The Volumetric Spectral Interference Functional

Volumetric Spectral Interference Functional (VSIF) Exact Certification, Spectral Graph Structure, and the Ihara–MaxCut Resonance Synopsis This work develops the Volumetric Spectral Interference Functional (VSIF) around one question: does the prime-cycle structure of a graph, read through its Ihara zeta function, carry information about the ground-state energy of the MaxCut Hamiltonian? Version History Version 2.0 (March 2026) posed the question as a heuristic estimator built from three sources: the prime explicit formula, the Ihara zeta function, and the spectral form factor. It reported approximate agreement on eighteen benchmark graphs. Version 2.1 (September 2026) rebuilds the construction from the ground up. Every object is defined and every property that is used is proved. Every measurement can be reproduced. The principal conjecture, the Ihara–MaxCut Resonance, is tested only on graphs whose ground-state energy is known exactly. The revised paper contains 104 proved statements: 15 theorems 63 propositions 15 lemmas 11 corollaries None of these statements assumes the conjecture or depends on its outcome. What the Reader Will Find 1. Exact Certification in Rational Arithmetic Upper bounds on the maximum cut arise in two forms: diagonal shifts; fractional loads on odd cycles. Both arise as values of a single operator, M(y,u). Every integer statement about these bounds is settled exactly on one lattice of levels, with no floating-point step. This determines the ground-state energy exactly on: nine deterministic benchmark graphs; all 84 integral random graphs with at most 20 vertices; all 50 random regular graphs with 40 to 100 vertices. An exact accounting of the remaining gap is also provided. 2. Exact Decomposition of the Certificate Gap The distance between any cut and any certificate splits exactly into: cycle deficits; spectral spread. When a certificate value is an integer, deciding whether a cut attains that value becomes a finite exact search. 3. Two Walkers on One Spectral Axis The certificate and the best cut move as two walkers that read the same leading eigenvector in different ways: one through its signs; the other through its squares. They meet exactly when a single exact count reaches zero. Gaussian fibers provide guidance but never determine the result. This structure has formal parallels with: Pollard's kangaroo method; Turing's completeness check; Platt's windowed isolation of zeros. 4. New Results on the Ihara Zeta Function 4.1 Regular Graphs On regular graphs, a single pole fixes the unshifted spectral bound. 4.2 General Graphs On every graph, the odd part of \log \zeta_G limits, through the odd girth, how far fractional certificates can reach. 5. Further Exact Tools and Structural Results 5.1 Co-Jointed Correction A co-jointed correction scores an estimate only after certification and certifies which side of the maximum cut it falls on. 5.2 Joint Barrier and Weak Duality A family of certificates has a joint barrier given by weak duality for the odd-cycle-tightened semidefinite relaxation, written as a Laplacian. 5.3 Cycle-Moment Unification A unification theorem expresses each cycle component as an exact function of cycle-length moments of one operator, together with a proved weighted Euler product. 5.4 Master Error Inequality A master error inequality separates: certified width; modelling error; discretization error. 5.5 Filter Theorems The paper develops filter theorems, including an exact endpoint theorem. 6. Optical and Propagator Interpretation The form factor, arm products, and walker guidance are derived as rear-plane intensities or propagator integrals of a lens system. No optical reading can alter a certificate. Negative Results 7. What the Construction Does Not Establish The revised work records several negative findings explicitly. 7.1 Prime and Riemann-Zeta Terms The terms constructed from primes and from zeros of \zeta do not see the graph and therefore cannot locate the ground state. 7.2 Ihara Contribution The Ihara term increases the position error rather than reducing it. 7.3 Status of the Principal Conjecture The principal Ihara–MaxCut Resonance conjecture is not supported by its own tests on graphs for which the answer is known exactly. The conjecture is therefore retained exactly as stated and reported as unsupported. Reproducibility and Verification 8. Reproducibility The paper provides: a reference implementation with a declared scope; a mechanically checked dependency graph; an independent verifier of every certificate; an independent validation of the mathematics; a protocol whose falsification criteria were fixed before any measurement. 9. Attribution and Prior Work A dedicated section separates results attributable to: Delorme and Poljak; Goemans and Williamson; Poljak and Rendl; Helmberg and Rendl; from results introduced in the present work. 10. Withdrawn Results from Version 2.0 Every result withdrawn from Version 2.0, including the estimator figures reported there, is recorded in an appendix together with the reason for its withdrawal. Importance and Classification of Results 11. Central Contribution The central contribution is a method that is not specific to MaxCut or to the Ihara zeta function. It turns a relaxation bound on an integer-valued objective into an exact integer decision. The construction rests on three facts: When the bound is the top eigenvalue of an operator, its levels form one integer lattice that exact rational tests can decide. The gap between the bound and any candidate solution splits into terms read directly from the certificate. When the bound is an integer, deciding whether it is attained reduces to a finite search. 12. Natural Extensions Natural next applications include: graph bisection; max-k-cut; unconstrained binary quadratic optimization; Ising ground states with couplings of both signs. Final Status of the Conjecture 13. Ihara–MaxCut Resonance The conjecture is kept exactly as stated and reported as unsupported. The tests are nevertheless results in their own right. They establish what the arithmetic and spectral components can and cannot carry, while the mathematical framework developed to carry out those tests stands independently of the conjecture. 14. Classification of the Work The work belongs to: exact combinatorial optimization; spectral graph theory; computational mathematics; mathematical verification. It also contains a documented negative finding concerning a cross-domain resonance hypothesis. The manuscript is a preprint, revised once and now left to rest. 15. Prospective Direction The preface identifies a flow-based computation of the fractional odd-cycle certificate as the most promising direction for later work.