
read the original abstract

We present a complete computational tabulation of all 961,619,972 P-positions in 4xn Chomp for n <= 3000, obtained via a new O(n^4) shadow-array sieve that replaces the O(n^5) hash-set approach of prior work. Three structural results are reported. First, we prove the Unique Extension property: for any triple (a,b,c), there is at most one value of d such that (a,b,c,d) is a P-position. The proof is a short contradiction using the move structure of Chomp and generalizes immediately to all k-row Chomp. Second, the P-positions exhibit a persistent bimodal decomposition into two subfamilies, HIGH and LOW, separated by a clean gap in the per-a median of d/a that grows monotonically from 0.040 at n=500 to 0.062 at n=3000, with the HIGH subfamily maintaining a stable density of 56.2% throughout. The previously conjectured global limit d/a -> 2/9 is shown to be a mixture artifact. Third, within each family the two larger row-length ratios satisfy an exact quadratic relation at machine precision, and numerical evidence suggests d/a -> 1/4 in the HIGH family, though a power-law convergence fit gives an asymptote of approximately 0.248 with exponent alpha ~ 0.05, leaving the exact limit open. The LOW family limit L3 ~ 0.183 is not well approximated by 3/16; the best rational with denominator at most 2000 is 20/109. Code and the n <= 500 dataset are available at https://github.com/gargarnav/chomp-4xn-v2.

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

0 major / 3 minor

Summary. The paper reports a complete tabulation of all 961,619,972 P-positions in 4xn Chomp for n ≤ 3000, computed via a new O(n^4) shadow-array sieve. It proves the Unique Extension property—that for any triple (a,b,c) there is at most one d making (a,b,c,d) a P-position—via a short contradiction argument based on Chomp move structure, which generalizes to arbitrary k-row Chomp. It further identifies a persistent bimodal decomposition of P-positions into HIGH and LOW subfamilies separated by a monotonically growing gap in median d/a (0.040 at n=500 to 0.062 at n=3000), with HIGH maintaining ~56.2% density; this decomposition shows that the previously conjectured global d/a → 2/9 limit is a mixture artifact. Within each subfamily the two larger row-length ratios satisfy an exact quadratic relation at machine precision, with numerical evidence and power-law fits suggesting d/a → ~0.248 (HIGH) and L3 ~ 0.183 (LOW). Code and the n ≤ 500 dataset are provided at a public repository.

Significance. If the results hold, the Unique Extension theorem is a load-bearing structural advance that applies immediately to all finite-row Chomp and supplies a new tool for analyzing P-position geometry in impartial games. The bimodal decomposition and explicit refutation of the 2/9 conjecture, grounded in exhaustive enumeration rather than prior parameter fits, open concrete avenues for asymptotic analysis. The availability of both the O(n^4) implementation and a verifiable n ≤ 500 subset constitutes a reproducible computational contribution that strengthens the observational claims.

minor comments (3)
  1. [Quadratic relations] In the description of the quadratic relations, state the precise quadratic equation satisfied by the ratios and the floating-point tolerance used to declare it 'exact at machine precision' (e.g., 10^{-12}).
  2. [Bimodal decomposition] The reported HIGH-family density of 56.2% should specify whether this is an average over all n ≤ 3000 or a stabilized value for large n; a short table of density versus n would clarify stability.
  3. [Asymptotic analysis] The power-law fit for the HIGH-family asymptote (0.248 with exponent ~0.05) should report the fitting range of n, the least-squares method, and residual statistics so readers can assess convergence quality.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The summary accurately captures the Unique Extension theorem, the bimodal HIGH/LOW decomposition, the refutation of the 2/9 conjecture, and the O(n^4) computational method.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The Unique Extension property is proven by an explicit short contradiction argument that relies only on the move structure of Chomp and does not reference the computational tabulation or any fitted quantities. The bimodal decomposition into HIGH and LOW subfamilies, the observed gap growth, stable density, quadratic relations at machine precision, and suggested asymptotic limits are all presented as direct empirical observations extracted from the n ≤ 3000 dataset generated by the shadow-array sieve; the authors supply both the O(n^4) implementation and the n ≤ 500 data at a public repository, allowing independent verification of completeness in the regime where the structural patterns first appear. No load-bearing step reduces by definition, by renaming a fitted input as a prediction, or by a self-citation chain whose cited result itself depends on the present claims; the derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The paper relies on standard combinatorial game theory axioms for Chomp and introduces fitted parameters for describing the observed asymptotic behaviors in the two families.

free parameters (3)
  • HIGH subfamily density = 56.2%
    Observed stable proportion of P-positions in the HIGH family across the computed range.
  • HIGH family d/a asymptote = 0.248
    Result of power-law convergence fit to the data for the HIGH subfamily.
  • LOW family L3 limit = 0.183
    Numerical value from the data, with best rational approximation 20/109 under denominator limit 2000.
axioms (1)
  • domain assumption Chomp moves consist of removing a rectangle from the top-right, preserving the bottom-left as the terminal position.
    Standard rule of the game used in the contradiction proof for unique extension.

pith-pipeline@v0.9.0 · 5857 in / 1606 out tokens · 62694 ms · 2026-05-22T10:12:57.529173+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

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/Foundation/Breath1024.lean eight-tick periodic micro-structure 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.

    period-112 = lcm(7,8)×2 ... period-8 signal of currently unknown origin. The solver uses 16-bit field encoding, ruling out the period-8 as a bitpacking artifact.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction 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.

    Unique Extension property: for any triple (a,b,c), there is at most one value of d such that (a,b,c,d) is a P-position. The proof is a short contradiction using the move structure of Chomp and generalizes immediately to all k-row Chomp.

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 1 Pith paper

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

  1. Unique Winning Opening Move in Three-Row Chomp

    math.CO 2026-05 unverdicted novelty 8.0

    Every 3 x n Chomp rectangle has exactly one winning opening move.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 1 Pith paper

  1. [1]

    Berlekamp, J

    E. Berlekamp, J. H. Conway, and R. K. Guy,Winning Ways for Your Mathematical Plays, 2nd ed., A K Peters, 2001

    work page 2001

  2. [2]

    On three-rowed Chomp,

    A. E. Brouwer, G. Horv´ ath, I. Moln´ ar-S´ aska, and C. Szab´ o, “On three-rowed Chomp,” Integers: Electronic Journal of Combinatorial Number Theory, vol. 5, no. 1, 2005, #G07

    work page 2005

  3. [3]

    A curious Nim-type game,

    D. Gale, “A curious Nim-type game,”American Mathematical Monthly, vol. 81, pp. 876– 879, 1974

    work page 1974

  4. [4]

    Mathematical Games: Sim, Chomp and Race Track,

    M. Gardner, “Mathematical Games: Sim, Chomp and Race Track,”Scientific American, vol. 228, no. 1, January 1973, pp. 108–115

    work page 1973

  5. [5]

    Three-rowed Chomp,

    D. Zeilberger, “Three-rowed Chomp,”Advances in Applied Mathematics, vol. 26, pp. 168– 179, 2001. 7

    work page 2001