--- pretty_name: "One-Sided Spectral Extremality Forces Maximal Degeneracy" license: other tags: - mathematics - spectral-theory - quantum-graphs - metric-graphs - eigenvalues - graph-laplacian - spectral-rigidity - nodal-theory - variational-analysis - reproducible-research - theorem-proving - research-candidate - mathematical-physics --- # One-Sided Spectral Extremality Forces Maximal Degeneracy **Complete Quantum-Graph Equality Classification, Direct Nodal Inertia, and Quantitative Port Obstructions** **Author credit:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki **Release:** v1.0.0 — 2026-09-25 **Status:** proof-complete **research candidate**; not independently peer reviewed; historical priority not certified. **Repository type:** standalone mathematical research artifact with manuscript, source, code, saved checks, proof audit, prior-art boundary, and machine-readable metadata. > **Expert-entry point:** start with [`MANUSCRIPT.pdf`](MANUSCRIPT.pdf), then read [`PRIOR_ART_AND_CLAIM_BOUNDARY.md`](PRIOR_ART_AND_CLAIM_BOUNDARY.md) and [`PROOF_AUDIT.md`](PROOF_AUDIT.md). > **AI-agent entry point:** read [`llms.txt`](llms.txt) and [`metadata/AI_AGENT_INDEX.json`](metadata/AI_AGENT_INDEX.json) before summarizing novelty or theorem status. ## Research claim in one paragraph For a finite connected compact metric graph with positive edge lengths, scalar operator `-u''`, standard Kirchhoff conditions at interior vertices, and Dirichlet/Neumann conditions at leaves, the release studies equality in the established high-index lower eigenvalue bound. In the stated regime and excluding the pure circle, the candidate theorem proves that **one lower-sharp eigenvalue already forces the full extremal structure**: the graph is a phase-locked lasso tree, a common-parity theta, or an even figure-eight; the threshold eigenvalue has multiplicity `D + N + 2β - 1`; and the same eigenvalue block is simultaneously upper-sharp. The release also proves a quantitative constrained-spectrum lower bound that turns incompatible closure constraints into an explicit positive eigenvalue penalty. ## Main theorem candidate Let `D`, `N`, `β`, and `L` denote the numbers of Dirichlet leaves, Neumann leaves, the cycle rank, and total length. In the high-index regime - `k >= max(N+β, 1)` when `D > 0`, - `k >= max(N+β, 2)` when `D = 0`, set `d = L / (k - (N+β)/2)` and `λ* = π²/d²`. The manuscript proves the equivalence ```text λ_k(G) = λ* ⇔ G is a phase-locked lasso tree, common-parity theta, or even figure-eight ⇔ mult(λ*) = D + N + 2β - 1 with top index k ⇔ the same threshold block is simultaneously lower- and upper-sharp. ``` ### Classified metric families - **Phase-locked lasso trees.** Each terminal loop has length `2 r d`, `r ∈ N`. After treating loop attachments as virtual Neumann leaves, each skeleton edge has length `(m + ν/2)d`, where `ν` counts virtual Neumann endpoints and `m` obeys the positivity rules stated in the manuscript. - **Common-parity theta graphs.** The three edge lengths are `m_i d` with positive integers `m_i` all of the same parity. - **Even figure-eight graphs.** Both loop lengths belong to `2d N`. The topology families themselves and the equivalence of **two-sided** extremality with maximal degeneracy are credited prior art. The candidate novelty asserted here is the **one-sided lower-sharpness implication**, its explicit metric completion, the direct nodal-inertia mechanism, and the quantitative constraint theorem. See [`PRIOR_ART_AND_CLAIM_BOUNDARY.md`](PRIOR_ART_AND_CLAIM_BOUNDARY.md) before making any priority claim. ## Additional proved results ### Direct nodal inertia on trees For a fully supported positive-frequency tree eigenfunction with `s` interior zero points and `r` nodal cells, ```text N_T(<λ) = s mult_T(λ)= r-s N_T(≤λ) = r. ``` The proof handles degenerate eigenvalues and branching zeros directly via an inertia calculation on the nodal-cell/zero incidence block. ### Residue-only sharpness criterion in the saturated-tree geometry For an admissibly opened saturated tree, ```text lower sharpness ⇔ R = 0, and R = 0 ⇒ K = 0. ``` This is explicitly **not** claimed for arbitrary finite-rank constraint problems; counterexamples to the unrestricted statement are included in the release. ### Quantitative constrained-spectrum lift For an energy-normalized constraint map, threshold `λ`, a lower bound `g>0` on the next unrestrained spectral gap, and squared threshold detection norm `ρ`, ```text λ_k(A_C) - λ ≥ gρ / (λ + g + ρ). ``` For tree openings, `ρ` is computable as a generalized eigenvalue involving the threshold residue matrix and the full path Gram matrix. The theorem is mathematical; it is not a hardware-performance guarantee. ## Reproducibility snapshot The archived final run records: | Check | Result | |---|---:| | Exact rational graph cases | 604 / 604 passed | | Independent exact ODE-nullity crosschecks | 604 / 604 passed | | Sharp instances in exact graph suite | 178 | | Finite-matrix quantitative-inequality tests | 200 / 200 passed | | Finite-element runs | 18 | | Recorded assertion failures | 0 | These are finite-instance checks and regression certificates, **not** a formal proof of the general theorem. The general proofs are in the manuscript. Run locally: ```bash python -m pip install -r requirements.txt python code/run_checks.py ``` See [`REPRODUCIBILITY.md`](REPRODUCIBILITY.md) for environment and interpretation details. ## File map for experts | File | Purpose | |---|---| | [`MANUSCRIPT.pdf`](MANUSCRIPT.pdf) | Typeset full paper and proofs | | [`MANUSCRIPT.tex`](MANUSCRIPT.tex) | LaTeX source | | [`MANUSCRIPT.txt`](MANUSCRIPT.txt) | Plain-text full manuscript for search/agents | | [`PUBLIC_SUMMARY.md`](PUBLIC_SUMMARY.md) | Accessible summary of the advance | | [`THEOREM_LEDGER.md`](THEOREM_LEDGER.md) | Claim-by-claim theorem status | | [`PROOF_AUDIT.md`](PROOF_AUDIT.md) | Adversarial proof audit | | [`EXPERT_REVIEW_GUIDE.md`](EXPERT_REVIEW_GUIDE.md) | Where an independent referee should attack the proof | | [`PRIOR_ART_AND_CLAIM_BOUNDARY.md`](PRIOR_ART_AND_CLAIM_BOUNDARY.md) | Prior art, novelty boundary, excluded claims | | [`REPRODUCIBILITY.md`](REPRODUCIBILITY.md) | Exact/numerical verification instructions | | [`metadata/AI_AGENT_INDEX.json`](metadata/AI_AGENT_INDEX.json) | Machine-readable entry points, claims, caveats, formulas | | [`llms.txt`](llms.txt) | Compact retrieval instructions for LLM/agent systems | | [`release/eve_spectral_extremality_amplification_release_v1.0.0.zip`](release/eve_spectral_extremality_amplification_release_v1.0.0.zip) | Immutable all-in-one source release | ## Search vocabulary Quantum graphs; metric graphs; compact metric graph Laplacian; Kirchhoff Laplacian; spectral graph theory; spectral geometry; eigenvalue inequalities; sharp eigenvalue bounds; equality cases; extremal eigenvalues; eigenvalue multiplicity; maximal degeneracy; spectral rigidity; nodal domains; nodal inertia; mixed Dirichlet-Neumann boundary conditions; lasso graph; theta graph; figure-eight graph; finite-rank constraints; form restrictions; spectral gap lower bound; reproducible mathematics. ## Scope and limitations The release does **not** claim a classification for the low-index branch, the pure circle under this bound, magnetic graph Laplacians, Robin conditions, potentials, higher-dimensional domains, or physical hardware. It does not claim peer review, proof-assistant verification, journal acceptance, a DOI, or certified historical priority. The author string above is project credit metadata and is not a legal identity determination. ## Licensing No new redistribution license is granted by this research package. The Hub metadata therefore uses `license: other`. See [`LICENSE_STATUS.md`](LICENSE_STATUS.md). Anyone redistributing or relicensing should first verify rights in all included materials. ## Citation Use [`CITATION.cff`](CITATION.cff) or [`CITATION.bib`](CITATION.bib). When citing, preserve the qualification that this is an independently unreviewed research candidate unless and until that status changes. ## Completeness **Deliverable completeness: 5/5 categories (100%).** This means the declared artifact categories are present—manuscript, executable implementations, saved results, proof/prior-art audits, and citation/provenance metadata. It is **not** a probability estimate for theorem correctness.