arxiv: 2604.25952 · v1 · submitted 2026-04-24 · 🧮 math.GM

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Structural Conjectures for 4 x n Chomp: Unique Extension, Asymptotic Ratios, and Period-112 Geometry

Arnav Garg

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Pith reviewed 2026-05-08 08:54 UTC · model grok-4.3

classification 🧮 math.GM

keywords chompp-positions4xn chompcombinatorial gamesunique extensionasymptotic ratiosperiodicitylosing positions

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The pith

P-positions in 4 by n Chomp exhibit unique fourth-row extensions, converging ratios, period-112 structure, and linear cone geometry.

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

The paper computes every P-position in 4 by n Chomp for all widths up to 500, producing a table of over four million losing positions. From these data it advances four conjectures about the underlying structure: any three row lengths admit at most one completing fourth length that yields a P-position, the four row-length ratios settle toward fixed asymptotic constants, the admissible triples obey a repeating modular pattern with period 112, and the entire set of P-positions lies inside a linear cone when viewed in three-dimensional space. A sympathetic reader cares because these regularities would turn a seemingly complex impartial game into one whose losing positions can be predicted by simple deterministic rules rather than exhaustive search.

Core claim

Exhaustive enumeration of 4,316,097 P-positions for n ≤ 500 shows that the set of losing positions obeys four structural rules: the unique-extension property that each (a, b, c) triple extends in at most one way to a P-position (a, b, c, d), convergence of row-length ratios to fixed constants, a period-112 modular constraint on extendable triples, and containment of all such positions inside a linear cone in (a, b, c)-space.

What carries the argument

The unique-extension property, which states that for any triple of row lengths there is at most one fourth-row length that completes a P-position.

Load-bearing premise

The four patterns observed in the computed P-positions for n up to 500 continue to hold without exception for all larger n.

What would settle it

Discovery of even one n > 500 in which a single (a, b, c) triple admits two distinct completing d values that both produce P-positions, or in which the observed row-length ratios deviate from the conjectured limits.

Figures

Figures reproduced from arXiv: 2604.25952 by Arnav Garg.

**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: 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

This paper presents a computational study of P-positions (losing positions for the player to move) in 4 x n Chomp, the combinatorial game played on a 4 x n rectangular grid. An optimized C++ solver with bitpacked state representation tabulates all 4,316,097 P-positions for boards with n <= 500, constituting the most extensive computational study of 4 x n Chomp to date. The analysis reveals four structural conjectures: (1) a Unique Extension property, stating that for any triple (a,b,c) of row lengths, there is at most one valid fourth-row length d completing a P-position; (2) convergence of row-length ratios to fixed asymptotic constants; (3) a period-112 modular structure governing the set of extendable triples; and (4) a linear cone geometry for the P-position set in (a,b,c)-space. These findings suggest a richer deterministic structure in 4 x n Chomp than previously suspected and raise new questions about the relationship between k-row games for consecutive k.

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 pushes 4xn Chomp computation to n=500 and extracts four conjectures from the data, but the claims rest on untested extrapolation beyond that range.

read the letter

The punchline is that this paper delivers the largest computation yet of losing positions in 4-row Chomp and turns the results into four specific conjectures about infinite behavior. That computation reaches n=500 and lists over four million P-positions, which is new ground. The work is strongest on the data side. The authors built an optimized solver that uses bitpacking to handle the states efficiently. From the output they identify a unique extension rule for completing P-positions, converging row ratios, a repeating period of 112 in the modular structure, and a linear cone shape in the three-dimensional space of row lengths. These are clean observations pulled straight from the enumeration. The soft spots are all around the jump from finite data to general claims. The conjectures are presented as holding for all n, yet the paper supplies no proof, no recurrence relation, and no run past 500 to see if the patterns survive. A single counterexample to unique extension or to the period would break the picture. There is also no mention of code release or independent verification of the solver, so the base numbers rest on trust. This paper is for researchers in combinatorial game theory who want to see what large-scale computation can reveal about Chomp. A reader interested in pattern hunting or in setting up their own verification would get something from it. It is not ready to be taken as established theory. I would bring this to the next reading group to talk about whether these patterns look provable. I would not cite the conjectures in my own work until they are checked further. The paper deserves a serious referee because the scale of the computation is a genuine step forward and the conjectures are stated clearly enough to be tested.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a computational enumeration of all P-positions in 4 × n Chomp for n ≤ 500, yielding 4,316,097 positions via an optimized C++ solver. From this dataset the authors extract four structural conjectures: (1) the Unique Extension property (at most one d completing any (a,b,c) to a P-position), (2) convergence of row-length ratios to fixed asymptotic constants, (3) a period-112 modular structure on extendable triples, and (4) a linear cone geometry for the P-position set in (a,b,c)-space.

Significance. If the conjectures hold, the work would supply the most detailed structural description yet of P-positions in a fixed-row Chomp variant, potentially enabling closed-form or geometric characterizations and illuminating the relationship between k-row and (k+1)-row games. The sheer scale of the enumeration (over four million positions) constitutes a concrete, reproducible dataset that future analytic work could exploit.

major comments (2)

[sections presenting the four structural conjectures] The four conjectures rest entirely on patterns observed in the finite set of 4,316,097 positions for n ≤ 500. No recurrence, invariance argument, or larger-scale verification is supplied to establish that the Unique Extension property, the period-112 structure, or the linear cone persist for all n > 500; a single counterexample beyond the computed range would falsify the central claims.
[computational method section] The manuscript supplies neither the source code nor a detailed error-analysis or verification protocol for the bit-packed solver. Consequently the correctness of the enumerated list itself—the sole empirical foundation for all conjectures—cannot be independently checked within the paper.

minor comments (2)

[abstract and results] The abstract and results sections would benefit from explicit statements that the conjectures are empirical extrapolations rather than proven theorems.
[introduction] Prior literature on 3 × n Chomp and general Chomp P-position geometry should be cited more systematically to clarify the novelty of the period-112 and cone observations.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the major comments point by point below, and have prepared revisions accordingly.

read point-by-point responses

Referee: The four conjectures rest entirely on patterns observed in the finite set of 4,316,097 positions for n ≤ 500. No recurrence, invariance argument, or larger-scale verification is supplied to establish that the Unique Extension property, the period-112 structure, or the linear cone persist for all n > 500; a single counterexample beyond the computed range would falsify the central claims.

Authors: We fully agree that the conjectures are empirical observations from the finite dataset up to n=500 and lack a general proof or recurrence relation. The paper presents these as conjectures rather than theorems. In the revised version, we will expand the discussion to explicitly state the finite nature of the evidence, include a caveat about possible counterexamples for larger n, and outline plans for further computational verification. This addresses the concern by clarifying the scope of the claims. revision: partial
Referee: The manuscript supplies neither the source code nor a detailed error-analysis or verification protocol for the bit-packed solver. Consequently the correctness of the enumerated list itself—the sole empirical foundation for all conjectures—cannot be independently checked within the paper.

Authors: We acknowledge this limitation in the current manuscript. To improve verifiability, the revised manuscript will include a more detailed description of the solver's implementation, including the bit-packing technique, and a verification protocol consisting of matching known P-positions for small n (n≤20) from prior literature and cross-validation with an independent Python implementation on a subset of the data. Furthermore, we will provide a link to the open-source C++ code repository. revision: yes

standing simulated objections not resolved

A rigorous analytic proof that the observed structural properties hold for all n > 500

Circularity Check

0 steps flagged

No circularity: conjectures are direct empirical observations from standard enumeration

full rationale

The paper performs exhaustive computation of all P-positions in 4xn Chomp for n≤500 using the standard recursive definition (P-positions are those with no move to another P-position). It then tabulates the 4,316,097 positions and extracts four observed patterns as conjectures about infinite behavior. No step defines a quantity in terms of itself, fits a parameter to data and renames the fit as a prediction, invokes self-citations for uniqueness theorems, or smuggles an ansatz. The computation relies only on the external, non-circular game-theoretic definition of P-positions; the conjectures are simply extrapolations from the finite list, not derivations that reduce to their inputs by construction. This is a standard, non-circular computational study.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard recursive definition of P-positions in impartial games and the assumption that the enumeration algorithm is correct; no additional free parameters, invented entities, or non-standard axioms are introduced.

axioms (1)

domain assumption P-positions are defined by the standard recursive rule in normal-play impartial games: a position is P if every move leads to an N-position.
The entire enumeration and subsequent pattern detection presuppose this definition.

pith-pipeline@v0.9.0 · 5491 in / 1394 out tokens · 83350 ms · 2026-05-08T08:54:53.125901+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references

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[2]

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[3]

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[4]

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[5]

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