Bonetto–Jardón Kojakhmetov Conjecture 1

Bonetto and Jardón Kojakhmetov, On the Eigenvalues of Graphs with Mixed Algebraic Structure, arXiv:2408.00487v1, Section 3, Conjecture 1 states:

The signature of the matrix M(ε) is a monotonic decreasing function of ε.

For every universe-polymorphic finite vertex type V with Fintype and DecidableEq instances, let F,G,H be simple undirected loopless graphs on V with disjoint G,H and G union H = F. Set M(u)=L_G+u A_H over the reals, using the ordinary combinatorial Laplacian and 0/1 adjacency matrix. For every real 0<s≤t, the entire assertion is

signature(M(t)) ≤ signature(M(s)).

Signature is the integer number of positive roots minus negative roots of the actual characteristic polynomial, counted with algebraic multiplicity; zero roots contribute zero. For these symmetric matrices, the full root multiset equals the indexed spectral eigenvalue multiset. Decreasing means nonincreasing, allowing equality and constant intervals. Singular matrices, persistent kernels, disconnected graphs, empty edge classes, the empty vertex type, s=t and the whole positive parameter axis remain included. Transition counts, limiting signatures, strict decrease and substitution of zero for a positive parameter are separate assertions and are not claimed.

The existing positive-subspace characterization of spectral inertia and the frozen negation identity for positive and negative indices provide an exact route that retains kernels and multiplicities. The source uses signature to compare stable and unstable directions in linear systems with diffusive and saddle interactions.

The source states the full signature assertion as Conjecture 1 and its Outlook asks for a proof or further understanding. Its perturbative results and numerical plots do not prove this assertion on the entire positive axis. The bounded source and registration searches found no other exact registration or posted complete settlement in their scope; they do not certify worldwide absence, priority or a new method.

The sole public theorem D5/S3/Combinatorics/SpectralGraph/BonettoSignature.result proves the complete quantified assertion above. The partition is retained for the original graph interpretation, although the proof uses only symmetry and positive semidefiniteness of L_G. For 0<s≤t,

M(s) = (s/t) M(t) + (1-s/t) L_G,
-M(t) = (t/s) (-M(s)) + (t/s-1) L_G.

The first coefficient in each identity is strictly positive and the second is nonnegative. Thus positive subspaces of M(t) remain positive for M(s), while positive subspaces of -M(s) remain positive for -M(t). The maximal-dimension characterization gives Nplus(M(t))≤Nplus(M(s)) and Nminus(M(s))≤Nminus(M(t)); subtracting in the integers gives the signature inequality. No inverse, eigenvalue-continuity argument, generic crossing, simple-spectrum assumption or kernel removal is used.

A different original matrix or signature convention, an omitted original case, an added hypothesis, failure of the algebraic-multiplicity bridge, or failure of either positive-subspace transfer would invalidate this resolution. An exact pre-existing settlement or another person's exact registration excludes the open-problem admission claim. A counterexample to a neighboring transition or limiting-signature assertion would not refute the stated monotonicity theorem.

The source note binds Definitions 1–3 and Remark 1, Definitions 7–8 and Remark 2, all of Section 3 including the conjecture and its transition discussion, Section 4 Outlook, and the version record. The primary HTML and record match the source hashes recorded in that note and the preregistration evidence. The native Lean result uses actual SimpleGraph matrices and strict-sign characteristic-root counts. Targeted Lean and scoped inspector evidence cover the whole result, standard axiom closure, empty dimension, singular repeated roots and equality. The corresponding Scribe resolution binds this exact theorem, with no companion theorem or replacement closed claim. Preregistration: https://github.com/the-omega-institute/trureturing/issues/23242.

theorem; Tier 1, a named external conjecture in an August 2024 paper with a complete short route from existing inertia results. Admission basis is open-problem-resolution, not a new spectral method. The result is conservatively proof_shape: bind-only. The definitions IsEdgePartition, mixedMatrix and signature are source-object bindings consumed by result. The private helpers mixedMatrix_isHermitian, signature_eq_indices and posIndex_le_of_form are bind-only and all consumed on the live result proof path. There are no unused wrappers, additional public theorems or new dependencies. Utility is none: this is a symbolic theorem over arbitrary finite types and all positive parameters, not a bounded enumeration or numerical certificate.

The four-slot source-bound audit is unfinished: https://github.com/the-omega-institute/trureturing/issues/23257. The scoped registration inventory is empty and no Reg mirror is delivered. Missing evidence comprises complete-telescope reconstruction and sourceSelection binding of both composite matrix operands; a lawful enrolled realization with the exact equivalence and actual-positive bridge; whole-family variation and sensitivity with other observations fixed; and actual observational dependence with current binding evidence. No declared_validated, designated-refutation exemption or interface incapability is claimed. The linked issue discloses this gap under CLAUDE.md §3.9 and supplies no audit proof. Mathematical freeze and the typed Proved resolution do not certify completion of this audit.

Worldwide prior-proof absence remains unverified beyond the bounded searches described in the source note and preregistration. Standalone scribe-content-checks exited 2 because its raw-v3-only reader rejects the scoped-v1 report before evaluating the candidate; issue 20682 records that contract boundary. This is not a passed-check claim. Full supported required CI on the final head and final independent CLI architecture, quality and tests reviews remain pending. Merge, the first dev push and solve credit remain caller-owned lifecycle steps.