README.md

metadata

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, then read PRIOR_ART_AND_CLAIM_BOUNDARY.md and PROOF_AUDIT.md.
AI-agent entry point: read llms.txt and 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

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 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,

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,

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,

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:

python -m pip install -r requirements.txt
python code/run_checks.py

See REPRODUCIBILITY.md for environment and interpretation details.

File map for experts

File	Purpose
MANUSCRIPT.pdf	Typeset full paper and proofs
MANUSCRIPT.tex	LaTeX source
MANUSCRIPT.txt	Plain-text full manuscript for search/agents
PUBLIC_SUMMARY.md	Accessible summary of the advance
THEOREM_LEDGER.md	Claim-by-claim theorem status
PROOF_AUDIT.md	Adversarial proof audit
EXPERT_REVIEW_GUIDE.md	Where an independent referee should attack the proof
PRIOR_ART_AND_CLAIM_BOUNDARY.md	Prior art, novelty boundary, excluded claims
REPRODUCIBILITY.md	Exact/numerical verification instructions
metadata/AI_AGENT_INDEX.json	Machine-readable entry points, claims, caveats, formulas
llms.txt	Compact retrieval instructions for LLM/agent systems
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. Anyone redistributing or relicensing should first verify rights in all included materials.

Citation

Use CITATION.cff or 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.