Successive new Laplacian layer maxima of token graphs

M. A. Reyes, C. Dalfó and M. À. Fiol, On the spectra and spectral radii of token graphs, arXiv:2310.16929v1, Section 3, Definition 3.1 and Conjecture 3.3; journal version, Boletín de la Sociedad Matemática Mexicana, volume 30, article 11 (2024), DOI 10.1007/s40590-023-00583-3, Section 3:

Conjecture 3.3. For any graph G, ρ₁(G) ≤ ρ₂(G) ≤ ⋯ ≤ ρ⌊n/2⌋(G).

Here G is a finite simple unweighted undirected graph of order n. The source uses the ordinary real Laplacian D−A. The vertices of F_j(G) are all j-element subsets of V(G), with an edge precisely when their symmetric difference is an edge of G. F_0(G) is the singleton empty configuration. Writing Sp_j for the full Laplacian spectral multiset, Definition 3.1 sets Δ_j = Sp_j − Sp_(j−1) by multiset subtraction and ρ_j = max Δ_j, for 1 ≤ j ≤ floor(n/2). Thus the conjecture quantifies every 1 ≤ k < floor(n/2) and asserts ρ_k ≤ ρ_(k+1). The source's connected-graph convention is satisfied by the counterexample.

The existing frozen tokenGraph definition models actual token subsets and moves. Full-sector spectral maxima and new-layer maxima are different quantities: inherited eigenvalues remain in the former and are removed, with their multiplicities, from the latter. This dossier addresses only Conjecture 3.3, not the neighboring minimum conjecture or a full-radius characterization.

Issue #23401 preregisters the whole original Conjecture 3.3. The bounded all-state qualification found no equivalent target registered to another owner and no complete posted answer in its searched scope. Closed #12457 and PR #12611 concern arXiv:2610.00500 Conjecture 1.1, a different assertion about full Laplacian radii. This is not a worldwide absence or priority claim.

Take the connected star K₁,₃ on four vertices and k=1. The actual zero-, one- and two-token spectral multisets are

| j | Sp_j | Δ_j | ρ_j | | --- | --- | --- | --- | | 0 | {0} | — | — | | 1 | {0,1,1,4} | {1,1,4} | 4 | | 2 | {0,1,1,3,3,4} | {3,3} | 3 |

The two-token graph is C₆. Explicit equivalences enumerate all token subsets; entrywise Laplacian identities and complete invertible rational eigenbases prove these entire spectra through characteristic-polynomial roots with multiplicity. Consecutive multiset subtraction then gives 4 > 3. Since 1 ≤ 1 < floor(4/2), this contradicts the original universal statement. The proof of nonempty layers covers every graph and every 1 ≤ j ≤ floor(n/2), using the strict binomial-cardinality increase, so ρ is an actual maximum throughout its source domain.

Dongsheng Wei, Generators of stability-preserving semigroups, spectral gaps, and classical ground-state computation, arXiv:2609.26802v1 §4.6 and v2 §7, supplies prior star/two-token-cycle and full-sector spectral ingredients. The source note attributes them; no new spectral method is claimed.

A mismatch in the token-subset enumeration, an incorrect D−A matrix, a missing eigenvalue multiplicity, set subtraction in place of multiset subtraction, or a source restriction excluding this connected star would invalidate this refutation. The matrices and full eigenbases are checked in Lean. In particular the full positive-level radii are both 4; comparing those instead would not establish the asserted new-layer decrease.

The settling endpoint is D5.S3.Combinatorics.Graph.TokenGraph.NewLayerMaximaRefutation.result. Its literal compiled type is Not claim; claim : Prop retains every n, every SimpleGraph (Fin n) with decidable adjacency, and every 1 ≤ k < n/2. There is no order cap, certificate premise or nonemptiness assumption. The source and journal readings agree on Definition 3.1 and Conjecture 3.3.

The module contains four authored public definitions (spectrum, layer, rho, claim), the sole designated public theorem result, ten private helper lemmas consumed by rho or result, and two compiler-generated companions. Its checked axiom closure is contained in {Classical.choice, Quot.sound, propext}, with no sorry or new axiom. The canonical Scribe attaches one OpenProblemResolutionClaim of kind Refuted to this endpoint and cites the Library source identity.

Tier 1, theorem; admission basis open-problem-resolution under #23401. The proof shape is bind-only and the escape witness is none: finite exact matrix calculations and existing spectral machinery discharge the entire external conjecture's negation. Every helper is consumed; no standalone method or positive-instance theorem is packaged. The designated utility refutation is exempt from four-slot registration; there are no other new public theorem audit targets. The utility is certified-instance with basis refutes, bound to this module's literal claim and result.

Source fact: immediately after Conjecture 3.3, the source observes from its equation (7) that, if the conjecture holds, then ρ_k(G) = ρ(F_k(G)) for k ≤ n/2. This is a conditional consequence of the source conjecture, not an additional unconditional premise of the refuted claim.

Compiled consequence: the K₁,₃ witness removes the inherited top eigenvalue 4 when forming the second new layer. Thus its residual layer is {3, 3} and has maximum 3, while the full two-token Laplacian spectrum still has radius 4. The full-spectrum positive-level radii therefore remain monotone in this witness (4 = 4); the decrease is specific to the new-layer maxima.

Source fact: neighboring Conjecture 3.2 concerns the minima α_k and is not settled by this counterexample. The refutation also does not settle the source's conditional equality consequence or any separate full-radius claim.

Open repair boundary: no condition restoring successive new-layer monotonicity has been proved here. Restricted graph classes, a different comparison of full-sector and residual spectra, and hypotheses under which the source's conditional equality can be established remain open questions for later work.

The supplied supplier-first and all-state qualification evidence has bounded search scope. It establishes neither worldwide novelty nor exhaustive absence of an earlier settlement. The star and cycle ingredients are attributed to Wei. Selected Lean/report checks establish only their selected modules and imports; final independent CLI architecture, quality and tests reviews and exact-head required CI remain separate publication gates. This dossier grants no solve credit, issue closure or merge authorization.