bibkey: chaithrarani2025marked authors: Chaithra P; Shushma Rani; R. Venkatesh year: 2025 title: “Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements” doi: 10.48550/arXiv.2507.20847 url: https://arxiv.org/html/2507.20847v1 claim: “Section 8.1: for a simple hypergraph on a positive finite number of vertices, the reciprocal of its signed ordinary independence polynomial has nonnegative rational coefficients at every multiindex if and only if every edge has even cardinality.” strata_touched:

• D5/S0/Certificates/Hypergraphs/EvenInversePositivityRefutation license: citation-only triage: anchor

Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements

Source wording and scope

Quotations below preserve the prose verbatim, with mathematical glyphs and whitespace normalized. Page numbers refer to the v1 PDF.

• Section 8.1, p. 23: “Let 𝒢 be a simple hypergraph. Then we have I(𝒢,−x)⁻¹ ≥ 0 if and only if all edges of 𝒢 must have even number of elements.”
• Section 3.1, p. 7: “Let n be a positive integer.”
• Definition 2(1), p. 7: “A hypergraph 𝒢 is called simple if for any e,f ∈ ℰ, we have |e| ≥ 2 and e ⊆ f implies e = f.”
• Definition 2(5), p. 7: “A subset I ⊆ 𝒱 is called independent if no edge of 𝒢 is entirely contained in I, i.e., e ⊈ I for all e ∈ ℰ.”

Definition 3 (p. 7) defines I(𝒢,x) as the sum of ∏_{v∈I} x_v over all independent sets, including the empty set. The ambient series have rational coefficients (p. 1); nonnegativity is coefficientwise (p. 2). The introductory Conjecture 1 is on p. 5; the full equivalence targeted here is the statement in Section 8.1 on p. 23. It concerns the ordinary polynomial, without marked vertices or a restriction to squarefree coefficients.

Relabel the source [n] as Fin n. Define indicator S to be the exponent vector equal to one on S and zero elsewhere, and

signedIndependence E : MvPowerSeries (Fin n) ℚ :=
  ∑ S ∈ Finset.univ.powerset.filter (fun S => ∀ e ∈ E, ¬ e ⊆ S),
    MvPowerSeries.monomial (indicator S) ((-1 : ℚ) ^ S.card)
SourceSimple E :=
  (∀ e ∈ E, 2 ≤ e.card) ∧ (∀ e ∈ E, ∀ f ∈ E, e ⊆ f → e = f)
claim := ∀ n : ℕ, 0 < n → ∀ E : Finset (Finset (Fin n)),
  SourceSimple E →
    ((∀ m : Fin n →₀ ℕ,
        0 ≤ MvPowerSeries.coeff m (signedIndependence E)⁻¹) ↔
      ∀ e ∈ E, Even e.card)

These are the semantic definitions and full proposition, written schematically with n implicit in the first two definitions. Simplicity ensures the empty set is independent, so the signed polynomial has constant coefficient one and its reciprocal is the actual formal power-series inverse.

Counterexample and attribution

The source supplies the conjecture and definitions. The repository refutation uses vertices 0,…,7, labeled a1,a2,a3,b1,b2,b3,s,t, and edges {6,7,0,1}, {6,7,1,2}, {6,7,2,0}, {6,7,3,4}, {6,7,4,5}, {6,7,5,3}. Six distinct four-element sets satisfy both simplicity clauses and have even size. Their full squarefree inverse coefficient is −14.

For a subset T, let i=|T∩{0,1,2}|, j=|T∩{3,4,5}|, and d=(1,1,0,−4). The certificate is c(T)=1 unless both 6,7 belong to T, in which case c(T)=2−d(i)d(j). With f(U)=(−1)^|U| for independent U and zero otherwise, it satisfies c(∅)=1 and c(T)=−∑_{∅≠U⊆T} f(U)c(T∖U) for nonempty T. Squarefree exponent decompositions are exactly subset/complement pairs. Induction on cardinality therefore identifies this certificate with coefficients of the actual inverse. At the full set, c(T)=2−(−4)(−4)=−14. This refutes the sufficient direction of the equivalence. The production Lean module D5/S0/Certificates/Hypergraphs/EvenInversePositivityRefutation.lean contains the complete finite certificate and public theorem result : Not claim. The production source compiles with compiler-checked result axiom closure [propext, Classical.choice, Quot.sound]; the certificate proves the actual inverse coefficient −14 at the full squarefree multiindex.

Verified locator

• DOI: 10.48550/arXiv.2507.20847
• URL: https://arxiv.org/html/2507.20847v1
• Version: arXiv:2507.20847v1; PDF pp. 1, 2, 5, 7, 23 as specified above.
• Retrieved primary HTML SHA-256: 39e8da49ecd6878f0cc909c3ec2b554d78ca66dda89f363a3450360ef36ca700.

Bounded literature status

The supplied bounded source check covers v1 and related work of Zhang–Dong (2020), DOI 10.1016/j.disc.2020.112134, and Foissy (2026), DOI 10.1016/j.aam.2026.103067. No later full resolution was located within that check. This does not establish exhaustive absence or worldwide priority. The source reports Sage checks through ten vertices, which conflicts with the eight-vertex witness. The tested families and code were unavailable to the supplied check; the cause of this discrepancy is undetermined.