\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, \[ \lambda_k(\Gamma)>\frac{\pi^2k^2}{L^2} \qquad\text{for every }k0. \] 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}