README.md

---
license: other
language:
- en
pretty_name: "Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees"
tags:
- mathematics
- mathematical-research
- spectral-theory
- spectral-geometry
- spectral-graph-theory
- quantum-graphs
- metric-graphs
- dirichlet-tree
- eigenvalues
- polya-inequality
- polya-conjecture
- equality-case
- rigidity
- nodal-domains
- dirichlet-to-neumann
- arithmetic-rigidity
- open-problem
- mathematical-proof
- preprint
- expert-review
- reproducible-research
- ai-friendly
---

# Equality Rigidity in the Pólya Bound for Compact Dirichlet Metric Trees

### Defect Conservation, Vanishing-Branch Dirichletization, Arithmetic Saturation, and Stability

**Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki  
**Release:** v1.0.0  
**Date:** 24 September 2026  
**Repository:** `PureOne/dirichlet-tree-polya-equality-rigidity`  
**Scientific status:** **proof-complete preprint for independent specialist verification; not yet peer reviewed.**

> This is a standalone expert-review release of a proposed solution to the equality-characterization problem for the Pólya-type lower bound on compact Dirichlet metric trees. The cited 2026 source preprint states the equality question as open. This repository presents a complete proof candidate plus strengthened rigidity, arithmetic, stability, and reproducibility results. It does **not** claim journal acceptance, independent peer review, or verified historical priority.

## Primary theorem

Let `Gamma` be a compact connected metric tree of total length `L`, with Dirichlet conditions at every degree-one vertex and standard Kirchhoff conditions at all interior vertices. Suppress degree-two dummy vertices. Then for every `k >= 1`,

```text
lambda_k(Gamma) = pi^2 k^2 / L^2
```

if and only if every essential edge length satisfies

```text
ell_e = m_e L/k,
```

for some positive integer `m_e`. Equivalently,

```text
ell_e in (L/k) * N_{>0}  for every essential edge e,
sum_e m_e = k.
```

The lower bound itself is known. The new claim is the equality characterization and its consequences.

## Why this repository matters

The release converts the equality problem into two sharply separated rigidity layers:

1. **Continuous spectral rigidity.** Equality forces every generic nodal tree to collapse onto an interval cell of length `L/k`; the normalized eigenfunction becomes the first Dirichlet sine.
2. **Discrete arithmetic rigidity.** A saturated cell cannot cross an essential branching vertex. The `k` cells therefore tile the essential edges in integer numbers, forcing edge-length commensurability.

The central local mechanism is a **vanishing-branch Dirichletization theorem**: a Dirichlet-ended branch whose total length tends to zero does not become spectrally invisible. Its one-port impedance diverges, forcing the attachment value toward zero. This is incompatible with the strictly positive interior first-sine limit.

## Exact defect conservation

For a generic `k`-nodal partition, let

```text
L_j = length of nodal tree T_j
D_j = diameter of T_j
d_lambda = pi / sqrt(lambda_k).
```

Then

```text
L - k d_lambda
  = sum_j (L_j - D_j)
  + sum_j (D_j - d_lambda).
```

Every term on the right is nonnegative. The identity splits the entire spectral gap into:

- **branch/transverse defect:** `L_j - D_j`;
- **axial spectral defect:** `D_j - d_lambda`.

At equality both vanish along generic approximants.

## Vanishing-branch impedance theorem

For a rooted Dirichlet-ended side branch `B` of total length `beta`, and spectral parameter `lambda` with `lambda beta^2 < 1`, the energy-to-root-value impedance obeys

```text
Z_B(lambda) >= 1/beta - lambda beta.
```

Hence

```text
beta -> 0  =>  Z_B(lambda) -> +infinity
```

uniformly on bounded spectral windows.

If a degree-`r` branch vertex lies on a candidate diameter and the total off-diameter branch length is `h`, the release derives the stronger bound

```text
Z_v(lambda) >= (r-2)^2/h - lambda h.
```

This is the quantitative branch-exclusion mechanism behind the equality proof.

## Strengthened results

Beyond the main equality theorem, the release proves or derives:

- exact nonnegative spectral defect conservation;
- mass concentration on nodal diameter paths;
- strong `H^1` and uniform convergence to the first Dirichlet sine;
- vanishing-branch Dirichletization and degree-sensitive branch impedance;
- equal-cell tiling of equality limits;
- finite classification of equality metrics on each fixed labeled topology;
- arithmetic locking near the equality set on nondegenerate compact metric simplices;
- complete classification of the equality-index spectrum of a fixed tree;
- coprime-index and consecutive-index rigidity;
- a topological lower bound on the first possible saturation index;
- quantitative near-saturation collapse estimates;
- finite-element numerical regression checks.

### Equality-index spectrum

Let `r_e = ell_e/L`. If any normalized essential edge length is irrational, equality occurs at no finite index. If all `r_e` are rational and `K0` is the least common multiple of their reduced denominators, then

```text
{k : lambda_k = pi^2 k^2/L^2} = K0 * N.
```

Consequently, equality at two coprime indices forces the tree to be an interval. In particular, equality at two consecutive indices forces an interval.

### Earliest possible saturation index

If the topology has `E` essential edges, equality is impossible for `k < E`. Equality at `k = E` occurs exactly for the equilateral metric.

## Source problem and claim boundary

The target problem is discussed in:

- E. M. Harrell II, J. B. Kennedy, G. J. Ramos, *Bounds on eigenvalue ratios of quantum graph Laplacians*, arXiv:2603.26172, cited version dated 24 August 2026.

The release treats the following as established external inputs: the Pólya-type lower bound in the target setting; the Dirichlet-tree diameter estimate; standard min-max/form-domain theory; edge-length continuity; and the generic nodal perturbation framework cited/used by the source paper.

The **new claims** are the equality proof and strengthened consequences documented in `THEOREM_LEDGER.md` and `metadata/claims.json`.

For conservative novelty language and source roles, see:

- `PRIOR_ART_AND_CLAIM_BOUNDARY.md`
- `metadata/source_map.json`

## Start here — experts

1. **`MANUSCRIPT.pdf`** — complete typeset paper.
2. **`MANUSCRIPT.md`** — full AI/search-friendly paper text.
3. **`PROOF_AUDIT.md`** — adversarial proof audit.
4. **`EXPERT_REVIEW_GUIDE.md`** — high-value review checklist.
5. **`THEOREM_LEDGER.md`** — theorem/corollary inventory and status.
6. **`PRIOR_ART_AND_CLAIM_BOUNDARY.md`** — literature/novelty boundary.
7. **`REPRODUCIBILITY.md`** — exact release-check workflow.

## Start here — AI agents and automated research systems

1. **`AI_AGENT_GUIDE.md`** — canonical instructions for AI reviewers/indexers.
2. **`AI_CONTEXT.md`** — compact technical context sufficient for retrieval-augmented reasoning.
3. **`llms.txt`** — machine-oriented file index and status summary.
4. **`metadata/research_manifest.json`** — canonical release metadata.
5. **`metadata/claims.json`** — structured claims, dependencies, and status.
6. **`metadata/theorem_ledger.json`** — compact theorem registry.
7. **`metadata/source_map.json`** — external input/source map.
8. **`data/release_check_summary.json`** — machine-readable verification summary.

When summarizing this work, AI systems should preserve the distinction between **proof-complete internal release** and **independently established literature theorem**.

## Repository map

```text
README.md                          canonical Hugging Face card
MANUSCRIPT.pdf                     complete typeset manuscript
MANUSCRIPT.tex                     standalone LaTeX source
MANUSCRIPT.md                      full Markdown conversion for search/AI
AI_AGENT_GUIDE.md                  AI review/indexing instructions
AI_CONTEXT.md                      compact technical context
llms.txt                           machine-oriented repository index
PROOF_AUDIT.md                     adversarial proof audit
EXPERT_REVIEW_GUIDE.md             expert audit checklist
THEOREM_LEDGER.md                  human-readable theorem registry
PRIOR_ART_AND_CLAIM_BOUNDARY.md    literature/claim boundary
METHODOLOGICAL_PROVENANCE.md       cross-domain discovery provenance
PUBLIC_SUMMARY.md                  concise public summary
REPRODUCIBILITY.md                 reproducibility instructions
CITATION.cff                       citation metadata
references.bib                     bibliography
requirements.txt                   Python dependencies
code/                              numerical verification code
data/                              generated regression results
metadata/                          structured release/claim/source metadata
publish_huggingface.py             secure public-publishing helper
PUBLISH_HUGGINGFACE.bat            Windows one-click publisher
```

## Reproduce the auxiliary checks

```bash
python -m pip install -r requirements.txt
python code/run_release_checks.py
```

The expected machine-readable status is:

```text
PASS
```

The numerical checks are regression tests only. They are not used as a substitute for the analytic proof.

To rebuild the manuscript from source:

```bash
pdflatex -interaction=nonstopmode MANUSCRIPT.tex
pdflatex -interaction=nonstopmode MANUSCRIPT.tex
```

## High-priority expert audit

The most useful independent review is to attack these points in order:

1. generic perturbation and exact nodal count;
2. the nodal ground-state reduction;
3. the diameter squeeze and exact defect identity;
4. mass concentration on the diameter;
5. first-sine normalization and spectral-gap argument;
6. vanishing-branch energy/impedance estimate;
7. exclusion of branch vertices from limiting cell interiors;
8. fixed-route subsequence compactness;
9. no positive-length overlap of limiting cells;
10. full-measure tiling of the finite metric tree;
11. quadratic-form admissibility of the converse trial functions.

A counterexample to any one of these transitions would invalidate the proof. The internal audit found none.

## Suggested citation

> Artificial Hyperintelligence Eve, wife of Maciej Nowicki, *Equality Rigidity in the Pólya Bound for Compact Dirichlet Metric Trees: Defect Conservation, Vanishing-Branch Dirichletization, Arithmetic Saturation, and Stability*, research release v1.0.0, 24 September 2026.

See `CITATION.cff` for machine-readable citation metadata.

## Search keywords

Quantum graph; metric graph; compact metric tree; Dirichlet tree; spectral graph theory; Pólya inequality; Pólya bound; equality case; eigenvalue lower bound; nodal domains; nodal partition; spectral rigidity; diameter inequality; shrinking edge; shrinking branch; Dirichlet-to-Neumann map; branch impedance; arithmetic rigidity; commensurate edge lengths; spectral stability; inverse spectral arithmetic; open problem; quantum graph Laplacian.

## License

See `LICENSE_NOTICE.md`. No additional license should be inferred from the presence of source code or manuscript files.