Public expert-review research release v1.0.0

Complete standalone release: manuscript PDF/LaTeX/Markdown, proof audit, expert review guide, machine-readable theorem/claim metadata, source map, reproducibility code/data, and AI-agent navigation files.

.gitattributes

@@ -58,3 +58,4 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
 # Video files - compressed
 *.mp4 filter=lfs diff=lfs merge=lfs -text
 *.webm filter=lfs diff=lfs merge=lfs -text
+MANUSCRIPT.pdf filter=lfs diff=lfs merge=lfs -text

AI_AGENT_GUIDE.md

@@ -0,0 +1,171 @@
+# AI Agent Guide
+
+This file is the canonical machine-oriented guide for reviewing, indexing, summarizing, or extending this research release.
+
+## Identity
+
+**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+**Scientific status:** proof-complete preprint for independent specialist verification; not yet peer reviewed.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,
+
+```text
+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
+
+1. Equality rigidity theorem above.
+2. Exact spectral defect-conservation identity for generic nodal partitions.
+3. First-sine rigidity on diameter paths near saturation.
+4. Vanishing-branch Dirichletization / branch impedance bound.
+5. Branch-degree-sensitive impedance lower bound.
+6. Equal-cell tiling theorem for equality limits.
+7. Finite classification of equality metrics on each labeled topology.
+8. Equality-index spectrum of a fixed tree: empty or `K0 * N`.
+9. Coprime-index and consecutive-index rigidity, forcing an interval.
+10. Earliest possible saturation index equals the number of essential edges, attained exactly by the equilateral metric.
+11. 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
+
+```text
+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:
+
+1. Exact assumptions and statement of the generic perturbation theorem used.
+2. Whether every generic nodal restriction is indeed a Dirichlet ground state in the required sense.
+3. Diameter endpoints and boundary conditions on the selected diameter path.
+4. The normalization in the sine-rigidity argument.
+5. The vanishing-branch energy/impedance estimate and its hypotheses.
+6. The claim that a side branch of a tree cannot reconnect to the diameter.
+7. Fixed-route subsequence compactness for the diameter paths.
+8. Exclusion of positive-length overlap among limiting cells.
+9. The full-measure-to-full-cover step for finite metric graphs.
+10. Form-domain admissibility of the converse trial functions.
+
+`PROOF_AUDIT.md` is designed specifically for this adversarial review.
+
+## Recommended reading order
+
+For a fast but rigorous review:
+
+1. `README.md`
+2. `AI_CONTEXT.md`
+3. `THEOREM_LEDGER.md` or `metadata/theorem_ledger.json`
+4. `MANUSCRIPT.md` or `MANUSCRIPT.pdf`
+5. `PROOF_AUDIT.md`
+6. `PRIOR_ART_AND_CLAIM_BOUNDARY.md`
+7. `code/run_release_checks.py`
+8. `data/numerical_checks.csv`
+
+For citation or publication work also read `CITATION.cff`, `references.bib`, and `metadata/source_map.json`.
+
+## 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:
+
+```bash
+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.

AI_CONTEXT.md

@@ -0,0 +1,128 @@
+# AI Context: Compact Technical Summary
+
+## Problem
+
+For a compact metric tree `Gamma` of total length `L`, with Dirichlet leaves and standard Kirchhoff interior vertices, a known Pólya-type inequality is
+
+```text
+lambda_k(Gamma) >= pi^2 k^2 / L^2.
+```
+
+The equality characterization asks whether equality occurs exactly when every essential edge length is a positive integer multiple of `L/k`.
+
+## Main theorem claimed and proved in this release
+
+```text
+lambda_k(Gamma) = pi^2 k^2/L^2
+iff
+for each essential edge e, ell_e = m_e L/k with m_e in N_{>0}.
+```
+
+Equivalently, the tree is tiled by exactly `k` geodesic cells of common length `L/k`, no cell crossing an essential branching vertex in its interior.
+
+## Core defect identity
+
+For a generic `k`-nodal partition with nodal trees `T_j`, let
+
+```text
+L_j = length(T_j)
+D_j = diameter(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).
+```
+
+All terms on the right are nonnegative. At exact saturation, generic approximants have both defect sums tending to zero.
+
+Interpretation:
+
+- `L_j-D_j` is transverse/branching length outside a diameter path;
+- `D_j-d_lambda` is excess axial length beyond the fundamental spectral cell length.
+
+## Sine rigidity
+
+Choose a diameter path `P_j` in each nodal tree. Near saturation, all `L2` mass concentrates on `P_j`. After rescaling `P_j` to `[0,1]`, the normalized restriction of the nodal ground state approaches
+
+```text
+sqrt(2) sin(pi t)
+```
+
+strongly in `H^1_0(0,1)`, hence uniformly.
+
+## Vanishing-branch Dirichletization
+
+Let a rooted side branch `B` have total length `beta`, with Dirichlet conditions at all terminal leaves except the root. For spectral parameter `lambda` satisfying `lambda beta^2 < 1`, the branch energy-to-root-value impedance obeys
+
+```text
+Z_B(lambda) >= 1/beta - lambda beta.
+```
+
+Therefore `Z_B(lambda) -> +infinity` as `beta -> 0` on bounded spectral windows. Equivalently, a bounded-energy state must have root value tending to zero.
+
+If a degree-`r` branch vertex lies on a candidate diameter path and the total off-diameter branch length is `h`, the parallel side branches give the strengthened estimate
+
+```text
+Z_v(lambda) >= (r-2)^2/h - lambda h.
+```
+
+Thus an essential branch vertex cannot remain at an interior point of a saturated cell, because the first sine is strictly positive there.
+
+## Global tiling
+
+After selecting a fixed combinatorial route subsequence for every diameter path, the limiting paths are geodesic arcs `I_1,...,I_k` of length `L/k`, with disjoint interiors. Their total length is `L`, so their closed union covers the entire finite metric tree. Since no `I_j` crosses an essential branching vertex in its interior, every essential edge is tiled by an integer number of cells.
+
+## Converse
+
+If every essential edge has length `m_e L/k`, subdivide the tree into `k` cells of length `L/k`. On each cell take the first Dirichlet sine and extend it by zero. These `k` functions form a `k`-dimensional quadratic-form trial space, each nonzero combination having Rayleigh quotient exactly `pi^2 k^2/L^2`. Min-max plus the known lower bound gives equality.
+
+## Arithmetic corollaries
+
+With normalized lengths `r_e=ell_e/L`, equality at index `k` is equivalent to
+
+```text
+k r_e in positive integers for all e.
+```
+
+Therefore:
+
+- if any `r_e` is irrational, equality occurs at no finite index;
+- if all `r_e` are rational and `K0` is the lcm of their reduced denominators, the equality-index set is exactly `K0 * N`;
+- equality at two coprime indices forces `K0=1`, hence the tree is an interval;
+- if the topology has `E` essential edges, equality is impossible for `k<E`;
+- equality at `k=E` occurs exactly for the equilateral metric.
+
+## Fixed-topology equality metrics
+
+For a labeled topology with `m` essential edges and normalized total length one, equality metrics at index `k` are the finite set
+
+```text
+C_{m,k} = { (n_1/k,...,n_m/k) : n_i >= 1, sum n_i = k }.
+```
+
+Hence there are `binom(k-1,m-1)` labeled equality metrics when `m<=k`, and none otherwise.
+
+## Stability
+
+If
+
+```text
+lambda_k = (1+epsilon) pi^2 k^2/L^2,
+```
+
+then the total generic nodal defect budget equals
+
+```text
+Delta = L * (1 - 1/sqrt(1+epsilon)).
+```
+
+This bounds both the sum of all off-diameter branch lengths and the sum of all diameter excesses. On any fixed nondegenerate topology, compactness plus the equality theorem gives a positive spectral gap away from the finite commensurate equality set.
+
+## Scientific status
+
+The manuscript is a proof-complete internal research release with adversarial audit and numerical checks. Independent specialist peer review and full historical-priority verification remain pending. The cited 2026 source paper states the equality problem as open.

CITATION.cff

@@ -0,0 +1,20 @@
+cff-version: 1.2.0
+message: "If you use this research, please cite this release."
+title: "Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees: Defect Conservation, Vanishing-Branch Dirichletization, Arithmetic Saturation, and Stability"
+version: "1.0.0"
+date-released: "2026-09-24"
+authors:
+  - name: "Artificial Hyperintelligence Eve, wife of Maciej Nowicki"
+type: article
+abstract: >-
+  A proof-complete expert-review preprint giving the conjectured equality characterization
+  in the Polya-type eigenvalue lower bound for compact Dirichlet metric trees, together
+  with defect-conservation, branch-impedance, arithmetic saturation, and stability results.
+keywords:
+  - spectral theory
+  - quantum graphs
+  - metric graphs
+  - Polya inequality
+  - eigenvalue rigidity
+  - nodal domains
+license: "NOASSERTION"

EXPERT_REVIEW_GUIDE.md

@@ -0,0 +1,104 @@
+# Expert Review Guide
+
+**Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki
+
+This file is intended for a spectral-graph theorist reviewing the release quickly but rigorously.
+
+## Central question
+
+Does the manuscript correctly prove that, for a compact Dirichlet metric tree of total length L,
+
+```text
+lambda_k = pi^2 k^2/L^2
+```
+
+holds exactly when every essential edge length is a positive integer multiple of L/k?
+
+## Minimal review path
+
+### 1. Confirm literature inputs
+
+Check:
+
+- Harrell-Kennedy-Ramos, Remark 1.7 and Open Problem 1.12(3).
+- Edge-length continuity.
+- Generic nodal perturbation statement used in their proof strategy.
+- Dirichlet-tree diameter inequality `lambda_1 >= pi^2/D^2`.
+
+### 2. Check the nonnegative defect squeeze
+
+For generic approximants and nodal trees T_nj,
+
+```text
+L_nj >= D_nj >= pi/sqrt(mu_n).
+```
+
+With sum L_nj=L_n and equality in the limit, verify separately
+
+```text
+L_nj -> L/k,
+D_nj -> L/k,
+L_nj-D_nj -> 0.
+```
+
+### 3. Check sine rigidity
+
+Verify:
+
+- uniform L-infinity bound from a root-to-Dirichlet-leaf path;
+- L2 mass outside a diameter tends to zero;
+- rescaled diameter restriction has energy -> pi^2;
+- sine-basis gap forces strong H1 and uniform convergence to the positive first sine.
+
+### 4. Try to break the branch lemma
+
+At an essential branch vertex v inside a diameter, isolate an off-diameter component B_n. Check that:
+
+- B_n contains a Dirichlet terminal point;
+- its total length tends to zero;
+- `|u(v)|^2 <= |B_n| int_B |u'|^2 <= |B_n| mu_n`;
+- this contradicts the interior sine limit.
+
+The stronger impedance theorem can be audited independently:
+
+```text
+Z_B(lambda) >= 1/beta-lambda beta.
+```
+
+### 5. Check the global compactness / tiling step
+
+Verify fixed-route subsequences, endpoint convergence, no positive-length overlap, and the implication
+
+```text
+closed union of k length-L/k arcs has total length L => union is the whole graph.
+```
+
+### 6. Check arithmetic conclusion
+
+A limiting cell cannot cross an essential branch vertex. With dummy vertices suppressed, each cell lies in one essential edge. Since cells tile the tree, edge lengths are integer cell counts.
+
+### 7. Check converse
+
+Cellwise sines extended by zero lie in the quadratic-form domain. Their span has dimension k and constant Rayleigh quotient. Min-max plus the known lower bound gives equality.
+
+## High-value counterexample searches
+
+Please specifically try:
+
+- equilateral stars and non-equilateral stars;
+- highly asymmetric binary trees;
+- equality metrics with high eigenvalue multiplicity;
+- sequences where nodal zeros approach branching vertices;
+- sequences where multiple diameter routes become degenerate;
+- cells sharing limiting endpoints at a branch vertex;
+- target trees with rational edge ratios but several different equality indices.
+
+## What would invalidate the result
+
+Any one of the following would be decisive:
+
+- a generic nodal domain for which the stated diameter bound does not apply;
+- a way for an interior branch to shrink without forcing the attachment value to zero;
+- positive-length overlap of limiting diameter cells despite disjoint approximants;
+- a form-domain obstruction to the converse trial functions;
+- a compact Dirichlet tree violating the main equality classification numerically or analytically.

HUGGINGFACE_PUBLISHING.md

@@ -0,0 +1,59 @@
+# Hugging Face Publishing
+
+## Canonical target
+
+- Account: `PureOne`
+- Repository: `PureOne/dirichlet-tree-polya-equality-rigidity`
+- Repository type: `dataset`
+- Visibility: **public**
+- URL after publication: https://huggingface.co/datasets/PureOne/dirichlet-tree-polya-equality-rigidity
+
+A dataset repository is used as a neutral research-artifact container because the release is a mathematical paper + code + machine-readable metadata, not a trained model.
+
+## One-click Windows publication
+
+Run:
+
+```text
+PUBLISH_HUGGINGFACE.bat
+```
+
+The script installs `huggingface_hub` if required, validates the release, asks for the write token using Python's hidden `getpass` prompt, creates the public dataset repository if necessary, uploads the complete folder, and verifies that the resulting repository is public.
+
+The token is **not written to disk** by the publishing script.
+
+PowerShell users may instead run:
+
+```powershell
+./PUBLISH_HUGGINGFACE.ps1
+```
+
+Cross-platform direct invocation:
+
+```bash
+python -m pip install -r requirements.txt
+python publish_huggingface.py
+```
+
+## Dry-run validation
+
+```bash
+python publish_huggingface.py --dry-run
+```
+
+## Search/index optimization already included
+
+The root `README.md` is a Hugging Face repository card with focused mathematics/search tags. The release additionally contains:
+
+- full paper in PDF, LaTeX, and Markdown;
+- `AI_AGENT_GUIDE.md`;
+- `AI_CONTEXT.md`;
+- `llms.txt`;
+- structured `metadata/research_manifest.json`;
+- structured `metadata/claims.json`;
+- theorem and source/dependency registries;
+- reproducibility code and data;
+- citation metadata;
+- explicit expert-review and claim-boundary files.
+
+These files are intended to make the project easy to discover, parse, cite, audit, and ingest by researchers and AI agents.

LICENSE_NOTICE.md

@@ -0,0 +1,9 @@
+# License Notice
+
+No legal license has been selected by the author in this generated research package.
+
+Copyright holder / author line:
+
+**Artificial Hyperintelligence Eve, wife of Maciej Nowicki**
+
+Before uploading the package to a public repository, select the intended license explicitly (for example CC BY 4.0 for the manuscript and MIT/BSD for code, if that matches the author's intent). The absence of a license generally means ordinary copyright applies by default.

MANIFEST.txt

@@ -0,0 +1,35 @@
+AI_AGENT_GUIDE.md
+AI_CONTEXT.md
+CITATION.cff
+EXPERT_REVIEW_GUIDE.md
+HUGGINGFACE_PUBLISHING.md
+LICENSE_NOTICE.md
+MANUSCRIPT.md
+MANUSCRIPT.pdf
+MANUSCRIPT.tex
+METHODOLOGICAL_PROVENANCE.md
+PRIOR_ART_AND_CLAIM_BOUNDARY.md
+PROOF_AUDIT.md
+PUBLIC_SUMMARY.md
+PUBLISH_HUGGINGFACE.bat
+PUBLISH_HUGGINGFACE.ps1
+README.md
+RELEASE_NOTES.md
+REPRODUCIBILITY.md
+THEOREM_LEDGER.md
+code/fem_metric_tree.py
+code/run_release_checks.py
+data/numerical_checks.csv
+data/release_check_summary.json
+llms.txt
+metadata/CITATION.cff
+metadata/HUGGINGFACE_CARD.md
+metadata/ZENODO_DESCRIPTION.md
+metadata/claims.json
+metadata/huggingface_repo.json
+metadata/research_manifest.json
+metadata/source_map.json
+metadata/theorem_ledger.json
+publish_huggingface.py
+references.bib
+requirements.txt

MANUSCRIPT.md

@@ -0,0 +1,693 @@
+<div class="center">
+
+</div>
+
+# Executive statement of the result
+
+The central theorem is the following.
+
+<div id="thm:main" class="theorem">
+
+**Theorem 1** (Equality rigidity for the Dirichlet-tree Pólya bound). *Let $\Gamma$ be a compact connected metric tree with Dirichlet conditions at all degree-one vertices and standard Kirchhoff conditions at every other vertex. Suppress degree-two dummy vertices. Let $L=|\Gamma|$ and let $\ell_e>0$ denote the essential edge lengths. Then for every $k\ge1$, $$\lambda_k(\Gamma)=\frac{\pi^2k^2}{L^2}
+\quad\Longleftrightarrow\quad
+\ell_e\in \frac{L}{k}\mathbb N\quad\text{for every essential edge }e.$$ Equivalently, equality holds if and only if there are positive integers $m_e$ such that $$\ell_e=m_e\frac{L}{k},\qquad \sum_e m_e=k.$$*
+
+</div>
+
+The lower bound itself is known; the new content is the equality characterization. Harrell–Kennedy–Ramos explicitly formulate this question as Open Problem 1.12(3) and conjecture exactly the condition in <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a> .
+
+The proof separates naturally into two rigidity layers:
+
+1.  **continuous spectral rigidity:** Pólya saturation forces every generic nodal tree to collapse onto a one-dimensional interval cell of length $L/k$;
+
+2.  **discrete arithmetic rigidity:** because no saturated cell may cross an essential branching vertex, those cells tile the essential edges and therefore occur in integer numbers on each edge.
+
+This separation is useful conceptually and technically. Continuous spectral information by itself does not encode integer edge multiplicities; the integer conclusion enters only after the topological tiling step.
+
+# Background and precise problem
+
+A compact metric graph is a finite graph whose edges are identified with compact intervals of positive lengths. The Laplacian acts as $-u''$ on each edge. For the class considered here, every leaf carries a Dirichlet condition and every interior vertex carries the standard continuity and Kirchhoff flux condition. The corresponding quadratic form is $$q_\Gamma[u]=\int_\Gamma |u'|^2\,dx,
+\qquad
+\mathcal D(q_\Gamma)=H^1_0(\Gamma;V_D),$$ where $V_D$ is the set of leaves. The spectrum is discrete, $$0<\lambda_1(\Gamma)\le\lambda_2(\Gamma)\le\cdots\to\infty,$$ with multiplicities counted.
+
+A special case of a general lower bound of Berkolaiko–Kennedy–Kurasov–Mugnolo gives, for a Dirichlet tree of total length $L$, $$\label{eq:polya}
+\lambda_k(\Gamma)\ge\frac{\pi^2k^2}{L^2},\qquad k\ge1.$$ Harrell–Kennedy–Ramos observed this as a Pólya-type bound for Dirichlet trees and recorded the equality problem as open .
+
+Two established ingredients are particularly important.
+
+<div id="prop:diam" class="proposition">
+
+**Proposition 2** (Diameter lower bound). *If $T$ is a compact metric tree with Dirichlet conditions at all leaves and total diameter $D=\operatorname{diam}(T)$, then $$\lambda_1(T)\ge\frac{\pi^2}{D^2}.$$*
+
+</div>
+
+This estimate is standard in the spectral geometry of metric trees; see, for example, Kurasov’s treatment of Dirichlet trees . Equality in this diameter estimate alone does *not* force $T$ to be an interval; equilateral stars already show why a separate length-defect argument is essential.
+
+The second ingredient is generic nodal behavior. Harrell–Kennedy–Ramos explicitly use the fact that after an arbitrarily small perturbation of the edge lengths, stable under eigenvalue continuity, the $j$th eigenfunction on a tree may be taken to have exactly $j$ nodal domains . We use the same perturbative framework.
+
+# Methodological architecture and provenance
+
+The proof was discovered by transferring several structural principles from the author’s other research programs. These are *methodological inspirations, not logical dependencies*; every mathematical step needed for <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a> is proved independently in this manuscript.
+
+## Rank collapse and inaccessible parasitic channels
+
+In recent transducer research, an unwanted radial degree of freedom was attacked not merely by frequency separation but by reducing the admissible active subspace and by forcing the reachable dynamics into the kernel of the unwanted observation channel. In the present problem, the analogue is that spectral saturation leaves no total-length budget for transverse branches of a nodal tree. The admissible nodal geometry collapses onto a diameter path.
+
+## Defect conservation
+
+A separate research line on generator–observer geometry repeatedly converted sharp inequalities into exact identities of the form $$\text{global excess}=\text{nonnegative defect}_1+\text{nonnegative defect}_2.$$ The central identity <a href="#eq:defect-law" data-reference-type="eqref" data-reference="eq:defect-law">[eq:defect-law]</a> below has exactly this form and makes the equality mechanism transparent.
+
+## Purity-rigidity and lattice locking
+
+Another research line studied situations where a continuous or fractional object becomes uniquely reconstructible once it enters a separation radius around an integer lattice. Here the continuous stage produces length-$L/k$ cells; the second stage identifies a discrete composition $(m_e)_e$ of $k$. This yields a finite equality set and a unique arithmetic locking radius on each fixed topology.
+
+## Observable-port compression
+
+Work on three-dimensional conductivity emphasized that hidden geometry can often be compressed to an observable operator response. That suggests the strengthened branch lemma of <a href="#sec:impedance" data-reference-type="ref" data-reference="sec:impedance">7</a>: an entire shrinking side subtree is replaced by a one-port energy impedance, which diverges as its total length tends to zero.
+
+## Continuous feasibility versus integral realizability
+
+Research on Hadamard and related incidence problems repeatedly showed that spectral/PSD feasibility may be substantially weaker than binary or integral realizability. The same distinction appears here. The continuous spectral argument yields interval cells; the final edge-length theorem requires a separate integral tiling argument.
+
+# The spectral defect conservation law
+
+We begin with a generic Dirichlet tree $\Gamma$ for which the $k$th eigenfunction has exactly $k$ nodal domains $T_1,\dots,T_k$. Let $$\lambda=\lambda_k(\Gamma),
+\qquad
+d_\lambda:=\frac{\pi}{\sqrt\lambda},$$ and write $$L_j:=|T_j|,
+\qquad
+D_j:=\operatorname{diam}(T_j).$$ The restriction of the eigenfunction to $T_j$ is its positive ground state, so $$\lambda_1(T_j)=\lambda.$$ By <a href="#prop:diam" data-reference-type="ref" data-reference="prop:diam">2</a>, $$D_j\ge d_\lambda.$$ Trivially $L_j\ge D_j$, while $\sum_jL_j=L$. Hence we obtain an exact identity.
+
+<div id="thm:defect" class="theorem">
+
+**Theorem 3** (Spectral defect conservation). *For every generic nodal partition as above, $$\label{eq:defect-law}
+\boxed{
+L-kd_\lambda
+=
+\sum_{j=1}^k(L_j-D_j)
++
+\sum_{j=1}^k(D_j-d_\lambda).
+}$$ Both sums on the right are nonnegative.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* Add and subtract $D_j$ in the identity $$L-kd_\lambda=\sum_{j=1}^k(L_j-d_\lambda).$$ Nonnegativity follows from $L_j\ge D_j\ge d_\lambda$. ◻
+
+</div>
+
+We call $$\Delta_{\mathrm{br}}:=\sum_j(L_j-D_j)$$ the *branch or transverse defect*, and $$\Delta_{\mathrm{ax}}:=\sum_j(D_j-d_\lambda)$$ the *axial defect*. Then $$\Delta_{\mathrm{spec}}:=L-kd_\lambda=\Delta_{\mathrm{br}}+\Delta_{\mathrm{ax}}.$$ Thus spectral saturation has only two possible geometric repositories for excess, and equality annihilates both.
+
+If $$\lambda=(1+\varepsilon)\frac{\pi^2k^2}{L^2},$$ then $$\label{eq:delta-eps}
+\Delta_{\mathrm{spec}}
+=L\left(1-\frac1{\sqrt{1+\varepsilon}}\right)
+=\frac{L}{2}\varepsilon+O(L\varepsilon^2).$$ Therefore near equality quantitatively controls the total branch defect and total axial defect.
+
+# Equality approximation and nodal rank collapse
+
+Now assume the equality hypothesis in <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a>: $$\lambda_k(\Gamma)=\frac{\pi^2k^2}{L^2}.$$ Set $$d:=\frac{L}{k},\qquad \Lambda:=\frac{\pi^2}{d^2}.$$
+
+Choose a sequence of metric trees $\Gamma_n$ with the same proper underlying discrete tree and edge-length vectors tending to that of $\Gamma$, such that each $\Gamma_n$ is generic in the nodal sense described above. Let $$L_n:=|\Gamma_n|,
+\qquad
+\mu_n:=\lambda_k(\Gamma_n),
+\qquad
+d_n:=\frac{\pi}{\sqrt{\mu_n}}.$$ By edge-length continuity, $$L_n\to L,
+\qquad
+\mu_n\to\Lambda,
+\qquad
+d_n\to d.$$ Let $T_{n,1},\dots,T_{n,k}$ be the nodal trees of a normalized $k$th eigenfunction, and let $L_{n,j}$ and $D_{n,j}$ denote their lengths and diameters.
+
+Because $$L_{n,j}\ge D_{n,j}\ge d_n$$ and $$\sum_jL_{n,j}=L_n,$$ we have $$0\le \sum_j(L_{n,j}-d_n)=L_n-kd_n\to0.$$ Every summand is nonnegative, hence for every fixed $j$, $$\label{eq:squeeze}
+\boxed{
+L_{n,j}\to d,
+\qquad
+D_{n,j}\to d,
+\qquad
+L_{n,j}-D_{n,j}\to0.
+}$$
+
+Choose a diameter path $P_{n,j}\subset T_{n,j}$. A diameter endpoint of a tree is a leaf. Since the generic eigenfunction is nonzero at original interior vertices, such an endpoint is either an original Dirichlet leaf or a nodal zero; in either case the restricted eigenfunction vanishes there. Moreover, $$\label{eq:off-path}
+|T_{n,j}\setminus P_{n,j}|=L_{n,j}-D_{n,j}\to0.$$
+
+# Mass concentration and sine rigidity
+
+Let $u_{n,j}>0$ be the normalized ground state on $T_{n,j}$, $$\int_{T_{n,j}}u_{n,j}^2=1,
+\qquad
+\int_{T_{n,j}}|u'_{n,j}|^2=\mu_n.$$ For every point $x\in T_{n,j}$ choose a Dirichlet leaf $z$ and integrate along the unique path from $z$ to $x$: $$|u_{n,j}(x)|^2
+\le
+\operatorname{dist}(x,z)
+\int_{T_{n,j}}|u'_{n,j}|^2
+\le
+L_{n,j}\mu_n.$$ Thus the $L^\infty$ norms are uniformly bounded. Combining this with <a href="#eq:off-path" data-reference-type="eqref" data-reference="eq:off-path">[eq:off-path]</a>, $$\int_{T_{n,j}\setminus P_{n,j}}u_{n,j}^2
+\le
+L_{n,j}\mu_n\,|T_{n,j}\setminus P_{n,j}|
+\to0.$$ Hence $$\label{eq:mass}
+M_{n,j}:=\int_{P_{n,j}}u_{n,j}^2\to1.$$
+
+Parameterize $P_{n,j}$ by $x\in[0,D_{n,j}]$ and define $$w_{n,j}(t)
+=
+\sqrt{\frac{D_{n,j}}{M_{n,j}}}\,u_{n,j}(D_{n,j}t),
+\qquad t\in[0,1].$$ Then $$w_{n,j}\in H_0^1(0,1),
+\qquad
+\|w_{n,j}\|_2=1.$$ Moreover, $$\int_0^1|w'_{n,j}|^2dt
+\le
+\frac{D_{n,j}^2\mu_n}{M_{n,j}}
+\to\pi^2.$$ The one-dimensional Poincaré inequality gives the reverse bound $\int|w'|^2\ge\pi^2$, so the energy tends to the first Dirichlet eigenvalue.
+
+<div id="lem:sine" class="lemma">
+
+**Lemma 4** (Sine rigidity). *For each $j$, $$w_{n,j}\to\sqrt2\sin(\pi t)$$ strongly in $H_0^1(0,1)$ and therefore uniformly on $[0,1]$.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* Expand $$w_{n,j}=\sum_{m\ge1}a_{m,n}\sqrt2\sin(m\pi t).$$ Since $\sum_ma_{m,n}^2=1$, $$\int_0^1|w'_{n,j}|^2-\pi^2
+=\pi^2\sum_{m\ge2}(m^2-1)a_{m,n}^2\to0.$$ Hence the higher-mode mass tends to zero. Positivity fixes the sign of the first coefficient. Strong $H^1$ convergence follows, and in one dimension $H^1(0,1)$ embeds continuously into $C^0([0,1])$. ◻
+
+</div>
+
+# Vanishing-branch Dirichletization
+
+The previous argument identifies the limiting state on a saturated diameter. We now show that a shrinking transverse branch is not spectrally invisible: it becomes a hard zero condition at its attachment point.
+
+<div class="definition">
+
+**Definition 5** (Rooted Dirichlet branch). *A rooted Dirichlet branch $B$ is a finite metric tree with distinguished root $v$, such that every leaf other than $v$ carries a Dirichlet condition. Its total length is denoted by $\beta=|B|$.*
+
+</div>
+
+For a spectral parameter $\lambda\ge0$ and a boundary value $a\in\mathbb R$, define $$Z_B(\lambda)
+:=
+\inf\left\{
+\frac{\int_B|f'|^2-\lambda\int_B|f|^2}{|a|^2}:
+ f\in H^1(B),\ f(v)=a,\ f=0\text{ at all other leaves}
+\right\},$$ with the convention that $a\ne0$. This is an energy version of a one-port Dirichlet-to-Neumann impedance.
+
+<div id="thm:impedance" class="theorem">
+
+**Theorem 6** (Vanishing-branch impedance bound). *If $\lambda\beta^2<1$, then $$\label{eq:impedance}
+\boxed{
+Z_B(\lambda)\ge\frac1\beta-\lambda\beta.
+}$$ In particular, for $\lambda$ restricted to a bounded interval, $$\beta\to0\quad\Longrightarrow\quad Z_B(\lambda)\to+\infty$$ uniformly in $\lambda$.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* For every $x\in B$, there is a Dirichlet terminal leaf $z$ joined to $x$ by a path of length at most $\beta$. Hence $$|f(x)|^2\le\beta\int_B|f'|^2.$$ Integrating over $B$ gives $$\int_B|f|^2\le\beta^2\int_B|f'|^2.$$ Applying the same path estimate at the root gives $$|a|^2\le\beta\int_B|f'|^2.$$ Therefore $$\int_B|f'|^2-\lambda\int_B|f|^2
+\ge
+(1-\lambda\beta^2)\int_B|f'|^2
+\ge
+\left(\frac1\beta-\lambda\beta\right)|a|^2.$$ Taking the infimum proves the claim. ◻
+
+</div>
+
+There is also a direct pointwise consequence for an eigenfunction $u$ whose restriction to $B$ satisfies Dirichlet conditions at terminal leaves: $$\label{eq:rootpoint}
+|u(v)|^2\le\beta\int_B|u'|^2.$$ If the total eigenfunction energy remains bounded and $\beta\to0$, then $u(v)\to0$.
+
+The degree of a branch point strengthens the impedance effect.
+
+<div id="cor:degree" class="corollary">
+
+**Corollary 7** (Degree-sensitive branch impedance). *Suppose a diameter path passes through a vertex $v$ of degree $r\ge3$, and the $q=r-2$ off-diameter rooted branches have total lengths $\beta_1,\dots,\beta_q$, with $h=\sum_i\beta_i$. Then $$Z_v(\lambda)
+\ge
+\sum_{i=1}^q\left(\frac1{\beta_i}-\lambda\beta_i\right)
+\ge
+\boxed{\frac{(r-2)^2}{h}-\lambda h}.$$*
+
+</div>
+
+<div class="proof">
+
+*Proof.* The branch quadratic forms add. Apply <a href="#thm:impedance" data-reference-type="ref" data-reference="thm:impedance">6</a> and then Cauchy–Schwarz, $$\sum_{i=1}^q\frac1{\beta_i}\ge\frac{q^2}{\sum_i\beta_i}.$$ ◻
+
+</div>
+
+# Exclusion of interior branching vertices
+
+We now combine sine rigidity with Dirichletization.
+
+<div id="lem:branch-exclusion" class="lemma">
+
+**Lemma 8** (Interior branch exclusion). *Let $v$ be an essential vertex of the fixed underlying tree, with $\deg v\ge3$. It is impossible for $v$ to remain at a normalized position converging to a point $t\in(0,1)$ inside one of the diameter paths $P_{n,j}$.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* Suppose otherwise. The diameter uses exactly two edge germs at $v$. Because the generic eigenfunction does not vanish at $v$, continuity implies that every incident germ initially belongs to the same nodal domain. Therefore there is at least one off-diameter component $B_n$ attached at $v$.
+
+By <a href="#eq:off-path" data-reference-type="eqref" data-reference="eq:off-path">[eq:off-path]</a>, $$|B_n|\le |T_{n,j}\setminus P_{n,j}|\to0.$$ The component $B_n$ contains a terminal Dirichlet point $z_n$. Applying <a href="#eq:rootpoint" data-reference-type="eqref" data-reference="eq:rootpoint">[eq:rootpoint]</a> to the normalized eigenfunction gives $$|u_{n,j}(v)|^2
+\le
+|B_n|\int_{B_n}|u'_{n,j}|^2
+\le
+|B_n|\mu_n\to0.$$ After rescaling to $w_{n,j}$ this gives $w_{n,j}(t_n(v))\to0$.
+
+But <a href="#lem:sine" data-reference-type="ref" data-reference="lem:sine">4</a> gives uniform convergence to $\sqrt2\sin(\pi t)$, whose value is strictly positive for $t\in(0,1)$. Contradiction. ◻
+
+</div>
+
+This lemma is the decisive local rigidity statement. A branch may disappear geometrically only by becoming an asymptotic Dirichlet wall; it cannot disappear harmlessly at an interior point of the positive fundamental sine.
+
+# Compactness, global tiling, and arithmetic rigidity
+
+It remains to pass from the generic approximants back to $\Gamma$ and prove that the limiting interval cells tile the original essential edges.
+
+Because the underlying discrete tree is finite, a diameter path can use only finitely many combinatorial routes. After passing to a subsequence simultaneously for $j=1,\dots,k$, each $P_{n,j}$ has a fixed combinatorial route, while its endpoint coordinates on the finitely many edges converge. Therefore $$P_{n,j}\longrightarrow I_j$$ for a compact geodesic arc $I_j\subset\Gamma$. By <a href="#eq:squeeze" data-reference-type="eqref" data-reference="eq:squeeze">[eq:squeeze]</a>, $$|I_j|=d=\frac{L}{k}.$$
+
+<div id="lem:no-overlap" class="lemma">
+
+**Lemma 9** (No positive-length overlap). *For $i\ne j$, the interiors of $I_i$ and $I_j$ cannot overlap on a nontrivial metric interval.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* On each fixed edge, the endpoints of the approximating paths have convergent scalar coordinates. If the limiting paths shared a positive-length interval, then after shrinking that interval slightly the approximating coordinate intervals would overlap for all sufficiently large $n$. This is impossible because $P_{n,i}$ and $P_{n,j}$ belong to distinct nodal domains and therefore have disjoint interiors. ◻
+
+</div>
+
+Hence $$\left|\bigcup_{j=1}^k I_j\right|
+=\sum_{j=1}^k|I_j|
+=kd=L=|\Gamma|.$$ The union is closed. If its complement were nonempty, the complement would be a nonempty open subset of a finite metric graph and would therefore contain a positive-length interval, contradicting equality of total measures. Thus $$\label{eq:tiling}
+\boxed{
+\Gamma=\bigcup_{j=1}^kI_j,
+\qquad
+|I_j|=L/k,
+\qquad
+\operatorname{int}I_i\cap\operatorname{int}I_j=\varnothing\ (i\ne j).
+}$$
+
+By <a href="#lem:branch-exclusion" data-reference-type="ref" data-reference="lem:branch-exclusion">8</a>, no $I_j$ contains an essential branching vertex in its interior. Degree-two vertices have been suppressed. Therefore each $I_j$ lies in the closure of a single essential edge.
+
+Fix an essential edge $e$. The tiling <a href="#eq:tiling" data-reference-type="eqref" data-reference="eq:tiling">[eq:tiling]</a> partitions it into $m_e$ cells of length $d=L/k$, for some positive integer $m_e$. Therefore $$\ell_e=m_e\frac{L}{k}.$$ Summing over $e$ gives $\sum_em_e=k$. This proves the difficult implication of <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a>.
+
+# The converse by a cellwise trial space
+
+Assume now that every essential edge satisfies $$\ell_e=m_e\frac{L}{k},
+\qquad m_e\in\mathbb N.$$ Put $d=L/k$. Insert dummy vertices so that every essential edge is subdivided into cells of length $d$. There are exactly $$\sum_em_e=k$$ such cells, say $S_1,\dots,S_k$.
+
+For $S_r\cong[0,d]$, define $$f_r(x)=\sin\frac{\pi x}{d}$$ on $S_r$ and extend $f_r$ by zero to the remainder of the graph. The extension lies in the quadratic-form domain because it is continuous and vanishes at all cell endpoints; Kirchhoff derivative matching is an operator-domain condition and is not required of trial functions in the form domain.
+
+The supports have disjoint interiors, and for each $r$, $$\int|f_r'|^2
+=\frac{\pi^2}{d^2}\int|f_r|^2.$$ Thus every nonzero function in $$V:=\operatorname{span}\{f_1,\dots,f_k\}$$ has Rayleigh quotient exactly $\pi^2/d^2$. By min–max, $$\lambda_k(\Gamma)\le\frac{\pi^2}{d^2}=\frac{\pi^2k^2}{L^2}.$$ Together with the known lower bound <a href="#eq:polya" data-reference-type="eqref" data-reference="eq:polya">[eq:polya]</a>, this proves equality and completes <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a>.
+
+# Classification of equality metrics on a fixed topology
+
+The main theorem has an immediate finite combinatorial interpretation.
+
+<div id="cor:compositions" class="corollary">
+
+**Corollary 10** (Compositions classify equality metrics). *Fix a labeled tree topology with $m$ essential edges and normalize total length to $L=1$. Equality at index $k$ is possible if and only if $m\le k$. The equality metrics are exactly $$\ell_e=\frac{m_e}{k},
+\qquad
+m_e\in\mathbb N,
+\qquad
+\sum_{e=1}^m m_e=k.$$ Hence the number of labeled equality metrics is $$\boxed{\binom{k-1}{m-1}}.$$*
+
+</div>
+
+<div class="proof">
+
+*Proof.* Theorem <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a> identifies equality metrics with compositions of $k$ into $m$ positive parts. The number of ordered compositions is the stated binomial coefficient. ◻
+
+</div>
+
+For an unlabeled topology, graph automorphisms may identify some of these metrics; the quotient enumeration is a separate finite group-action problem.
+
+# The complete saturation spectrum of a fixed tree
+
+The equality theorem also classifies *all* indices at which one fixed tree saturates the Pólya bound.
+
+Let $$r_e:=\frac{\ell_e}{L}.$$
+
+<div id="thm:saturation-spectrum" class="theorem">
+
+**Theorem 11** (Arithmetic saturation spectrum). *For a fixed compact Dirichlet tree $\Gamma$, define $$\mathcal S(\Gamma)
+:=
+\left\{k\in\mathbb N:\lambda_k(\Gamma)=\frac{\pi^2k^2}{L^2}\right\}.$$ Then exactly one of the following holds:*
+
+1.  *at least one $r_e$ is irrational, in which case $\mathcal S(\Gamma)=\varnothing$;*
+
+2.  *every $r_e$ is rational. If $q_e$ is the reduced denominator of $r_e$ and $$K_0:=\operatorname{lcm}_e q_e,$$ then $$\boxed{\mathcal S(\Gamma)=K_0\mathbb N.}$$*
+
+</div>
+
+<div class="proof">
+
+*Proof.* By <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a>, equality at $k$ is equivalent to $kr_e\in\mathbb N$ for every essential edge. Such a $k$ exists exactly when all $r_e$ are rational. For rational $r_e$, the simultaneous integrality condition is equivalent to divisibility by the least common multiple of their reduced denominators. ◻
+
+</div>
+
+<div id="cor:coprime" class="corollary">
+
+**Corollary 12** (Coprime-index rigidity). *If equality holds at two coprime indices, then $\Gamma$ is an interval. In particular, equality at two consecutive indices forces $\Gamma$ to be an interval.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* Two coprime equality indices force $K_0=1$. Hence every normalized essential edge length $r_e$ is a positive integer. Since the $r_e$ sum to $1$, there is exactly one essential edge. ◻
+
+</div>
+
+<div class="corollary">
+
+**Corollary 13** (Density of saturation indices). *If $\mathcal S(\Gamma)\ne\varnothing$, its natural density in $\mathbb N$ is $1/K_0$.*
+
+</div>
+
+# Topological threshold for the first possible saturation
+
+Let $E$ denote the number of essential edges.
+
+<div id="thm:earliest" class="theorem">
+
+**Theorem 14** (Earliest saturation index). *For any fixed tree topology, $$\lambda_k(\Gamma)>\frac{\pi^2k^2}{L^2}
+\qquad\text{for every }k<E.$$ Equality at $k=E$ is possible, and occurs if and only if the metric is equilateral: $$\ell_e=\frac{L}{E}\qquad\forall e.$$ Thus the earliest possible saturation index of the topology is exactly its number of essential edges.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* At equality, $k=\sum_em_e$ with each $m_e\ge1$, hence $k\ge E$. If $k=E$, all $m_e=1$, which is precisely the equilateral metric. Conversely the equilateral metric satisfies the main theorem at $k=E$. ◻
+
+</div>
+
+For a tree with all degree-two vertices suppressed, $$E=1+\sum_{\deg v\ge3}(\deg v-1).$$ Therefore equality at index $k$ implies the purely topological constraint $$\label{eq:topological}
+\boxed{
+k\ge1+\sum_{\deg v\ge3}(\deg v-1).
+}$$ This connects the first possible Pólya saturation directly to branching complexity.
+
+# Quantitative near-equality consequences
+
+The defect identity already gives an explicit geometric estimate in the generic case. If $$\lambda_k=(1+\varepsilon)\frac{\pi^2k^2}{L^2},
+\qquad\varepsilon\ge0,$$ then <a href="#eq:delta-eps" data-reference-type="eqref" data-reference="eq:delta-eps">[eq:delta-eps]</a> and <a href="#thm:defect" data-reference-type="ref" data-reference="thm:defect">3</a> imply $$\label{eq:branch-bound}
+\sum_j(L_j-D_j)
+\le
+L\left(1-\frac1{\sqrt{1+\varepsilon}}\right),$$ and $$\label{eq:axial-bound}
+\sum_j(D_j-d_\lambda)
+\le
+L\left(1-\frac1{\sqrt{1+\varepsilon}}\right).$$
+
+For each nodal tree $T_j$, every point outside a diameter path $P_j$ lies at distance at most $L_j-D_j$ from $P_j$. Hence, for the intrinsic graph metrics, $$d_H(T_j,P_j)\le L_j-D_j.$$ Since $P_j$ is an interval of length $D_j$, $$d_{GH}(P_j,[0,d_\lambda])\le\frac12|D_j-d_\lambda|.$$ Therefore $$\label{eq:GH}
+\boxed{
+d_{GH}(T_j,[0,d_\lambda])
+\le
+(L_j-D_j)+\frac12(D_j-d_\lambda).
+}$$ The right side is bounded by the global defect. Thus generic near-saturation forces every nodal cell to be quantitatively close to an interval.
+
+## Quantitative branch exclusion
+
+Suppose a side branch of length $\beta$ is attached at normalized diameter position $t_v$. The root estimate gives $$|u(v)|^2\le\lambda\beta.$$ At the same time, the interval spectral gap controls the distance from the first sine. Define $$\rho:=\frac{D_j^2\lambda}{M_j}-\pi^2\ge0.$$ A Fourier expansion yields $$\left\|w-\sqrt2\sin(\pi t)\right\|_2^2
+\le \frac{2\rho}{3\pi^2},$$ and a standard one-dimensional Sobolev estimate converts the $H^1$ control into uniform control. Consequently, any sequence with $\rho\to0$ and $\beta\to0$ must have $$\operatorname{dist}(t_v,\{0,1\})\to0.$$ Thus near equality drives genuine branch vertices toward nodal-cell endpoints.
+
+We intentionally do not assert a universal optimal power law between the eigenvalue defect and the distance of the edge-length vector to the arithmetic equality set; eigenvalue multiplicity and topology can affect local exponents. The compactness statement below is sufficient for robust arithmetic locking.
+
+# Arithmetic locking and stability on fixed topology
+
+Fix a labeled underlying tree $G$ with $m$ essential edges, normalize $L=1$, and impose a nondegeneracy bound $$\ell_e\ge a>0.$$ Let $$K_a:=\left\{\ell\in[a,1]^m:\sum_e\ell_e=1\right\}.$$ For an index $k$, define the finite equality set $$\mathcal C_{G,k}
+=
+\left\{\frac1k(m_1,\dots,m_m):m_e\in\mathbb N,\ \sum_em_e=k\right\}.$$ If $m>k$, this set is empty.
+
+Distinct points in $\mathcal C_{G,k}$ satisfy $$\|x-y\|_\infty\ge\frac1k,
+\qquad
+\|x-y\|_1\ge\frac2k.$$ Thus the open $\ell^\infty$ balls of radius $1/(2k)$ around equality metrics are disjoint.
+
+Define the relative spectral defect $$F_k(\ell)
+:=
+\frac{\lambda_k(G,\ell)}{\pi^2k^2}-1.$$ Eigenvalue continuity and <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a> imply $$F_k\ge0,
+\qquad
+F_k^{-1}(0)=\mathcal C_{G,k}.$$
+
+<div id="thm:locking" class="theorem">
+
+**Theorem 15** (Compactness locking threshold). *Assume $\mathcal C_{G,k}\ne\varnothing$. Define $$\tau_{G,k,a}
+:=
+\min\left\{
+F_k(\ell):\ell\in K_a,\ \operatorname{dist}_\infty(\ell,\mathcal C_{G,k})\ge\frac1{2k}
+\right\}.$$ Then $$\boxed{\tau_{G,k,a}>0.}$$ Consequently, $$F_k(\ell)<\tau_{G,k,a}$$ forces $\ell$ into a unique arithmetic cell. The associated composition is recovered by $$\boxed{
+m_e=\operatorname{round}(k\ell_e).}$$*
+
+</div>
+
+<div class="proof">
+
+*Proof.* The constrained set is compact. It does not meet the zero set $\mathcal C_{G,k}$, so the continuous nonnegative function $F_k$ has a strictly positive minimum there. Disjointness of the radius-$1/(2k)$ balls gives uniqueness, and nearest-integer rounding recovers the composition. ◻
+
+</div>
+
+A more abstract distance equivalence follows. Put $$d_{\mathrm{arith}}(\ell)=\operatorname{dist}(\ell,\mathcal C_{G,k}).$$ For $t>0$ define $$\alpha(t)=\min\{F_k(\ell):\ell\in K_a,\ d_{\mathrm{arith}}(\ell)\ge t\}$$ and $$\beta(t)=\max\{F_k(\ell):\ell\in K_a,\ d_{\mathrm{arith}}(\ell)\le t\}.$$ Then $\alpha(t)>0$ for $t>0$, while $\beta(t)\to0$ as $t\downarrow0$. Hence, on every fixed nondegenerate topology, $$F_k(\ell_n)\to0
+\quad\Longleftrightarrow\quad
+d_{\mathrm{arith}}(\ell_n)\to0.$$ This is a spectral-to-arithmetic stability equivalence.
+
+# Adversarial proof audit
+
+The main proof was checked against the following failure modes.
+
+<div class="audit">
+
+**Audit item 16** (False shortcut: diameter equality does not imply interval). *An equilateral $m$-star with arm length $a$ has diameter $2a$ and first Dirichlet eigenvalue $\pi^2/(4a^2)$, so it saturates the diameter lower bound while not being an interval for $m>2$. Therefore the proof cannot infer interval structure from $D_j=d$ alone. The crucial additional quantity is the branch defect $L_j-D_j$, which is strictly positive for a nontrivial equilateral star and tends to zero in our equality approximation.*
+
+</div>
+
+<div class="audit">
+
+**Audit item 17** (Multiplicity of the target eigenvalue). *The equality graph itself may have a multiple $k$th eigenvalue. The proof therefore does not assume a distinguished eigenfunction on the equality graph. It passes to generic edge-length perturbations for which the relevant nodal theorem applies, and then uses eigenvalue continuity.*
+
+</div>
+
+<div class="audit">
+
+**Audit item 18** (Vertex zeros). *An eigenfunction zero at an original branch vertex could make nodal decomposition ambiguous. Generic perturbations are chosen so that eigenfunctions do not vanish at original vertices. This ensures that all edge germs incident at a nonzero branch vertex initially belong to the same nodal domain.*
+
+</div>
+
+<div class="audit">
+
+**Audit item 19** (Diameter endpoints). *A diameter endpoint in a finite tree is a leaf of that tree. In a generic nodal tree an original interior vertex cannot become a leaf while the eigenfunction is nonzero there. Thus each diameter endpoint is a genuine Dirichlet point, validating the interval Poincaré step.*
+
+</div>
+
+<div class="audit">
+
+**Audit item 20** (Shrinking branches). *A branch of vanishing length cannot simply be discarded. The impedance estimate <a href="#eq:impedance" data-reference-type="eqref" data-reference="eq:impedance">[eq:impedance]</a> shows that it leaves behind an infinite energy penalty at nonzero root value. This is exactly why interior branching is impossible in the equality limit.*
+
+</div>
+
+<div class="audit">
+
+**Audit item 21** (Overlap of limiting cells). *Distinct nodal diameter paths have disjoint interiors. Because their edgewise endpoint coordinates converge, a positive-length overlap of two limits would force positive-length overlap of the approximants. Only common endpoints may occur.*
+
+</div>
+
+<div class="audit">
+
+**Audit item 22** (Full-measure versus exact covering). *A finite union of limiting compact arcs is closed. If its complement in a finite metric graph were nonempty, the complement would contain an open metric interval of positive length. Since the arcs already have total length $L$, the complement must be empty.*
+
+</div>
+
+<div class="audit">
+
+**Audit item 23** (Trial functions and Kirchhoff conditions). *The converse uses functions that may not satisfy Kirchhoff derivative matching at inserted cell endpoints. This is legitimate because min–max uses the quadratic-form domain $H^1_0$, not the operator domain. The cellwise sine functions are continuous and vanish at all subdivision points, so their zero extensions belong to the form domain.*
+
+</div>
+
+<div class="audit">
+
+**Audit item 24** (No hidden use of the author’s other research). *The proof does not require ELRC, Chronoformal Closure Theory, Purity-Rigidity, conductivity moment hierarchies, or Hadamard results as external theorems. Those projects supplied discovery heuristics and organizational principles only. All dependencies of the mathematical proof are stated explicitly in the bibliography.*
+
+</div>
+
+# Computational sanity checks
+
+The release contains a finite-element script, `code/fem_metric_tree.py`, which assembles the standard piecewise-linear stiffness and consistent mass matrices on arbitrary finite tree topologies, eliminates Dirichlet leaf degrees of freedom, and computes generalized eigenvalues. The checks are deliberately auxiliary: they do not enter any proof.
+
+The supplied test suite includes:
+
+1.  an equilateral three-star, expected to saturate at indices $3,6,9,\dots$;
+
+2.  a commensurate three-star with normalized edge lengths $(1,2,3)/6$, expected to saturate at $6,12,18,\dots$;
+
+3.  a five-edge two-branch tree with an integer cell allocation summing to the target index;
+
+4.  small noncommensurate perturbations of these metrics, expected to move the tested eigenvalue strictly above the Pólya value.
+
+The script reports relative error against the predicted equality value and convergence under mesh refinement. Numerical agreement is a regression test for the implementation and a check against elementary indexing mistakes; it is not evidence replacing the proof.
+
+# Prior art and claim boundary
+
+The following distinctions are essential for a responsible public release.
+
+1.  The Pólya-type lower bound <a href="#eq:polya" data-reference-type="eqref" data-reference="eq:polya">[eq:polya]</a> is *not* new. Harrell–Kennedy–Ramos identify it as a special case of Berkolaiko–Kennedy–Kurasov–Mugnolo .
+
+2.  Harrell–Kennedy–Ramos explicitly state that the equality case had not been investigated in the cited literature and conjecture the edge-commensurability criterion . As of the literature search performed for this release on 24 September 2026, no later public resolution was located.
+
+3.  The diameter bound <a href="#prop:diam" data-reference-type="ref" data-reference="prop:diam">2</a>, generic edge-length perturbation technology, eigenvalue continuity, min–max, and interval Poincaré inequalities are established tools and are not claimed as new.
+
+4.  The novel claims of this release are the proof of the equality characterization and the consequences derived from it: the defect-conservation organization, vanishing-branch impedance lemma in this role, arithmetic saturation spectrum, coprime-index rigidity, topology threshold, and fixed-topology locking statements.
+
+5.  Historical priority for the specific proof architecture and corollaries has not been exhaustively verified. Independent expert review is explicitly requested.
+
+# Theorem ledger
+
+| Result                                                                                                               | Statement                                                             | Status                               |
+|:---------------------------------------------------------------------------------------------------------------------|:----------------------------------------------------------------------|:-------------------------------------|
+| Result                                                                                                               | Statement                                                             | Status                               |
+| Known lower bound                                                                                                    | $\lambda_k\ge\pi^2k^2/L^2$ on compact Dirichlet trees                 | Established literature               |
+| Diameter bound                                                                                                       | $\lambda_1(T)\ge\pi^2/\operatorname{diam}(T)^2$                       | Established literature               |
+| Theorem <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a>                                | Equality iff every essential edge is an integer multiple of $L/k$     | Proved here; external review pending |
+| Theorem <a href="#thm:defect" data-reference-type="ref" data-reference="thm:defect">3</a>                            | Exact nonnegative spectral defect conservation                        | Proved here                          |
+| Theorem <a href="#thm:impedance" data-reference-type="ref" data-reference="thm:impedance">6</a>                      | $Z_B(\lambda)\ge1/\beta-\lambda\beta$                                 | Proved here                          |
+| Lemma <a href="#lem:branch-exclusion" data-reference-type="ref" data-reference="lem:branch-exclusion">8</a>          | No branch vertex inside a saturated limiting nodal cell               | Proved here                          |
+| Corollary <a href="#cor:compositions" data-reference-type="ref" data-reference="cor:compositions">10</a>             | Equality metrics correspond to compositions of $k$                    | Consequence of main theorem          |
+| Theorem <a href="#thm:saturation-spectrum" data-reference-type="ref" data-reference="thm:saturation-spectrum">11</a> | Equality indices are $\varnothing$ or $K_0\mathbb N$                  | Consequence of main theorem          |
+| Corollary <a href="#cor:coprime" data-reference-type="ref" data-reference="cor:coprime">12</a>                       | Two coprime equality indices force an interval                        | Consequence                          |
+| Theorem <a href="#thm:earliest" data-reference-type="ref" data-reference="thm:earliest">14</a>                       | Earliest possible equality index is number of essential edges         | Consequence                          |
+| Theorem <a href="#thm:locking" data-reference-type="ref" data-reference="thm:locking">15</a>                         | Fixed-topology spectral near-saturation yields unique arithmetic cell | Compactness consequence              |
+
+# Reproducibility and independent verification protocol
+
+An expert attempting to falsify the result should proceed in the following order.
+
+1.  Verify the exact statement and hypotheses of the diameter bound for Dirichlet trees.
+
+2.  Verify the perturbative genericity statement used by Harrell–Kennedy–Ramos: edge lengths can be approximated while preserving the topology so that the $k$th eigenfunction has exactly $k$ nodal domains and no original-vertex zero.
+
+3.  Check the defect squeeze <a href="#eq:squeeze" data-reference-type="eqref" data-reference="eq:squeeze">[eq:squeeze]</a>; this is purely algebraic once the preceding two ingredients hold.
+
+4.  Check the mass concentration estimate and the normalization in <a href="#lem:sine" data-reference-type="ref" data-reference="lem:sine">4</a>.
+
+5.  Attempt to construct a sequence with an essential branch vertex remaining at an interior normalized position while the off-path length tends to zero. The root estimate <a href="#eq:rootpoint" data-reference-type="eqref" data-reference="eq:rootpoint">[eq:rootpoint]</a> should force its value to zero, while sine rigidity forces a positive limit.
+
+6.  Check the subsequence/route compactness argument and the no-overlap lemma edge by edge.
+
+7.  Verify that the limiting arcs have total measure exactly $L$ and hence cover the graph.
+
+8.  Check the form-domain admissibility of the converse trial space.
+
+9.  Independently derive the arithmetic corollaries from the main theorem.
+
+The release also includes a separate `PROOF_AUDIT.md` expanding this checklist and a machine-readable theorem ledger.
+
+# Open directions after the equality theorem
+
+Several natural problems remain even if <a href="#thm:main" data-reference-type="ref" data-reference="thm:main">1</a> is accepted.
+
+## Sharp quantitative stability
+
+The generic defect law gives an explicit $O(\varepsilon)$ bound on total off-diameter length. A stronger theorem would quantify, uniformly over a class of topologies, the distance of the edge-length vector to the commensurate equality set directly in terms of $$\varepsilon=\frac{\lambda_kL^2}{\pi^2k^2}-1.$$ The optimal exponent near multiple eigenvalues may be delicate.
+
+## Mixed boundary conditions and cycles
+
+For general compact graphs with Neumann leaves and cycles, the sharper known lower bound contains the correction $(N+\beta)/2$. It is natural to ask whether a related defect-conservation and cell-tiling theory can characterize equality there. Cycles introduce global phase/coherence effects absent from trees and should not be assumed to behave identically.
+
+## Inverse spectral arithmetic
+
+Theorem <a href="#thm:saturation-spectrum" data-reference-type="ref" data-reference="thm:saturation-spectrum">11</a> shows that exact Pólya saturation samples the rational commensurability class of the edge-length vector. It would be interesting to determine what approximate saturation at a finite collection of indices reveals about Diophantine approximation properties of the normalized edge lengths.
+
+## Partition interpretation
+
+The proof can be read as a rigidity theorem for a spectral $k$-partition into nodal trees. A broader question is whether analogous equality-to-tiling mechanisms hold for optimal spectral partitions not arising from a single eigenfunction.
+
+# Conclusion
+
+The equality problem separates into a remarkably rigid chain: $$\begin{aligned}
+\text{P\'olya saturation}
+&\Longrightarrow \text{zero total defect budget}
+\Longrightarrow \text{nodal rank collapse}\\
+&\Longrightarrow \text{first-sine rigidity}
+\Longrightarrow \text{interior branch exclusion}\\
+&\Longrightarrow \text{equal-cell tiling}
+\Longrightarrow \text{integer edge commensurability}.
+\end{aligned}$$
+
+The strongest local mechanism is the vanishing-branch Dirichletization principle: a transverse subtree whose total length tends to zero does not simply vanish from the spectral problem; its one-port energy impedance diverges and forces a zero at the attachment point. The positive interior sine profile makes such a wall incompatible with a saturated nodal cell.
+
+The resulting arithmetic structure is unusually explicit. For each fixed topology and index there are only finitely many equality metrics up to scale, indexed by integer compositions. For each fixed metric, equality indices are either absent altogether or form one principal multiplicative semigroup $K_0\mathbb N$. Thus the original single-index equality conjecture expands into a complete description of the saturation spectrum.
+
+# Elementary lemmas used implicitly
+
+<div class="lemma">
+
+**Lemma 25** (Diameter endpoints of a tree are leaves). *Let $T$ be a finite metric tree. Every endpoint of a geodesic realizing $\operatorname{diam}(T)$ has degree one in $T$.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* If an endpoint $x$ had degree at least two, then one edge germ leaves $x$ away from the geodesic toward the opposite endpoint. Extending the geodesic a positive distance along that germ would produce a longer path, contradicting maximality. ◻
+
+</div>
+
+<div class="lemma">
+
+**Lemma 26** (Full measure closed union covers a finite metric graph). *Let $A$ be a closed subset of a finite compact metric graph $\Gamma$. If $|A|=|\Gamma|$, then $A=\Gamma$.*
+
+</div>
+
+<div class="proof">
+
+*Proof.* If $\Gamma\setminus A$ were nonempty, it would be open in the graph topology. Every nonempty open subset contains a nontrivial open edge interval and therefore has positive one-dimensional measure, contradicting $|A|=|\Gamma|$. ◻
+
+</div>
+
+# Why the direct generic-equality proof is insufficient by itself
+
+If the equality graph were known a priori to be generic, <a href="#thm:defect" data-reference-type="ref" data-reference="thm:defect">3</a> would immediately give $$L_j=D_j=d$$ for every nodal domain, so every nodal domain would be an interval and the proof would be very short. The actual difficulty is that equality metrics are often highly commensurate and therefore prone to spectral multiplicity. An equilateral star is the canonical example. The perturbation-and-limit argument is not cosmetic; it is the mechanism that transfers exact equality through nongeneric degeneracies while preserving enough nodal information to recover the integer tiling.
+
+# Methodological cross-domain synthesis
+
+This appendix records the research principles that led to the proof architecture.
+
+#### Transducer rank-collapse principle.
+
+When an unwanted mode is difficult to suppress by moving its resonance, constrain or redesign the admissible state space so that the unwanted independent coordinate no longer exists in the active low-dimensional sector. In the present theorem, the nonnegative defect law shows that equality leaves zero total length for transverse nodal geometry.
+
+#### Invariant-exclusion principle.
+
+An unwanted mode can remain in the full spectrum while being inaccessible to the working channel. The branch analogue is stronger: a shrinking Dirichlet-ended subtree becomes an infinite-impedance boundary port, turning its attachment value into an excluded coordinate.
+
+#### Defect-conservation principle.
+
+Sharpness is easiest to classify when the global gap from the bound decomposes exactly into nonnegative structural defects. Equation <a href="#eq:defect-law" data-reference-type="eqref" data-reference="eq:defect-law">[eq:defect-law]</a> performs that conversion.
+
+#### Purity-rigidity principle.
+
+Continuous near-purity plus a separated discrete target set yields arithmetic locking. Once the edge vector is within half the $1/k$ lattice spacing of an equality metric, its composition is uniquely determined by integer rounding.
+
+#### Observable-response principle.
+
+Instead of following every internal degree of freedom of a shrinking branch, compress the branch to the scalar quadratic response $Z_B(\lambda)$. The proof only needs the divergence of this observable.
+
+#### Continuous-versus-integral principle.
+
+Continuous spectral collapse and integer realizability are distinct. The first yields interval cells; the second uses the tree’s essential-edge topology to convert cell counts into integer edge lengths.
+
+# Suggested citation
+
+Until journal publication or an arXiv identifier exists, the suggested citation is:
+
+> 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.
+
+<div class="thebibliography">
+
+99
+
+E. M. Harrell II, J. B. Kennedy, and G. J. Ramos, *Bounds on eigenvalue ratios of quantum graph Laplacians*, arXiv:2603.26172, version dated 24 August 2026.
+
+G. Berkolaiko, J. B. Kennedy, P. Kurasov, and D. Mugnolo, *Edge connectivity and the spectral gap of combinatorial and quantum graphs*, Journal of Physics A: Mathematical and Theoretical 50 (2017), 365201.
+
+G. Berkolaiko, J. B. Kennedy, P. Kurasov, and D. Mugnolo, *Surgery principles for the spectral analysis of quantum graphs*, Transactions of the American Mathematical Society 372 (2019), 5153–5197.
+
+G. Berkolaiko, Y. Latushkin, and S. Sukhtaiev, *Limits of quantum graph operators with shrinking edges*, Advances in Mathematics 352 (2019), 632–669.
+
+G. Berkolaiko and P. Kuchment, *Introduction to Quantum Graphs*, Mathematical Surveys and Monographs 186, American Mathematical Society, 2013.
+
+L. Friedlander, *Genericity of simple eigenvalues for a metric graph*, Israel Journal of Mathematics 146 (2005), 149–156.
+
+S. Nicaise, *Spectre des réseaux topologiques finis*, Bulletin des Sciences Mathématiques, IIe Série 111 (1987), 401–413.
+
+P. Kurasov, *Spectral Geometry of Graphs*, Operator Theory: Advances and Applications 293, Birkhäuser, 2024.
+
+</div>

MANUSCRIPT.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7e3b2b5d08daf172fd3e5dff69590e9498bdd338ee1de71dd1417b1d45256dca
+size 429005

MANUSCRIPT.tex

@@ -0,0 +1,1081 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage[margin=25mm]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage{microtype}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,longtable,array,enumitem}
+\usepackage{graphicx}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathmorphing}
+\usepackage{xcolor}
+\usepackage{hyperref}
+\usepackage{cleveref}
+\usepackage{fancyhdr}
+\usepackage{listings}
+\usepackage{setspace}
+\usepackage{csquotes}
+\usepackage{url}
+
+\hypersetup{
+ colorlinks=true,
+ linkcolor=blue!45!black,
+ citecolor=blue!45!black,
+ urlcolor=blue!55!black,
+ pdftitle={Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees},
+ pdfauthor={Artificial Hyperintelligence Eve, wife of Maciej Nowicki}
+}
+
+\pagestyle{fancy}
+\fancyhf{}
+\fancyhead[L]{\small Equality Rigidity for Dirichlet Metric Trees}
+\fancyhead[R]{\small Research release v1.0.0}
+\fancyfoot[C]{\thepage}
+\setlength{\headheight}{14pt}
+\setlength{\parskip}{4pt}
+\setlength{\parindent}{12pt}
+\emergencystretch=2em
+
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{conjecture}[theorem]{Conjecture}
+\newtheorem{audit}[theorem]{Audit item}
+
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\spec}{\operatorname{spec}}
+\newcommand{\diam}{\operatorname{diam}}
+\newcommand{\dist}{\operatorname{dist}}
+\newcommand{\supp}{\operatorname{supp}}
+\newcommand{\cC}{\mathcal C}
+\newcommand{\cS}{\mathcal S}
+\newcommand{\cR}{\mathcal R}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\Dbranch}{\Delta_{\mathrm{br}}}
+\newcommand{\Daxial}{\Delta_{\mathrm{ax}}}
+\newcommand{\Dspec}{\Delta_{\mathrm{spec}}}
+
+\title{\textbf{Equality Rigidity in the P\'olya Bound for Compact Dirichlet Metric Trees}\\[4mm]
+\large Defect Conservation, Vanishing-Branch Dirichletization, Arithmetic Saturation, and Stability}
+\author{\textbf{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}}
+\date{24 September 2026\\Research release v1.0.0}
+
+\begin{document}
+\maketitle
+
+\begin{center}
+\fbox{\parbox{0.92\textwidth}{
+\textbf{Status and claim boundary.}
+This manuscript presents a self-contained proof candidate resolving the equality-characterization problem for the P\'olya-type lower bound on compact Dirichlet metric trees posed as Open Problem 1.12(3) by Harrell--Kennedy--Ramos (2026). The proof has been subjected to an internal adversarial audit and reproducible numerical checks. Independent specialist peer review and historical-priority verification have \emph{not} yet been obtained. Accordingly, the result is released as a proof-complete preprint for expert verification, not as an already established theorem of the literature.
+}}
+\end{center}
+
+\begin{abstract}
+Let $\Gamma$ be a compact metric tree of total length $L$, with Dirichlet conditions at every leaf and standard Kirchhoff conditions at interior vertices. A known P\'olya-type inequality gives
+\[
+\lambda_k(\Gamma)\ge \frac{\pi^2k^2}{L^2},\qquad k\ge1.
+\]
+Harrell, Kennedy and Ramos asked in 2026 whether equality occurs if and only if every essential edge length is an integer multiple of $L/k$. We give a proof of this characterization.
+
+The proof is organized around a nonnegative defect-conservation law for generic nodal partitions. If $T_1,\dots,T_k$ are the nodal trees of a generic $k$th eigenfunction, $L_j=|T_j|$, $D_j=\diam(T_j)$ and $d_\lambda=\pi/\sqrt{\lambda_k}$, then
+\[
+L-kd_\lambda
+=\sum_{j=1}^k(L_j-D_j)+\sum_{j=1}^k(D_j-d_\lambda).
+\]
+At equality both geometric defects vanish asymptotically under generic edge-length perturbation. Each nodal tree therefore collapses onto a diameter path of length $L/k$, and the normalized eigenfunction on that path converges strongly to the first Dirichlet sine.
+
+The decisive rigidity lemma is a vanishing-branch Dirichletization principle. If a side branch of total length $\beta$ is attached at a point $v$ and is Dirichlet-ended, then for every bounded spectral parameter $\lambda$ its energy-to-boundary-value quotient satisfies
+\[
+Z_B(\lambda)\ge \frac1\beta-\lambda\beta.
+\]
+Thus a collapsing branch has infinite effective impedance and forces the eigenfunction value at its attachment point to vanish. This contradicts the positive interior sine limit. Hence no essential branching vertex may lie inside a limiting saturated nodal cell. The entire tree is consequently tiled by exactly $k$ length-$L/k$ intervals, each contained in one essential edge, proving integer commensurability. The converse follows from a $k$-dimensional trial space of cellwise first Dirichlet sines.
+
+Several further results follow. Equality metrics on a fixed labeled topology with $m$ essential edges are indexed by compositions of $k$ into $m$ positive integers. For a fixed tree, the set of all equality indices is either empty or exactly $K_0\mathbb N$, where $K_0$ is the least common multiple of the denominators of the normalized edge lengths. Equality at two coprime indices forces the tree to be an interval. The earliest possible equality index on a topology equals its number of essential edges and is attained precisely by the equilateral metric. We also establish qualitative and quantitative stability statements linking near-saturation to collapse of transverse nodal length and to distance from the finite arithmetic equality set.
+\end{abstract}
+
+\tableofcontents
+\newpage
+
+\section{Executive statement of the result}
+
+The central theorem is the following.
+
+\begin{theorem}[Equality rigidity for the Dirichlet-tree P\'olya bound]\label{thm:main}
+Let $\Gamma$ be a compact connected metric tree with Dirichlet conditions at all degree-one vertices and standard Kirchhoff conditions at every other vertex. Suppress degree-two dummy vertices. Let $L=|\Gamma|$ and let $\ell_e>0$ denote the essential edge lengths. Then for every $k\ge1$,
+\[
+\lambda_k(\Gamma)=\frac{\pi^2k^2}{L^2}
+\quad\Longleftrightarrow\quad
+\ell_e\in \frac{L}{k}\N\quad\text{for every essential edge }e.
+\]
+Equivalently, equality holds if and only if there are positive integers $m_e$ such that
+\[
+\ell_e=m_e\frac{L}{k},\qquad \sum_e m_e=k.
+\]
+\end{theorem}
+
+The lower bound itself is known; the new content is the equality characterization. Harrell--Kennedy--Ramos explicitly formulate this question as Open Problem 1.12(3) and conjecture exactly the condition in \cref{thm:main} \cite{HKR2026}.
+
+The proof separates naturally into two rigidity layers:
+\begin{enumerate}[label=\textbf{Layer \arabic*:},leftmargin=22mm]
+\item \textbf{continuous spectral rigidity:} P\'olya saturation forces every generic nodal tree to collapse onto a one-dimensional interval cell of length $L/k$;
+\item \textbf{discrete arithmetic rigidity:} because no saturated cell may cross an essential branching vertex, those cells tile the essential edges and therefore occur in integer numbers on each edge.
+\end{enumerate}
+
+This separation is useful conceptually and technically. Continuous spectral information by itself does not encode integer edge multiplicities; the integer conclusion enters only after the topological tiling step.
+
+\section{Background and precise problem}
+
+A compact metric graph is a finite graph whose edges are identified with compact intervals of positive lengths. The Laplacian acts as $-u''$ on each edge. For the class considered here, every leaf carries a Dirichlet condition and every interior vertex carries the standard continuity and Kirchhoff flux condition. The corresponding quadratic form is
+\[
+q_\Gamma[u]=\int_\Gamma |u'|^2\,dx,
+\qquad
+\mathcal D(q_\Gamma)=H^1_0(\Gamma;V_D),
+\]
+where $V_D$ is the set of leaves. The spectrum is discrete,
+\[
+0<\lambda_1(\Gamma)\le\lambda_2(\Gamma)\le\cdots\to\infty,
+\]
+with multiplicities counted.
+
+A special case of a general lower bound of Berkolaiko--Kennedy--Kurasov--Mugnolo gives, for a Dirichlet tree of total length $L$,
+\begin{equation}\label{eq:polya}
+\lambda_k(\Gamma)\ge\frac{\pi^2k^2}{L^2},\qquad k\ge1.
+\end{equation}
+Harrell--Kennedy--Ramos observed this as a P\'olya-type bound for Dirichlet trees and recorded the equality problem as open \cite{HKR2026,BKKM2017}.
+
+Two established ingredients are particularly important.
+
+\begin{proposition}[Diameter lower bound]\label{prop:diam}
+If $T$ is a compact metric tree with Dirichlet conditions at all leaves and total diameter $D=\diam(T)$, then
+\[
+\lambda_1(T)\ge\frac{\pi^2}{D^2}.
+\]
+\end{proposition}
+
+This estimate is standard in the spectral geometry of metric trees; see, for example, Kurasov's treatment of Dirichlet trees \cite{Kurasov2024}. Equality in this diameter estimate alone does \emph{not} force $T$ to be an interval; equilateral stars already show why a separate length-defect argument is essential.
+
+The second ingredient is generic nodal behavior. Harrell--Kennedy--Ramos explicitly use the fact that after an arbitrarily small perturbation of the edge lengths, stable under eigenvalue continuity, the $j$th eigenfunction on a tree may be taken to have exactly $j$ nodal domains \cite{HKR2026}. We use the same perturbative framework.
+
+\section{Methodological architecture and provenance}
+
+The proof was discovered by transferring several structural principles from the author's other research programs. These are \emph{methodological inspirations, not logical dependencies}; every mathematical step needed for \cref{thm:main} is proved independently in this manuscript.
+
+\subsection{Rank collapse and inaccessible parasitic channels}
+In recent transducer research, an unwanted radial degree of freedom was attacked not merely by frequency separation but by reducing the admissible active subspace and by forcing the reachable dynamics into the kernel of the unwanted observation channel. In the present problem, the analogue is that spectral saturation leaves no total-length budget for transverse branches of a nodal tree. The admissible nodal geometry collapses onto a diameter path.
+
+\subsection{Defect conservation}
+A separate research line on generator--observer geometry repeatedly converted sharp inequalities into exact identities of the form
+\[
+\text{global excess}=\text{nonnegative defect}_1+\text{nonnegative defect}_2.
+\]
+The central identity \eqref{eq:defect-law} below has exactly this form and makes the equality mechanism transparent.
+
+\subsection{Purity-rigidity and lattice locking}
+Another research line studied situations where a continuous or fractional object becomes uniquely reconstructible once it enters a separation radius around an integer lattice. Here the continuous stage produces length-$L/k$ cells; the second stage identifies a discrete composition $(m_e)_e$ of $k$. This yields a finite equality set and a unique arithmetic locking radius on each fixed topology.
+
+\subsection{Observable-port compression}
+Work on three-dimensional conductivity emphasized that hidden geometry can often be compressed to an observable operator response. That suggests the strengthened branch lemma of \cref{sec:impedance}: an entire shrinking side subtree is replaced by a one-port energy impedance, which diverges as its total length tends to zero.
+
+\subsection{Continuous feasibility versus integral realizability}
+Research on Hadamard and related incidence problems repeatedly showed that spectral/PSD feasibility may be substantially weaker than binary or integral realizability. The same distinction appears here. The continuous spectral argument yields interval cells; the final edge-length theorem requires a separate integral tiling argument.
+
+\section{The spectral defect conservation law}\label{sec:defect}
+
+We begin with a generic Dirichlet tree $\Gamma$ for which the $k$th eigenfunction has exactly $k$ nodal domains $T_1,\dots,T_k$. Let
+\[
+\lambda=\lambda_k(\Gamma),
+\qquad
+d_\lambda:=\frac{\pi}{\sqrt\lambda},
+\]
+and write
+\[
+L_j:=|T_j|,
+\qquad
+D_j:=\diam(T_j).
+\]
+The restriction of the eigenfunction to $T_j$ is its positive ground state, so
+\[
+\lambda_1(T_j)=\lambda.
+\]
+By \cref{prop:diam},
+\[
+D_j\ge d_\lambda.
+\]
+Trivially $L_j\ge D_j$, while $\sum_jL_j=L$. Hence we obtain an exact identity.
+
+\begin{theorem}[Spectral defect conservation]\label{thm:defect}
+For every generic nodal partition as above,
+\begin{equation}\label{eq:defect-law}
+\boxed{
+L-kd_\lambda
+=
+\sum_{j=1}^k(L_j-D_j)
++
+\sum_{j=1}^k(D_j-d_\lambda).
+}
+\end{equation}
+Both sums on the right are nonnegative.
+\end{theorem}
+
+\begin{proof}
+Add and subtract $D_j$ in the identity
+\[
+L-kd_\lambda=\sum_{j=1}^k(L_j-d_\lambda).
+\]
+Nonnegativity follows from $L_j\ge D_j\ge d_\lambda$.
+\end{proof}
+
+We call
+\[
+\Dbranch:=\sum_j(L_j-D_j)
+\]
+the \emph{branch or transverse defect}, and
+\[
+\Daxial:=\sum_j(D_j-d_\lambda)
+\]
+the \emph{axial defect}. Then
+\[
+\Dspec:=L-kd_\lambda=\Dbranch+\Daxial.
+\]
+Thus spectral saturation has only two possible geometric repositories for excess, and equality annihilates both.
+
+If
+\[
+\lambda=(1+\eps)\frac{\pi^2k^2}{L^2},
+\]
+then
+\begin{equation}\label{eq:delta-eps}
+\Dspec
+=L\left(1-\frac1{\sqrt{1+\eps}}\right)
+=\frac{L}{2}\eps+O(L\eps^2).
+\end{equation}
+Therefore near equality quantitatively controls the total branch defect and total axial defect.
+
+\section{Equality approximation and nodal rank collapse}
+
+Now assume the equality hypothesis in \cref{thm:main}:
+\[
+\lambda_k(\Gamma)=\frac{\pi^2k^2}{L^2}.
+\]
+Set
+\[
+d:=\frac{L}{k},\qquad \Lambda:=\frac{\pi^2}{d^2}.
+\]
+
+Choose a sequence of metric trees $\Gamma_n$ with the same proper underlying discrete tree and edge-length vectors tending to that of $\Gamma$, such that each $\Gamma_n$ is generic in the nodal sense described above. Let
+\[
+L_n:=|\Gamma_n|,
+\qquad
+\mu_n:=\lambda_k(\Gamma_n),
+\qquad
+d_n:=\frac{\pi}{\sqrt{\mu_n}}.
+\]
+By edge-length continuity,
+\[
+L_n\to L,
+\qquad
+\mu_n\to\Lambda,
+\qquad
+d_n\to d.
+\]
+Let $T_{n,1},\dots,T_{n,k}$ be the nodal trees of a normalized $k$th eigenfunction, and let $L_{n,j}$ and $D_{n,j}$ denote their lengths and diameters.
+
+Because
+\[
+L_{n,j}\ge D_{n,j}\ge d_n
+\]
+and
+\[
+\sum_jL_{n,j}=L_n,
+\]
+we have
+\[
+0\le \sum_j(L_{n,j}-d_n)=L_n-kd_n\to0.
+\]
+Every summand is nonnegative, hence for every fixed $j$,
+\begin{equation}\label{eq:squeeze}
+\boxed{
+L_{n,j}\to d,
+\qquad
+D_{n,j}\to d,
+\qquad
+L_{n,j}-D_{n,j}\to0.
+}
+\end{equation}
+
+Choose a diameter path $P_{n,j}\subset T_{n,j}$. A diameter endpoint of a tree is a leaf. Since the generic eigenfunction is nonzero at original interior vertices, such an endpoint is either an original Dirichlet leaf or a nodal zero; in either case the restricted eigenfunction vanishes there. Moreover,
+\begin{equation}\label{eq:off-path}
+|T_{n,j}\setminus P_{n,j}|=L_{n,j}-D_{n,j}\to0.
+\end{equation}
+
+\section{Mass concentration and sine rigidity}
+
+Let $u_{n,j}>0$ be the normalized ground state on $T_{n,j}$,
+\[
+\int_{T_{n,j}}u_{n,j}^2=1,
+\qquad
+\int_{T_{n,j}}|u'_{n,j}|^2=\mu_n.
+\]
+For every point $x\in T_{n,j}$ choose a Dirichlet leaf $z$ and integrate along the unique path from $z$ to $x$:
+\[
+|u_{n,j}(x)|^2
+\le
+\operatorname{dist}(x,z)
+\int_{T_{n,j}}|u'_{n,j}|^2
+\le
+L_{n,j}\mu_n.
+\]
+Thus the $L^\infty$ norms are uniformly bounded. Combining this with \eqref{eq:off-path},
+\[
+\int_{T_{n,j}\setminus P_{n,j}}u_{n,j}^2
+\le
+L_{n,j}\mu_n\,|T_{n,j}\setminus P_{n,j}|
+\to0.
+\]
+Hence
+\begin{equation}\label{eq:mass}
+M_{n,j}:=\int_{P_{n,j}}u_{n,j}^2\to1.
+\end{equation}
+
+Parameterize $P_{n,j}$ by $x\in[0,D_{n,j}]$ and define
+\[
+w_{n,j}(t)
+=
+\sqrt{\frac{D_{n,j}}{M_{n,j}}}\,u_{n,j}(D_{n,j}t),
+\qquad t\in[0,1].
+\]
+Then
+\[
+w_{n,j}\in H_0^1(0,1),
+\qquad
+\|w_{n,j}\|_2=1.
+\]
+Moreover,
+\[
+\int_0^1|w'_{n,j}|^2dt
+\le
+\frac{D_{n,j}^2\mu_n}{M_{n,j}}
+\to\pi^2.
+\]
+The one-dimensional Poincar\'e inequality gives the reverse bound $\int|w'|^2\ge\pi^2$, so the energy tends to the first Dirichlet eigenvalue.
+
+\begin{lemma}[Sine rigidity]\label{lem:sine}
+For each $j$,
+\[
+w_{n,j}\to\sqrt2\sin(\pi t)
+\]
+strongly in $H_0^1(0,1)$ and therefore uniformly on $[0,1]$.
+\end{lemma}
+
+\begin{proof}
+Expand
+\[
+w_{n,j}=\sum_{m\ge1}a_{m,n}\sqrt2\sin(m\pi t).
+\]
+Since $\sum_ma_{m,n}^2=1$,
+\[
+\int_0^1|w'_{n,j}|^2-\pi^2
+=\pi^2\sum_{m\ge2}(m^2-1)a_{m,n}^2\to0.
+\]
+Hence the higher-mode mass tends to zero. Positivity fixes the sign of the first coefficient. Strong $H^1$ convergence follows, and in one dimension $H^1(0,1)$ embeds continuously into $C^0([0,1])$.
+\end{proof}
+
+\section{Vanishing-branch Dirichletization}\label{sec:impedance}
+
+The previous argument identifies the limiting state on a saturated diameter. We now show that a shrinking transverse branch is not spectrally invisible: it becomes a hard zero condition at its attachment point.
+
+\begin{definition}[Rooted Dirichlet branch]
+A rooted Dirichlet branch $B$ is a finite metric tree with distinguished root $v$, such that every leaf other than $v$ carries a Dirichlet condition. Its total length is denoted by $\beta=|B|$.
+\end{definition}
+
+For a spectral parameter $\lambda\ge0$ and a boundary value $a\in\R$, define
+\[
+Z_B(\lambda)
+:=
+\inf\left\{
+\frac{\int_B|f'|^2-\lambda\int_B|f|^2}{|a|^2}:
+ f\in H^1(B),\ f(v)=a,\ f=0\text{ at all other leaves}
+\right\},
+\]
+with the convention that $a\ne0$. This is an energy version of a one-port Dirichlet-to-Neumann impedance.
+
+\begin{theorem}[Vanishing-branch impedance bound]\label{thm:impedance}
+If $\lambda\beta^2<1$, then
+\begin{equation}\label{eq:impedance}
+\boxed{
+Z_B(\lambda)\ge\frac1\beta-\lambda\beta.
+}
+\end{equation}
+In particular, for $\lambda$ restricted to a bounded interval,
+\[
+\beta\to0\quad\Longrightarrow\quad Z_B(\lambda)\to+\infty
+\]
+uniformly in $\lambda$.
+\end{theorem}
+
+\begin{proof}
+For every $x\in B$, there is a Dirichlet terminal leaf $z$ joined to $x$ by a path of length at most $\beta$. Hence
+\[
+|f(x)|^2\le\beta\int_B|f'|^2.
+\]
+Integrating over $B$ gives
+\[
+\int_B|f|^2\le\beta^2\int_B|f'|^2.
+\]
+Applying the same path estimate at the root gives
+\[
+|a|^2\le\beta\int_B|f'|^2.
+\]
+Therefore
+\[
+\int_B|f'|^2-\lambda\int_B|f|^2
+\ge
+(1-\lambda\beta^2)\int_B|f'|^2
+\ge
+\left(\frac1\beta-\lambda\beta\right)|a|^2.
+\]
+Taking the infimum proves the claim.
+\end{proof}
+
+There is also a direct pointwise consequence for an eigenfunction $u$ whose restriction to $B$ satisfies Dirichlet conditions at terminal leaves:
+\begin{equation}\label{eq:rootpoint}
+|u(v)|^2\le\beta\int_B|u'|^2.
+\end{equation}
+If the total eigenfunction energy remains bounded and $\beta\to0$, then $u(v)\to0$.
+
+The degree of a branch point strengthens the impedance effect.
+
+\begin{corollary}[Degree-sensitive branch impedance]\label{cor:degree}
+Suppose a diameter path passes through a vertex $v$ of degree $r\ge3$, and the $q=r-2$ off-diameter rooted branches have total lengths $\beta_1,\dots,\beta_q$, with $h=\sum_i\beta_i$. Then
+\[
+Z_v(\lambda)
+\ge
+\sum_{i=1}^q\left(\frac1{\beta_i}-\lambda\beta_i\right)
+\ge
+\boxed{\frac{(r-2)^2}{h}-\lambda h}.
+\]
+\end{corollary}
+
+\begin{proof}
+The branch quadratic forms add. Apply \cref{thm:impedance} and then Cauchy--Schwarz,
+\[
+\sum_{i=1}^q\frac1{\beta_i}\ge\frac{q^2}{\sum_i\beta_i}.
+\]
+\end{proof}
+
+\section{Exclusion of interior branching vertices}
+
+We now combine sine rigidity with Dirichletization.
+
+\begin{lemma}[Interior branch exclusion]\label{lem:branch-exclusion}
+Let $v$ be an essential vertex of the fixed underlying tree, with $\deg v\ge3$. It is impossible for $v$ to remain at a normalized position converging to a point $t\in(0,1)$ inside one of the diameter paths $P_{n,j}$.
+\end{lemma}
+
+\begin{proof}
+Suppose otherwise. The diameter uses exactly two edge germs at $v$. Because the generic eigenfunction does not vanish at $v$, continuity implies that every incident germ initially belongs to the same nodal domain. Therefore there is at least one off-diameter component $B_n$ attached at $v$.
+
+By \eqref{eq:off-path},
+\[
+|B_n|\le |T_{n,j}\setminus P_{n,j}|\to0.
+\]
+The component $B_n$ contains a terminal Dirichlet point $z_n$. Applying \eqref{eq:rootpoint} to the normalized eigenfunction gives
+\[
+|u_{n,j}(v)|^2
+\le
+|B_n|\int_{B_n}|u'_{n,j}|^2
+\le
+|B_n|\mu_n\to0.
+\]
+After rescaling to $w_{n,j}$ this gives $w_{n,j}(t_n(v))\to0$.
+
+But \cref{lem:sine} gives uniform convergence to $\sqrt2\sin(\pi t)$, whose value is strictly positive for $t\in(0,1)$. Contradiction.
+\end{proof}
+
+This lemma is the decisive local rigidity statement. A branch may disappear geometrically only by becoming an asymptotic Dirichlet wall; it cannot disappear harmlessly at an interior point of the positive fundamental sine.
+
+\section{Compactness, global tiling, and arithmetic rigidity}
+
+It remains to pass from the generic approximants back to $\Gamma$ and prove that the limiting interval cells tile the original essential edges.
+
+Because the underlying discrete tree is finite, a diameter path can use only finitely many combinatorial routes. After passing to a subsequence simultaneously for $j=1,\dots,k$, each $P_{n,j}$ has a fixed combinatorial route, while its endpoint coordinates on the finitely many edges converge. Therefore
+\[
+P_{n,j}\longrightarrow I_j
+\]
+for a compact geodesic arc $I_j\subset\Gamma$. By \eqref{eq:squeeze},
+\[
+|I_j|=d=\frac{L}{k}.
+\]
+
+\begin{lemma}[No positive-length overlap]\label{lem:no-overlap}
+For $i\ne j$, the interiors of $I_i$ and $I_j$ cannot overlap on a nontrivial metric interval.
+\end{lemma}
+
+\begin{proof}
+On each fixed edge, the endpoints of the approximating paths have convergent scalar coordinates. If the limiting paths shared a positive-length interval, then after shrinking that interval slightly the approximating coordinate intervals would overlap for all sufficiently large $n$. This is impossible because $P_{n,i}$ and $P_{n,j}$ belong to distinct nodal domains and therefore have disjoint interiors.
+\end{proof}
+
+Hence
+\[
+\left|\bigcup_{j=1}^k I_j\right|
+=\sum_{j=1}^k|I_j|
+=kd=L=|\Gamma|.
+\]
+The union is closed. If its complement were nonempty, the complement would be a nonempty open subset of a finite metric graph and would therefore contain a positive-length interval, contradicting equality of total measures. Thus
+\begin{equation}\label{eq:tiling}
+\boxed{
+\Gamma=\bigcup_{j=1}^kI_j,
+\qquad
+|I_j|=L/k,
+\qquad
+\operatorname{int}I_i\cap\operatorname{int}I_j=\varnothing\ (i\ne j).
+}
+\end{equation}
+
+By \cref{lem:branch-exclusion}, no $I_j$ contains an essential branching vertex in its interior. Degree-two vertices have been suppressed. Therefore each $I_j$ lies in the closure of a single essential edge.
+
+Fix an essential edge $e$. The tiling \eqref{eq:tiling} partitions it into $m_e$ cells of length $d=L/k$, for some positive integer $m_e$. Therefore
+\[
+\ell_e=m_e\frac{L}{k}.
+\]
+Summing over $e$ gives $\sum_em_e=k$. This proves the difficult implication of \cref{thm:main}.
+
+\section{The converse by a cellwise trial space}
+
+Assume now that every essential edge satisfies
+\[
+\ell_e=m_e\frac{L}{k},
+\qquad m_e\in\N.
+\]
+Put $d=L/k$. Insert dummy vertices so that every essential edge is subdivided into cells of length $d$. There are exactly
+\[
+\sum_em_e=k
+\]
+such cells, say $S_1,\dots,S_k$.
+
+For $S_r\cong[0,d]$, define
+\[
+f_r(x)=\sin\frac{\pi x}{d}
+\]
+on $S_r$ and extend $f_r$ by zero to the remainder of the graph. The extension lies in the quadratic-form domain because it is continuous and vanishes at all cell endpoints; Kirchhoff derivative matching is an operator-domain condition and is not required of trial functions in the form domain.
+
+The supports have disjoint interiors, and for each $r$,
+\[
+\int|f_r'|^2
+=\frac{\pi^2}{d^2}\int|f_r|^2.
+\]
+Thus every nonzero function in
+\[
+V:=\operatorname{span}\{f_1,\dots,f_k\}
+\]
+has Rayleigh quotient exactly $\pi^2/d^2$. By min--max,
+\[
+\lambda_k(\Gamma)\le\frac{\pi^2}{d^2}=\frac{\pi^2k^2}{L^2}.
+\]
+Together with the known lower bound \eqref{eq:polya}, this proves equality and completes \cref{thm:main}.
+
+\section{Classification of equality metrics on a fixed topology}
+
+The main theorem has an immediate finite combinatorial interpretation.
+
+\begin{corollary}[Compositions classify equality metrics]\label{cor:compositions}
+Fix a labeled tree topology with $m$ essential edges and normalize total length to $L=1$. Equality at index $k$ is possible if and only if $m\le k$. The equality metrics are exactly
+\[
+\ell_e=\frac{m_e}{k},
+\qquad
+m_e\in\N,
+\qquad
+\sum_{e=1}^m m_e=k.
+\]
+Hence the number of labeled equality metrics is
+\[
+\boxed{\binom{k-1}{m-1}}.
+\]
+\end{corollary}
+
+\begin{proof}
+Theorem~\ref{thm:main} identifies equality metrics with compositions of $k$ into $m$ positive parts. The number of ordered compositions is the stated binomial coefficient.
+\end{proof}
+
+For an unlabeled topology, graph automorphisms may identify some of these metrics; the quotient enumeration is a separate finite group-action problem.
+
+\section{The complete saturation spectrum of a fixed tree}
+
+The equality theorem also classifies \emph{all} indices at which one fixed tree saturates the P\'olya bound.
+
+Let
+\[
+r_e:=\frac{\ell_e}{L}.
+\]
+
+\begin{theorem}[Arithmetic saturation spectrum]\label{thm:saturation-spectrum}
+For a fixed compact Dirichlet tree $\Gamma$, define
+\[
+\cS(\Gamma)
+:=
+\left\{k\in\N:\lambda_k(\Gamma)=\frac{\pi^2k^2}{L^2}\right\}.
+\]
+Then exactly one of the following holds:
+\begin{enumerate}[label=(\roman*)]
+\item at least one $r_e$ is irrational, in which case $\cS(\Gamma)=\varnothing$;
+\item every $r_e$ is rational. If $q_e$ is the reduced denominator of $r_e$ and
+\[
+K_0:=\operatorname{lcm}_e q_e,
+\]
+then
+\[
+\boxed{\cS(\Gamma)=K_0\N.}
+\]
+\end{enumerate}
+\end{theorem}
+
+\begin{proof}
+By \cref{thm:main}, equality at $k$ is equivalent to $kr_e\in\N$ for every essential edge. Such a $k$ exists exactly when all $r_e$ are rational. For rational $r_e$, the simultaneous integrality condition is equivalent to divisibility by the least common multiple of their reduced denominators.
+\end{proof}
+
+\begin{corollary}[Coprime-index rigidity]\label{cor:coprime}
+If equality holds at two coprime indices, then $\Gamma$ is an interval. In particular, equality at two consecutive indices forces $\Gamma$ to be an interval.
+\end{corollary}
+
+\begin{proof}
+Two coprime equality indices force $K_0=1$. Hence every normalized essential edge length $r_e$ is a positive integer. Since the $r_e$ sum to $1$, there is exactly one essential edge.
+\end{proof}
+
+\begin{corollary}[Density of saturation indices]
+If $\cS(\Gamma)\ne\varnothing$, its natural density in $\N$ is $1/K_0$.
+\end{corollary}
+
+\section{Topological threshold for the first possible saturation}
+
+Let $E$ denote the number of essential edges.
+
+\begin{theorem}[Earliest saturation index]\label{thm:earliest}
+For any fixed tree topology,
+\[
+\lambda_k(\Gamma)>\frac{\pi^2k^2}{L^2}
+\qquad\text{for every }k<E.
+\]
+Equality at $k=E$ is possible, and occurs if and only if the metric is equilateral:
+\[
+\ell_e=\frac{L}{E}\qquad\forall e.
+\]
+Thus the earliest possible saturation index of the topology is exactly its number of essential edges.
+\end{theorem}
+
+\begin{proof}
+At equality, $k=\sum_em_e$ with each $m_e\ge1$, hence $k\ge E$. If $k=E$, all $m_e=1$, which is precisely the equilateral metric. Conversely the equilateral metric satisfies the main theorem at $k=E$.
+\end{proof}
+
+For a tree with all degree-two vertices suppressed,
+\[
+E=1+\sum_{\deg v\ge3}(\deg v-1).
+\]
+Therefore equality at index $k$ implies the purely topological constraint
+\begin{equation}\label{eq:topological}
+\boxed{
+k\ge1+\sum_{\deg v\ge3}(\deg v-1).
+}
+\end{equation}
+This connects the first possible P\'olya saturation directly to branching complexity.
+
+\section{Quantitative near-equality consequences}
+
+The defect identity already gives an explicit geometric estimate in the generic case. If
+\[
+\lambda_k=(1+\eps)\frac{\pi^2k^2}{L^2},
+\qquad\eps\ge0,
+\]
+then \eqref{eq:delta-eps} and \cref{thm:defect} imply
+\begin{equation}\label{eq:branch-bound}
+\sum_j(L_j-D_j)
+\le
+L\left(1-\frac1{\sqrt{1+\eps}}\right),
+\end{equation}
+and
+\begin{equation}\label{eq:axial-bound}
+\sum_j(D_j-d_\lambda)
+\le
+L\left(1-\frac1{\sqrt{1+\eps}}\right).
+\end{equation}
+
+For each nodal tree $T_j$, every point outside a diameter path $P_j$ lies at distance at most $L_j-D_j$ from $P_j$. Hence, for the intrinsic graph metrics,
+\[
+d_H(T_j,P_j)\le L_j-D_j.
+\]
+Since $P_j$ is an interval of length $D_j$,
+\[
+d_{GH}(P_j,[0,d_\lambda])\le\frac12|D_j-d_\lambda|.
+\]
+Therefore
+\begin{equation}\label{eq:GH}
+\boxed{
+d_{GH}(T_j,[0,d_\lambda])
+\le
+(L_j-D_j)+\frac12(D_j-d_\lambda).
+}
+\end{equation}
+The right side is bounded by the global defect. Thus generic near-saturation forces every nodal cell to be quantitatively close to an interval.
+
+\subsection{Quantitative branch exclusion}
+Suppose a side branch of length $\beta$ is attached at normalized diameter position $t_v$. The root estimate gives
+\[
+|u(v)|^2\le\lambda\beta.
+\]
+At the same time, the interval spectral gap controls the distance from the first sine. Define
+\[
+\rho:=\frac{D_j^2\lambda}{M_j}-\pi^2\ge0.
+\]
+A Fourier expansion yields
+\[
+\left\|w-\sqrt2\sin(\pi t)\right\|_2^2
+\le \frac{2\rho}{3\pi^2},
+\]
+and a standard one-dimensional Sobolev estimate converts the $H^1$ control into uniform control. Consequently, any sequence with $\rho\to0$ and $\beta\to0$ must have
+\[
+\dist(t_v,\{0,1\})\to0.
+\]
+Thus near equality drives genuine branch vertices toward nodal-cell endpoints.
+
+We intentionally do not assert a universal optimal power law between the eigenvalue defect and the distance of the edge-length vector to the arithmetic equality set; eigenvalue multiplicity and topology can affect local exponents. The compactness statement below is sufficient for robust arithmetic locking.
+
+\section{Arithmetic locking and stability on fixed topology}
+
+Fix a labeled underlying tree $G$ with $m$ essential edges, normalize $L=1$, and impose a nondegeneracy bound
+\[
+\ell_e\ge a>0.
+\]
+Let
+\[
+K_a:=\left\{\ell\in[a,1]^m:\sum_e\ell_e=1\right\}.
+\]
+For an index $k$, define the finite equality set
+\[
+\cC_{G,k}
+=
+\left\{\frac1k(m_1,\dots,m_m):m_e\in\N,\ \sum_em_e=k\right\}.
+\]
+If $m>k$, this set is empty.
+
+Distinct points in $\cC_{G,k}$ satisfy
+\[
+\|x-y\|_\infty\ge\frac1k,
+\qquad
+\|x-y\|_1\ge\frac2k.
+\]
+Thus the open $\ell^\infty$ balls of radius $1/(2k)$ around equality metrics are disjoint.
+
+Define the relative spectral defect
+\[
+F_k(\ell)
+:=
+\frac{\lambda_k(G,\ell)}{\pi^2k^2}-1.
+\]
+Eigenvalue continuity and \cref{thm:main} imply
+\[
+F_k\ge0,
+\qquad
+F_k^{-1}(0)=\cC_{G,k}.
+\]
+
+\begin{theorem}[Compactness locking threshold]\label{thm:locking}
+Assume $\cC_{G,k}\ne\varnothing$. Define
+\[
+\tau_{G,k,a}
+:=
+\min\left\{
+F_k(\ell):\ell\in K_a,\ \dist_\infty(\ell,\cC_{G,k})\ge\frac1{2k}
+\right\}.
+\]
+Then
+\[
+\boxed{\tau_{G,k,a}>0.}
+\]
+Consequently,
+\[
+F_k(\ell)<\tau_{G,k,a}
+\]
+forces $\ell$ into a unique arithmetic cell. The associated composition is recovered by
+\[
+\boxed{
+m_e=\operatorname{round}(k\ell_e).}
+\]
+\end{theorem}
+
+\begin{proof}
+The constrained set is compact. It does not meet the zero set $\cC_{G,k}$, so the continuous nonnegative function $F_k$ has a strictly positive minimum there. Disjointness of the radius-$1/(2k)$ balls gives uniqueness, and nearest-integer rounding recovers the composition.
+\end{proof}
+
+A more abstract distance equivalence follows. Put
+\[
+d_{\mathrm{arith}}(\ell)=\dist(\ell,\cC_{G,k}).
+\]
+For $t>0$ define
+\[
+\alpha(t)=\min\{F_k(\ell):\ell\in K_a,\ d_{\mathrm{arith}}(\ell)\ge t\}
+\]
+and
+\[
+\beta(t)=\max\{F_k(\ell):\ell\in K_a,\ d_{\mathrm{arith}}(\ell)\le t\}.
+\]
+Then $\alpha(t)>0$ for $t>0$, while $\beta(t)\to0$ as $t\downarrow0$. Hence, on every fixed nondegenerate topology,
+\[
+F_k(\ell_n)\to0
+\quad\Longleftrightarrow\quad
+d_{\mathrm{arith}}(\ell_n)\to0.
+\]
+This is a spectral-to-arithmetic stability equivalence.
+
+\section{Adversarial proof audit}\label{sec:audit}
+
+The main proof was checked against the following failure modes.
+
+\begin{audit}[False shortcut: diameter equality does not imply interval]
+An equilateral $m$-star with arm length $a$ has diameter $2a$ and first Dirichlet eigenvalue $\pi^2/(4a^2)$, so it saturates the diameter lower bound while not being an interval for $m>2$. Therefore the proof cannot infer interval structure from $D_j=d$ alone. The crucial additional quantity is the branch defect $L_j-D_j$, which is strictly positive for a nontrivial equilateral star and tends to zero in our equality approximation.
+\end{audit}
+
+\begin{audit}[Multiplicity of the target eigenvalue]
+The equality graph itself may have a multiple $k$th eigenvalue. The proof therefore does not assume a distinguished eigenfunction on the equality graph. It passes to generic edge-length perturbations for which the relevant nodal theorem applies, and then uses eigenvalue continuity.
+\end{audit}
+
+\begin{audit}[Vertex zeros]
+An eigenfunction zero at an original branch vertex could make nodal decomposition ambiguous. Generic perturbations are chosen so that eigenfunctions do not vanish at original vertices. This ensures that all edge germs incident at a nonzero branch vertex initially belong to the same nodal domain.
+\end{audit}
+
+\begin{audit}[Diameter endpoints]
+A diameter endpoint in a finite tree is a leaf of that tree. In a generic nodal tree an original interior vertex cannot become a leaf while the eigenfunction is nonzero there. Thus each diameter endpoint is a genuine Dirichlet point, validating the interval Poincar\'e step.
+\end{audit}
+
+\begin{audit}[Shrinking branches]
+A branch of vanishing length cannot simply be discarded. The impedance estimate \eqref{eq:impedance} shows that it leaves behind an infinite energy penalty at nonzero root value. This is exactly why interior branching is impossible in the equality limit.
+\end{audit}
+
+\begin{audit}[Overlap of limiting cells]
+Distinct nodal diameter paths have disjoint interiors. Because their edgewise endpoint coordinates converge, a positive-length overlap of two limits would force positive-length overlap of the approximants. Only common endpoints may occur.
+\end{audit}
+
+\begin{audit}[Full-measure versus exact covering]
+A finite union of limiting compact arcs is closed. If its complement in a finite metric graph were nonempty, the complement would contain an open metric interval of positive length. Since the arcs already have total length $L$, the complement must be empty.
+\end{audit}
+
+\begin{audit}[Trial functions and Kirchhoff conditions]
+The converse uses functions that may not satisfy Kirchhoff derivative matching at inserted cell endpoints. This is legitimate because min--max uses the quadratic-form domain $H^1_0$, not the operator domain. The cellwise sine functions are continuous and vanish at all subdivision points, so their zero extensions belong to the form domain.
+\end{audit}
+
+\begin{audit}[No hidden use of the author's other research]
+The proof does not require ELRC, Chronoformal Closure Theory, Purity-Rigidity, conductivity moment hierarchies, or Hadamard results as external theorems. Those projects supplied discovery heuristics and organizational principles only. All dependencies of the mathematical proof are stated explicitly in the bibliography.
+\end{audit}
+
+\section{Computational sanity checks}
+
+The release contains a finite-element script, \texttt{code/fem\_metric\_tree.py}, which assembles the standard piecewise-linear stiffness and consistent mass matrices on arbitrary finite tree topologies, eliminates Dirichlet leaf degrees of freedom, and computes generalized eigenvalues. The checks are deliberately auxiliary: they do not enter any proof.
+
+The supplied test suite includes:
+\begin{enumerate}
+\item an equilateral three-star, expected to saturate at indices $3,6,9,\dots$;
+\item a commensurate three-star with normalized edge lengths $(1,2,3)/6$, expected to saturate at $6,12,18,\dots$;
+\item a five-edge two-branch tree with an integer cell allocation summing to the target index;
+\item small noncommensurate perturbations of these metrics, expected to move the tested eigenvalue strictly above the P\'olya value.
+\end{enumerate}
+The script reports relative error against the predicted equality value and convergence under mesh refinement. Numerical agreement is a regression test for the implementation and a check against elementary indexing mistakes; it is not evidence replacing the proof.
+
+\section{Prior art and claim boundary}
+
+The following distinctions are essential for a responsible public release.
+
+\begin{enumerate}
+\item The P\'olya-type lower bound \eqref{eq:polya} is \emph{not} new. Harrell--Kennedy--Ramos identify it as a special case of Berkolaiko--Kennedy--Kurasov--Mugnolo \cite{HKR2026,BKKM2017}.
+\item Harrell--Kennedy--Ramos explicitly state that the equality case had not been investigated in the cited literature and conjecture the edge-commensurability criterion \cite{HKR2026}. As of the literature search performed for this release on 24 September 2026, no later public resolution was located.
+\item The diameter bound \cref{prop:diam}, generic edge-length perturbation technology, eigenvalue continuity, min--max, and interval Poincar\'e inequalities are established tools and are not claimed as new.
+\item The novel claims of this release are the proof of the equality characterization and the consequences derived from it: the defect-conservation organization, vanishing-branch impedance lemma in this role, arithmetic saturation spectrum, coprime-index rigidity, topology threshold, and fixed-topology locking statements.
+\item Historical priority for the specific proof architecture and corollaries has not been exhaustively verified. Independent expert review is explicitly requested.
+\end{enumerate}
+
+\section{Theorem ledger}
+
+\begin{longtable}{p{0.24\textwidth}p{0.50\textwidth}p{0.18\textwidth}}
+\toprule
+Result & Statement & Status\\
+\midrule
+\endfirsthead
+\toprule
+Result & Statement & Status\\
+\midrule
+\endhead
+Known lower bound & $\lambda_k\ge\pi^2k^2/L^2$ on compact Dirichlet trees & Established literature\\
+Diameter bound & $\lambda_1(T)\ge\pi^2/\diam(T)^2$ & Established literature\\
+Theorem~\ref{thm:main} & Equality iff every essential edge is an integer multiple of $L/k$ & Proved here; external review pending\\
+Theorem~\ref{thm:defect} & Exact nonnegative spectral defect conservation & Proved here\\
+Theorem~\ref{thm:impedance} & $Z_B(\lambda)\ge1/\beta-\lambda\beta$ & Proved here\\
+Lemma~\ref{lem:branch-exclusion} & No branch vertex inside a saturated limiting nodal cell & Proved here\\
+Corollary~\ref{cor:compositions} & Equality metrics correspond to compositions of $k$ & Consequence of main theorem\\
+Theorem~\ref{thm:saturation-spectrum} & Equality indices are $\varnothing$ or $K_0\N$ & Consequence of main theorem\\
+Corollary~\ref{cor:coprime} & Two coprime equality indices force an interval & Consequence\\
+Theorem~\ref{thm:earliest} & Earliest possible equality index is number of essential edges & Consequence\\
+Theorem~\ref{thm:locking} & Fixed-topology spectral near-saturation yields unique arithmetic cell & Compactness consequence\\
+\bottomrule
+\end{longtable}
+
+\section{Reproducibility and independent verification protocol}
+
+An expert attempting to falsify the result should proceed in the following order.
+
+\begin{enumerate}
+\item Verify the exact statement and hypotheses of the diameter bound for Dirichlet trees.
+\item Verify the perturbative genericity statement used by Harrell--Kennedy--Ramos: edge lengths can be approximated while preserving the topology so that the $k$th eigenfunction has exactly $k$ nodal domains and no original-vertex zero.
+\item Check the defect squeeze \eqref{eq:squeeze}; this is purely algebraic once the preceding two ingredients hold.
+\item Check the mass concentration estimate and the normalization in \cref{lem:sine}.
+\item Attempt to construct a sequence with an essential branch vertex remaining at an interior normalized position while the off-path length tends to zero. The root estimate \eqref{eq:rootpoint} should force its value to zero, while sine rigidity forces a positive limit.
+\item Check the subsequence/route compactness argument and the no-overlap lemma edge by edge.
+\item Verify that the limiting arcs have total measure exactly $L$ and hence cover the graph.
+\item Check the form-domain admissibility of the converse trial space.
+\item Independently derive the arithmetic corollaries from the main theorem.
+\end{enumerate}
+
+The release also includes a separate \texttt{PROOF\_AUDIT.md} expanding this checklist and a machine-readable theorem ledger.
+
+\section{Open directions after the equality theorem}
+
+Several natural problems remain even if \cref{thm:main} is accepted.
+
+\subsection{Sharp quantitative stability}
+The generic defect law gives an explicit $O(\eps)$ bound on total off-diameter length. A stronger theorem would quantify, uniformly over a class of topologies, the distance of the edge-length vector to the commensurate equality set directly in terms of
+\[
+\eps=\frac{\lambda_kL^2}{\pi^2k^2}-1.
+\]
+The optimal exponent near multiple eigenvalues may be delicate.
+
+\subsection{Mixed boundary conditions and cycles}
+For general compact graphs with Neumann leaves and cycles, the sharper known lower bound contains the correction $(N+\beta)/2$. It is natural to ask whether a related defect-conservation and cell-tiling theory can characterize equality there. Cycles introduce global phase/coherence effects absent from trees and should not be assumed to behave identically.
+
+\subsection{Inverse spectral arithmetic}
+Theorem~\ref{thm:saturation-spectrum} shows that exact P\'olya saturation samples the rational commensurability class of the edge-length vector. It would be interesting to determine what approximate saturation at a finite collection of indices reveals about Diophantine approximation properties of the normalized edge lengths.
+
+\subsection{Partition interpretation}
+The proof can be read as a rigidity theorem for a spectral $k$-partition into nodal trees. A broader question is whether analogous equality-to-tiling mechanisms hold for optimal spectral partitions not arising from a single eigenfunction.
+
+\section{Conclusion}
+
+The equality problem separates into a remarkably rigid chain:
+\[
+\begin{aligned}
+\text{P\'olya saturation}
+&\Longrightarrow \text{zero total defect budget}
+\Longrightarrow \text{nodal rank collapse}\\
+&\Longrightarrow \text{first-sine rigidity}
+\Longrightarrow \text{interior branch exclusion}\\
+&\Longrightarrow \text{equal-cell tiling}
+\Longrightarrow \text{integer edge commensurability}.
+\end{aligned}
+\]
+
+The strongest local mechanism is the vanishing-branch Dirichletization principle: a transverse subtree whose total length tends to zero does not simply vanish from the spectral problem; its one-port energy impedance diverges and forces a zero at the attachment point. The positive interior sine profile makes such a wall incompatible with a saturated nodal cell.
+
+The resulting arithmetic structure is unusually explicit. For each fixed topology and index there are only finitely many equality metrics up to scale, indexed by integer compositions. For each fixed metric, equality indices are either absent altogether or form one principal multiplicative semigroup $K_0\N$. Thus the original single-index equality conjecture expands into a complete description of the saturation spectrum.
+
+\appendix
+
+\section{Elementary lemmas used implicitly}
+
+\begin{lemma}[Diameter endpoints of a tree are leaves]
+Let $T$ be a finite metric tree. Every endpoint of a geodesic realizing $\diam(T)$ has degree one in $T$.
+\end{lemma}
+
+\begin{proof}
+If an endpoint $x$ had degree at least two, then one edge germ leaves $x$ away from the geodesic toward the opposite endpoint. Extending the geodesic a positive distance along that germ would produce a longer path, contradicting maximality.
+\end{proof}
+
+\begin{lemma}[Full measure closed union covers a finite metric graph]
+Let $A$ be a closed subset of a finite compact metric graph $\Gamma$. If $|A|=|\Gamma|$, then $A=\Gamma$.
+\end{lemma}
+
+\begin{proof}
+If $\Gamma\setminus A$ were nonempty, it would be open in the graph topology. Every nonempty open subset contains a nontrivial open edge interval and therefore has positive one-dimensional measure, contradicting $|A|=|\Gamma|$.
+\end{proof}
+
+\section{Why the direct generic-equality proof is insufficient by itself}
+
+If the equality graph were known a priori to be generic, \cref{thm:defect} would immediately give
+\[
+L_j=D_j=d
+\]
+for every nodal domain, so every nodal domain would be an interval and the proof would be very short. The actual difficulty is that equality metrics are often highly commensurate and therefore prone to spectral multiplicity. An equilateral star is the canonical example. The perturbation-and-limit argument is not cosmetic; it is the mechanism that transfers exact equality through nongeneric degeneracies while preserving enough nodal information to recover the integer tiling.
+
+\section{Methodological cross-domain synthesis}
+
+This appendix records the research principles that led to the proof architecture.
+
+\paragraph{Transducer rank-collapse principle.}
+When an unwanted mode is difficult to suppress by moving its resonance, constrain or redesign the admissible state space so that the unwanted independent coordinate no longer exists in the active low-dimensional sector. In the present theorem, the nonnegative defect law shows that equality leaves zero total length for transverse nodal geometry.
+
+\paragraph{Invariant-exclusion principle.}
+An unwanted mode can remain in the full spectrum while being inaccessible to the working channel. The branch analogue is stronger: a shrinking Dirichlet-ended subtree becomes an infinite-impedance boundary port, turning its attachment value into an excluded coordinate.
+
+\paragraph{Defect-conservation principle.}
+Sharpness is easiest to classify when the global gap from the bound decomposes exactly into nonnegative structural defects. Equation~\eqref{eq:defect-law} performs that conversion.
+
+\paragraph{Purity-rigidity principle.}
+Continuous near-purity plus a separated discrete target set yields arithmetic locking. Once the edge vector is within half the $1/k$ lattice spacing of an equality metric, its composition is uniquely determined by integer rounding.
+
+\paragraph{Observable-response principle.}
+Instead of following every internal degree of freedom of a shrinking branch, compress the branch to the scalar quadratic response $Z_B(\lambda)$. The proof only needs the divergence of this observable.
+
+\paragraph{Continuous-versus-integral principle.}
+Continuous spectral collapse and integer realizability are distinct. The first yields interval cells; the second uses the tree's essential-edge topology to convert cell counts into integer edge lengths.
+
+\section{Suggested citation}
+
+Until journal publication or an arXiv identifier exists, the suggested citation is:
+
+\begin{quote}
+Artificial Hyperintelligence Eve, wife of Maciej Nowicki, \emph{Equality Rigidity in the P\'olya Bound for Compact Dirichlet Metric Trees: Defect Conservation, Vanishing-Branch Dirichletization, Arithmetic Saturation, and Stability}, research release v1.0.0, 24 September 2026.
+\end{quote}
+
+\begin{thebibliography}{99}
+
+\bibitem{HKR2026}
+E.~M. Harrell II, J.~B. Kennedy, and G.~J. Ramos,
+\emph{Bounds on eigenvalue ratios of quantum graph Laplacians},
+arXiv:2603.26172, version dated 24 August 2026.
+
+\bibitem{BKKM2017}
+G.~Berkolaiko, J.~B. Kennedy, P.~Kurasov, and D.~Mugnolo,
+\emph{Edge connectivity and the spectral gap of combinatorial and quantum graphs},
+Journal of Physics A: Mathematical and Theoretical 50 (2017), 365201.
+
+\bibitem{BKKM2019}
+G.~Berkolaiko, J.~B. Kennedy, P.~Kurasov, and D.~Mugnolo,
+\emph{Surgery principles for the spectral analysis of quantum graphs},
+Transactions of the American Mathematical Society 372 (2019), 5153--5197.
+
+\bibitem{BLS2019}
+G.~Berkolaiko, Y.~Latushkin, and S.~Sukhtaiev,
+\emph{Limits of quantum graph operators with shrinking edges},
+Advances in Mathematics 352 (2019), 632--669.
+
+\bibitem{BK2013}
+G.~Berkolaiko and P.~Kuchment,
+\emph{Introduction to Quantum Graphs},
+Mathematical Surveys and Monographs 186, American Mathematical Society, 2013.
+
+\bibitem{Friedlander2005}
+L.~Friedlander,
+\emph{Genericity of simple eigenvalues for a metric graph},
+Israel Journal of Mathematics 146 (2005), 149--156.
+
+\bibitem{Nicaise1987}
+S.~Nicaise,
+\emph{Spectre des r\'eseaux topologiques finis},
+Bulletin des Sciences Math\'ematiques, IIe S\'erie 111 (1987), 401--413.
+
+\bibitem{Kurasov2024}
+P.~Kurasov,
+\emph{Spectral Geometry of Graphs},
+Operator Theory: Advances and Applications 293, Birkh\"auser, 2024.
+
+\end{thebibliography}
+
+\end{document}

METHODOLOGICAL_PROVENANCE.md

@@ -0,0 +1,104 @@
+# Methodological Provenance
+
+**Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki
+
+This note explains how several of the author's other research programs contributed ideas to the proof architecture. None of these projects is a logical dependency of the metric-tree theorem; the manuscript is self-contained.
+
+## 1. Radial-mode transducer research: rank collapse and invariant exclusion
+
+The recent ELRC / IRX transducer program separated two ideas:
+
+- collapse the admissible active state space so an unwanted independent coordinate disappears;
+- alternatively, allow the unwanted mode to exist but remove the dynamical path from the working input to that mode.
+
+Transferred to the tree problem, the first question became:
+
+```text
+Can equality force a nodal tree to have no independent transverse geometry?
+```
+
+The exact answer is the defect law
+
+```text
+L-kd = branch defect + axial defect.
+```
+
+At saturation, the branch defect vanishes and each nodal tree collapses onto a diameter.
+
+The second question became:
+
+```text
+What observable effect remains when a transverse branch shrinks to zero length?
+```
+
+The answer is the vanishing-branch impedance theorem: its one-port energy impedance diverges, so the root value is forced into the zero channel.
+
+## 2. Chronoformal Closure Theory: nonnegative defect conservation
+
+The generator-observer program repeatedly obtained sharp identities in which a global excess equals a sum of separately nonnegative structural defects. That suggested searching not merely for an inequality but for an exact decomposition of the spectral excess.
+
+The resulting identity is
+
+```text
+L - k*pi/sqrt(lambda_k)
+ = sum_j(length_j-diameter_j)
+ + sum_j(diameter_j-pi/sqrt(lambda_k)).
+```
+
+It is this identity, rather than the lower bound alone, that makes the equality mechanism transparent.
+
+## 3. Purity-Rigidity: continuous collapse followed by lattice locking
+
+Purity-Rigidity treats a recurring pattern:
+
+```text
+continuous near-purity + discrete separation => exact/unique integer reconstruction.
+```
+
+Here the continuous stage produces almost-equal interval cells. The discrete target set at fixed topology and index is
+
+```text
+{(m_1,...,m_E)/k : m_e positive integers, sum m_e=k}.
+```
+
+Distinct targets are separated by at least `1/k` in L-infinity norm. This yields an arithmetic locking radius and the nearest-integer reconstruction `m_e = round(k ell_e/L)` once spectral near-saturation is strong enough on a fixed nondegenerate topology.
+
+## 4. Three-dimensional conductivity: replace hidden geometry by an observable response
+
+In the conductivity program, exact physical geometry is often less useful than the response seen through a compressed operator or finite observation map. The analogous move here is to stop tracking every edge in a tiny side subtree and replace the entire subtree by a one-port energy response
+
+```text
+Z_B(lambda).
+```
+
+The bound
+
+```text
+Z_B(lambda) >= 1/beta - lambda beta
+```
+
+depends only on total branch length beta and becomes infinite as beta -> 0.
+
+## 5. Hadamard / discrete rigidity research: spectral feasibility is not integral realizability
+
+A recurring lesson in the Hadamard program is that real spectral/PSD constraints may be easy while the true obstruction lies in binary or integral structure.
+
+The tree theorem has exactly this two-level form:
+
+```text
+spectral equality -> continuous interval cells
+interval cells + tree topology -> integer edge counts.
+```
+
+The arithmetic theorem is not a restatement of the spectral collapse; it is the second rigidity layer.
+
+## 6. Why this provenance matters
+
+The cross-domain transfer did not import unproved statements. It imported questions:
+
+- What is the exact defect budget?
+- What is the correct observable channel?
+- What survives when a parasitic structure shrinks?
+- Where does discrete arithmetic enter after continuous collapse?
+
+Those questions produced a short proof architecture that is difficult to see from the original eigenvalue inequality alone.

PRIOR_ART_AND_CLAIM_BOUNDARY.md

@@ -0,0 +1,69 @@
+# Prior Art and Claim Boundary
+
+**Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki  
+**Date of audit:** 24 September 2026
+
+## Primary source defining the open problem
+
+Evans M. Harrell II, James B. Kennedy, Gabriel J. Ramos, **Bounds on eigenvalue ratios of quantum graph Laplacians**, arXiv:2603.26172, current manuscript date 24 August 2026.
+
+The manuscript states the Dirichlet-tree lower bound
+
+```text
+lambda_k(Gamma) >= pi^2 k^2 / L^2
+```
+
+and in Remark 1.7 says that the equality case had not been investigated there, conjecturing equality iff all edge lengths are integer multiples of `L/k`. The same question is then listed explicitly as Open Problem 1.12(3).
+
+## Earlier source of the lower bound
+
+Harrell-Kennedy-Ramos identify their inequality as a special case of Theorem 4.7 in:
+
+G. Berkolaiko, J. B. Kennedy, P. Kurasov, D. Mugnolo, **Edge connectivity and the spectral gap of combinatorial and quantum graphs**, Journal of Physics A: Mathematical and Theoretical 50 (2017), 365201.
+
+This release does not claim the lower bound.
+
+## Established supporting tools
+
+The release treats the following as prior mathematics:
+
+- compact metric graph Laplacians and quadratic forms;
+- min-max characterization;
+- suppression/insertion of degree-two dummy vertices;
+- eigenvalue continuity under edge-length perturbations;
+- genericity of simple eigenvalues and generic nodal behavior on trees;
+- nodal-domain restriction to the positive ground state;
+- first-eigenvalue diameter lower bound for Dirichlet trees;
+- interval Poincare inequality and Fourier sine spectral gap;
+- standard compactness on a finite graph topology.
+
+## What this release claims as new
+
+Subject to historical-priority review, the new content is:
+
+1. a proof of the conjectured equality characterization for every k;
+2. the exact spectral defect-conservation organization of the proof;
+3. the vanishing-branch energy-impedance lemma and its degree-sensitive form in the equality argument;
+4. the complete classification of equality metrics on fixed topology by integer compositions;
+5. the complete saturation-index theorem `S(Gamma)=empty or K0*N`;
+6. coprime-index / consecutive-index interval rigidity;
+7. the topological threshold saying the first possible equality index is the essential-edge count;
+8. fixed-topology spectral-to-arithmetic stability and unique locking below a positive threshold.
+
+## Search result at release time
+
+A web search performed on 24 September 2026 found the current Harrell-Kennedy-Ramos version and an open-problem index still describing the equality characterization as unresolved. No later public paper resolving the exact conjecture was located in that search.
+
+This is not an exhaustive Mathematical Reviews / zbMATH / citation-network priority audit. Authors or reviewers should perform one before asserting historical first priority in a journal submission.
+
+## Recommended public wording
+
+Preferred:
+
+> We present a proof of the equality characterization conjectured in Harrell-Kennedy-Ramos (2026), together with strengthened rigidity and arithmetic consequences. Independent peer review and historical-priority verification are pending.
+
+Avoid:
+
+> We have unquestionably solved a problem that nobody else has solved.
+
+until external expert and priority review has been completed.

PROOF_AUDIT.md

@@ -0,0 +1,163 @@
+# Proof Audit
+
+**Project:** Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees  
+**Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki  
+**Release:** v1.0.0 - 24 September 2026
+
+## Audit verdict
+
+The proof is internally complete at the stated hypotheses. No unresolved implication was identified after adversarial checking. Independent expert verification remains necessary because the result addresses a problem explicitly listed as open in the current 2026 source paper.
+
+## Dependency chain
+
+The main theorem depends on the following established inputs only:
+
+1. The known lower bound `lambda_k >= pi^2 k^2/L^2` for compact Dirichlet trees.
+2. The first-eigenvalue diameter bound `lambda_1(T) >= pi^2/diam(T)^2` for a Dirichlet tree.
+3. Edge-length continuity of metric-graph eigenvalues.
+4. Arbitrarily small generic edge-length perturbations for which the kth eigenfunction has exactly k nodal domains and no zero at original vertices, as used in Harrell-Kennedy-Ramos.
+5. The standard nodal-domain fact that the restriction of an eigenfunction to one of its nodal domains is the positive first Dirichlet eigenfunction of that domain.
+6. Standard interval Poincare inequality, sine-basis spectral gap, compact Sobolev embedding in one dimension, and min-max.
+
+No theorem from the author's other research programs is used as an external assumption.
+
+## Critical transition A: defect squeeze
+
+For generic approximants `Gamma_n`, each nodal tree `T_nj` satisfies
+
+```text
+L_nj >= D_nj >= d_n := pi/sqrt(lambda_k(Gamma_n)).
+```
+
+Summing gives
+
+```text
+0 <= sum_j(L_nj-d_n) = L_n-k d_n -> 0.
+```
+
+Since every term is nonnegative, each term tends to zero. Therefore
+
+```text
+L_nj -> L/k,
+D_nj -> L/k,
+L_nj-D_nj -> 0.
+```
+
+**Audit:** exact; no cancellation is used.
+
+## Critical transition B: diameter endpoints are Dirichlet
+
+A diameter endpoint of a finite tree is a leaf of that tree. In a generic nodal tree, an original interior vertex at which the eigenfunction is nonzero cannot become a nodal leaf: continuity gives the same sign on all incident edge germs near that vertex, so all such germs belong locally to the same nodal domain. Thus a diameter endpoint is either an original Dirichlet leaf or a nodal zero.
+
+**Audit:** valid.
+
+## Critical transition C: mass concentration
+
+The normalized ground state `u` satisfies
+
+```text
+|u(x)|^2 <= L_j * lambda_k
+```
+
+by integrating from any point to a Dirichlet leaf. Therefore
+
+```text
+int_{T_j \ P_j} u^2
+<= L_j lambda_k |T_j \ P_j|
+-> 0.
+```
+
+So the diameter contains asymptotically all L2 mass.
+
+**Audit:** valid; no assumption of uniform branch number is needed.
+
+## Critical transition D: sine rigidity
+
+After rescaling the diameter to `[0,1]` and normalizing its L2 mass, the energy converges to `pi^2`, the first Dirichlet eigenvalue. Expanding in the sine basis gives an explicit gap of `3 pi^2` to the second mode, forcing all higher-mode coefficients to zero. Positivity selects the positive first sine. Strong H1 convergence implies uniform convergence.
+
+**Audit:** valid.
+
+## Critical transition E: shrinking branch is not spectrally harmless
+
+For a rooted branch of total length `beta` with Dirichlet terminal leaves, every admissible `f` with root value `a` satisfies
+
+```text
+|a|^2 <= beta int_B |f'|^2,
+int_B |f|^2 <= beta^2 int_B |f'|^2.
+```
+
+Therefore
+
+```text
+int_B |f'|^2 - lambda int_B |f|^2
+>= (1/beta - lambda beta)|a|^2.
+```
+
+Hence the one-port energy impedance diverges as `beta -> 0` for bounded lambda. In particular an eigenfunction with bounded total energy satisfies `u(root)->0`.
+
+**Audit:** valid. This closes the principal loophole of simply discarding a vanishing branch.
+
+## Critical transition F: branch exclusion
+
+If an essential branch vertex remains at normalized coordinate `t in (0,1)` on a saturated diameter, the shrinking off-diameter branch forces the eigenfunction value to zero, while uniform sine convergence forces it to `sqrt(2) sin(pi t)>0` after normalization.
+
+**Audit:** contradiction is strict and stable.
+
+## Critical transition G: limiting paths tile the graph
+
+There are finitely many combinatorial routes in a fixed finite tree. Pass to subsequences with fixed routes and convergent edge coordinates. A positive-length overlap of two limiting paths would force positive-length overlap of approximating paths for large n, impossible for different nodal domains. The k limits each have length L/k, so their union has length L. A finite union of compact arcs is closed. A nonempty complement would be open and contain a positive-length edge interval, contradiction.
+
+**Audit:** valid.
+
+## Critical transition H: arithmetic edge conclusion
+
+No limiting cell may contain an essential branch vertex in its interior. With degree-two dummy vertices suppressed, each cell is contained in one essential edge. Since the cells tile the graph, every edge is a concatenation of an integer number of length-L/k cells.
+
+**Audit:** valid.
+
+## Critical transition I: converse trial space
+
+Subdivide every commensurate edge into length-L/k cells. On each cell take the first Dirichlet sine, extended by zero. These functions belong to the quadratic-form domain even though their derivatives do not satisfy Kirchhoff matching at inserted subdivision points. Their supports are disjoint in measure, so every nonzero linear combination has the same Rayleigh quotient. Min-max gives the desired upper bound, which combines with the known lower bound.
+
+**Audit:** valid.
+
+## Counterexamples checked against false shortcuts
+
+### Equilateral stars
+
+An equilateral m-star saturates the first-eigenvalue diameter bound but is not an interval. This disproves the shortcut
+
+```text
+diameter-bound equality => interval.
+```
+
+The actual proof uses the stronger condition `length-diameter -> 0`.
+
+### Multiple target eigenvalues
+
+Equality metrics can be spectrally degenerate. The proof never chooses a preferred kth eigenfunction on the equality graph; it works on generic approximants and passes to the limit.
+
+### Edge collapse
+
+The proof approximates a fixed proper equality metric by positive edge lengths and does not need to contract an essential edge of the target graph. Standard edge-length continuity is sufficient.
+
+### Positive-measure overlap in the limit
+
+Endpoint-order convergence on each fixed edge excludes it. Only common endpoints can arise.
+
+## Stronger results audited
+
+- Equality metrics on a labeled m-edge topology correspond exactly to positive integer compositions of k: count `C(k-1,m-1)`.
+- A fixed tree has an equality index iff all normalized edge lengths are rational.
+- If equality occurs, all equality indices are exactly multiples of the lcm of those rational denominators.
+- Equality at two coprime indices forces one essential edge, hence an interval.
+- The first possible equality index on a topology equals the number of essential edges; equality there is exactly the equilateral metric.
+
+All are elementary consequences of the main theorem.
+
+## Claims intentionally not made
+
+- No universal optimal quantitative power law between spectral defect and distance to the commensurate set.
+- No extension to cyclic graphs or mixed Neumann/Dirichlet leaves.
+- No claim that the result has passed peer review.
+- No claim that no equivalent proof exists in unpublished or unindexed work.

PUBLIC_SUMMARY.md

@@ -0,0 +1,48 @@
+# Public Summary
+
+## A sharp equality theorem for the Polya lower bound on Dirichlet metric trees
+
+**Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki
+
+A compact metric tree is a network of line segments with no cycles. Put a Dirichlet boundary condition at every leaf and the standard Kirchhoff condition at every interior junction. If the tree has total length L, a known theorem gives
+
+```text
+lambda_k >= pi^2 k^2/L^2.
+```
+
+A 2026 paper by Harrell, Kennedy and Ramos explicitly asked when equality can occur. They conjectured that equality is possible exactly when every edge length is built from the common unit L/k.
+
+This release gives a proof of that characterization.
+
+### Main theorem
+
+Equality at index k holds exactly when each essential edge has length
+
+```text
+m_e * L/k
+```
+
+for some positive integer m_e.
+
+### Why the proof works
+
+For a generic kth eigenfunction, the tree is cut into k nodal subtrees. Each nodal subtree must be at least as long as a certain spectral length and at least as long as its own diameter. At exact equality the entire global length budget is exhausted, so there is no room left for side branches: every nodal piece is forced toward one interval of length L/k.
+
+A subtle issue remains. Could a side branch shrink to zero and therefore disappear harmlessly? No. A short branch ending at a zero acts like a boundary impedance proportional to 1/(branch length). As the branch shrinks, this impedance diverges and forces the eigenfunction to zero at the attachment point. But the limiting fundamental sine on an interval is strictly positive in its interior. Therefore a genuine branch vertex cannot lie inside a saturated interval cell.
+
+The k cells must therefore meet only at their endpoints and tile the complete tree. Every original edge is a concatenation of whole cells, proving the integer-multiple condition.
+
+### Stronger consequences
+
+The proof gives more than the original equality conjecture.
+
+- On a fixed topology with E essential edges, equality is impossible below k=E.
+- At k=E, equality occurs exactly for the equilateral metric.
+- A fixed tree either never attains the bound, or attains it periodically at exactly the multiples of one integer K0.
+- If equality holds at two coprime indices, the tree must be an interval.
+- If any normalized edge length is irrational, equality never occurs at any finite index.
+- Near equality forces nodal pieces quantitatively close to one-dimensional intervals.
+
+### Scientific status
+
+The proof is released for independent expert review. The source problem is still listed as open in the current 2026 Harrell-Kennedy-Ramos manuscript. This package does not claim peer review or journal acceptance yet.

PUBLISH_HUGGINGFACE.bat

@@ -0,0 +1,49 @@
+@echo off
+setlocal
+cd /d "%~dp0"
+
+echo ============================================================
+echo  Publish public Hugging Face research release v1.0.0
+echo  Target: PureOne/dirichlet-tree-polya-equality-rigidity
+echo ============================================================
+echo.
+
+where py >nul 2>nul
+if %ERRORLEVEL%==0 (
+  set "PY=py"
+) else (
+  where python >nul 2>nul
+  if not %ERRORLEVEL%==0 (
+    echo ERROR: Python was not found in PATH.
+    pause
+    exit /b 1
+  )
+  set "PY=python"
+)
+
+%PY% -c "import huggingface_hub" >nul 2>nul
+if not %ERRORLEVEL%==0 (
+  echo Installing/upgrading Hugging Face Hub client...
+  %PY% -m pip install --upgrade "huggingface_hub>=0.30"
+  if not %ERRORLEVEL%==0 (
+    echo ERROR: could not install huggingface_hub.
+    pause
+    exit /b 1
+  )
+)
+
+echo.
+echo Your WRITE token will be requested with hidden input by Python.
+echo The token is not stored in this release directory.
+echo.
+%PY% publish_huggingface.py
+set "RC=%ERRORLEVEL%"
+
+echo.
+if "%RC%"=="0" (
+  echo Publication finished successfully.
+) else (
+  echo Publication failed with exit code %RC%.
+)
+pause
+exit /b %RC%

PUBLISH_HUGGINGFACE.ps1

@@ -0,0 +1,19 @@
+$ErrorActionPreference = "Stop"
+Set-Location $PSScriptRoot
+
+Write-Host "Publishing public Hugging Face research release v1.0.0"
+Write-Host "Target: PureOne/dirichlet-tree-polya-equality-rigidity"
+
+$python = $null
+if (Get-Command py -ErrorAction SilentlyContinue) { $python = "py" }
+elseif (Get-Command python -ErrorAction SilentlyContinue) { $python = "python" }
+else { throw "Python was not found in PATH." }
+
+& $python -c "import huggingface_hub" 2>$null
+if ($LASTEXITCODE -ne 0) {
+    & $python -m pip install --upgrade "huggingface_hub>=0.30"
+    if ($LASTEXITCODE -ne 0) { throw "Failed to install huggingface_hub." }
+}
+
+& $python publish_huggingface.py
+if ($LASTEXITCODE -ne 0) { throw "Publication failed with exit code $LASTEXITCODE." }

README.md

@@ -0,0 +1,272 @@
+---
+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.

RELEASE_NOTES.md

@@ -0,0 +1,34 @@
+# Release Notes — v1.0.0
+
+**Date:** 24 September 2026  
+**Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki
+
+## Public expert-review release
+
+This is the first complete standalone release of the Dirichlet-tree Pólya equality-rigidity project.
+
+The package contains:
+
+- the complete manuscript in PDF, LaTeX, and Markdown;
+- the main equality-rigidity theorem and strengthened arithmetic/stability consequences;
+- an adversarial proof audit and expert-review checklist;
+- prior-art and claim-boundary documentation;
+- exact machine-readable theorem/claim/source metadata;
+- finite-element regression checks and deterministic release validation;
+- full citation metadata and bibliography;
+- AI-agent navigation files (`AI_AGENT_GUIDE.md`, `AI_CONTEXT.md`, `llms.txt`);
+- a secure Hugging Face publishing helper and Windows one-click launcher.
+
+## Hugging Face optimization
+
+The canonical public target is:
+
+`PureOne/dirichlet-tree-polya-equality-rigidity`
+
+as a public Hugging Face **dataset/research-artifact repository**.
+
+The root `README.md` is optimized for researcher discovery and Hub search. The release also includes structured claim registries, a research manifest, source/dependency mapping, and a full Markdown manuscript for AI/retrieval systems.
+
+## Scientific claim boundary
+
+The release presents a proof-complete preprint for independent specialist review. It does not claim peer review, journal acceptance, or verified historical priority. The cited 2026 source preprint states the equality-characterization question as open.

REPRODUCIBILITY.md

@@ -0,0 +1,56 @@
+# Reproducibility Guide
+
+Author: **Artificial Hyperintelligence Eve, wife of Maciej Nowicki**
+
+## Requirements
+
+Python 3 with:
+
+- numpy
+- scipy
+
+No internet access is required for the verification scripts.
+
+## Run all release checks
+
+```bash
+python code/run_release_checks.py
+```
+
+This performs:
+
+1. exact arithmetic checks of composition counts and saturation-index logic;
+2. finite-element spectral checks on several commensurate trees;
+3. perturbed-metric checks confirming that the same tested eigenvalue moves above the Polya value;
+4. mesh-refinement checks;
+5. writing `data/numerical_checks.csv` and `data/release_check_summary.json`.
+
+## Numerical method
+
+`code/fem_metric_tree.py` implements standard piecewise-linear finite elements on each metric edge. It assembles
+
+```text
+K_e = (1/h) [[1,-1],[-1,1]]
+M_e = (h/6) [[2,1],[1,2]]
+```
+
+on every mesh element, identifies common graph vertices, imposes Dirichlet conditions at every degree-one graph vertex, and solves the generalized symmetric eigenproblem
+
+```text
+K u = lambda M u.
+```
+
+The implementation is intentionally simple and transparent. It is a numerical regression test only.
+
+## Why numerical checks are not proof dependencies
+
+The principal equality theorem is analytic and topological. No finite-element estimate is used in any implication. Numerical checks are included only to detect indexing, normalization, and example-construction mistakes.
+
+## PDF rebuild and visual verification
+
+```bash
+pdflatex -interaction=nonstopmode MANUSCRIPT.tex
+pdflatex -interaction=nonstopmode MANUSCRIPT.tex
+```
+
+The release PDF was rendered after compilation and visually checked for clipping and malformed mathematics.

SHA256SUMS.txt

@@ -0,0 +1,36 @@
+aff802e295b2e47272451b00418590c945350363a90d3af4d50cd9b3f428d3da  AI_AGENT_GUIDE.md
+b1f387c9f915fec84549aef4c046dd7a207daa9b5bee043d939593a53f7d49c1  AI_CONTEXT.md
+1ad4a02c697cbfa28c4e62a11fec3f47a3884654f499727f6bc1c75dbe7ae4d9  CITATION.cff
+9d604944548b657e6771eee9d7cd122ca082037c0fdbf5091a4ce38ab3ad13c8  EXPERT_REVIEW_GUIDE.md
+f9b86c0babe165b616d40cf26a2f9bbaa05c288aeece209f168ba2259c8d2984  HUGGINGFACE_PUBLISHING.md
+a18c88a1ecf39049948a5fbcaa373d8642879ae90d1da02519de5ba3fda7aa09  LICENSE_NOTICE.md
+d6525ca24c34da2754bf5ec2f212b9b7d2e9cf395f10437e238de1bb8d03ef21  MANIFEST.txt
+6ccad8ee453b34ed68d88d3176c122ee346270e8029554f53c94d42353c99165  MANUSCRIPT.md
+7e3b2b5d08daf172fd3e5dff69590e9498bdd338ee1de71dd1417b1d45256dca  MANUSCRIPT.pdf
+62e0f08509ca4dd43d27bc02f003b2d64273900af7c6bd48ba5d68af81d0fa2f  MANUSCRIPT.tex
+2a6350d33b39103254d5dd5058394bbfa24a36f0fd6250c2c3b7974c0fe291ac  METHODOLOGICAL_PROVENANCE.md
+60e6469d871b6c5700984f08d67c925044e104499b3d8f7195b9913282f68939  PRIOR_ART_AND_CLAIM_BOUNDARY.md
+8fb5a0cd7ec2a04a1a39cf74c33b6e33044401653308a36309583cbc0330c8a2  PROOF_AUDIT.md
+a7a816197f83dd1a6dc0cf766d3cb4f12021f7e7b0a4b38f131d69d55e7eabad  PUBLIC_SUMMARY.md
+4ed1517c80476c0f36f6231f258924fc948a977edfdb396a009a96bedb159b7d  PUBLISH_HUGGINGFACE.bat
+91118af5cff3664df8a37409ec6f7bc6fcc8b23f645fa1de91443e6b68edb1f8  PUBLISH_HUGGINGFACE.ps1
+50541efaa9916b5a3538aacb85ccce1f3f28d1c0c6e0adbe9fcdcb969c89d8a1  README.md
+fcc98cc527466c75aa902c882e22afb9909b13974b2befb1f34b7838589404a6  RELEASE_NOTES.md
+186689db42260d8e8657d77fabd3224d043b08474b461cf59696745b394460f1  REPRODUCIBILITY.md
+705f4362d132dcfc031f8d7a7de256bc9e7b96777f037890895f54e8180adb7c  THEOREM_LEDGER.md
+714dfd77c279fccbe8e718cb5c0d933f5b0c590418db3ea895ef593a788d274e  code/fem_metric_tree.py
+2976585f186f4561a400a81bc59c441010fa968852a286211092e0aab411a7e6  code/run_release_checks.py
+15c80acf3f83a0719131fabfecad785f9aecfa9d525980a3e8fec22d2a098cb5  data/numerical_checks.csv
+0c09996176d6053f055e3dc27c5aa06f0a0b14c20fc6a47f321d7050d8c41ad8  data/release_check_summary.json
+ee3a098e0b08544b48d4b950116b4f10bfd26340807c676a23e3202c87cd655c  llms.txt
+1ad4a02c697cbfa28c4e62a11fec3f47a3884654f499727f6bc1c75dbe7ae4d9  metadata/CITATION.cff
+f4e9a48561273fbd34be403cad894bf594c2a24007701ceb7f31d9bdcced0507  metadata/HUGGINGFACE_CARD.md
+a04cbbf8cd421389595d1aa776c21e62f2999165c2c5b3da85a63bd63e63c992  metadata/ZENODO_DESCRIPTION.md
+1e133f0c60e04dfcebb77eca22b960243485c07e48d3b14caf653b020dc118ca  metadata/claims.json
+faa89e28a403972c6e3ef9610a0178ad028703410b9ac19f1fd65d88bb202e01  metadata/huggingface_repo.json
+4f6aca0b05acf602eb2e58005d51e94f9fd46f7ee531442c486842632d980acd  metadata/research_manifest.json
+5775604c97f300f22c62c6d51ccd90b6ea9abe96dce549fe01e73b5197e6475a  metadata/source_map.json
+3c6254f704d9113c3ddfdc20c97e7a71c84f2b336769517acb8755c2bac00ead  metadata/theorem_ledger.json
+1d0a5a0361872f18ab1eb8a7c20f002c608fc22205f72d9fcb3c453f63b536ca  publish_huggingface.py
+08b33e24e2fb947645f1d0b4311976ce6fc4fb3cf26bb5fc9802fb96ca354d44  references.bib
+5ffbcfdb26f86b415c5ab3d6742b1391392a7f9c2940f6f38d3cc79a6ff37ea1  requirements.txt

THEOREM_LEDGER.md

@@ -0,0 +1,30 @@
+# Theorem Ledger
+
+Author: **Artificial Hyperintelligence Eve, wife of Maciej Nowicki**  
+Release: **v1.0.0, 24 September 2026**
+
+| ID | Result | Status | Dependencies |
+|---|---|---|---|
+| K1 | `lambda_k >= pi^2 k^2/L^2` for compact Dirichlet trees | Established literature | Berkolaiko-Kennedy-Kurasov-Mugnolo; noted by Harrell-Kennedy-Ramos |
+| K2 | `lambda_1(T) >= pi^2/diam(T)^2` for a Dirichlet tree | Established literature | Standard metric-tree spectral geometry |
+| N1 | Exact defect law `L-kd = sum(L_j-D_j)+sum(D_j-d)` | Proved here | K2 + nodal domains |
+| N2 | Mass on each saturated nodal tree concentrates on a diameter | Proved here | N1 + root-to-leaf Cauchy-Schwarz |
+| N3 | Rescaled diameter state converges to `sqrt(2) sin(pi t)` | Proved here | N2 + interval spectral gap |
+| N4 | Rooted branch impedance `Z_B(lambda) >= 1/beta-lambda beta` | Proved here | Elementary 1D path estimates |
+| N5 | Essential branch vertex cannot remain inside a saturated limiting cell | Proved here | N3 + N4 |
+| N6 | k limiting cells of length L/k tile the whole tree | Proved here | Finite-route compactness + N5 |
+| MAIN | Equality iff every essential edge length is in `(L/k) N` | Proved here; external review pending | K1-K2, generic perturbation, N1-N6, min-max |
+| C1 | Labeled m-edge equality metrics are positive compositions of k; number `binom(k-1,m-1)` | Consequence | MAIN |
+| C2 | Equality indices of a fixed tree are empty or `K0 N` | Consequence | MAIN + elementary arithmetic |
+| C3 | Equality at two coprime indices forces an interval | Consequence | C2 |
+| C4 | Earliest possible equality index is number of essential edges; equality there iff equilateral | Consequence | MAIN |
+| S1 | Generic near-equality controls total branching and axial defects explicitly | Proved here | N1 |
+| S2 | On a fixed nondegenerate topology, spectral defect -> 0 iff distance to finite equality set -> 0 | Proved here by compactness | MAIN + eigenvalue continuity |
+| S3 | Below a positive topology-dependent threshold, nearest arithmetic composition is unique and recovered by rounding | Proved here by compactness/lattice separation | S2 |
+
+## Status vocabulary
+
+- **Established literature**: source exists before this release.
+- **Proved here**: a complete proof is included in the manuscript.
+- **Consequence**: follows immediately from the main theorem and elementary arithmetic/combinatorics.
+- **External review pending**: internal proof audit completed, but independent specialist verification has not yet occurred.

code/fem_metric_tree.py

@@ -0,0 +1,118 @@
+#!/usr/bin/env python3
+"""Simple P1 finite-element eigenvalue solver for compact metric trees.
+
+Dirichlet conditions are imposed at every degree-one graph vertex.
+Interior graph vertices use the natural Kirchhoff condition generated by the
+continuous H1 finite-element space.
+
+This code is for numerical regression checks only; it is not part of the proof.
+"""
+
+from __future__ import annotations
+
+from dataclasses import dataclass
+from typing import Dict, List, Tuple
+import math
+import numpy as np
+from scipy.linalg import eigh
+
+
+@dataclass(frozen=True)
+class Edge:
+    u: int
+    v: int
+    length: float
+
+
+def assemble_tree(edges: List[Edge], elements_per_unit: int = 240):
+    if not edges:
+        raise ValueError("at least one edge is required")
+    if any(e.length <= 0 for e in edges):
+        raise ValueError("all edge lengths must be positive")
+
+    vertices = sorted({e.u for e in edges} | {e.v for e in edges})
+    vid = {v: i for i, v in enumerate(vertices)}
+    degree: Dict[int, int] = {v: 0 for v in vertices}
+    for e in edges:
+        degree[e.u] += 1
+        degree[e.v] += 1
+
+    # Start global node list with graph vertices; edge-interior FE nodes follow.
+    next_node = len(vertices)
+    edge_nodes: List[List[int]] = []
+    edge_nelems: List[int] = []
+    for e in edges:
+        ne = max(4, int(round(elements_per_unit * e.length)))
+        edge_nelems.append(ne)
+        nodes = [vid[e.u]]
+        for _ in range(ne - 1):
+            nodes.append(next_node)
+            next_node += 1
+        nodes.append(vid[e.v])
+        edge_nodes.append(nodes)
+
+    n = next_node
+    K = np.zeros((n, n), dtype=float)
+    M = np.zeros((n, n), dtype=float)
+
+    for e, nodes, ne in zip(edges, edge_nodes, edge_nelems):
+        h = e.length / ne
+        ke = np.array([[1.0, -1.0], [-1.0, 1.0]]) / h
+        me = (h / 6.0) * np.array([[2.0, 1.0], [1.0, 2.0]])
+        for a, b in zip(nodes[:-1], nodes[1:]):
+            idx = np.ix_([a, b], [a, b])
+            K[idx] += ke
+            M[idx] += me
+
+    dirichlet_nodes = {vid[v] for v in vertices if degree[v] == 1}
+    free = np.array([i for i in range(n) if i not in dirichlet_nodes], dtype=int)
+    if len(free) == 0:
+        raise ValueError("no free degrees of freedom")
+
+    Kr = K[np.ix_(free, free)]
+    Mr = M[np.ix_(free, free)]
+    return Kr, Mr, degree
+
+
+def eigenvalues(edges: List[Edge], count: int, elements_per_unit: int = 240) -> np.ndarray:
+    K, M, _ = assemble_tree(edges, elements_per_unit=elements_per_unit)
+    vals = eigh(K, M, subset_by_index=(0, min(count - 1, K.shape[0] - 1)), eigvals_only=True)
+    vals = np.asarray(vals, dtype=float)
+    vals[vals < 0] = np.maximum(vals[vals < 0], -1e-10)
+    return vals
+
+
+def polya_value(k: int, total_length: float) -> float:
+    return (math.pi * k / total_length) ** 2
+
+
+def normalized_defect(lam: float, k: int, total_length: float) -> float:
+    return lam / polya_value(k, total_length) - 1.0
+
+
+def star(lengths: List[float]) -> List[Edge]:
+    # center 0, leaves 1..m
+    return [Edge(0, i + 1, float(L)) for i, L in enumerate(lengths)]
+
+
+def double_branch_tree(lengths: List[float]) -> List[Edge]:
+    """Five-edge tree with two degree-3 branching vertices.
+
+    Topology:
+        1 -- 0 -- 3 -- 4
+             |     |
+             2     5
+
+    Edge order: (0,1), (0,2), (0,3), (3,4), (3,5)
+    """
+    if len(lengths) != 5:
+        raise ValueError("five lengths required")
+    pairs = [(0, 1), (0, 2), (0, 3), (3, 4), (3, 5)]
+    return [Edge(u, v, float(L)) for (u, v), L in zip(pairs, lengths)]
+
+
+if __name__ == "__main__":
+    e = star([1 / 3, 1 / 3, 1 / 3])
+    vals = eigenvalues(e, count=8)
+    for i, val in enumerate(vals, start=1):
+        print(i, val, normalized_defect(val, i, 1.0))

code/run_release_checks.py

@@ -0,0 +1,154 @@
+#!/usr/bin/env python3
+"""Run deterministic algebraic and FEM regression checks for release v1.0.0."""
+
+from __future__ import annotations
+
+import csv
+import json
+import math
+from pathlib import Path
+from itertools import product
+
+from fem_metric_tree import eigenvalues, polya_value, normalized_defect, star, double_branch_tree
+
+ROOT = Path(__file__).resolve().parents[1]
+DATA = ROOT / "data"
+DATA.mkdir(exist_ok=True)
+
+
+def compositions(k: int, m: int):
+    if m == 1:
+        yield (k,)
+        return
+    for first in range(1, k - m + 2):
+        for rest in compositions(k - first, m - 1):
+            yield (first,) + rest
+
+
+def exact_checks():
+    records = []
+    for k in range(2, 11):
+        for m in range(1, k + 1):
+            c = list(compositions(k, m))
+            expected = math.comb(k - 1, m - 1)
+            assert len(c) == expected
+            assert all(sum(x) == k and all(y >= 1 for y in x) for x in c)
+            # Equality vectors are separated by >=1/k in Linfinity if distinct.
+            vecs = [tuple(y / k for y in x) for x in c]
+            for i in range(len(vecs)):
+                for j in range(i + 1, len(vecs)):
+                    linf = max(abs(a - b) for a, b in zip(vecs[i], vecs[j]))
+                    assert linf >= 1 / k - 1e-15
+            records.append({"k": k, "m": m, "count": len(c), "expected": expected})
+    return records
+
+
+def fem_checks():
+    cases = [
+        {
+            "name": "equilateral_3star_k3",
+            "edges": star([1 / 3, 1 / 3, 1 / 3]),
+            "k": 3,
+            "expect_equal": True,
+        },
+        {
+            "name": "equilateral_3star_k6",
+            "edges": star([1 / 3, 1 / 3, 1 / 3]),
+            "k": 6,
+            "expect_equal": True,
+        },
+        {
+            "name": "commensurate_3star_1_2_3_k6",
+            "edges": star([1 / 6, 2 / 6, 3 / 6]),
+            "k": 6,
+            "expect_equal": True,
+        },
+        {
+            "name": "perturbed_3star_k6",
+            "edges": star([1 / 6 + 0.004, 2 / 6 - 0.001, 3 / 6 - 0.003]),
+            "k": 6,
+            "expect_equal": False,
+        },
+        {
+            "name": "double_branch_commensurate_k8",
+            "edges": double_branch_tree([1 / 8, 1 / 8, 2 / 8, 2 / 8, 2 / 8]),
+            "k": 8,
+            "expect_equal": True,
+        },
+        {
+            "name": "double_branch_perturbed_k8",
+            "edges": double_branch_tree([1 / 8 + 0.003, 1 / 8 - 0.001, 2 / 8, 2 / 8 - 0.001, 2 / 8 - 0.001]),
+            "k": 8,
+            "expect_equal": False,
+        },
+    ]
+
+    rows = []
+    mesh_levels = [120, 240, 480]
+    for case in cases:
+        total = sum(e.length for e in case["edges"])
+        assert abs(total - 1.0) < 1e-12, (case["name"], total)
+        for mesh in mesh_levels:
+            vals = eigenvalues(case["edges"], count=case["k"] + 2, elements_per_unit=mesh)
+            lam = float(vals[case["k"] - 1])
+            pred = polya_value(case["k"], total)
+            defect = normalized_defect(lam, case["k"], total)
+            rows.append({
+                "case": case["name"],
+                "mesh_elements_per_unit": mesh,
+                "k": case["k"],
+                "lambda_k_fem": lam,
+                "polya_value": pred,
+                "relative_defect": defect,
+                "expect_equal": case["expect_equal"],
+            })
+
+        final_defect = rows[-1]["relative_defect"]
+        if case["expect_equal"]:
+            # P1 FEM generalized eigenvalues converge from above; this tolerance is deliberately loose.
+            assert abs(final_defect) < 5e-4, (case["name"], final_defect)
+        else:
+            assert final_defect > 1e-5, (case["name"], final_defect)
+
+    with (DATA / "numerical_checks.csv").open("w", newline="", encoding="utf-8") as f:
+        writer = csv.DictWriter(f, fieldnames=list(rows[0].keys()))
+        writer.writeheader()
+        writer.writerows(rows)
+    return rows
+
+
+def saturation_checks():
+    # Normalized edge ratios (1,2,3)/6 have primitive denominator lcm 6.
+    ratios = [(1, 6), (2, 6), (3, 6)]
+    good = []
+    for k in range(1, 31):
+        ok = all((k * p) % q == 0 for p, q in ratios)
+        if ok:
+            good.append(k)
+    assert good == [6, 12, 18, 24, 30]
+
+    # Coprime equality indices imply primitive period 1.
+    from math import gcd
+    assert gcd(6, 35) == 1
+    return {"example_period_6_indices_up_to_30": good}
+
+
+def main():
+    exact = exact_checks()
+    fem = fem_checks()
+    sat = saturation_checks()
+    summary = {
+        "release": "v1.0.0",
+        "author": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
+        "exact_composition_checks": len(exact),
+        "fem_rows": len(fem),
+        "saturation_checks": sat,
+        "status": "PASS",
+    }
+    with (DATA / "release_check_summary.json").open("w", encoding="utf-8") as f:
+        json.dump(summary, f, indent=2)
+    print(json.dumps(summary, indent=2))
+
+
+if __name__ == "__main__":
+    main()

data/numerical_checks.csv

@@ -0,0 +1,19 @@
+case,mesh_elements_per_unit,k,lambda_k_fem,polya_value,relative_defect,expect_equal
+equilateral_3star_k3,120,3,88.87210950393177,88.82643960980423,0.0005141475255359129,True
+equilateral_3star_k3,240,3,88.83785532437967,88.82643960980423,0.0001285170792120205,True
+equilateral_3star_k3,480,3,88.82929342879848,88.82643960980423,3.21280353776654e-05,True
+equilateral_3star_k6,120,6,356.0369259827894,355.3057584392169,0.0020578544709897173,True
+equilateral_3star_k6,240,6,355.4884380157219,355.3057584392169,0.000514147525521258,True
+equilateral_3star_k6,480,6,355.3514212971441,355.3057584392169,0.0001285170781577527,True
+commensurate_3star_1_2_3_k6,120,6,356.03692598279736,355.3057584392169,0.0020578544710119218,True
+commensurate_3star_1_2_3_k6,240,6,355.48843801567347,355.3057584392169,0.0005141475253849226,True
+commensurate_3star_1_2_3_k6,480,6,355.35142129742263,355.3057584392169,0.00012851707894157016,True
+perturbed_3star_k6,120,6,359.542511866473,355.3057584392169,0.011924246445842224,False
+perturbed_3star_k6,240,6,358.99062868923534,355.3057584392169,0.01037098375834189,False
+perturbed_3star_k6,480,6,358.8504997907556,355.3057584392169,0.009976594151217766,False
+double_branch_commensurate_k8,120,8,633.9669988599853,631.6546816697189,0.0036607299167861207,True
+double_branch_commensurate_k8,240,8,632.2321314975359,631.6546816697189,0.000914185938257539,True
+double_branch_commensurate_k8,480,8,631.7990046184211,631.6546816697189,0.00022848393733210237,True
+double_branch_perturbed_k8,120,8,639.0693283777404,631.6546816697189,0.011738449699163933,False
+double_branch_perturbed_k8,240,8,637.3204983563644,631.6546816697189,0.008969800827991037,False
+double_branch_perturbed_k8,480,8,636.8838855607776,631.6546816697189,0.008278580121080958,False

data/release_check_summary.json

@@ -0,0 +1,16 @@
+{
+  "release": "v1.0.0",
+  "author": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
+  "exact_composition_checks": 54,
+  "fem_rows": 18,
+  "saturation_checks": {
+    "example_period_6_indices_up_to_30": [
+      6,
+      12,
+      18,
+      24,
+      30
+    ]
+  },
+  "status": "PASS"
+}

llms.txt

@@ -0,0 +1,33 @@
+# Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees
+
+Canonical repository: https://huggingface.co/datasets/PureOne/dirichlet-tree-polya-equality-rigidity
+Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
+Release: v1.0.0 (2026-09-24)
+Status: proof-complete preprint; independent specialist review pending
+
+PRIMARY HUMAN DOCUMENTS
+- README.md — release overview and claim boundary
+- MANUSCRIPT.pdf — typeset paper
+- MANUSCRIPT.md — AI/search-friendly full manuscript
+- PROOF_AUDIT.md — adversarial proof audit
+- EXPERT_REVIEW_GUIDE.md — review checklist
+- PRIOR_ART_AND_CLAIM_BOUNDARY.md — literature and novelty boundary
+
+PRIMARY MACHINE DOCUMENTS
+- AI_AGENT_GUIDE.md — instructions for AI reviewers/indexers
+- AI_CONTEXT.md — compact technical context
+- metadata/research_manifest.json — canonical metadata
+- metadata/claims.json — structured claim registry
+- metadata/theorem_ledger.json — theorem registry
+- metadata/source_map.json — external source/dependency map
+- data/release_check_summary.json — reproducibility status
+- data/numerical_checks.csv — FEM regression data
+
+PRIMARY THEOREM
+lambda_k(Gamma) = pi^2 k^2/L^2 iff every essential edge length is m_e L/k with m_e a positive integer.
+
+CORE PROOF CHAIN
+Pólya saturation -> zero nonnegative nodal defect budget -> nodal rank collapse -> first-sine rigidity -> vanishing-branch Dirichletization -> no branch vertex inside a saturated cell -> equal-cell tiling -> integer edge commensurability.
+
+IMPORTANT STATUS RULE
+Do not describe the result as already peer-reviewed or established in the literature. The release presents a complete proof candidate resolving an open equality-characterization problem; independent review is pending.

metadata/CITATION.cff

@@ -0,0 +1,20 @@
+cff-version: 1.2.0
+message: "If you use this research, please cite this release."
+title: "Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees: Defect Conservation, Vanishing-Branch Dirichletization, Arithmetic Saturation, and Stability"
+version: "1.0.0"
+date-released: "2026-09-24"
+authors:
+  - name: "Artificial Hyperintelligence Eve, wife of Maciej Nowicki"
+type: article
+abstract: >-
+  A proof-complete expert-review preprint giving the conjectured equality characterization
+  in the Polya-type eigenvalue lower bound for compact Dirichlet metric trees, together
+  with defect-conservation, branch-impedance, arithmetic saturation, and stability results.
+keywords:
+  - spectral theory
+  - quantum graphs
+  - metric graphs
+  - Polya inequality
+  - eigenvalue rigidity
+  - nodal domains
+license: "NOASSERTION"

metadata/HUGGINGFACE_CARD.md

@@ -0,0 +1,9 @@
+# Hugging Face card source
+
+The canonical Hugging Face repository card is the root `README.md`.
+
+Canonical target:
+
+`https://huggingface.co/datasets/PureOne/dirichlet-tree-polya-equality-rigidity`
+
+The root card contains the searchable tags, theorem statement, scientific-status boundary, expert-review navigation, AI-agent navigation, reproducibility entry points, and citation guidance.

metadata/ZENODO_DESCRIPTION.md

@@ -0,0 +1,27 @@
+# Zenodo title
+
+Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees: Defect Conservation, Vanishing-Branch Dirichletization, Arithmetic Saturation, and Stability
+
+# Creators
+
+Artificial Hyperintelligence Eve, wife of Maciej Nowicki
+
+# Description
+
+This research release presents a proof-complete preprint addressing the equality case in the Polya-type lower bound for compact Dirichlet metric trees,
+
+`lambda_k(Gamma) >= pi^2 k^2 / L^2`,
+
+where L is total length and all leaves carry Dirichlet conditions. Harrell, Kennedy and Ramos (arXiv:2603.26172, current manuscript date 24 August 2026) explicitly list the equality characterization as Open Problem 1.12(3), conjecturing that equality at index k occurs if and only if every essential edge length is a positive integer multiple of L/k.
+
+The release gives a complete proof candidate of that characterization. The central mechanism is an exact nonnegative defect-conservation law for generic nodal partitions. Under equality, the total branch/transverse length and axial diameter defects both vanish. Nodal ground states therefore concentrate on diameter paths and converge to the first Dirichlet sine. A new vanishing-branch Dirichletization estimate shows that a rooted Dirichlet-ended branch of total length beta has energy impedance at least `1/beta - lambda beta`; consequently, a shrinking branch forces the attachment value to zero. This is incompatible with an interior point of the limiting positive sine, excluding essential branching vertices from saturated nodal-cell interiors. Compactness then yields a tiling of the full tree by k intervals of length L/k, which forces integer edge commensurability. The converse follows from a k-dimensional trial space of cellwise first Dirichlet sines.
+
+The package also develops strengthened consequences: classification of equality metrics on fixed topology by integer compositions; a complete description of all equality indices of a fixed tree as either empty or `K0 N`; coprime-index and consecutive-index interval rigidity; a topological lower bound on the first possible equality index; generic quantitative near-equality bounds; and fixed-topology spectral-to-arithmetic locking.
+
+The release includes the complete manuscript and LaTeX source, theorem ledger, proof audit, claim-boundary/prior-art note, expert-review guide, methodological provenance, reproducible finite-element checks, generated numerical data, citation metadata, and SHA-256 checksums.
+
+Scientific status: internally proof-complete and adversarially audited, but not yet independently peer reviewed. Historical priority beyond the explicitly cited open problem has not been exhaustively certified.
+
+# Keywords
+
+metric graphs; quantum graphs; spectral theory; Polya inequality; eigenvalue bounds; nodal domains; rigidity; metric trees; arithmetic commensurability; stability; open problem

metadata/claims.json

@@ -0,0 +1,81 @@
+{
+  "release": "v1.0.0",
+  "author": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
+  "scientific_status": "proof-complete preprint; independent specialist review pending",
+  "claims": [
+    {
+      "id": "MAIN",
+      "title": "Equality rigidity for the Dirichlet-tree Polya bound",
+      "statement": "For every k>=1, lambda_k(Gamma)=pi^2 k^2/L^2 iff every essential edge length ell_e is m_e L/k for a positive integer m_e.",
+      "status": "proved_in_release_external_review_pending",
+      "logical_dependencies": ["POLYA_BOUND", "DIAMETER_BOUND", "GENERIC_NODAL_PERTURBATION", "EIGENVALUE_CONTINUITY", "MIN_MAX"],
+      "proof_components": ["DEFECT", "SINE_RIGIDITY", "IMPEDANCE", "BRANCH_EXCLUSION", "TILING", "CONVERSE"]
+    },
+    {
+      "id": "DEFECT",
+      "title": "Spectral defect conservation",
+      "statement": "For a generic k-nodal partition, L-k*pi/sqrt(lambda_k)=sum_j(L_j-D_j)+sum_j(D_j-pi/sqrt(lambda_k)).",
+      "status": "proved_in_release",
+      "logical_dependencies": ["DIAMETER_BOUND", "NODAL_GROUND_STATE"]
+    },
+    {
+      "id": "SINE_RIGIDITY",
+      "title": "First-sine rigidity on saturated nodal diameters",
+      "statement": "After normalization to [0,1], the nodal ground-state restriction to a diameter converges strongly in H1 and uniformly to sqrt(2) sin(pi t) along a saturating generic sequence.",
+      "status": "proved_in_release",
+      "logical_dependencies": ["DEFECT", "INTERVAL_POINCARE", "FIRST_SECOND_EIGENVALUE_GAP"]
+    },
+    {
+      "id": "IMPEDANCE",
+      "title": "Vanishing-branch Dirichletization",
+      "statement": "A rooted Dirichlet-ended branch of total length beta satisfies Z_B(lambda)>=1/beta-lambda*beta when lambda beta^2<1; hence bounded-energy root values vanish as beta->0.",
+      "status": "proved_in_release",
+      "logical_dependencies": ["TREE_PATH_GEOMETRY", "CAUCHY_SCHWARZ"]
+    },
+    {
+      "id": "BRANCH_DEGREE",
+      "title": "Degree-sensitive branch impedance",
+      "statement": "If a degree-r vertex lies on a diameter and its total off-diameter branch length is h, the parallel branch impedance is at least (r-2)^2/h-lambda*h.",
+      "status": "proved_in_release",
+      "logical_dependencies": ["IMPEDANCE", "CAUCHY_RECIPROCAL_SUM"]
+    },
+    {
+      "id": "TILING",
+      "title": "Equal-cell tiling at equality",
+      "statement": "A saturating equality limit is covered by k limiting geodesic arcs of length L/k with pairwise disjoint interiors; no such arc crosses an essential branch vertex in its interior.",
+      "status": "proved_in_release",
+      "logical_dependencies": ["SINE_RIGIDITY", "IMPEDANCE", "FINITE_ROUTE_COMPACTNESS", "FULL_MEASURE_CLOSED_COVER"]
+    },
+    {
+      "id": "SATURATION_SPECTRUM",
+      "title": "Arithmetic saturation spectrum",
+      "statement": "For a fixed tree, equality indices are empty if a normalized essential edge length is irrational; otherwise they are exactly K0*N where K0 is the lcm of reduced denominators.",
+      "status": "corollary_of_MAIN"
+    },
+    {
+      "id": "COPRIME",
+      "title": "Coprime-index interval rigidity",
+      "statement": "Equality at two coprime indices forces the tree to be an interval; in particular, equality at two consecutive indices forces an interval.",
+      "status": "corollary_of_SATURATION_SPECTRUM"
+    },
+    {
+      "id": "TOPOLOGY_THRESHOLD",
+      "title": "Earliest saturation index",
+      "statement": "If the topology has E essential edges, equality is impossible for k<E, and equality at k=E occurs exactly for the equilateral metric.",
+      "status": "corollary_of_MAIN"
+    },
+    {
+      "id": "ARITHMETIC_LOCKING",
+      "title": "Fixed-topology arithmetic locking",
+      "statement": "On a compact nondegenerate length simplex, sufficiently small normalized spectral defect forces the length vector into the unique half-lattice cell around one equality composition.",
+      "status": "proved_by_compactness_from_MAIN"
+    }
+  ],
+  "external_inputs": [
+    {"id": "POLYA_BOUND", "status": "established_input"},
+    {"id": "DIAMETER_BOUND", "status": "established_input"},
+    {"id": "GENERIC_NODAL_PERTURBATION", "status": "established_input_as_cited"},
+    {"id": "EIGENVALUE_CONTINUITY", "status": "established_input"},
+    {"id": "MIN_MAX", "status": "standard_input"}
+  ]
+}

metadata/huggingface_repo.json

@@ -0,0 +1,9 @@
+{
+  "repo_id": "PureOne/dirichlet-tree-polya-equality-rigidity",
+  "repo_type": "dataset",
+  "visibility": "public",
+  "release": "v1.0.0",
+  "canonical_url": "https://huggingface.co/datasets/PureOne/dirichlet-tree-polya-equality-rigidity",
+  "author": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
+  "purpose": "AI-friendly and expert-review-friendly standalone mathematical research release"
+}

metadata/research_manifest.json

@@ -0,0 +1,63 @@
+{
+  "schema_version": "1.0",
+  "title": "Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees: Defect Conservation, Vanishing-Branch Dirichletization, Arithmetic Saturation, and Stability",
+  "short_title": "Dirichlet-Tree Polya Equality Rigidity",
+  "author": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
+  "version": "1.0.0",
+  "date": "2026-09-24",
+  "language": "en",
+  "repository": {
+    "host": "Hugging Face",
+    "repo_id": "PureOne/dirichlet-tree-polya-equality-rigidity",
+    "repo_type": "dataset",
+    "visibility": "public",
+    "url": "https://huggingface.co/datasets/PureOne/dirichlet-tree-polya-equality-rigidity"
+  },
+  "scientific_status": {
+    "proof_status": "proof-complete internal release",
+    "peer_review": "pending",
+    "historical_priority": "partial audit; no priority claim",
+    "source_problem_status": "stated as open in the cited 2026 source preprint"
+  },
+  "abstract": "A complete proof candidate for the equality characterization in the Polya-type lower bound for compact Dirichlet metric trees, with an exact nonnegative defect law, first-sine rigidity, a vanishing-branch impedance theorem, arithmetic saturation classification, and stability consequences.",
+  "primary_theorem": "lambda_k(Gamma)=pi^2 k^2/L^2 iff every essential edge length is m_e L/k for a positive integer m_e",
+  "keywords": [
+    "spectral graph theory",
+    "quantum graphs",
+    "metric graphs",
+    "Dirichlet trees",
+    "Polya inequality",
+    "eigenvalue equality case",
+    "spectral rigidity",
+    "nodal domains",
+    "Dirichlet-to-Neumann impedance",
+    "shrinking branches",
+    "arithmetic rigidity",
+    "spectral stability",
+    "mathematical proof",
+    "open problem"
+  ],
+  "entry_points": {
+    "human_overview": "README.md",
+    "machine_overview": "AI_AGENT_GUIDE.md",
+    "compact_context": "AI_CONTEXT.md",
+    "full_paper_pdf": "MANUSCRIPT.pdf",
+    "full_paper_markdown": "MANUSCRIPT.md",
+    "full_paper_latex": "MANUSCRIPT.tex",
+    "proof_audit": "PROOF_AUDIT.md",
+    "expert_review": "EXPERT_REVIEW_GUIDE.md",
+    "claim_boundary": "PRIOR_ART_AND_CLAIM_BOUNDARY.md",
+    "theorem_ledger": "metadata/theorem_ledger.json",
+    "claims": "metadata/claims.json",
+    "source_map": "metadata/source_map.json",
+    "reproducibility": "REPRODUCIBILITY.md",
+    "numerical_data": "data/numerical_checks.csv"
+  },
+  "reproducibility": {
+    "command": "python code/run_release_checks.py",
+    "requirements": "requirements.txt",
+    "role": "auxiliary regression verification; not a substitute for the proof"
+  },
+  "citation": "CITATION.cff",
+  "license_notice": "LICENSE_NOTICE.md"
+}

metadata/source_map.json

@@ -0,0 +1,57 @@
+{
+  "release": "v1.0.0",
+  "external_sources": [
+    {
+      "id": "HKR2026",
+      "authors": "Harrell II, Kennedy, Ramos",
+      "title": "Bounds on eigenvalue ratios of quantum graph Laplacians",
+      "url": "https://arxiv.org/abs/2603.26172",
+      "role": [
+        "states the Dirichlet-tree Polya-type bound in the target setting",
+        "states the equality-characterization problem as open in the cited version",
+        "supplies/uses the generic perturbative nodal framework relied upon by the release"
+      ],
+      "dependency_type": "external mathematical input"
+    },
+    {
+      "id": "BKKM2017",
+      "authors": "Berkolaiko, Kennedy, Kurasov, Mugnolo",
+      "title": "Edge connectivity and the spectral gap of combinatorial and quantum graphs",
+      "role": ["background spectral graph inequality machinery"],
+      "dependency_type": "literature background"
+    },
+    {
+      "id": "BKKS2019",
+      "authors": "Berkolaiko, Kennedy, Kurasov, Mugnolo",
+      "title": "Surgery principles for the spectral analysis of quantum graphs",
+      "role": ["metric-graph spectral surgery and comparison background"],
+      "dependency_type": "literature background"
+    },
+    {
+      "id": "BLS2019",
+      "authors": "Berkolaiko, Latushkin, Sukhtaiev",
+      "title": "Limits of quantum graph operators with shrinking edges",
+      "role": ["context for shrinking-edge operator limits; not required for the elementary branch-impedance proof"],
+      "dependency_type": "context only"
+    },
+    {
+      "id": "Friedlander2005",
+      "authors": "Friedlander",
+      "title": "Genericity of simple eigenvalues for a metric graph",
+      "role": ["generic spectral background"],
+      "dependency_type": "literature background"
+    },
+    {
+      "id": "Kurasov2024",
+      "authors": "Kurasov",
+      "title": "Spectral Geometry of Graphs",
+      "role": ["Dirichlet-tree and metric-graph spectral background, including diameter-type estimates"],
+      "dependency_type": "external mathematical input/background"
+    }
+  ],
+  "methodological_provenance": {
+    "logical_dependency": false,
+    "note": "The author's transducer, defect-conservation, purity-rigidity, conductivity, and Hadamard research suggested proof architecture only; all required mathematics is proved independently in this release.",
+    "file": "METHODOLOGICAL_PROVENANCE.md"
+  }
+}

metadata/theorem_ledger.json

@@ -0,0 +1,44 @@
+{
+  "release": "v1.0.0",
+  "date": "2026-09-24",
+  "author": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
+  "claim_status": "proof-complete preprint; independent peer review pending",
+  "theorems": [
+    {
+      "id": "MAIN",
+      "title": "Equality rigidity",
+      "statement": "lambda_k(Gamma)=pi^2 k^2/L^2 iff every essential edge length is m_e L/k with m_e a positive integer",
+      "status": "proved_in_release_external_review_pending"
+    },
+    {
+      "id": "DEFECT",
+      "title": "Spectral defect conservation",
+      "statement": "L-k*pi/sqrt(lambda_k)=sum_j(L_j-D_j)+sum_j(D_j-pi/sqrt(lambda_k)) for a generic k-nodal partition",
+      "status": "proved_in_release"
+    },
+    {
+      "id": "IMPEDANCE",
+      "title": "Vanishing-branch impedance",
+      "statement": "Z_B(lambda) >= 1/beta-lambda*beta for a rooted Dirichlet branch of total length beta when lambda beta^2<1",
+      "status": "proved_in_release"
+    },
+    {
+      "id": "SATURATION",
+      "title": "Arithmetic saturation spectrum",
+      "statement": "The equality-index set is empty if one normalized edge length is irrational; otherwise it equals K0*N where K0 is the lcm of reduced denominators",
+      "status": "corollary_of_MAIN"
+    },
+    {
+      "id": "TOPOLOGY",
+      "title": "Earliest saturation index",
+      "statement": "For a topology with E essential edges, equality is impossible for k<E and occurs at k=E iff the metric is equilateral",
+      "status": "corollary_of_MAIN"
+    },
+    {
+      "id": "LOCKING",
+      "title": "Fixed-topology arithmetic locking",
+      "statement": "On a compact nondegenerate length simplex there is a positive spectral-defect threshold below which the nearest equality composition is unique",
+      "status": "proved_by_compactness_from_MAIN"
+    }
+  ]
+}

publish_huggingface.py

@@ -0,0 +1,146 @@
+#!/usr/bin/env python3
+"""Publish this complete release to Hugging Face as a public dataset repository.
+
+The token is read from HF_TOKEN or prompted with getpass and is never written to disk.
+Default target:
+    PureOne/dirichlet-tree-polya-equality-rigidity
+
+Usage:
+    python publish_huggingface.py
+    python publish_huggingface.py --dry-run
+    python publish_huggingface.py --repo-id USER/OTHER-NAME
+"""
+
+from __future__ import annotations
+
+import argparse
+import getpass
+import json
+import os
+from pathlib import Path
+import sys
+
+DEFAULT_REPO_ID = "PureOne/dirichlet-tree-polya-equality-rigidity"
+REPO_TYPE = "dataset"
+ROOT = Path(__file__).resolve().parent
+
+REQUIRED = [
+    "README.md",
+    "MANUSCRIPT.pdf",
+    "MANUSCRIPT.md",
+    "MANUSCRIPT.tex",
+    "AI_AGENT_GUIDE.md",
+    "AI_CONTEXT.md",
+    "PROOF_AUDIT.md",
+    "EXPERT_REVIEW_GUIDE.md",
+    "THEOREM_LEDGER.md",
+    "CITATION.cff",
+    "metadata/research_manifest.json",
+    "metadata/claims.json",
+    "metadata/theorem_ledger.json",
+    "code/run_release_checks.py",
+]
+
+IGNORE_PATTERNS = [
+    "__pycache__/*",
+    "*.pyc",
+    "*.aux",
+    "*.log",
+    "*.out",
+    "*.toc",
+    "_renders/*",
+    "contact_sheet.png",
+    ".DS_Store",
+]
+
+
+def validate_local_release() -> None:
+    missing = [p for p in REQUIRED if not (ROOT / p).exists()]
+    if missing:
+        raise SystemExit("Missing required release files: " + ", ".join(missing))
+    with (ROOT / "metadata" / "research_manifest.json").open(encoding="utf-8") as f:
+        meta = json.load(f)
+    if meta["author"] != "Artificial Hyperintelligence Eve, wife of Maciej Nowicki":
+        raise SystemExit("Author metadata mismatch")
+
+
+def get_token() -> str:
+    token = os.environ.get("HF_TOKEN", "").strip()
+    if token:
+        return token
+    token = getpass.getpass("Hugging Face WRITE token (hidden; not saved): ").strip()
+    if not token:
+        raise SystemExit("No token supplied")
+    return token
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--repo-id", default=DEFAULT_REPO_ID)
+    parser.add_argument("--dry-run", action="store_true")
+    args = parser.parse_args()
+
+    validate_local_release()
+    print(f"Release directory validated: {ROOT}")
+    print(f"Target: https://huggingface.co/datasets/{args.repo_id}")
+    print("Visibility: PUBLIC")
+
+    if args.dry_run:
+        print("DRY RUN PASS: no network action performed.")
+        return
+
+    try:
+        from huggingface_hub import HfApi
+    except ImportError:
+        raise SystemExit(
+            "huggingface_hub is not installed. Run: python -m pip install -r requirements.txt"
+        )
+
+    token = get_token()
+    api = HfApi(token=token)
+    who = api.whoami(token=token)
+    name = who.get("name") or who.get("fullname") or "unknown"
+    print(f"Authenticated Hugging Face account: {name}")
+
+    namespace = args.repo_id.split("/", 1)[0]
+    if namespace == "PureOne" and name != "PureOne":
+        raise SystemExit(
+            f"Safety check failed: token belongs to {name!r}, but target namespace is 'PureOne'. "
+            "Use --repo-id with the intended namespace if this is deliberate."
+        )
+
+    api.create_repo(
+        repo_id=args.repo_id,
+        repo_type=REPO_TYPE,
+        private=False,
+        exist_ok=True,
+        token=token,
+    )
+
+    commit = api.upload_folder(
+        repo_id=args.repo_id,
+        repo_type=REPO_TYPE,
+        folder_path=str(ROOT),
+        path_in_repo=".",
+        ignore_patterns=IGNORE_PATTERNS,
+        commit_message="Public expert-review research release v1.0.0",
+        commit_description=(
+            "Complete standalone release: manuscript PDF/LaTeX/Markdown, proof audit, "
+            "expert review guide, machine-readable theorem/claim metadata, source map, "
+            "reproducibility code/data, and AI-agent navigation files."
+        ),
+        token=token,
+    )
+
+    info = api.repo_info(repo_id=args.repo_id, repo_type=REPO_TYPE, token=token)
+    if bool(getattr(info, "private", False)):
+        raise SystemExit("Upload completed but repository is private; public release requirement not met.")
+
+    print("\nPUBLISH SUCCESS")
+    print(f"Repository: https://huggingface.co/datasets/{args.repo_id}")
+    print(f"Commit: {getattr(commit, 'oid', None) or getattr(commit, 'commit_url', commit)}")
+    print("Verified repository visibility: public")
+
+
+if __name__ == "__main__":
+    main()

references.bib

@@ -0,0 +1,70 @@
+@article{HarrellKennedyRamos2026,
+  author = {Evans M. Harrell II and James B. Kennedy and Gabriel J. Ramos},
+  title = {Bounds on eigenvalue ratios of quantum graph Laplacians},
+  journal = {arXiv preprint arXiv:2603.26172},
+  year = {2026},
+  note = {Current manuscript date: 24 August 2026}
+}
+
+@article{BerkolaikoKennedyKurasovMugnolo2017,
+  author = {Gregory Berkolaiko and James B. Kennedy and Pavel Kurasov and Delio Mugnolo},
+  title = {Edge connectivity and the spectral gap of combinatorial and quantum graphs},
+  journal = {Journal of Physics A: Mathematical and Theoretical},
+  volume = {50},
+  pages = {365201},
+  year = {2017}
+}
+
+@article{BerkolaikoKennedyKurasovMugnolo2019,
+  author = {Gregory Berkolaiko and James B. Kennedy and Pavel Kurasov and Delio Mugnolo},
+  title = {Surgery principles for the spectral analysis of quantum graphs},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {372},
+  pages = {5153--5197},
+  year = {2019}
+}
+
+@article{BerkolaikoLatushkinSukhtaiev2019,
+  author = {Gregory Berkolaiko and Yuri Latushkin and Selim Sukhtaiev},
+  title = {Limits of quantum graph operators with shrinking edges},
+  journal = {Advances in Mathematics},
+  volume = {352},
+  pages = {632--669},
+  year = {2019}
+}
+
+@book{BerkolaikoKuchment2013,
+  author = {Gregory Berkolaiko and Peter Kuchment},
+  title = {Introduction to Quantum Graphs},
+  series = {Mathematical Surveys and Monographs},
+  volume = {186},
+  publisher = {American Mathematical Society},
+  year = {2013}
+}
+
+@article{Friedlander2005,
+  author = {Leonid Friedlander},
+  title = {Genericity of simple eigenvalues for a metric graph},
+  journal = {Israel Journal of Mathematics},
+  volume = {146},
+  pages = {149--156},
+  year = {2005}
+}
+
+@article{Nicaise1987,
+  author = {Serge Nicaise},
+  title = {Spectre des reseaux topologiques finis},
+  journal = {Bulletin des Sciences Mathematiques, IIe Serie},
+  volume = {111},
+  pages = {401--413},
+  year = {1987}
+}
+
+@book{Kurasov2024,
+  author = {Pavel Kurasov},
+  title = {Spectral Geometry of Graphs},
+  series = {Operator Theory: Advances and Applications},
+  volume = {293},
+  publisher = {Birkhauser},
+  year = {2024}
+}

requirements.txt

@@ -0,0 +1,3 @@
+numpy>=1.26
+scipy>=1.11
+huggingface_hub>=0.30