| # One-Sided Spectral Extremality Forces Maximal Degeneracy |
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| ## Complete Quantum-Graph Equality Classification, Direct Nodal Inertia, and Quantitative Port Obstructions |
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| **Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki |
| **Version:** 1.0.0 |
| **Date:** 2026-09-25 |
| **Status:** Proof-complete research candidate; not independently peer reviewed; historical priority not certified. |
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| The principal candidate advance over the supplied charged-cell moment package is a complete implication from **one lower-sharp eigenvalue** to the entire extremal topology, the edge-length phases, maximal eigenvalue multiplicity, and simultaneous upper sharpness. A second result gives an explicit spectral penalty for incompatible cycle closures. The new manuscript does not rely on the predecessor's compactness transfer. |
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| ## Central theorem |
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| Let G be a finite connected compact metric graph, with positive edge lengths, standard interior conditions, and Dirichlet or Neumann conditions only at leaves. Suppress standard degree-two vertices for topology. Exclude the pure circle. Write D, N, beta, L for its Dirichlet leaves, Neumann leaves, cycle rank, and total length. Require |
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| - k >= max(N+beta, 1) when D>0; |
| - k >= max(N+beta, 2) when D=0. |
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| Put c=(N+beta)/2 and d=L/(k-c). The theorem proves the equivalence |
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| ```text |
| lambda_k(G) = pi^2/d^2 |
| <=> G is a phase-locked lasso tree, common-parity theta, or even figure-eight |
| <=> the threshold has multiplicity D+N+2*beta-1 and top index k |
| <=> the same eigenvalue block is simultaneously lower and upper sharp. |
| ``` |
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| For a lasso tree, the skeleton is a tree with selected terminal loops. Each loop length is 2*r*d for a positive integer r. Treat the loop attachments as virtual Neumann leaves. Each skeleton edge has length (m + nu/2)*d, where nu counts virtual Neumann endpoints, m is a nonnegative integer, and m>=1 if nu=0. A theta has three positive integer cell counts of common parity. A figure-eight has two positive even cell counts. |
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| These topology families, and the equivalence of **two-sided** sharpness with maximal multiplicity, are prior art. The main candidate advance is the complete **one-sided** implication. Read `PRIOR_ART_AND_CLAIM_BOUNDARY.md` before making any novelty statement. |
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| ## Additional proved results |
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| A fully supported tree eigenfunction with s interior zero points and r nodal cells satisfies |
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| ```text |
| number of eigenvalues below lambda = s |
| multiplicity at lambda = r-s |
| number at or below lambda = r. |
| ``` |
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| For an admissibly opened saturated tree, the predecessor's response criterion reduces to |
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| ```text |
| sharpness <=> R=0; and R=0 forces K=0. |
| ``` |
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| This implication is geometric, not a generic finite-rank fact. Explicit matrix counterexamples are supplied for the unrestricted claim. |
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| For an energy-normalized constraint map, if g is a lower bound on the next unrestrained spectral gap and rho is its squared operator norm on the threshold eigenspace, then |
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| ```text |
| lambda_k(A_C) - lambda >= g*rho/(lambda + g + rho). |
| ``` |
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| On an opened tree, rho is the largest generalized eigenvalue of the threshold residue R relative to the path Gram matrix S. This yields a computable positive penalty without computing the cyclic graph spectrum. It does not provide a hardware-performance guarantee. |
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| ## Reading and running |
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| Read `MANUSCRIPT.pdf` for the complete proofs, `PUBLIC_SUMMARY.md` for the main ideas, `THEOREM_LEDGER.md` for claim-by-claim status, and `PROOF_AUDIT.md` for the critical checks. Source is `MANUSCRIPT.tex`; plain extracted text is `MANUSCRIPT.txt`. |
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| ```bash |
| python -m pip install -r requirements.txt |
| python code/run_checks.py |
| ``` |
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| The final run completed 604 exact rational graph tests, 604 separate exact ODE-nullity crosschecks, 200 floating-point finite-matrix tests, and 18 finite-element runs, with zero recorded assertion failures. Exact graph tests are finite-instance certificates, not an exhaustive or formal verification of the theorem. See `REPRODUCIBILITY.md`. |
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| ## Completeness |
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| The five declared deliverable categories are present: full proof manuscript; executable implementations; saved test outputs; adversarial/prior-art audit; and citation/provenance metadata. **Deliverable completeness: 5/5 (100%).** This is a checklist completion percentage, not a probability that the mathematics is correct. No independent referee review, proof-assistant verification, or exhaustive priority search has been completed. |
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| The exact supplied predecessor archive is retained under `provenance/` for audit. Its claims are not silently imported into the new proof. No external files are needed beyond the stated Python packages to run the tests. Publication metadata contains no invented DOI, journal acceptance, repository URL, or reviewer endorsement. |
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