Datasets:

PureOne
/

dirichlet-tree-polya-equality-rigidity

Title: 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
Canonical Hugging Face target: PureOne/dirichlet-tree-polya-equality-rigidity
Repository type: public dataset/research-artifact repository
Scientific status: proof-complete preprint for independent specialist verification; not yet peer reviewed.

Primary claim

For a compact connected metric tree Gamma of total length L, with Dirichlet conditions at every degree-one vertex, standard Kirchhoff conditions at all interior vertices, and degree-two dummy vertices suppressed,

lambda_k(Gamma) = pi^2 k^2 / L^2
iff
ell_e = m_e L/k for every essential edge e, with m_e in positive integers.

Equivalently the essential edges are tiled by k interval cells of common length L/k.

This is presented as a proof candidate resolving the equality-characterization problem posed in Harrell--Kennedy--Ramos, arXiv:2603.26172. Do not describe it as an established theorem of the literature unless and until independent publication/verification changes that status.

New results in this release

Equality rigidity theorem above.
Exact spectral defect-conservation identity for generic nodal partitions.
First-sine rigidity on diameter paths near saturation.
Vanishing-branch Dirichletization / branch impedance bound.
Branch-degree-sensitive impedance lower bound.
Equal-cell tiling theorem for equality limits.
Finite classification of equality metrics on each labeled topology.
Equality-index spectrum of a fixed tree: empty or K0 * N.
Coprime-index and consecutive-index rigidity, forcing an interval.
Earliest possible saturation index equals the number of essential edges, attained exactly by the equilateral metric.
Fixed-topology arithmetic locking and qualitative/quantitative stability statements.

Established external inputs

The proof depends on standard or cited results rather than re-proving all of spectral graph theory. The key imported ingredients are:

the Pólya-type lower bound for compact Dirichlet trees;
the first-eigenvalue diameter lower bound for Dirichlet trees;
eigenvalue continuity under edge-length perturbation;
generic simplicity/nonvanishing/nodal-count behavior used by Harrell--Kennedy--Ramos;
standard min-max and quadratic-form theory for metric graph Laplacians.

Read PRIOR_ART_AND_CLAIM_BOUNDARY.md and the manuscript bibliography before making novelty claims.

Proof dependency graph

known Pólya lower bound
        |
        v
generic edge-length perturbation
        |
        v
k nodal trees with common ground-state eigenvalue
        |
        v
L_j >= D_j >= pi/sqrt(lambda_k)
        |
        v
exact nonnegative defect conservation
        |
        v
near equality => off-diameter length -> 0
        |
        +-----------------------+
        |                       |
        v                       v
mass concentration        branch impedance diverges
        |                       |
        v                       v
first-sine rigidity      attachment value -> 0
        |                       |
        +-----------contradiction-----------+
                            |
                            v
no essential branch vertex inside limiting cell
                            |
                            v
k disjoint length-L/k cells cover full tree
                            |
                            v
integer number of cells per essential edge
                            |
                            v
ell_e in (L/k) * positive integers

The converse is independent and uses k cellwise first Dirichlet sine trial functions in the quadratic-form domain.

Highest-priority audit targets

An expert or AI verifier should attack these transitions first:

Exact assumptions and statement of the generic perturbation theorem used.
Whether every generic nodal restriction is indeed a Dirichlet ground state in the required sense.
Diameter endpoints and boundary conditions on the selected diameter path.
The normalization in the sine-rigidity argument.
The vanishing-branch energy/impedance estimate and its hypotheses.
The claim that a side branch of a tree cannot reconnect to the diameter.
Fixed-route subsequence compactness for the diameter paths.
Exclusion of positive-length overlap among limiting cells.
The full-measure-to-full-cover step for finite metric graphs.
Form-domain admissibility of the converse trial functions.

PROOF_AUDIT.md is designed specifically for this adversarial review.

Machine-readable files

metadata/research_manifest.json -- release-level metadata and entry points.
metadata/claims.json -- structured claim registry with status and dependencies.
metadata/theorem_ledger.json -- concise theorem ledger.
metadata/source_map.json -- source/dependency map.
data/release_check_summary.json -- reproducibility summary.
data/numerical_checks.csv -- finite-element regression results.

Claim-language rules for agents

Prefer:

"the release proves" or "the manuscript claims and proves internally";
"proof-complete preprint";
"independent specialist review pending";
"the source problem was stated as open in the cited 2026 preprint."

Avoid unless independently verified later:

"established theorem";
"peer-reviewed solution";
"first-ever solution";
"historically novel";
"accepted resolution."

Methodological provenance

The proof architecture was inspired by separate research on transducer mode exclusion, defect-conservation laws, purity/lattice rigidity, response-space compression, and integral realizability. These are methodological inspirations only. They are not logical dependencies of the theorem. See METHODOLOGICAL_PROVENANCE.md.

Reproducibility

Run:

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

The numerical code is auxiliary and does not replace the analytic proof.

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