README.md

---
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.