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Paper metadata

Original submission
June 1, 2025
Latest revision
October 23, 2025 · version 2
Claimed MSC
arXiv categories
math.CO
Comments
30 pages, 12 figures. v2: Added Section 6 generalizing to subarrangements of Type B Coxeter arrangement and signed graphs. Extended abstract of this paper was accepted for FPSAC 2025

Abstract

A recent result of Lofano and Paolini expresses the characteristic polynomial of a real hyperplane arrangement in terms of a projection statistic on the regions of the arrangement. We use this result to give an alternative proof for Greene and Zaslavsky's interpretation for the coefficients of the chromatic polynomial of a graph and further generalize this interpretation to signed graphs. We also show that this projection statistic has a nice combinatorial interpretation in the case of the braid arrangement, which generalizes to graphical arrangements of natural unit interval graphs.

Dependence graphs

Proof lineage

Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements

Statement-restricted proof-dependence graph for the projection formula, the braid, graphical and natural-unit-interval interpretations, and the type-B signed-source-component extension. It traces the two independent projection sources and the signed-graph arrangement inputs to exact published claims, while stopping at published books and monographs. Context, comparison, historical citations and results reproved internally are excluded. The type-A and natural-unit-interval results check, but an exact rank-two counterexample disproves the type-B projection and coefficient branch.

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AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 23, 2026
01Statements6 reported findingsContains wrong statements

The generic projection formula and the braid, graphical, and natural-unit-interval interpretations are correct. The type-BB extension is false as stated: Definition 6.7 can fail to produce a partition, and an exact rank-two example contradicts Theorem 6.17, Corollary 6.18, and Theorem 6.8. Separately, Remark 6.4's claimed bijection with simple signed graphs is false when both opposite-sign edge orbits occur. Allowing signed multigraphs repairs that remark but not the type-BB projection theorem.

Theorem 2.5Correct

The generic projection formula is stated correctly

PDF pages 4–5 · Theorem 2.5 · arXiv:2506.00941v2

For a real arrangement and a generic point, the number of regions whose metric projection lies in a kk-dimensional face equals the absolute kk-th characteristic coefficient. Kabluchko's exact Theorem 1.2 proves this dimension-indexed form directly; Lofano–Paolini Corollary 5.14 independently gives the codimension-indexed Betti form. The paper's sign convention is consistent with both.

Exact arXiv version 2 ↗
Theorems 3.3 and 4.4Correct after harmless local notation repair

The braid and graphical source-component interpretations are correct

PDF pages 6–14 · statements and complete proofs of Theorems 3.3 and 4.4 · arXiv:2506.00941v2

In the braid arrangement, the block containing each right-to-left minimum is exactly the next block exposed by the projection. In a graphical arrangement, the analogous blocks are the source components of the region's acyclic orientation. The strong coordinate separation makes the canonical face uniquely closest. In the type-AA lower-bound calculation, one occurrence of pip_i where the comparison uses the competing projection qiq_i is a uniquely determined symbol slip; the preceding equal-block identity makes the repaired line valid.

Theorem 5.2 and Corollary 5.3Correct

The natural-unit-interval specialization and factorization are correct

PDF pages 15–17 · Theorem 5.2, Corollary 5.3, and proofs · arXiv:2506.00941v2

For a natural unit interval graph, the permitted descents force each source component to begin at a right-to-left minimum, and every such minimum begins one component. Counting the eligible choices gives the displayed product for the characteristic polynomial. The endpoint and isolated-vertex cases agree with the graph and arrangement dimensions.

Remark 6.4Incorrect · verified scope repair

Simple symmetric graphs do not biject with simple signed graphs

PDF pages 18–19 · Remark 6.4 and its signed-graph construction · arXiv:2506.00941v2

On signed vertices {±1,±2}\{\pm1,\pm2\}, a simple symmetric graph may contain both orbits {{1,2},{−1,−2}}\{\{1,2\},\{-1,-2\}\} and {{1,−2},{−1,2}}\{\{1,-2\},\{-1,2\}\}. The proposed quotient then needs both a positive and a negative edge between vertices 1 and 2, which a simple graph with one sign function on its edge set cannot encode. Zaslavsky's cited framework allows parallel edges of opposite signs. Thus the remark is repaired by allowing signed multigraphs, or by forbidding simultaneous opposite-sign orbits.

Definition 6.7Incorrect definition · explicit counterexample

The signed-source construction need not produce a partition

PDF page 19 · Definition 6.7 · arXiv:2506.00941v2

In the branch −m∈Rm-m\in R_m, the definition updates the zero block BB but never defines SkS_k, although the next recursive stage uses it. Even with the obvious repair Sk=∅S_k=\varnothing, take the symmetric graph with edges {1,−1}\{1,-1\}, {1,−2}\{1,-2\}, and {−1,2}\{-1,2\} and the region oriented 1→−11\to-1, 1→−21\to-2, 2→−12\to-1. The first stage gives B={±1}B=\{\pm1\}; the second gives S2={2,−1}S_2=\{2,-1\}. Thus S2∩B={−1}S_2\cap B=\{-1\}, so the displayed antipalindromic sequence is not a partition.

Theorem 6.17, Corollary 6.18, and Theorem 6.8Incorrect · explicit rank-two counterexample

The type-BB projection and coefficient theorems are false as stated

PDF pages 20–23 and 25–28 · Theorems 6.8 and 6.17, Corollary 6.18, Remark 6.20, and proof · arXiv:2506.00941v2

Use the symmetric graph with edges {1,−1}\{1,-1\}, {1,−2}\{1,-2\}, and {−1,2}\{-1,2\}, whose arrangement is x1=0x_1=0 and x1+x2=0x_1+x_2=0. For the region R={x1≤0, x1+x2≤0}R=\{x_1\le0,\ x_1+x_2\le0\} and v=(58,1)v=(58,1), the separation hypothesis holds because 58>(14⋅22+1)⋅1=5758>(14\cdot2^2+1)\cdot1=57. The KKT identity (58,1)=57(1,0)+(1,1)(58,1)=57(1,0)+(1,1) shows that proj⁡R(v)=(0,0)\operatorname{proj}_R(v)=(0,0), a zero-dimensional face. Definition 6.7 instead gives one signed source component, so Theorem 6.17 predicts dimension one. Across all four regions, the signed-source counts are (1,1,1,2)(1,1,1,2) while the projection dimensions are (0,1,1,2)(0,1,1,2); hence the distribution also contradicts Corollary 6.18 and Theorem 6.8, since χA(q)=(q−1)2\chi_{\mathcal A}(q)=(q-1)^2.

02Proofs6 reported findingsContains incorrect or incomplete proofs

Sections 2–5 have complete proofs after one harmless symbol repair. The proof of the type-BB main theorem is not valid as written, and the exact rank-two counterexample shows that the theorem cannot be repaired only by filling its gaps. Definition 6.7 can fail before Lemma 6.24 applies; separately, the lemma omits its universal signed-containment argument, the split-block contradiction uses the wrong face and reachability closure, and the two distance cases contain several incorrect estimates.

Lemma 6.24Incomplete · unresolved

The universal signed prefix containment is asserted but not proved

PDF page 25 · Lemma 6.24 and its final proof sentence · arXiv:2506.00941v2

The lemma first says that the canonical signed-source partition labels a face, but Definition 6.7 does not even produce a partition in the rank-two counterexample recorded above. The natural disjointness repair S2:=R2∖B={2}S_2:=R_2\setminus B=\{2\} gives the sequence ({2},{±1},{−2})(\{2\},\{\pm1\},\{-2\}), whose face has x1=0x_1=0 and x2>0x_2>0 and is not a face of the region x1≤0x_1\le0, x1+x2≤0x_1+x_2\le0. Its stronger universal-prefix clause is also asserted only by analogy with type AA. Reiner identifies type-BB flats but supplies neither a zero-block saturation rule nor this ordered-face containment, so both clauses remain defective.

Split-block claim in Theorem 6.17Incorrect and incomplete as written

The higher-dimensional face and reachability argument do not yield the stated contradiction

PDF pages 25–26 · first claim in the proof of Theorem 6.17 · arXiv:2506.00941v2

The proof starts from the hypothesized nearest face F′F' and splits one of its blocks, so the refined face must contain F′F' in its closure for the strict distance contradiction. The text instead says twice that the higher-dimensional face contains the canonical face FF. More seriously, Dt′D_t' is defined as the vertices reachable from bi′b_{i'}, while moving that block later requires the opposite closure condition: with the paper's orientation, a path β→α\beta\to\alpha gives xα≥xβx_\alpha\ge x_\beta, contrary to the ordering inference printed in the subclaim. A backward-closed or component-based refinement might repair the strategy, but the manuscript does not prove that the resulting split labels a face of the region.

Error bounds in Theorem 6.17Incorrect typo · uniquely repairable

The tail denominator changes from 14n2+114n^2+1 to 6n2+16n^2+1

PDF page 27 · displayed bound for ε3\varepsilon_3 in the proof of Theorem 6.17 · arXiv:2506.00941v2

The separation hypothesis and the preceding pointwise estimate use 14n2+114n^2+1, but the next display bounds ε3\varepsilon_3 with (6n2+1)2(6n^2+1)^2. The following combined estimate returns to (14n2+1)2(14n^2+1)^2. Replacing 6 by 14 is forced and only strengthens the stated bound.

Case 1 of Theorem 6.17Local lower bound verified; upstream dependence unresolved

The strict coefficient gap is valid once the decomposition is valid

PDF page 27 · case B0=D0B_0=D_0 in the proof of Theorem 6.17 · arXiv:2506.00941v2

For integers b<d≤nb<d\le n, one has 1/b−1/d≥1/(n−1)−1/n=1/(n(n−1))>1/n21/b-1/d\ge1/(n-1)-1/n=1/(n(n-1))>1/n^2. Thus the final displayed strict inequality in Case 1 is sound after the denominator repair. It does not close the theorem because obtaining the first differing blocks still uses Lemma 6.24 and the failed split-block claim.

Case 2 of Theorem 6.17Incorrect and incomplete as written · unresolved

Zero-block sums and the special-coordinate estimate are not a valid distance comparison

PDF pages 27–28 · case B0≠D0B_0\ne D_0 in the proof of Theorem 6.17 · arXiv:2506.00941v2

The Euclidean norm sums once over coordinates 1,…,n1,\ldots,n, but a zero block contains both ii and −i-i; sums over all i∈B0i\in B_0 therefore double count unless restricted to one representative, such as B0∩[n]B_0\cap[n]. The element β\beta is chosen as the least signed element rather than least in absolute value, so the later minimality claims do not follow. If the asserted conclusion β=−α\beta=-\alpha is used, the displayed square has the wrong sign: it should involve (v∣β∣+qβ)2(v_{|\beta|}+q_\beta)^2, not a difference. Finally, the coordinate β\beta is already part of the later-block error estimate for the competing projection, yet its lower bound is added again afterward. The numerical fact 1/b−1/d>1/n21/b-1/d>1/n^2 remains correct, but the preceding upper and lower bounds do not justify applying it. A new disjoint coordinate decomposition is needed.

Sections 3–5Correct after one symbol repair

The type-AA and natural-unit-interval proofs close

PDF pages 6–17 · proofs of Theorems 3.3, 4.4, and 5.2 · arXiv:2506.00941v2

The ordered-partition projection formula, prefix containment, and first-differing-block estimate are complete in type AA. In one lower-bound line pip_i must be qiq_i, but all earlier blocks are equal, so the unique repair preserves the equality used next. The natural-unit-interval argument then identifies the source components with the permitted right-to-left minima and requires no type-BB claim.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

04Sources5 reported findingsContains incorrect or incomplete source use

The exact projection, graphical-arrangement, type-BB flat, signed-orientation, and signed-characteristic source claims were traced, and their marked theorems are correct. The source-use verdict is adverse because the signed-graph sources allow parallel opposite-sign edges while the focal text narrows them to a simple-graph bijection. Reiner is correctly cited only for unordered type-BB flats; the later ordered-face claims are presented internally, not attributed to Reiner. The inaccessible Lofano–Paolini publisher PDF is explicitly handled by an arXiv-version fallback and accepted-manuscript comparison.

Projection formulaCorrectly invoked with one access limitation

Kabluchko is exact; Lofano–Paolini is an explicit fallback

PDF pages 4–5 · Theorem 2.5 and citations [5]–[6] · arXiv:2506.00941v2; Kabluchko journal page 1479; Lofano–Paolini arXiv:1809.02476v2 pages 20–21

Kabluchko Theorem 1.2 matches the focal formula exactly and was checked in an open version of record. Lofano–Paolini Corollary 5.14 gives the equivalent codimension and Betti-number count; exact arXiv version 2 and the institutional accepted manuscript agree, but the publisher PDF was not freely retrievable. Either source independently suffices, so that access limitation does not weaken the focal theorem.

Graphical inputsCorrectly invoked

Stanley's two marked results have exact applicability

PDF pages 9–10 and 15–17 · graphical and natural-unit-interval sections · arXiv:2506.00941v2; Stanley printed pages 414–418 · Theorem 2.7 and Proposition 2.5

Stanley's region and acyclic-orientation bijection applies to the same simple graphical arrangement, and his polynomial identity identifies its characteristic polynomial with the graph chromatic polynomial. The focal source-component and right-to-left-minimum refinements are then proved internally rather than attributed to Stanley.

Type-$B$ flatsCorrectly invoked; limited to flats

Reiner correctly supports the flat identification; it does not supply the later internal claims

PDF pages 21–25 · discussion before Definition 6.16 and Lemma 6.24 · arXiv:2506.00941v2; Reiner journal pages 197–199

Reiner's exact article identifies type-BB intersection flats with sign-stable partitions and one possible zero block, exactly as cited. The focal paper then introduces the antipalindromic ordering and universal prefix-containment claim internally; it does not attribute those later claims to Reiner. Their defects are therefore internal proof defects rather than misuse of this source.

Signed-graph arrangement inputsIncorrectly narrowed in the focal translation

The exact sources use a wider multigraph convention

PDF pages 18–23 · Remark 6.4, Theorems 6.8 and 6.12, and Lemma 6.14 · arXiv:2506.00941v2; Zaslavsky 1991 Theorem 4.4 and Zaslavsky 2012 Lemma 4.3

Zaslavsky's orientation theorem and signed chromatic and characteristic identity are correct, but the source framework permits multiple edges, including positive and negative links with the same endpoints. The focal construction defines a simple signed graph with a single sign function, so it cannot represent every simple symmetric graph. Allowing signed multigraphs restores applicability of the source theorems; it does not resolve the separate projection proof gaps.

Recursive source closureComplete apart from the explicit limitations above

Every marked non-book source was checked through its proof-critical chain

PDF pages 4–5, 9–10, and 18–25 · all externally sourced steps in the audited claims · arXiv:2506.00941v2

Kabluchko was traced through KVZ and Stanley; Lofano–Paolini through Orlik–Solomon, Orlik–Terao, and Zaslavsky's region monograph; the signed orientation and characteristic branches through Zaslavsky's 1982 vector theorem. Recursion stops at published books or chapters. Contextual citations, independently reproved results, and comparison papers are recorded separately rather than turned into proof edges.

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Paper
arXiv:2506.00941v2
Authors listed
Neha Goregaokar
Audit date
August 23, 2026
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