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A local Boolean condition on every triangle decides exactly which weighted tournaments are affine inversion graphs.

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2026-07-13 05:41 UTC pith:O657UKDP

load-bearing objection Clean local characterization of affine inversion graphs that genuinely simplifies Papi and yields a better recognition algorithm; the ChatGPT story is incidental.

arxiv 2607.08931 v1 pith:O657UKDP submitted 2026-07-09 math.CO

A ChatGPT-assisted Triangle Characterization of Affine Permutation Inversion Graphs

Sara C. Billey , Herman Chau , Kevin Liu This is my paper
classification math.CO MSC 05E1620F5505C20
keywords affine permutationaffine symmetric groupinversion graphinversion settournament graphBoolean triangle conditionweak orderCoxeter group
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.

Affine permutations can be encoded by weighted tournaments on n vertices whose edge weights record how many inversions sit between each pair of residue classes. An earlier global characterization of which tournaments arise this way is replaced here by a single arithmetic check on every directed path of length two: after a one-line shift of the weights, the quantity rt(a,c) - rt(a,b) - rt(b,c) must lie in {0,1}. The paper proves the two characterizations are equivalent, shows that the local test forces the tournament to be acyclic, and derives three immediate consequences: an O(n^{3}) recognition algorithm for inversion graphs, path-weight inequalities, and the fact that induced subtournaments remain inversion graphs after label standardization. The same local test yields a new O(|R| + n^{3}) algorithm that decides whether an arbitrary finite set R is the inversion set of some affine permutation.

Core claim

A weighted tournament T on [n] that satisfies the zero-weight condition is the completed affine inversion graph of a unique affine permutation if and only if its shifted adjacency matrix obeys the Boolean triangle condition rt(a,c) - rt(a,b) - rt(b,c) ∈ {0,1} for every directed path a → b → c.

What carries the argument

The Boolean triangle condition on the shifted weight matrix rt: for every directed path of length two the difference of the three shifted weights is forced to be either 0 or 1; this single local arithmetic constraint replaces the two global conditions of the earlier characterization.

Load-bearing premise

The proof that the local triangle condition implies the earlier global characterization rests on a three-case analysis of where a third vertex sits in the total order of the tournament; any gap in those cases would leave the equivalence incomplete.

What would settle it

Exhibit a weighted tournament on [n] that satisfies the zero-weight condition and the Boolean triangle condition on every length-two path, yet fails one of the two conditions of the earlier global characterization (or fails to arise from any affine permutation).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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 / 5 minor

Summary. The paper gives a new characterization of affine inversion graphs of elements of the affine symmetric group: a weighted tournament on [n] with nonnegative integer weights satisfying the zero-weight condition is the completed affine inversion graph of a unique w if and only if the shifted weights obey rt(a,c)-rt(a,b)-rt(b,c)∈{0,1} for every directed path a o b o c (Theorem 1.1, the Boolean triangle condition). Necessity is obtained from an explicit triangle identity on window notation (Lemma 3.2 / Corollary 3.3). Sufficiency is proved by showing that the Boolean condition implies both of Papi’s conditions (Theorem 2.7). Consequences include an O(n^{3}) recognition algorithm for graphs, path inequalities, a hereditary property under induced subtournaments, and an O(|R|+n^{3}) algorithm that decides whether a finite set R is an affine inversion set.

Significance. The result is a genuine simplification of Papi’s 1997 characterization: a single local numerical condition on every length-2 path replaces a global inequality plus a quantified composition condition. The equivalence is proved carefully and yields immediately usable algorithms whose complexity improves on the naïve O(|R|^{2}) test of the classical triple axioms when |R| is large. The path inequalities and hereditary property are clean corollaries that may be useful for the authors’ ongoing work on affine higher Bruhat orders. The transparent AI-assisted discovery narrative is a secondary but welcome contribution to the growing literature on machine-assisted combinatorics. The manuscript is self-contained, the proofs are elementary, and the algorithmic consequences are concrete and falsifiable.

minor comments (5)
  1. In the acknowledgments the date of the ChatGPT session is given as February 9, 2026, while the arXiv stamp is July 2026; a brief clarification of the timeline would avoid reader confusion.
  2. Figures 1 and 2 would be clearer if the zero-weight edges were drawn with a distinct style (already dashed) and the nonzero weights were placed consistently next to the arrows rather than at midpoints that can be ambiguous for longer edges.
  3. The lookup table in Lemma 3.2 is helpful, but the subsequent derivation of Corollary 3.3 via δ_odd could be condensed by one sentence that simply records the four parity cases; the present expansion is correct but slightly repetitive.
  4. In Step 2 of the algorithm in Theorem 3.7 the phrase “initial segment of the form {0,1,…,q-1}” should explicitly allow the empty set (already handled by the “at most one nonempty” check) so that implementers do not mis-code the zero-weight case.
  5. A short remark after Corollary 3.5 noting that the upper bound k-1 is sharp (e.g., a directed path of length k with all consecutive shifted weights 0 and the long edge of weight k-1) would make the path inequality more informative.

Circularity Check

0 steps flagged

No circularity: Boolean triangle condition is independently derived from definitions and shown equivalent to Papi's external 1997 characterization by direct case analysis.

full rationale

The derivation chain is self-contained and non-circular. Forward direction of Theorem 1.1 follows from the definition of affine inversion graphs (Definition 2.2) via explicit formulas in Lemmas 3.1–3.2 and Corollary 3.3, which compute the shifted-weight difference on any triangle from the window notation and residue permutations; these are first-principles calculations, not assumptions of the target. Converse direction assumes only the zero-weight condition plus Boolean triangle condition, proves acyclicity/nonnegativity of shifted weights (Lemma 3.4), then verifies Papi's two conditions (I)–(II) by exhaustive three-case analysis on the position of the third vertex relative to the total order induced by the tournament, using only the Boolean condition and the elementary bound rt(p,q)+rt(q,p)≥-1. Equivalence to an affine inversion graph is then imported from Papi's external Theorem 2.7 (1997), not from any self-citation. Self-citations [3,4,7] appear solely as motivation for higher Bruhat orders and are unused in any proof. No parameters are fitted, no quantity is defined in terms of the claimed characterization, and the O(|R|+n^{3}) recognition algorithm is a direct consequence rather than an input. The paper therefore contains no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The work sits entirely inside the standard theory of the affine symmetric group and its inversion graphs. It relies on two published characterizations (Björner–Brenti uniqueness and Papi’s graph conditions) and ordinary tournament combinatorics; no free parameters or ungrounded physical entities are introduced. The Boolean triangle condition is a new combinatorial predicate, not a postulated object requiring independent experimental evidence.

axioms (4)
  • domain assumption Affine permutations are uniquely determined by their inversion sets (Björner–Brenti, Thm. 4.6 / Cor. 5.2).
    Used to guarantee that a graph satisfying the triangle condition corresponds to at most one affine permutation.
  • domain assumption Papi’s two conditions (I)–(II) completely characterize affine inversion graphs (Papi 1997, Thm. 4).
    The converse half of the main theorem is proved by showing the Boolean condition implies Papi’s conditions.
  • standard math A tournament is acyclic if and only if it contains no directed 3-cycle.
    Invoked in Lemma 3.4 to conclude acyclicity from the Boolean condition.
  • domain assumption Classical triple axioms for inversion sets of affine permutations (Thm. 2.1).
    Used to justify correctness of the residue-string compression algorithm in Thm. 3.7.
invented entities (1)
  • Boolean triangle condition independent evidence
    purpose: Local numerical predicate on every directed path of length two that replaces Papi’s global conditions.
    Defined in equation (1.2) and shown equivalent to the existing notion of affine inversion graph; the equivalence supplies the independent handle.

pith-pipeline@v1.1.0-grok45 · 16984 in / 2358 out tokens · 41227 ms · 2026-07-13T05:41:46.524565+00:00 · methodology

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read the original abstract

Inversion sets of permutations in the affine symmetric group $\widetilde{S}_n$ were studied extensively by Bj\"orner and Brenti. One of their methods for encoding an inversion set is through an affine inversion graph, which is a certain weighted graph on vertex set $[n]=\{1,2,\ldots,n\}$. Subsequent work by Papi characterized which graphs arise as affine inversion graphs. In this paper, we provide an alternative characterization in terms of a simple local condition on each triangle in a weighted tournament graph. This new characterization was produced with the assistance of ChatGPT, which suggested several key insights that simplified portions of Papi's original characterization. Consequences of our characterization include efficient algorithms for recognizing inversion graphs and inversion sets. Furthermore, we give bounds on the weights along directed paths, and we show that standardizing the labels on an induced subgraph results in another inversion graph. We conclude with a new order $O(|R|+n^{3})$ algorithm for testing if a given set $R$ is the inversion set of an affine permutation.

Figures

Figures reproduced from arXiv: 2607.08931 by Herman Chau, Kevin Liu, Sara C. Billey.

Figure 1
Figure 1. Figure 1: The affine inversion graph Gw for w “ p4, 5, ´1, 2q P Sr4. Dashed edges have weight 0. Example 2.6. In [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The affine inversion graph Gw for w “ p4, 9, ´5, 2q P Sr4. In [5, §5], Bj¨orner and Brenti asked if affine inversion graphs could be characterized directly. In 1997, Papi answered this in the affirmative [15, Thm. 4]. To state Papi’s characterization, recall the shifted weight matrix rtpa, bq from (1.1), given by (2.2) rtpa, bq :“ # tpa, bq, a ď b, tpa, bq ´ 1, a ą b for any weighted directed graph G on rn… view at source ↗

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Reference graph

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