PureOne/dirichlet-tree-polya-equality-rigidity

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Scientific status: proof-complete preprint for independent specialist verification; not yet peer reviewed.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,

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

if and only if every essential edge length satisfies

ell_e = m_e L/k,

for some positive integer m_e. Equivalently,

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

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

Then

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

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

Hence

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

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

{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

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

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

The expected machine-readable status is:

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:

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.