Rank Access and Closure Depth in the Relation-Layer Cohomology of Token Arenas
Abstract
Rank Access and Closure Depth in the Relation-Layer Cohomology of Token Arenas A refuted conjecture, a refuted mechanism, and two species of invariant harmonic residue Driven by Dean Kulik October 2026 Abstract A Kdisj-type relation layer over a token arena assigns cohomology to a graph by declaring which closed loops count as relations. We settle an open problem about such layers and then identify what actually governs them. First, the conjecture that two vertex orbits force εharm(K4) = 0 is refuted by an exact 12-vertex counterexample with |Aut| = 24, reproduced by two implementations whose constructions of the invariant cochain space share no logic. Second, we localize the failure: it is not a shortage of cells. On the witness's two-dimensional residue, all 1083 arena 4-cycles supply rank 1, while 20 arena 3-cycles of an excluded shape supply rank 2, and the two nonzero square families have the same image, a single line. Third, we refute the natural mechanism for this — that the residue is an odd-cycle blind spot — by exhibiting bipartite arenas, containing no odd cycle of any length, that still carry residue. The operative property is omission from the declared family, not parity. Fourth, we prove that εharm(K3,4) > 0 on an infinite family — every cycle graph with n ≥ 5 and 2 ≤ K ≤ n−1, under the trivial group and the rotation subgroup, and zero under the full dihedral group — answering a question the odd-cycle reading had posed. Fifth, we define closure depth Lclosure, prove it exceeds 4 for every member of that family, and measure it as exactly n for Cn and 4 for the witness, so no short vocabulary closes the family. This separates two populated regimes: residue a longer reader dissolves, and residue carried by configuration-space topology. Finally we identify the second kind. At trivial group the framework's own quantity is simplicial H1 over ℚ, and the declared cells are the 2-skeleton of Abrams' discretized configuration space; the cell counts and Euler characteristic are verified against closed form, Abrams' subdivision hypothesis for Cn reduces to n ≥ K + 1, the measured rank is 1 on all 33 admissible rows and 0 on all 8 excluded boundary rows, and the equivariant degree is solved exactly: +1 for rotations, −1 for reflections. All arithmetic is exact rational. Keywords: graph configuration space, graph braid group, token graph, discrete Morse theory, equivariant cohomology, cube complex, relation layer, closure depth MSC 2020: 55R80, 05C10, 20F36, 55N91, 05C85 License: CC BY-NC 4.0. Source code and executed notebook accompany this paper; see Appendix B. 1. Introduction A graph does not come with a cohomology. One is obtained by declaring a relation layer: a set of closed loops in some derived space that are stipulated to bound. Change the declaration and the cohomology changes while the graph does not. This paper is about what that dependence actually is, measured rather than described. The derived space here is a token arena. Fix a finite simple graph X, an integer K ≥ 1, and a group G ≤ Aut(X). The arena AK(X) has the K-element subsets of V(X) as vertices, with an edge whenever two subsets differ by moving one token along one edge of X. A relation layer declares a family of arena loops as 2-cells; the G-invariant cochain complex of the resulting 2-complex has a first cohomology whose dimension we write εharm. The quantity is relative to the declaration, and that is the point. Two declarations are natural and were already in use. Kdisj admits only commuting squares: two token moves along base edges with disjoint closed supports, which therefore commute. K4 admits every 4-cycle of the arena. Since Kdisj ⊆ K4, monotonicity gives εharm(K4) ≤ εharm(Kdisj), and the interesting question is when the larger family closes what the smaller one leaves open. 1.1 What this paper settles The conjecture under test asserted that a substrate with exactly two vertex orbits forces εharm(K4) = 0. It is false. Section 3 exhibits a 12-vertex witness satisfying the hypothesis with εharm(K4) = 1, certified in exact rational arithmetic and reproduced independently. Refuting a conjecture is the cheap part. The substance is in sections 4 through 8, which answer: what is the residue, and what reads it? Three successive mechanisms were proposed during the investigation and two were killed by their own predictions. What survives is a statement about rank, a statement about closure depth, and an identification of one residue species against a cited theorem whose hypothesis we verify and whose boundary we test. 1.2 Summary of results § Statement Status A §3 The two-orbit conjecture is false: a 12-vertex, |Aut| = 24, two-orbit witness has εharm(K4) = 1. refuted by exact witness B §4 On the witness's 2-dimensional residue, 1083 arena 4-cycles have rank 1; 20 arena 3-cycles have rank 2. The spectator and rotation square families share one image line. computed C §5 The odd-cycle mechanism is false: bipartite arenas with no odd cycle of any length still carry residue. refuted by witness family D §6 εharm(K3,4) > 0 on an infinite family: Cn for every n ≥ 5 and 2 ≤ K ≤ n−1, under G = 1 and G = Zn; zero under D2n. PROVED (Thm 6.3), given [1] E §7 Lclosure = 4 for the witness; = n for Cn on every tested row. Lclosure > 4 for all n ≥ 5 is proved. computed; partly proved F §8 At trivial G the computed quantity is dim H¹(UDK(X); ℚ); Abrams' hypothesis for Cn is n ≥ K+1; measured 1 on 33 admissible rows, 0 on 8 boundary rows; equivariant degree +1 / −1. cited theorem, hypothesis verified, prediction confirmed Table 1. The results, with the epistemic status of each. No entry is a theorem proved here; A and C are refutations by explicit witness, B, D and E are exact finite computations, and F is a cited equivalence whose hypothesis is checked and whose prediction is confirmed across its admissible range and contradicted at its first excluded row. 1.3 Background, and what is new here The toolkit is established and no part of it is claimed here. Unordered K-token states on a graph, collision-free token moves, the cubical model in which independent moves supply cells, H1 measuring exchange and traversal loops, the homotopy type UConfK(S1) ≃ S1, and the fact that a rotation preserves while a reflection reverses an orientation class on a circle — all of this is standard graph-configuration-space and graph-braid-group topology [1, 3, 4, 5, 7]. Nothing below asserts otherwise, and §8.7 itemizes the provenance of every ingredient in the one place where a cited theorem does load-bearing work. What is new is the treatment of the relation family as a variable rather than as background. The datum studied here is the four-part object ℜ = (X, K, G, 𝒦) ⟼ H1G(AK(X), 𝒦), in which the arena and the symmetry sector are held fixed while the declared vocabulary varies over Kdisj, K4, K3, K3,4, K≤L. The standard model takes the commuting cells and stops; the question here is comparative — which global circulations survive which local language. The distinctive object is the closure profile L ⟼ εharm(K≤L), and its threshold Lclosure. One identification places the comparative picture inside existing literature and we make it explicitly. Grigor’yan, Lin, Muranov and Yau prove that the first path homology of a graph is the cycle space modulo the subspace generated by all 3- and 4-cycles [9, §6]. At G = 1 that is exactly our K3,4: εharm(K3,4 | G = 1) = dim H1path(AK(X); ℚ). So the strongest declared vocabulary considered here is not an invention: in the non-equivariant case it computes first path homology of the arena. Measured, this sharpens the dichotomy of §7.3 into standard language. The 66-vertex arena of the §3 witness has cycle rank 235 and rank d1(K3,4) = 235, so its first path homology vanishes; every arena of Cn in the tested range has first path homology of dimension 1. Vocabulary residue is residue the arena does not actually carry; traversal residue is residue it does. A bounded novelty search was run for the closure profile. Scope: English-language search of arXiv and the open web, October 2026, on cycle-length filtrations of graph configuration spaces, short-cycle fillings in graph braid complexes, cohomology under selective 2-cell attachment, and path-homology filtrations by cycle length, together with backward citation from [9] and [2]. Result: [9] supplies the fixed 3-and-4-cycle quotient, on a graph rather than an arena, non-equivariantly, and defines no bounded-length family and no vanishing threshold. We found no source defining the parametrized family K≤L or an invariant of Lclosure type. We therefore classify the closure profile as apparently new within that search scope, which is a scoped statement and not a novelty theorem; a single overlooked reference would retract it. The underlying ingredients remain established as above. 1.3 Methodological commitments Three commitments shaped what is claimed. First, every quantity reported here was executed; no number in this paper is quoted from an unrun expression. Second, every structural quantity was computed twice, by two engines sharing no logic at the point where an error would be most consequential (§9.1). Third, a killed claim is reported, not discarded. Section 9.4 lists seven claims refuted during the investigation, five of them by the author's own controls. Their negative information is part of the result: the reason the earlier search population looked so clean is itself a finding (§3.4). 2. The invariant relation-layer complex 2.1 Token arenas Definition 2.1. (token arena) Let X be a finite simple graph with vertex set V, and K ≥ 1. The arena AK(X) is the graph whose vertices are the K-element subsets S ⊆ V, with S adjacent to T whenever T = (S \ {u}) ∪ {w} for some u ∈ S, w ∉ S with uw ∈ E(X). Arena vertices are collision-free K-to
