slug: zhang-2026-threshold-hypergraph-independence-polynomial
bibkey: zhang2023thresholdindependence
doi: 10.1016/j.dam.2025.07.017
url: https://arxiv.org/abs/2303.01024v1
triage: theorem
motivation_gids:
  - D5/S3/Combinatorics/Hypergraphs/ZhangThresholdIndependenceRefutation.result

Independence polynomials do not determine T2-threshold hypergraphs

Problem

E. Zhang, On the Independence Polynomial and Threshold of an Antiregular k-Hypergraph, arXiv:2303.01024v1, Section 5, asks:

Alternatively, a more general question is, given two T2-threshold k-hypergraphs with the same independence polynomial, are they isomorphic? These are left for future work.

The universal positive answer is false. The settling declaration is D5/S3/Combinatorics/Hypergraphs/ZhangThresholdIndependenceRefutation.result. The preregistration is #19406. The formal result uses five vertices and $k=3$; no arbitrary-$N$ theorem is delivered.

Statement echo

The encoding permits $k=0,1,2$, whereas the source question concerns $k\ge 3$; the counterexample lies within the source range and all its edges are nonempty. The finite-type instances enumerate vertices and decide equality; they add no mathematical restriction to the finite source objects. The carrier Polynomial ℤ is the formal integer-coefficient expression of Equation (8).

Motivation

An independence polynomial records the number of independent subsets of each size. The pair here has equal size profiles but different vertex-incidence counts. Thus the polynomial loses information needed for isomorphism even inside the T2-threshold class.

Gap

Example 3 in the source gives non-isomorphic hypergraphs with equal independence polynomials, but establishes T2 only for one of them. The present pair satisfies T2 on both sides. The narrower antiregular identification is not established.

Route

Index the vertices by $0,1,2,3,4$. Let $H$ contain every triple except $012,013,014$, and let $K$ contain every triple except $012,013,023$. The labelings are $(0,0,1,1,1)$ with threshold $1$ for $H$, and $(0,1,1,1,3)$ with threshold $2$ for $K$. The private helpers h5_threshold and k5_threshold check these certificates.

The private helper polynomials_equal enumerates the independent subsets and proves equality of the two polynomials. The private helper degree_transport uses Finset.card_bij to transport incident-edge counts across a vertex bijection. The private helper not_isomorphic uses degree $5$ at vertex $2$ of $H$ and absence of degree $5$ in $K$. All five helpers occur on the live proof path of result and are classified bind-only.

Falsifier

An isomorphism of the displayed pair would preserve the number of edges incident to each vertex. It would send vertex $2$ of $H$ to a vertex of $K$ of degree $5$, contradicting the checked degree obstruction. A source-reading change that excludes this pair from T2 would invalidate the correspondence; Definition 4.1(T2) is represented with its real labels and strict inequality.

Evidence

The kernel-checked result refutes the universal claim, using the private threshold, polynomial and non-isomorphism certificates. Its axiom closure is contained in $\{\mathrm{propext},Classical.choice,Quot.sound\}$. No atom or coverage edge is used; the admission basis is open-problem-resolution (#19406; Refuted).

The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9). There are no other public theorems or published owner declarations in this delivery.

Experiment

The independent finite computation is the pinned experiment directory. Its script is check.py, SHA-256 72434578a87af92ee58b5f03c081f20e7e2e83a3527fb6a275ae6b4d842e4824. Command: git -C ../trureturing-experiments show a2c892b2d83ecae613ebfdbe7d4b628faddb2f58:docs/reports/zhang-2023-threshold-hypergraph-independence/check.py | python3 (exit $0$). For exactly $N=5,6,7,8$, it checks both T2 labelings on every triple, the coefficient lists $(1,N,(\genfrac{}{}{0ex}{}{N}{2}),3,0,\dots ,0)$, and non-isomorphism by all $N!$ vertex permutations. The coefficient lists are respectively $(1,5,10,3,0,0)$, $(1,6,15,3,0,0,0)$, $(1,7,21,3,0,0,0,0)$ and $(1,8,28,3,0,0,0,0,0)$. The program reports RESULT PASS; its tested scope is finite.

Triage

What the settlement shows

The mechanism for this pair is that the independent-set size profile retains three independent triples but loses their intersection pattern. Polynomial equality and the non-isomorphism obstruction are checked on the same five-vertex objects; the explicit profiles are computed by the experiment. The degree-$5$ obstruction supplies the structural distinction.

The nearest surviving statements are the source’s independently proved polynomial formulas and threshold results: they do not require a positive answer to its closing question. No stronger uniqueness theorem, T1 analogue, arbitrary-size formal family or antiregular identification is established here. The source’s separate antiregular uniqueness question remains open within this delivery. No other theorem of the source is asserted to be refuted.

Literature status

The source and literature readings attributed to the orchestrator in #19406 cover Equation (8), Definition 4.1(T2), Example 3 and Section 5 of arXiv v1. The search seat screened the author’s works 2306.15180, 2502.02097 and 2609.12293 and the related paper 2602.21382; it reported no settlement and zero citations in OpenAlex and Crossref. MathDB was not located. These are the recorded search bounds, not a proof of absence of prior work.

ASSUMED-UNVERIFIED

The final subscription journal text has not been read. The literature screen’s completeness, worldwide priority and the antiregular identification remain ASSUMED-UNVERIFIED. The computed $N=5,6,7,8$ checks do not establish an arbitrary-$N$ formal family. The journal DOI locates the source; the quoted statements are from arXiv:2303.01024v1.