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\title{Prime-Star Localization and Second-Order Spectra of Finite Divisibility Cover Graphs} \author{Idris Ali Shaik} \date{Version: v1.2 -- referee draft -- preprint · 7 September 2026} \begin{document} \maketitle \begin{center} \small Independent researcher, Dubai, United Arab Emirates\\ Email: \href{mailto:shaikidris@apache.org}{shaikidris@apache.org}\quad ORCID: \href{https://orcid.org/0009-0009-9699-9712}{0009-0009-9699-9712}\\[0.6em] \textbf{Preprint DOI (all versions):} \href{https://doi.org/10.5281/zenodo.22145761}{10.5281/zenodo.22145761}\\[0.6em] \textbf{2020 Mathematics Subject Classification:} Primary 05C50; Secondary 11N05\\ \textbf{Keywords:} arithmetic graph; divisibility cover graph; adjacency spectrum; prime-counting function; eigenvector localization; Feshbach--Schur map\\ \textbf{Paper license:} \href{https://creativecommons.org/licenses/by/4.0/}{Creative Commons Attribution 4.0 International (CC BY 4.0)} \end{center} \section*{Abstract}\label{abstract} \addcontentsline{toc}{section}{Abstract} Let \(G_X\) be the graph on the positive integers \(n\le X\) in which \(n\) is joined to \(np\) whenever \(p\) is prime and \(np\le X\). Let \(A_X\) be its adjacency matrix, let \(\lambda_1(A_X)\) be its spectral radius, and let \(\pi(X)\) count the primes up to \(X\). Then \[ \lambda_1(A_X)^2 =\pi(X)+4\sum_{p\le X}\frac1{p-1}+o(1). \] More generally, fix a finite set \(S\) of primes and retain only the positive integers divisible by none of the primes in \(S\). Let \(a_15\), and leaves it unchanged when \(q=5\). Multiplicities do not matter: \(c(a)\) depends only on the distinct prime divisors of \(a\). The proof gives three further pieces of information. First, if \(H_{S,X}\) denotes the adjacency formed by labels at most \(\sqrt X\), then \[ \|H_{S,X}\|\asymp_S\frac{X^{1/4}}{\sqrt{\log X}}. \] Second, the eigenvector associated with a fixed \(a_j\) localizes on its large-prime star at rate \(O_{S,j}(X^{-1/4})\). Third, before using the prime number theorem and Mertens estimates, the squared eigenvalue correction is an explicit finite sum over the neighbouring stars reached by multiplying or dividing \(a_j\) by a small prime, with error \(O_{S,j}(1/\log X)\). Section 8 derives this finite identity, and its evaluation gives the displayed constant-order formula. The paired positive and negative modes also give an adjacency-evolution corollary: a state begun at a fixed centre oscillates, uniformly in time, between that centre and the normalized shell of its large-prime leaves. The explicit finite remainder determines the phase of this oscillation for times \(o(\sqrt{X\log X})\). The same formula remains uniform when the centre varies with \(X\): for every fixed \(\varepsilon>0\), it holds for all retained \[ a\le(\log X)^{1/2-\varepsilon}. \] At the preceding finite-sum stage the uniform spectral error is \(O_{S,\varepsilon}(a^2/\log X)\). Consequently the entire growing block of positive eigenvalues indexed by centres in this window is eventually simple and ordered by those centres. The fixed-mode localization and centre--shell evolution also extend uniformly to this block, with error \(O_{S,\varepsilon}(a^{3/2}X^{-1/4})\) at centre \(a\). Replacing the exact frequency by the explicit first-exit frequency retains uniform \(o(1)\) accuracy for \(|t|=o(\sqrt{X\log X}/a^{5/2})\). Three results from this package are mechanically checked in a companion Lean 4 development: the uniform finite-prime-sum form of Theorem 9.6, its undeleted root-centred largest-eigenvalue specialization, and the uniform explicit-frequency centre--shell dynamics bound underlying Corollary 9.9. They are registered in the Palomar Registry as PALOMAR-2026-08-29-000017, version 2 {[}22{]}. The exact declaration names, immutable source commit, pinned dependencies, and reproduction commands are recorded after Section 10. Two natural comparison frequencies occur in the proof. The exact star frequency is \(\sqrt{d_{S,X}(a)}=\sqrt{\pi_S(X/a)-\pi_S(\sqrt X)}\); its leading displacement restores the missing boundary count \(\pi_S(\sqrt X)\) after squaring. The finer first-exit calculation instead compares the squared eigenvalue with \(\pi_S(X/a)\) and exposes the logarithmic and constant layers. Other arithmetic graph constructions retain broader divisibility relations or restrict the vertices to the divisors of one integer {[}4,5{]}. Cover-matrix spectra have been computed for structured product posets by Anđelić and da Fonseca {[}6{]}. Graph-index changes under local perturbations have been studied by Rowlinson {[}7{]} and by Dalfó, Fiol, and Garriga {[}8{]}. Sparse random graphs also exhibit extreme eigenvalues governed by high-degree vertices and eigenvectors localized near them {[}1,2{]}. Frieze and Tsourakakis obtain such a law from a star-forest decomposition in a random Apollonian model {[}3{]}. The supremum norm and inverse participation ratio used below make this deterministic localization quantitative; compare the localized edge eigenvectors in {[}1,15{]}. Continuous-time adjacency evolution on exact stars, and transfer from a vertex to a superposition over a vertex subset, are classical {[}16,17{]}; the adjacency-evolution corollary below shows that the coherent centre--shell motion persists in the full arithmetic graph. If \(\Delta\) denotes the maximum degree, the classical induced-star argument gives \(\lambda_1\ge\sqrt{\Delta}\), while Hong's bound gives \(\lambda_1\le\sqrt{2m-n+1}\) for a connected graph with \(n\) vertices and \(m\) edges {[}13{]}. For the present graph family the first bound is asymptotically sharp, whereas the second is larger than the true scale by a factor of order \(\sqrt{\log X\log\log X}\). The closest arithmetic-matrix lineage begins with the Redheffer matrix. Barrett, Forcade, and Pollington proved that its spectral radius is asymptotic to \(\sqrt X\) {[}18{]}, and Barrett and Jarvis obtained precise expansions for its two dominant eigenvalues {[}19{]}. Kline then introduced an \(O(X)\)-nonzero descendant with the same determinantal representation {[}20{]}. Writing \(P(i)\) for the largest prime divisor of \(i>1\), Kline's matrix \(\mathcal R_X\) has \[ (\mathcal R_X)_{ij}=1 \quad\Longleftrightarrow\quad i=j,\quad i=1,\quad\text{or}\quad \bigl(i\text{ is squarefree and }i/j=P(i)\bigr). \] Thus \(\mathcal R_X\) is nonsymmetric, has unit diagonal and a distinguished first row, and otherwise keeps only the largest-prime arrow from each squarefree row. It has \((2+6/\pi^2)X+o(X)\) nonzero entries. By contrast, \(A_X\) is the symmetric, zero-diagonal cover adjacency, keeps every prime-labelled cover edge, and has order \(X\log\log X\) nonzero entries. Kline proved {[}21{]} that the two dominant eigenvalues of \(\mathcal R_X\) satisfy \[ r_\pm=\pm\sqrt{\pi(X)}+\frac12\log\log X+O(1). \] The spectrum of \(A_X\) is exactly symmetric about zero, and the present result gives \[ \lambda_1(A_X) =\sqrt{\pi(X)}+ \frac{2\log\log X+2\kappa}{\sqrt{\pi(X)}} +o\!\left(\frac1{\sqrt{\pi(X)}}\right). \] The different next-order scales are already visible at \(X=10^7\), where \(\pi(X)=664579\). Kline reports \(r_+=817.6477\), while the full cover adjacency gives \(\lambda_1=815.226460312\): {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedleft\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedleft\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright matrix \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft positive excess above \(\sqrt{\pi(X)}\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft squared excess above \(\pi(X)\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot Kline's \(\mathcal R_X\) & \(2.4305\) & \(3968.76\) \\ cover adjacency \(A_X\) & \(0.009311\) & \(15.1816\) \\ \end{longtable} } Asymptotically, Kline's squared excess has order \(\sqrt{\pi(X)}\log\log X\), whereas the cover graph has squared excess \(4\log\log X+4\kappa+o(1)\). These are expansions of different operators, not competing estimates for one matrix. Together the results form a natural sequence: the Redheffer radius \(\sqrt X\), its two-term dominant spectrum, Kline's prime-sparse \(\sqrt{\pi(X)}\) spectrum, and the constant-order squared expansion for the symmetric cover adjacency. In the last case the correction is generated by the large-prime star boundary and its first-return resolvents. The proof uses weighted Schur tests and Weyl inequalities {[}11{]}, isolated eigenvalue perturbation {[}9,10{]}, the Feshbach--Schur complement {[}12{]}, and the prime number theorem and Mertens estimates {[}14{]}. Section 2 records the isolated-mode comparison. Sections 3--4 define the arithmetic graph and its large-prime star forest, Section 5 proves the sharp small-prime norm, and Sections 6--7 establish the fixed-mode spectral comparison and localization. Section 8 derives and evaluates the finite remainder. Section 9 proves the uniform logarithmic-window form. \section{2. A standard isolated-mode comparison}\label{a-standard-isolated-mode-comparison} We record the perturbative mechanism separately from its arithmetic realization. It is a finite-dimensional form of standard isolated-eigenvalue perturbation and the Feshbach--Schur map; see Kato {[}10{]}, Davis--Kahan {[}9{]}, and Dusson--Sigal--Stamm {[}12{]}. \subsection{Proposition 2.1 --- Vanishing first order forces second order}\label{proposition-2.1-vanishing-first-order-forces-second-order} Let \(L\) and \(H\) be Hermitian matrices on a finite-dimensional Hilbert space. Suppose \[ Lu=\mu u,\qquad \|u\|=1, \] \(\mu\) is a simple eigenvalue of \(L\), and, with \(Q=I-uu^*\), \[ \operatorname{dist}\bigl(\mu,\sigma(QLQ)\bigr)\ge\gamma>0. \] If \[ \langle u,Hu\rangle=0, \qquad \|H\|\le\frac{\gamma}{4}, \] then the eigenvalue \(\lambda\) of \(L+H\) matched to \(\mu\) is simple and satisfies \[ |\lambda-\mu| \le 2\frac{\|Hu\|^2}{\gamma}. \] If \(v\) is its unit eigenvector, with phase chosen so that \(\langle u,v\rangle\ge0\), then \[ \|v-u\| \le 3\frac{\|Hu\|}{\gamma}. \] \subsubsection{Proof}\label{proof} Let \(I=(\mu-\gamma/2,\mu+\gamma/2)\). The matrix \(L\) has exactly one eigenvalue in \(I\), namely \(\mu\). Weyl's inequality places one eigenvalue of \(L+H\) within \(\gamma/4\) of \(\mu\), while every other eigenvalue remains at distance at least \(3\gamma/4\) from \(\mu\). Thus \(L+H\) has exactly one eigenvalue in \(I\); this is the matched eigenvalue \(\lambda\), and it is simple. The orthogonal decomposition \(\mathbb Cu\oplus u^\perp\) gives \[ L+H= \begin{pmatrix} \mu&b^*\\ b&C \end{pmatrix}, \qquad b=QHu, \qquad C=Q(L+H)Q. \] Weyl's inequality places the matched eigenvalue within \(\|H\|\) of \(\mu\), while the spectrum of \(C\) lies within \(\|H\|\) of \(\sigma(QLQ)\). Hence \[ \operatorname{dist}\bigl(\lambda,\sigma(C)\bigr) \ge\gamma-2\|H\| \ge\frac{\gamma}{2}. \] The one-dimensional Schur complement is therefore exact: \[ \lambda-\mu =b^*(\lambda-C)^{-1}b. \] Since \(\|b\|\le\|Hu\|\), the eigenvalue estimate follows. Writing \(v=\alpha u+w\), with \(w\perp u\), the lower block equation gives \[ w=(\lambda-C)^{-1}b\,\alpha. \] Thus \(\|w\|/|\alpha|\le2\|Hu\|/\gamma\le1/2\). Normalization and the chosen phase imply \(\|v-u\|\le3\|Hu\|/\gamma\). \(\square\) In the arithmetic graph below, \(L\) is the large-prime star forest and \(H\) is the small-prime adjacency. No small-prime edge has both endpoints in the support of one star mode, so the graph-specific identity \[ \langle u_a,H_{S,X}u_a\rangle=0 \] supplies the hypothesis that removes the first-order displacement. \section{3. The finite prime-cover graph}\label{the-finite-prime-cover-graph} Put \[ P_S=\prod_{p\in S}p, \qquad V_S(X)=\{n\le X:(n,P_S)=1\}. \] Write \[ \pi_S(y)=\#\{p\le y:p\notin S\}. \] The graph \(G_{S,X}\) has vertex set \(V_S(X)\) and undirected edges \[ n\longleftrightarrow np \] whenever \(p\notin S\) is prime and \(np\le X\). Let \(A_{S,X}\) be its adjacency matrix. Every \(n\in V_S(X)\) factors entirely over primes outside \(S\), so removing its prime factors gives a path to \(1\); hence \(G_{S,X}\) is connected. The coloring \[ n\longmapsto(-1)^{\Omega(n)} \] makes the graph bipartite, so its spectrum is symmetric about zero. Split the adjacency at the radial square-root boundary: \[ A_{S,X}=L_{S,X}+H_{S,X}, \] where \(L_{S,X}\) contains prime labels \(p>\sqrt X\), and \(H_{S,X}\) contains labels \(p\le\sqrt X\). For every positive integer \(n\), define \[ d_{S,X}(n) = \bigl(\pi_S(X/n)-\pi_S(\sqrt X)\bigr)_+, \qquad (x)_+=\max\{x,0\}. \] \begin{figure} \centering \includegraphics[width=0.98\linewidth,height=\textheight,keepaspectratio,alt={The prime-cover graph G\_\{16\}. Solid edges have prime label p\textgreater4=\textbackslash sqrt\{16\} and form a disjoint star forest, with nonempty stars at 1, 2, and 3, shown shaded. Dashed edges have label p\textbackslash le 4. No dashed edge has both endpoints in one large-prime star.}]{fig-cover-split.pdf} \caption{The prime-cover graph \(G_{16}\). Solid edges have prime label \(p>4=\sqrt{16}\) and form a disjoint star forest, with nonempty stars at \(1\), \(2\), and \(3\), shown shaded. Dashed edges have label \(p\le 4\). No dashed edge has both endpoints in one large-prime star.}\label{fig:cover-split} \end{figure} \section{4. The exact large-prime star forest}\label{the-exact-large-prime-star-forest} \subsection{Proposition 4.1}\label{proposition-4.1} The graph with adjacency \(L_{S,X}\) is a disjoint union of stars. Its nonempty star centered at \(n<\sqrt X\) has leaves \[ \mathcal L_{n,X} = \{np:\sqrt X\sqrt X\) has lower endpoint below \(\sqrt X\) and upper endpoint above it. A number at most \(X\) cannot have two distinct prime divisors exceeding \(\sqrt X\). Hence a large-prime leaf has a unique center and cannot itself be a center. The components are stars, and the spectral formulas are those of \(K_{1,d}\). \(\square\) Because \(d_{S,X}(n)\) is nonincreasing in \(n\), the \(j\)-th positive eigenvalue of \(L_{S,X}\) is \(\sqrt{d_{S,X}(a_j)}\) whenever this degree is positive. \section{5. The sharp norm of the small-prime graph}\label{the-sharp-norm-of-the-small-prime-graph} \subsection{Theorem 5.1}\label{theorem-5.1} For every fixed finite \(S\), \[ \|H_{S,X}\| \asymp_S \frac{X^{1/4}}{\sqrt{\log X}}. \] \subsubsection{Proof}\label{proof-2} Write \(Y=\sqrt X\). The principal subgraph on \[ \{1\}\cup\{p\le Y:p\notin S\} \] is \(K_{1,\pi_S(Y)}\). Therefore \[ \|H_{S,X}\| \ge\sqrt{\pi_S(Y)} \asymp \frac{X^{1/4}}{\sqrt{\log X}}. \] For the upper bound, take the positive weight \(w_\theta(n)=n^{-\theta}\). The weighted Schur row ratio is \[ \frac{(H_{S,X}w_\theta)(n)}{w_\theta(n)} \le \sum_{p\le Y}p^{-\theta} + \sum_{\substack{p\mid n\\p\le Y}}p^\theta. \] Uniformly for \(\theta\) in a fixed neighborhood of \(1/2\), partial summation and the Chebyshev bound give \[ \sum_{p\le Y}p^{-\theta} \ll\frac{Y^{1-\theta}}{\log Y}. \] In the incoming sum, prime divisors at most \(Y^{1/2}\) contribute at most \[ 2\log_2Y\,Y^{\theta/2}=o(Y^\theta). \] At most three distinct prime divisors of \(n\le Y^2\) exceed \(Y^{1/2}\); four would have product greater than \(Y^2\). Hence \[ \sum_{\substack{p\mid n\\Y^{1/2}0\), \[ \left| \lambda_j(A_{S,X})-\sqrt{d_{S,X}(a_j)} \right| \le\|H_{S,X}\| \ll\frac{X^{1/4}}{\sqrt{\log X}}. \] For fixed \(j\), the prime number theorem gives \[ d_{S,X}(a_j) \sim\frac{X}{a_j\log X}. \] The distinct fixed-center star eigenvalues are therefore eventually simple and separated by \[ \gamma_{j,X}\asymp_{S,j}\sqrt{X/\log X}. \] In the form \(\sin\angle(v_{j,X}^{+},u_{a_j,X}^{+}) \le \|H_{S,X}\|/\gamma_{j,X}\), Davis--Kahan and Theorem 5.1 give the eigenvector estimate \[ \|v_{j,X}^{+}-u_{a_j,X}^{+}\|_2 =O_{S,j}(X^{-1/4}). \] Let \(J\delta_n=(-1)^{\Omega(n)}\delta_n\). Then \(JA_{S,X}J=-A_{S,X}\). After fixing the sign of \(v_{j,X}^{+}\) as above, put \[ v_{j,X}^{-}=(-1)^{\Omega(a_j)}Jv_{j,X}^{+}. \] It is a unit eigenvector of eigenvalue \(-\lambda_j(A_{S,X})\), and \[ \|v_{j,X}^{-}-u_{a_j,X}^{-}\|_2 =O_{S,j}(X^{-1/4}). \] \subsection{Corollary 6.1 --- Hybrid hub--shell localization}\label{corollary-6.1-hybrid-hubshell-localization} For each fixed \(j\) and either sign, \[ \boxed{ \|v_{j,X}^{\pm}\|_\infty =\frac1{\sqrt2}+O_{S,j}(X^{-1/4}) } \] and, with \[ \operatorname{IPR}(v)=\sum_n|v(n)|^4, \] \[ \boxed{ \operatorname{IPR}(v_{j,X}^{\pm}) =\frac14+O_{S,j}(X^{-1/4}). } \] If \(P_{a_j,X}\) is the coordinate projection onto \(\{a_j\}\cup\mathcal L_{a_j,X}\), then \[ \boxed{ \|(I-P_{a_j,X})v_{j,X}^{\pm}\|_2^2 =O_{S,j}(X^{-1/2}). } \] \subsubsection{Proof}\label{proof-3} Put \(d=d_{S,X}(a_j)\). The exact star mode satisfies \[ \|u_{a_j,X}^{\pm}\|_\infty=\frac1{\sqrt2}, \qquad \operatorname{IPR}(u_{a_j,X}^{\pm}) =\frac14+\frac1{4d}. \] For unit vectors \(x,y\), the reverse triangle inequality gives \[ |\|x\|_4-\|y\|_4| \le\|x-y\|_4 \le\|x-y\|_2. \] Since \(\|x\|_4,\|y\|_4\le1\), it follows that \[ |\|x\|_4^4-\|y\|_4^4| \le4\|x-y\|_2. \] The first two claims now follow from the paired localization estimates and \(d^{-1}=O_{S,j}(\log X/X)\). The exact star mode vanishes off the star, so the squared off-star mass is at most \(\|v_{j,X}^{\pm}-u_{a_j,X}^{\pm}\|_2^2\). \(\square\) The limit \(\operatorname{IPR}(v_{j,X}^{\pm})\to1/4\) describes a hybrid profile: asymptotically half the mass lies at the centre, while the other half is spread coherently over an expanding large-prime shell. Figure~\ref{fig:star-mode} shows the unperturbed profile on the star at \(1\) in \(G_{16}\). \begin{figure} \centering \includegraphics[width=0.52\linewidth,height=\textheight,keepaspectratio,alt={The positive unit mode of the large-prime star at 1 in G\_\{16\}, of degree d=4. Half the \textbackslash ell\^{}2 mass sits at the centre; the remaining half is spread equally over the four leaves.}]{fig-star-mode.pdf} \caption{The positive unit mode of the large-prime star at \(1\) in \(G_{16}\), of degree \(d=4\). Half the \(\ell^2\) mass sits at the centre; the remaining half is spread equally over the four leaves.}\label{fig:star-mode} \end{figure} \subsection{Corollary 6.2 --- Centre--shell adjacency evolution}\label{corollary-6.2-centreshell-adjacency-evolution} Define the normalized shell state \[ \psi_{a_j,X} =\frac1{\sqrt{d_{S,X}(a_j)}} \sum_{m\in\mathcal L_{a_j,X}}\delta_m. \] Then, for every fixed \(j\), \[ \boxed{ \sup_{t\in\mathbb R} \left\| e^{-itA_{S,X}}\delta_{a_j} -\cos(\lambda_jt)\delta_{a_j} +i\sin(\lambda_jt)\psi_{a_j,X} \right\|_2 =O_{S,j}(X^{-1/4}), } \] where \(\lambda_j=\lambda_j(A_{S,X})\). Consequently, \[ \sup_{t\in\mathbb R} \|(I-P_{a_j,X})e^{-itA_{S,X}}\delta_{a_j}\|_2^2 =O_{S,j}(X^{-1/2}), \] and \[ \lim_{T\to\infty}\frac1T\int_0^T \left| \langle\delta_{a_j},e^{-itA_{S,X}}\delta_{a_j}\rangle \right|^2dt =\frac12+O_{S,j}(X^{-1/4}). \] \subsubsection{Proof}\label{proof-4} Write \(u^\pm=u_{a_j,X}^{\pm}\), \(v^\pm=v_{j,X}^{\pm}\), and \(\varepsilon_X=\max_\pm\|v^\pm-u^\pm\|_2\). Since \[ \delta_{a_j}=\frac{u^++u^-}{\sqrt2}, \qquad \psi_{a_j,X}=\frac{u^+-u^-}{\sqrt2}, \] the vector \(w=(v^++v^-)/\sqrt2\) satisfies \(\|w-\delta_{a_j}\|_2\le\sqrt2\,\varepsilon_X\). Moreover, \[ e^{-itA_{S,X}}w =\frac{e^{-it\lambda_j}v^++e^{it\lambda_j}v^-}{\sqrt2}. \] Replacing \(v^\pm\) by \(u^\pm\) and using unitarity gives, uniformly in \(t\), \[ \left\| e^{-itA_{S,X}}\delta_{a_j} -\cos(\lambda_jt)\delta_{a_j} +i\sin(\lambda_jt)\psi_{a_j,X} \right\|_2 \le2\sqrt2\,\varepsilon_X. \] The confinement estimate follows because the comparison state is supported on the star. Taking its inner product with \(\delta_{a_j}\), squaring, and averaging \(\cos^2(\lambda_jt)\) proves the last assertion. \(\square\) The global norm bound is not the correct scale for the fixed eigenvalue displacement. The next section identifies the exact cancellation that removes its first-order contribution. \section{7. Second-order spectral readout}\label{second-order-spectral-readout} \subsection{Lemma 7.1}\label{lemma-7.1} For fixed \(j\), put \[ \mu_{j,X}=\sqrt{d_{S,X}(a_j)}, \qquad u=u_{a_j,X}^{+}. \] Then \[ \langle u,H_{S,X}u\rangle=0 \] and \[ \|H_{S,X}u\|_2^2 =O_{S,j}\!\left(\frac{\sqrt X}{\log X}\right). \] \subsubsection{Proof}\label{proof-5} No small-prime edge joins two vertices in the support of \(u\). The center \(a_j\) and a leaf \(a_jp\) differ by \(p>\sqrt X\), while two distinct leaves are incomparable. This proves the first identity. Unique factorization also implies that distinct support vertices have disjoint \(H_{S,X}\)-neighbor sets. Consequently, \[ \|H_{S,X}u\|_2^2 = \frac12\deg_H(a_j) + \frac1{2d_{S,X}(a_j)} \sum_{\sqrt X\|C_{m,X}\|\), define \[ \Sigma_{m,X}(z) =K_{m,X}^*(z-C_{m,X})^{-1}K_{m,X} =\sum_{r\ge0}z^{-r-1}K_{m,X}^*C_{m,X}^rK_{m,X}. \] Since \(\|K_{m,X}\|\le\|H_{S,X}\|\), the self-energy has norm \(O_{S,j,m}(1/\sqrt{\log X})\). The fixed boundary star eigenvalues are separated from \(\mu_{j,X}\) by \(\asymp_{S,j,m}\sqrt{X/\log X}\). Applying the one-dimensional Feshbach reduction inside the boundary, using the zero diagonal of \(R_{m,X}\), yields \[ \lambda_j-\mu_{j,X} =\langle u_{j,X}, \Sigma_{m,X}(\mu_{j,X})u_{j,X}\rangle +o_{S,j}(1/\sqrt{\log X}). \] Indeed, boundary-mode mixing contributes \(O_{S,j,m}(\sqrt{\log X/X})\), and the resolvent identity replaces \(\lambda_j\) by \(\mu_{j,X}\) with still smaller error. Put \(b_{j,X}=K_{m,X}u_{j,X}\). The exact residual calculation of Lemma 7.1, followed by removal of the bounded boundary component, gives \[ \|b_{j,X}\|_2^2 =\frac12\pi_S(Y)+O_{S,j}((\log X)^2) \sim\frac{\sqrt X}{\log X}. \] Indeed, the centre coordinate of \(u_{j,X}\) has squared mass \(1/2\) and has \(\pi_S(Y)+O_{S,j}(1)\) small-prime neighbours. This contributes \(\tfrac12\pi_S(Y)+O_{S,j}(1)\). The normalized large-prime leaves contribute \(O_{S,j}((\log X)^2)\) by Lemma 7.1, and removing the finite boundary changes only the stated error. We next bound longer excursions. Write \(f=H_{S,X}u_{j,X}\), and split the complement again into its large-prime and small-prime parts. Because \(P\) is a union of complete \(L_{S,X}\)-star components, it commutes with \(L_{S,X}\), and so does \(Q=I-P\). Since \(b_{j,X}=Qf\), this gives \[ QL_{S,X}Qb_{j,X}=QL_{S,X}f, \qquad \|QL_{S,X}Qb_{j,X}\|\le\|L_{S,X}f\|, \qquad \|QH_{S,X}Qb_{j,X}\|\le\|H_{S,X}\|\|f\|. \] On the large-prime star centered at \(c\), \[ \|(L_{S,X}f)|_{\{c\}\cup\mathcal L(c)}\|^2 =\left|\sum_{v\in\mathcal L(c)}f(v)\right|^2 +d_{S,X}(c)|f(c)|^2. \] The center exits have forms \(a_jq\) and \(a_j/q\); the leaf exits have forms \(a_jpq\) and \((a_j/q)p\). Using \[ d_{S,X}(c)\ll_S\frac{X}{c\log X}, \qquad \sum_{q\le Y}\frac1q=O(\log\log X), \qquad \sum_q\frac1{q^2}<\infty, \] and \(d_{S,X}(a_j)\asymp_{S,j}X/\log X\), direct grouping by the exit prime gives the following bound. For an up-exit \(q\), the centre \(a_jq\) and all vertices \(a_jpq\) lie in the large-prime star centred at \(a_jq\). The finitely many down-exits \(q\mid a_j\) similarly lie in the star centred at \(a_j/q\). Distinct exit primes give disjoint stars: an equality \(a_jqp=a_jq'p'\), with \(p,p'>Y\) and \(q,q'\le Y\), forces \(q=q'\) and \(p=p'\) by unique factorization; an up/down equality \(a_jqp=(a_j/r)p'\), with \(r\mid a_j\) prime, would give \(rqp=p'\), which is impossible; and a down/down equality reduces to \(r'p=rp'\), forcing \(r=r'\) and \(p=p'\). Moreover, \[ d_{S,X}(a_jq)\ll_{S,j}\frac{d_{S,X}(a_j)}q \] uniformly for \(q\le Y\). Consequently, \[ \sum_{q\le Y}d_{S,X}(a_jq) \ll_{S,j}d_{S,X}(a_j)\sum_{q\le Y}\frac1q, \] and \[ \sum_{q\le Y}\frac{d_{S,X}(a_jq)^2}{d_{S,X}(a_j)} \ll_{S,j}d_{S,X}(a_j)\sum_q\frac1{q^2}. \] The finitely many down-exit stars contribute \(O_{S,j}(d_{S,X}(a_j))\). Therefore \[ \sum_{q\le Y} \left(d_{S,X}(a_jq)+ \frac{d_{S,X}(a_jq)^2}{d_{S,X}(a_j)}\right) +O_{S,j}(d_{S,X}(a_j)) \ll_{S,j}d_{S,X}(a_j)\log\log X. \] The preceding star identity therefore gives \[ \|C_{m,X}b_{j,X}\|_2^2 =O_{S,j,m}\!\left(\frac{X\log\log X}{\log X}\right). \] Since \(\|C_{m,X}\|/\mu_{j,X}\le\rho<1\), the positive-length Neumann tail is at most \[ \frac{\|b_{j,X}\|_2\,\|C_{m,X}b_{j,X}\|_2} {\mu_{j,X}^2(1-\rho)} =O_{S,j}\!\left(X^{-1/4}\sqrt{\log\log X}\right) =o_{S,j}(1/\sqrt{\log X}). \] The zero-step first return is therefore the full leading term. Finally, \[ \frac{\|b_{j,X}\|_2^2}{\mu_{j,X}} \sim \frac{\sqrt X/\log X}{\sqrt{X/(a_j\log X)}} =\frac{\sqrt{a_j}}{\sqrt{\log X}}, \] which proves the theorem. \(\square\) \subsection{Corollary 7.4 --- Spectral readout consequence}\label{corollary-7.4-spectral-readout-consequence} For every fixed finite deletion set \(S\) and fixed \(j\), \[ \boxed{ \lambda_j(A_{S,X})^2 =\pi_S(X/a_j)+o_{S,j}\!\left(\frac{\sqrt X}{\log X}\right). } \] \subsubsection{Proof}\label{proof-8} Write \(\lambda_j=\mu_{j,X}+\Delta_{j,X}\). Theorem 7.3 and the PNT give \[ 2\mu_{j,X}\Delta_{j,X} =\left(2+o_{S,j}(1)\right)\frac{\sqrt X}{\log X} =\pi_S(\sqrt X)+o_S(\sqrt X/\log X). \] Since \(\Delta_{j,X}^2=o(\sqrt X/\log X)\), expanding the square proves the claim. The cancellation is universal in fixed \(j\): the factor \(\sqrt{a_j}\) in the displacement cancels the factor \(a_j^{-1/2}\) in the star scale. \(\square\) \section{8. The explicit fixed-mode remainder}\label{the-explicit-fixed-mode-remainder} The cancellation in the preceding readout can be continued one layer further for every fixed mode. A single-star boundary at \(a=a_j\) is enough: larger stars remain in the interior and are controlled by two-sided gaps. Write \(Y=\sqrt X\) and \(d_c=d_{S,X}(c)\). Put \(\mu=\sqrt{d_a}\) and \[ r_{q,X}=\frac{d_{aq}}{d_a} \qquad(q\notin S,\ q\le Y\text{ prime}), \qquad s_{q,X}=\frac{d_{a/q}}{d_a} \qquad(q\mid a,\ q\text{ prime}). \] \subsection{Proposition 8.1 --- First-exit decomposition}\label{proposition-8.1-first-exit-decomposition} Let \(u\) be the positive unit mode of the large-prime star at \(a\), let \(\mathcal B=\{a\}\cup\{ap:Y0\). With \(c=a/q\), \(s=d_c/d_a>1\), and \(\psi\) the normalized symmetric leaf vector of the star at \(c\), the leaf part of \(b_{a/q}\) has a component \(\sqrt{1/(2s)}\) along \(\psi\) and a kernel component of squared mass \((s-1)/(2s)\). The two-dimensional resolvent together with the kernel contribution \(1/\mu\) gives \[ 2\mu\,\langle b_{a/q},(\mu-L_{a/q})^{-1}b_{a/q}\rangle =\frac{5-s}{1-s}. \] This quantity is negative for \(15\); no uniform sign assertion is used below. As \(X\to\infty\), \(s_{q,X}\to q\) and the term tends to \((q-5)/(q-1)\). \subsubsection{Proof}\label{proof-9} On the centre / normalized-leaf basis the up-star is the matrix with off-diagonal entry \(\mu\sqrt{r}\), while \(b_{aq}\) has coordinates \(2^{-1/2}(1,\sqrt{r})\). Direct inversion of \(\mu I-L_{aq}\) yields the first identity. For the down-star, \(\mu I-L_c=\mu\begin{pmatrix}1&-\sqrt{s}\\-\sqrt{s}&1\end{pmatrix}\), so \[ (\mu-L_c)^{-1} =\frac1{\mu(1-s)} \begin{pmatrix}1&\sqrt{s}\\\sqrt{s}&1\end{pmatrix} \] on \(\operatorname{span}\{\delta_c,\psi\}\). The two-dimensional quadratic form is \((3s+1)/(2\mu s(1-s))\). The kernel contributes \((s-1)/(2s\mu)\). Multiplying by \(2\mu\) and simplifying gives \((5-s)/(1-s)\). \(\square\) Summing the leading \(1\) in the up-star identity over allowed \(q\le Y\) produces exactly \(\pi_S(Y)\). On the compressed direct sum \(L_U=\bigoplus L_c\) over the centres of Proposition 8.1 one therefore has \[ 2\mu\,\langle b,(\mu-L_U)^{-1}b\rangle-\pi_S(Y) =4\sum_{\substack{q\le Y\\q\notin S}}\frac{r_{q,X}}{1-r_{q,X}} +\sum_{\substack{q\mid a\\q\text{ prime}}}\frac{5-s_{q,X}}{1-s_{q,X}}. \] This identity is for the compressed direct sum only. \subsection{Lemma 8.3 --- Two-sided interior separation}\label{lemma-8.3-two-sided-interior-separation} Relative to \(\ell^2(\mathcal B)\oplus\ell^2(\mathcal B^c)\), write \[ A_{S,X} = \begin{pmatrix}L_B&K^*\\K&C\end{pmatrix}, \] so that \(L_Bu=\mu u\) and \(b=Ku\). All edges leaving \(\mathcal B\) carry small-prime labels, hence \(\|K\|\ll_S X^{1/4}/\sqrt{\log X}\). There is \(\delta_{S,j}>0\) such that, for all large \(X\), \[ \operatorname{dist}\bigl(\mu,\sigma(C)\bigr)\ge\delta_{S,j}\,\mu, \qquad \operatorname{dist}\bigl(\lambda_j,\sigma(C)\bigr)\ge\tfrac12\delta_{S,j}\,\mu. \] \subsubsection{Proof}\label{proof-10} Write \(C=L_C+H_C\), where \(L_C\) is the star forest with the \(a\)-star deleted. The spectrum of \(L_C\) is \(\{\pm\sqrt{d_c}:c\ne a\}\cup\{0\}\). For allowed \(c>a\), \(d_c\le d_{a'}=(a/a'+o(1))d_a\) with \(a'\) the least allowed integer exceeding \(a\). For the finitely many allowed \(c0\) and all large \(X\). \subsubsection{Proof}\label{proof-12} Proposition 8.1 gives \(b\in U\). The compression of \(C\) to \(U\) equals \(L_U\). To see this, write every vertex of \(U\) either as a listed centre \(c\), or as \(cp\) with \(p>Y\). A small-prime edge between two centres would require \(c_2=c_1\ell\) for a prime \(\ell\le Y\). Comparing the possibilities \(c=aq\) and \(c=a/r\) forces a prime to equal a product of two primes. A centre-leaf edge would require \(c_1\ell=c_2p\); for fixed \(a\) and large \(X\), the large prime \(p>Y\) would then divide \(c_1\ell\), whose prime factors are all at most \(Y\), which is impossible. Finally, a leaf-leaf edge would give \(c_1p_1\ell=c_2p_2\). If \(p_1\ne p_2\), unique factorization forces one large prime to divide the small-prime side; if \(p_1=p_2\), it reduces to the excluded centre-centre case. Thus no small-prime edge joins two vertices of \(U\). Large-prime edges inside \(U\) beyond the star edges are impossible because a large-prime leaf has a unique centre. Every \(U\)-to-\(W\) edge therefore carries a small-prime label. For the gap, write \(C_W=L_W+H_W\) on the remaining centres. Those centres are a fixed ratio away from \(a\), so the argument of Lemma 8.3 applies, and \(\|H_W\|\le\|H_{S,X}\|=o(\mu)\). \(\square\) \subsection{Lemma 8.6 --- Quadratic-form replacement}\label{lemma-8.6-quadratic-form-replacement} \[ \langle b,(\mu-C)^{-1}b\rangle =\sum_c\langle b_c,(\mu-L_c)^{-1}b_c\rangle +O_{S,j}\bigl(1/\sqrt{X\log X}\bigr). \] \subsubsection{Proof}\label{proof-13} Lemma 8.5 makes the Schur complement exact: \[ \langle b,(\mu-C)^{-1}b\rangle =\langle b,(\mu-L_U-T)^{-1}b\rangle, \qquad T=J^*(\mu-C_W)^{-1}J, \] with \(\|T\|=O_{S,j}(1/\sqrt{\log X})\). Put \(R_U=(\mu-L_U)^{-1}\). Chebyshev gives \(r_{q,X}\ll_S1/q\). Choose a finite prime cutoff so the tail is below \(1/2\); on the remaining finitely many primes the PNT limit \(r_{q,X}\to1/q\) supplies a common constant \(\rho_S<1\). Inside \(U\), up-stars are separated from \(\mu\) by \(\mu(1-\sqrt{r_{q,X}})\ge(1-\sqrt{\rho_S})\mu\) with \(\sup_q r_{q,X}\le\rho_S<1\), down-stars by \(\mu(\sqrt{s_{q,X}}-1)\ge(\sqrt2-1+o(1))\mu\), and the kernel and negative branches by \(\mu\). Hence \(\|R_U\|=O_{S,j}(1/\mu)\). Also \[ \|b\|^2 =\tfrac12\pi_S(Y)+\tfrac12\sum_q r_{q,X}+\omega(a) =O_{S,j}(\sqrt X/\log X). \] For each up-star, direct inversion gives \[ \|R_Ub_{aq}\|^2 =\frac{1+6r_{q,X}+r_{q,X}^2} {2\mu^2(1-r_{q,X})^2} =O_{S,j}(\mu^{-2}). \] The finitely many down-stars satisfy the same bound. Summing the \(\pi_S(Y)\) up-stars therefore gives \[ \|R_Ub\|^2 =O_{S,j}\!\left(\frac{\pi_S(Y)}{\mu^2}\right) =O_{S,j}(X^{-1/2}). \] Moreover, after decreasing the limiting gap if necessary, \[ \operatorname{dist}(\mu,\sigma(L_U+T)) \ge \delta_{U,S,j}\mu-\|T\|\gg_{S,j}\mu, \qquad \delta_{U,S,j} =\tfrac12\min\{1-\sqrt{\rho_S},\sqrt2-1,1\}>0. \] In particular, \(\|R_UT\|=o(1)\), so \(\|R_UT\|<1/2\) for all sufficiently large \(X\). The resolvent identity is \[ (\mu-L_U-T)^{-1}-R_U =R_UT(\mu-L_U-T)^{-1}, \] and \[ (\mu-L_U-T)^{-1}=(I-R_UT)^{-1}R_U. \] Consequently, \[ \left| \langle b,(\mu-L_U-T)^{-1}b\rangle-\langle b,R_Ub\rangle \right| \le 2\|T\|\,\|R_Ub\|^2 =O_{S,j}(1/\sqrt{X\log X}). \] This is the required quadratic-form replacement. \(\square\) \subsection{Theorem 8.7 --- Explicit fixed-mode prime-star correction}\label{theorem-8.7-explicit-fixed-mode-prime-star-correction} For every fixed finite deletion set \(S\) and fixed index \(j\), \[ \boxed{ \lambda_j(A_{S,X})^2-\pi_S(X/a_j) =4\sum_{\substack{q\le\sqrt X\\q\notin S}} \frac{r_{q,X}}{1-r_{q,X}} +\sum_{\substack{q\mid a_j\\q\text{ prime}}}\frac{5-s_{q,X}}{1-s_{q,X}} +O_{S,j}(1/\log X). } \] \subsubsection{Proof}\label{proof-14} Insert Lemma 8.6 into Lemma 8.4 and evaluate the direct sum by Lemma 8.2. The count \(\pi_S(Y)\) cancels. The quadratic-form error, multiplied by \(2\mu=O_{S,j}(\sqrt{X/\log X})\), is \(O_{S,j}(1/\log X)\). \(\square\) \subsection{Corollary 8.8 --- Arithmetic control of the oscillation phase}\label{corollary-8.8-arithmetic-control-of-the-oscillation-phase} For all sufficiently large \(X\), let \[ F_{S,j}(X) =\pi_S(X/a_j) +4\sum_{\substack{q\le\sqrt X\\q\notin S}} \frac{r_{q,X}}{1-r_{q,X}} +\sum_{\substack{q\mid a_j\\q\text{ prime}}} \frac{5-s_{q,X}}{1-s_{q,X}}, \qquad \widehat\lambda_{j,X}=\sqrt{F_{S,j}(X)}. \] For every fixed finite \(S\) and fixed \(j\), uniformly for real \(t\), \[ \boxed{ \begin{aligned} \bigl\|e^{-itA_{S,X}}\delta_{a_j} &-\cos(\widehat\lambda_{j,X}t)\delta_{a_j} +i\sin(\widehat\lambda_{j,X}t)\psi_{a_j,X}\bigr\|_2\\ &\ll_{S,j}X^{-1/4}+\frac{|t|}{\sqrt{X\log X}}. \end{aligned} } \] In particular, the explicit prime-star remainder determines the coherent centre--shell evolution with \(o(1)\) error throughout every time window \(|t|=o(\sqrt{X\log X})\). \subsubsection{Proof}\label{proof-15} Theorem 8.7 and \(\lambda_j\asymp_{S,j}\sqrt{X/\log X}\) give \[ |\lambda_j-\widehat\lambda_{j,X}| =\frac{|\lambda_j^2-F_{S,j}(X)|} {\lambda_j+\widehat\lambda_{j,X}} \ll_{S,j}\frac1{\sqrt{X\log X}}. \] Combine Corollary 6.2 with the Lipschitz bounds for sine and cosine. \(\square\) \subsection{Corollary 8.9 --- Fixed-mode constant layer}\label{corollary-8.9-fixed-mode-constant-layer} Let \(B_1\) be the prime Meissel--Mertens constant, defined by \(\sum_{p\le x}1/p=\log\log x+B_1+o(1)\), and put \[ \kappa_S =B_1-\sum_{p\in S}\frac1p+\sum_{p\notin S}\frac1{p(p-1)}, \qquad c(a)=\sum_{\substack{q\mid a\\q\text{ prime}}}\frac{q-5}{q-1}. \] Then, for every fixed finite \(S\) and fixed \(j\), \[ \boxed{ \lambda_j(A_{S,X})^2 =\pi_S(X/a_j)+4\log\log X+4\kappa_S+c(a_j)+o_{S,j}(1). } \] The case \(j=1\) is \(c(1)=0\). The correction \(c(a)\) is additive over distinct prime divisors and ignores multiplicity, so for example \(c(2)=c(4)=-3\), \(c(3)=-1\), \(c(5)=0\), and \(c(6)=-4\). \subsubsection{Proof}\label{proof-16} Put \(L=\log X\) and \(\widetilde L=L-\log a\). We use the standard quantitative consequence of the PNT \[ \pi_S(t)=\frac{t}{\log t}+O_S\!\left(\frac{t}{\log^2t}\right) \qquad(t\ge Y/a), \] uniformly in this range. When \(q\le Y/a\), it gives \[ r_{q,X} =\frac{\widetilde L}{q(\widetilde L-\log q)} +O\!\left(\frac1{qL}+\frac a{\sqrt X}\right) \] for allowed \(q\). When \(q>Y/a\), the truncated degree \(d_{aq}\) and therefore \(r_{q,X}\) vanish, while the displayed main term is \(O(a/Y)\). The errors sum to \(o(1)\). The main term splits as \(1/q\) plus a logarithmic increment. Mertens' theorem yields \(\sum_{q\le Y,\,q\notin S}1/q =\log L-\log 2+B_1-\sum_{p\in S}1/p+o(1)\). Partial summation with \(\sum_{p\le t}(\log p)/p=\log t+O(1)\) converts the increment into \[ \int_2^Y\frac{d\log t}{\widetilde L-\log t} =\log\frac{\widetilde L-\log 2}{\widetilde L-L/2}+o(1) =\log 2+o(1), \] because \(a\) is fixed. Thus \(\sum r_{q,X}=\log\log X+B_1-\sum_{p\in S}1/p+o(1)\), exactly as for \(a=1\). Since \[ \frac{r}{1-r}=r+\frac{r^2}{1-r}, \] extend \(r_{q,X}=0\) for primes \(q>Y\). Chebyshev gives \(r_{q,X}\ll_S1/q\) and \(\sup_q r_{q,X}\le\rho_S<1\), hence \(r_{q,X}^2/(1-r_{q,X})\ll_Sq^{-2}\) on a fixed prime index set. Dominated convergence produces the convergent sum \(\sum_{q\notin S}1/(q(q-1))\). The up-sum is therefore \(4\log\log X+4\kappa_S+o(1)\). The finitely many down-star terms tend to \(c(a)\) by the PNT. Theorem 8.7 completes the proof. \(\square\) The logarithmic increment is essential here: pointwise convergence \(r_{q,X}\to1/q\) cannot be inserted directly into the divergent first-order prime sum. The \(-\log2\) from truncating the reciprocal-prime sum at \(\sqrt X\) is exactly restored by the increment integral. \textbf{Finite-prime interpretation.} The universal correction can equivalently be written without a separately named constant. Indeed, \[ \frac1{p-1}=\frac1p+\frac1{p(p-1)}, \] so, for fixed \(S\), \[ 4\sum_{\substack{p\le X\\p\notin S}}\frac1{p-1} =4\log\log X+4\kappa_S+o_S(1). \] Thus \(\kappa_S\) is the convergent correction that converts the prime harmonic sum into the truncated \((p-1)^{-1}\)-sum. The truncation at \(p\le X\) is essential. \section{9. Uniformity in a logarithmic window}\label{uniformity-in-a-logarithmic-window} The fixed-mode proof hides its dependence on the centre inside the \(O_{S,j}\)-constants. We now track that dependence on the range needed for the same explicit expansion, including the centre-dependent term \(c(a)\). Throughout this section \(S\) and \(\varepsilon>0\) are fixed, \[ a=a(X)\le(\log X)^{1/2-\varepsilon} \] is an allowed centre, and \(j(a)\) is the number of allowed positive integers at most \(a\). Put \(Y=\sqrt X\), \[ d_c=d_{S,X}(c), \qquad \mu=\sqrt{d_a}, \] and let \(u\) be the positive unit mode of the star at \(a\). Define \(r_{q,X}=d_{aq}/d_a\), \(s_{q,X}=d_{a/q}/d_a\), and \[ M_{S,a,X} =4\sum_{\substack{q\le Y\\q\notin S}} \frac{r_{q,X}}{1-r_{q,X}} +\sum_{\substack{q\mid a\\q\text{ prime}}} \frac{5-s_{q,X}}{1-s_{q,X}}. \] All errors below are uniform over this range; no uniformity in \(S\) or \(\varepsilon\) is claimed. \subsection{Lemma 9.1 --- Uniform degrees, residuals, and matching}\label{lemma-9.1-uniform-degrees-residuals-and-matching} Uniformly on the stated range, \[ d_a =\frac{X}{a\log(X/a)} +O_S\!\left(\frac{X}{a(\log X)^2}\right), \qquad \mu\asymp_S\sqrt{\frac{X}{a\log X}}, \] \[ \langle u,H_{S,X}u\rangle=0, \qquad \|H_{S,X}u\|_2^2 =\frac12\pi_S(Y) +\frac12\sum_{\substack{q\le Y\\q\notin S}}r_{q,X} +\omega(a) \asymp_S\frac{\sqrt X}{\log X}. \] The degrees \(d_c\) are strictly decreasing near \(a\), consecutive allowed centres \(a_\pm\) satisfy \[ \left|\mu-\sqrt{d_{a_\pm}}\right| \ge\frac{\mu}{4a}, \] and \[ \left|\lambda_{j(a)}(A_{S,X})-\mu\right| \le\|H_{S,X}\| \le\frac{\mu}{40a} \] for all sufficiently large \(X\). Hence \(\lambda_{j(a)}\) is the unique eigenvalue matched to the star at \(a\). \subsubsection{Proof}\label{proof-17} The PNT is uniform on \([Y/a,X/a]\), whose lower endpoint is \(\sqrt X/(\log X)^{O(1)}\), and gives the degree formula. The exact first-exit decomposition of Proposition 8.1 gives the residual identity; its first-order term vanishes by the same unique-factorization argument as Lemma 7.1. The up-sum is \(O_S(\log\log X)\), while \(\omega(a)=O(\log a/\log\log a)\), so both are absorbed by \(\pi_S(Y)\). The allowed integers are periodic modulo \(P_S\), so consecutive allowed integers differ by at most \(P_S\). If \(a_+\) is the next allowed integer, subtracting the two uniform degree estimates gives \[ d_a-d_{a_+} =\frac{X(a_+-a)}{aa_+\log X} \left(1+O_S\!\left(\frac{a}{\log X}\right)\right). \] The analogous formula holds for \(d_{a_-}-d_a\). For the closest possible successor, the limiting coefficient after division by the two square roots satisfies \[ a\left(1-\sqrt{\frac{a}{a+1}}\right) =\frac1{1+1/a+\sqrt{1+1/a}} \ge1-\frac1{\sqrt2}>\frac14. \] A more distant retained successor only increases the separation, the predecessor separation is larger, and the uniform PNT error is \(o(1)\) on the logarithmic window. Hence \[ \left|\mu-\sqrt{d_{a_\pm}}\right| \ge\frac{\mu}{4a} \] for all sufficiently large \(X\), uniformly in the stated window. Since the degrees are nonincreasing, every earlier star lies above this gap and every later star below it. Finally Theorem 5.1 and \(a\le(\log X)^{1/2-\varepsilon}\) imply \(\|H_{S,X}\|=o(\mu/a)\). Weyl's inequality completes the matching. \(\square\) \subsection{Lemma 9.2 --- Uniform three-block reduction}\label{lemma-9.2-uniform-three-block-reduction} Let \(\mathcal B\) be the support of the star at \(a\). Let \(U\) be the span of the full large-prime stars centred at \[ \{aq:q\le Y,\ q\notin S\} \cup \{a/q:q\mid a,\ q\text{ prime}\}, \] isolated centres included, and let \(W=(\mathcal B\cup\operatorname{supp}U)^\perp\). Relative to \(\mathcal B\oplus U\oplus W\), \[ A_{S,X} = \begin{pmatrix} L_{\mathcal B}&K^*&0\\ K&L_U&J^*\\ 0&J&C_W \end{pmatrix}, \qquad L_U=\bigoplus_c L_c, \qquad b=Ku\in U. \] The compression to \(U\) is exactly the displayed star direct sum, and \(J\) is a subblock of \(H_{S,X}\). \subsubsection{Proof}\label{proof-18} Every small-prime neighbour of \(a\) is an \(aq\) or \(a/q\). Every small-prime neighbour of a leaf \(ap\) is an \(apq\) or \((a/q)p\); the cutoff \(apq\le X\) puts the first vertex in the star at \(aq\), and the second lies in the star at \(a/q\). Thus there is no \(\mathcal B\)--\(W\) edge. Unique factorization gives the remaining claims exactly as in Proposition 8.1 and Lemma 8.5. \(\square\) \subsection{Lemma 9.3 --- Uniform gaps and complement self-energy}\label{lemma-9.3-uniform-gaps-and-complement-self-energy} There are constants \(\rho_S<1\) and \(\delta_S>0\) such that \[ \sup_q r_{q,X}\le\rho_S, \qquad \operatorname{dist}\bigl(\mu,\sigma(L_U)\bigr)\ge\delta_S\mu \] uniformly on the stated range. If \(|z-\mu|\le\|H_{S,X}\|\), then \[ \operatorname{dist}\bigl(z,\sigma(C_W)\bigr)\ge\frac{\mu}{5a}, \] and, for \[ T(z)=J^*(z-C_W)^{-1}J, \] \[ \|T(z)\| \ll_S\frac{a^{3/2}}{\sqrt{\log X}}, \qquad \frac{\|T(z)\|}{\mu}\ll_S\frac{a^2}{\sqrt X}=o(1). \] \subsubsection{Proof}\label{proof-19} For \(q\le Y/a\), the uniform degree estimate gives \[ r_{q,X} =\frac{\log(X/a)} {q(\log(X/a)-\log q)}+o_S(1); \] for larger \(q\), \(r_{q,X}=0\). The least allowed prime is therefore the only possible extremizer, and one may take, for example, \(\rho_S=3/4\) for large \(X\). Down-stars have \(s_{q,X}=q+o(1)\ge2+o(1)\), while the kernel and negative branches are at distance at least \(\mu\). This proves the star-union gap. Let \(L_W\) be the compression of the large-prime star forest to \(W\). The centres left in \(W\) are at least one allowed step from \(a\), so monotonicity of the star degrees and Lemma 9.1 give \[ \operatorname{dist}\bigl(\mu,\sigma(L_W)\bigr)\ge\frac{\mu}{4a}. \] Weyl's inequality gives \(\operatorname{dist}(\sigma(C_W),\sigma(L_W))\le\|H_{S,X}\|\). Since also \(|z-\mu|\le\|H_{S,X}\|\) and \(\|H_{S,X}\|=o(\mu/a)\), for sufficiently large \(X\) we may take \(2\|H_{S,X}\|\le\mu/(20a)\). Hence \[ \operatorname{dist}\bigl(z,\sigma(C_W)\bigr) \ge\frac{\mu}{4a}-2\|H_{S,X}\| \ge\frac{\mu}{5a}. \] Therefore \[ \|T(z)\| \le\frac{5a\|H_{S,X}\|^2}{\mu}, \] and Theorem 5.1 gives the two displayed estimates. \(\square\) \subsection{Lemma 9.4 --- Uniform quadratic-form comparison}\label{lemma-9.4-uniform-quadratic-form-comparison} With \(R_U=(\mu-L_U)^{-1}\), \[ \lambda_{j(a)}-\mu =\langle b,R_Ub\rangle +O_S\!\left( \frac{a^{5/2}}{\sqrt X\,\sqrt{\log X}} \right), \] and \[ \bigl|\lambda_{j(a)}-\mu\bigr| \ll_S\sqrt{\frac{a}{\log X}}. \] \subsubsection{Proof}\label{proof-20} First reduce the \(W\)-block at energy \(\lambda_{j(a)}\). Lemma 9.3 keeps the reduced lower block \(L_U+T(\lambda_{j(a)})\) a fixed multiple of \(\mu\) away from the target. A second Schur reduction on \(\mathcal B\) is therefore exact. Since the star spectrum is \(\{\mu,0,-\mu\}\), ordinary isolated-eigenvalue expansion about the one-dimensional \(\mu\)-eigenspace inside this star block applies. Indeed, for \(|z-\mu|\le\|H_{S,X}\|\), put \[ \Sigma_{\mathrm{eff}}(z) =K^*\bigl(z-L_U-T(z)\bigr)^{-1}K, \] then the uniform star-union gap gives, for all sufficiently large \(X\), \[ \|\Sigma_{\mathrm{eff}}(z)\| \le \frac{2\|H_{S,X}\|^2}{\delta_S\mu} \ll_S\sqrt{\frac{a}{\log X}}. \] The mixing error is therefore \(O_S(\|\Sigma_{\mathrm{eff}}(z)\|^2/\mu) =O_S(a^{3/2}/(\sqrt X\sqrt{\log X}))\), and the expansion gives \[ \lambda_{j(a)}-\mu = \left\langle b,\bigl(\lambda_{j(a)}-L_U-T(\lambda_{j(a)})\bigr)^{-1}b \right\rangle +O_S\!\left( \frac{a^{3/2}}{\sqrt X\,\sqrt{\log X}} \right). \] The star-union gap and Lemma 9.1 make the quadratic form \(O_S(\sqrt{a/\log X})\). It remains to replace the reduced resolvent by \(R_U\). Direct inversion on each up-star and down-star, using Lemma 8.2, gives \[ \|R_Ub\|_2^2 \ll_S\frac{\pi_S(Y)+\omega(a)}{\mu^2} \ll_S\frac{a}{\sqrt X}. \] Here \(\omega(a)\le\log_2 a=o(\pi_S(Y))\) uniformly in the stated logarithmic window, so the growing number of down-stars is absorbed by the up-star contribution. Put \(D=\mu-\lambda_{j(a)}+T(\lambda_{j(a)})\). Lemma 9.3 and the preceding bound imply \(\|R_UD\|\ll_Sa^2/\sqrt X=o(1)\), so \[ (\lambda_{j(a)}-L_U-T)^{-1} =(I-R_UD)^{-1}R_U. \] Estimating the resulting quadratic-form difference by \[ \|b\|\,\|R_U\|\,\|D\|\,\|R_Ub\| \] and using \(\|b\|\ll_SX^{1/4}/\sqrt{\log X}\), \(\|R_U\|\ll_S\mu^{-1}\), and \(\|D\|\ll_Sa^{3/2}/\sqrt{\log X}\) gives the stated error. \(\square\) \subsection{Theorem 9.5 --- Uniform prime-star remainder}\label{theorem-9.5-uniform-prime-star-remainder} For fixed \(S\) and \(\varepsilon>0\), uniformly over allowed \(a\le(\log X)^{1/2-\varepsilon}\), \[ \boxed{ \lambda_{j(a)}(A_{S,X})^2-\pi_S(X/a) =M_{S,a,X} +O_{S,\varepsilon}\!\left(\frac{a^2}{\log X}\right). } \] \subsubsection{Proof}\label{proof-21} The direct-sum identities of Lemma 8.2 give \[ 2\mu\langle b,R_Ub\rangle-\pi_S(Y)=M_{S,a,X}. \] Insert Lemma 9.4 into \[ \lambda_{j(a)}^2-\mu^2 =2\mu(\lambda_{j(a)}-\mu) +(\lambda_{j(a)}-\mu)^2. \] The quadratic-form error contributes \(O_S(a^2/\log X)\); the star-block mixing and the square contribute \(O_S(a/\log X)\), which is absorbed by the former because \(a\ge1\). The implied constant is independent of the moving centre. Since \(a^2/\log X\le(\log X)^{-2\varepsilon}\), this error is also \(o_{S,\varepsilon}(1)\). Finally \(\mu^2=\pi_S(X/a)-\pi_S(Y)\). \(\square\) \subsection{Theorem 9.6 --- Uniform constant layer}\label{theorem-9.6-uniform-constant-layer} Under the same hypotheses, \[ \boxed{ \lambda_{j(a)}(A_{S,X})^2 =\pi_S(X/a)+4\log\log X+4\kappa_S+c(a) +o_{S,\varepsilon}(1). } \] \subsubsection{Proof}\label{proof-22} Put \(L=\log X\) and \(\widetilde L=\log(X/a)\). Uniform PNT estimates give, for \(q\le Y/a\), \[ r_{q,X} =\frac{\widetilde L} {q(\widetilde L-\log q)} +O_S\!\left(\frac1{qL}+\frac a{\sqrt X}\right), \] and \(r_{q,X}=0\) for \(q>Y/a\). The errors sum to \(o(1)\) uniformly. Moreover, \[ \frac{\widetilde L-\log 2}{\widetilde L-L/2} =2\,\frac{1-(\log a+\log 2)/L}{1-2\log a/L} =2+O\!\left(\frac{1+\log a}{L}\right) =2+o_{\varepsilon}(1), \] uniformly on \(a\le(\log X)^{1/2-\varepsilon}\). Thus the logarithmic increment in the partial-summation calculation of Corollary 8.9 remains \(\log 2+o_{\varepsilon}(1)\), and Mertens' theorem gives \[ \sum_{\substack{q\le Y\\q\notin S}}r_{q,X} =\log\log X+B_1-\sum_{p\in S}\frac1p+o(1). \] The displayed estimate also gives \(r_{q,X}\ll_S1/q\) uniformly. Writing \(r/(1-r)=r+r^2/(1-r)\), Lemma 9.3 therefore permits dominated convergence in the quadratic tail and produces \[ \sum_{q\notin S}\frac1{q(q-1)}. \] Thus the up-star contribution is \(4\log\log X+4\kappa_S+o(1)\). For a prime \(q\mid a\), the same uniform PNT estimate gives \[ s_{q,X} =q+O_S\!\left(\frac{q(1+\log q)}{\log X}\right). \] For \(f(s)=(5-s)/(1-s)\), one has \(f'(s)=4/(1-s)^2\). Since \(s_{q,X}\ge(q+1)/2\) uniformly for all large \(X\), \[ f(s_{q,X})-f(q) \ll_S\frac{1+\log q}{q\log X}. \] Summing over the distinct prime divisors of \(a\), and using \(\sum_{q\mid a}\log q\le\log a\), shows that the total down-star error is \(O_S(\log a/\log X)=o(1)\) uniformly on the stated range. Theorem 9.5 completes the proof. \(\square\) \subsection{Corollary 9.7 --- Uniform ordering and simplicity}\label{corollary-9.7-uniform-ordering-and-simplicity} Fix \(S\) and \(\varepsilon>0\), and put \[ J_X(\varepsilon) =\#\bigl\{a:\ a\text{ is retained and } a\le(\log X)^{1/2-\varepsilon}\bigr\}. \] For all sufficiently large \(X\), the top \(J_X(\varepsilon)\) positive eigenvalues are simple and are indexed in decreasing order by the retained centres in this window. Uniformly for retained \(a2}\frac1{p(p-1)}. \] The graph is bipartite under \((-1)^{\Omega(n)}\), so every negative extreme eigenvalue is the negative of its positive partner and satisfies the same squared law. The error in Theorem 9.5 also records the present method boundary. The single-centre first-exit reduction gives constant-order spectral precision when \(a^2/\log X=o(1)\), hence throughout \(a=o(\sqrt{\log X})\). Extending the constant-layer expansion farther would require a smaller complement-replacement error or a larger retained block; no obstruction beyond this reduction is asserted here. \subsection{9.1 Comparison with classical spectral-radius bounds}\label{comparison-with-classical-spectral-radius-bounds} For \(n\ge2\), \[ \deg_{G_{S,X}}(n) \le\pi_S(X/2)+\omega(n) \le\pi_S(X/2)+\log_2X. \] Since \(\pi_S(X)-\pi_S(X/2)\asymp X/\log X\), the root is the unique maximum-degree vertex for all sufficiently large \(X\), with \[ \Delta(G_{S,X})=\deg(1)=\pi_S(X). \] The induced root star gives the classical lower bound \(\lambda_1\ge\sqrt\Delta\). Corollary 8.9 determines its exact second-order excess: \[ \boxed{ \lambda_1-\sqrt\Delta = \frac{4\log\log X+4\kappa_S+o_S(1)} {\lambda_1+\sqrt{\pi_S(X)}} = \frac{4\log\log X+4\kappa_S+o_S(1)} {2\sqrt{\pi_S(X)}}. } \] At the other extreme, let \(N_S(X)=|V_S(X)|\) and \(m_{S,X}\) be the number of edges. Elementary summation gives \[ N_S(X)\sim\delta_SX, \qquad m_{S,X} =\sum_{\substack{p\le X\\p\notin S}}N_S(X/p) \sim\delta_SX\log\log X, \] where \(\delta_S=\prod_{p\in S}(1-1/p)\). Hong's general bound \[ \lambda_1\le\sqrt{2m_{S,X}-N_S(X)+1} \] is therefore of order \(\sqrt{X\log\log X}\), whereas the true scale is \(\lambda_1\asymp\sqrt{X/\log X}\). The generic upper bound is larger by order \(\sqrt{\log X\log\log X}\); it does not see the prime-star cancellation. \subsection{9.2 Finite illustration}\label{finite-illustration} For \(S=\varnothing\), put \[ D_{a,X} =\lambda_{j(a)}(A_X)^2-\pi(X/a)-4\log\log X-4\kappa. \] Corollary 8.9 predicts \(D_{a,X}\to c(a)\). The following contrasts test three features of the centre correction while cancelling the common constant layer: \[ D_{4,X}-D_{2,X}\longrightarrow0, \qquad D_{5,X}-D_{1,X}\longrightarrow0, \] \[ D_{6,X}-D_{2,X}-D_{3,X}+D_{1,X}\longrightarrow0. \] They respectively test insensitivity to prime-power multiplicity, the zero contribution of the prime \(5\), and additivity over distinct prime divisors. The full adjacency matrix was assembled in compressed sparse row format, and its largest eigenvalues were computed by the symmetric Lanczos method through ARPACK with tolerance \(10^{-10}\); no star-compressed eigenproblem was used. The value of \(\kappa\) cancels identically from each contrast. {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedleft\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2500}} >{\raggedleft\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2500}} >{\raggedleft\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2500}} >{\raggedleft\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2500}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedleft \(X\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(D_{4,X}-D_{2,X}\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(D_{5,X}-D_{1,X}\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(D_{6,X}-D_{2,X}-D_{3,X}+D_{1,X}\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(10^5\) & \(-0.347015\) & \(-0.758249\) & \(-0.218576\) \\ \(10^6\) & \(-0.257172\) & \(-0.706250\) & \(-0.012184\) \\ \(10^7\) & \(-0.188205\) & \(-0.596186\) & \(-0.025043\) \\ \end{longtable} } All three contrasts are closer to the predicted value zero at \(X=10^7\) than at \(X=10^5\); the additive contrast has magnitude below \(0.03\) at both \(10^6\) and \(10^7\). The individual \(D_{a,X}\) retain a visibly centre-dependent \(1/\log X\) drift. For the displayed nontrivial centres, the coefficient \((\log X)(D_{a,X}-c(a))\) is empirically approximately proportional to \(\omega(a)\). The contrasts cancel much of that drift. This is a finite-size observation, not an additional asymptotic assertion. For a direct test of the finite remainder, let \(M_{a,X}\) denote the explicit up-star plus down-star sum on the right-hand side of Theorem 8.7, and put \[ E_{a,X} =\lambda_{j(a)}(A_X)^2-\pi(X/a)-M_{a,X}. \] Theorem 8.7 gives \(E_{a,X}=O_a(1/\log X)\). The following direct numerical values test this finite-sum statement for the first two centres. The case \(a=2\) includes the nonzero down-star contribution. {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}} >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}} >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}} >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}} >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2000}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedleft \(X\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(E_{1,X}\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \((\log X)E_{1,X}\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(E_{2,X}\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \((\log X)E_{2,X}\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(10^4\) & \(1.345224\) & \(12.389966\) & \(2.103328\) & \(19.372365\) \\ \(10^5\) & \(0.990617\) & \(11.404905\) & \(1.599306\) & \(18.412693\) \\ \(10^6\) & \(0.753793\) & \(10.414028\) & \(1.248529\) & \(17.249064\) \\ \(10^7\) & \(0.609552\) & \(9.824822\) & \(1.025544\) & \(16.529817\) \\ \(10^8\) & \(0.527548\) & \(9.717795\) & \(0.884951\) & \(16.301398\) \\ \end{longtable} } These finite values illustrate the theorem's normalization; they are not used in its proof. \section{10. Conclusion}\label{conclusion} Theorem 8.7 identifies the correction to every fixed squared extreme eigenvalue as an explicit sum over up-stars and down-stars. Corollary 8.8 shows that this finite remainder controls the coherent centre--shell phase on time windows \(o(\sqrt{X\log X})\), and Corollary 8.9 evaluates it as \(4\log\log X+4\kappa_S+c(a_j)+o_{S,j}(1)\). Theorems 9.5--9.6 give the quantitative finite-sum error \(O_{S,\varepsilon}(a^2/\log X)\) and keep the evaluated constant layer uniformly through the logarithmic window \(a\le(\log X)^{1/2-\varepsilon}\). Corollary 9.7 identifies the resulting growing block as simple and ordered by its arithmetic centres. Corollaries 9.8--9.9 show that the same block is uniformly localized on its prime stars and supports exact-frequency centre--shell evolution, uniformly for all real times; on the stated growing time window, its frequency may be replaced by the explicit first-exit expression \(\sqrt{\pi_S(X/a)+M_{S,a,X}}\). Corollary 9.10 packages these individual estimates as operator-norm convergence of the whole logarithmic-window spectral projection to the corresponding prime-star projection. Thus the arithmetic cutoff produces an exact prime-star boundary, first-order coupling vanishes, and the surviving self-energy is evaluated through constant order. The extreme spectrum is therefore determined by the star degrees, the universal up-star term, and the divisor correction \(c(a_j)\). The proved uniform range is the family of fixed-power windows \(a\le(\log X)^{1/2-\varepsilon}\) with fixed \(\varepsilon>0\). The larger condition \(a=o(\sqrt{\log X})\) is suggested by the scalar remainder but is not asserted here; neither range is claimed as a barrier for other spectral methods. \section*{Declarations and code availability}\label{declarations-and-code-availability} \addcontentsline{toc}{section}{Declarations and code availability} \subsection*{Formalization and software availability}\label{formalization-and-software-availability} \addcontentsline{toc}{subsection}{Formalization and software availability} The accompanying Lean 4 development formalizes three selected results from the manuscript. Their registered declarations are \[ \begin{aligned} &\texttt{PrimeStar.palomar\_uniform\_finite\_prime\_sum},\\ &\texttt{PrimeStar.palomar\_top\_finite\_prime\_sum},\\ &\texttt{PrimeStar.palomar\_uniform\_explicit\_frequency\_dynamics}. \end{aligned} \] The first is the uniform finite-prime-sum form of Theorem 9.6 on \(a\le(\log X)^{1/2-\varepsilon}\); the second is its \(S=\varnothing\), \(a=1\) largest-eigenvalue specialization displayed in the abstract; and the third is the finite localization-plus-phase estimate at frequency \(\sqrt{\pi_S(X/a)+M_{S,a,X}}\) underlying Corollary 9.9. The development also contains supporting results for the star decomposition, spectral matching, first-exit identities, eigenvector localization, bipartite pairing, and exact-frequency adjacency evolution. Corollary 9.10's collective projection statement is proved in the manuscript but is not one of the three declarations selected by the Palomar record. The formalization is registered as \href{https://palomar-registry.org/entry?id=PALOMAR-2026-08-29-000017\&version=2}{PALOMAR-2026-08-29-000017, version 2} {[}22{]}. The registered source is the public repository \url{https://github.com/shaikidris/prime-star-spectra-formalization} at immutable commit \path{fe3fe9fa41f5c69d9e1e16abc6da144511f778b0}; Palomar also preserves that source and its pinned dependencies. The project uses Lean 4.32.0, Mathlib commit \path{3dffaf2f18b47d11948f6390838ea6f2ae662aaf}, and \path{PrimeNumberTheoremAnd} commit \path{7715064f690d0689f30889846f4e2c5e7ec0c47e}. A local reproduction is \begin{Shaded} \begin{Highlighting}[] \NormalTok{lake exe cache get} \NormalTok{lake build} \NormalTok{lake build PrimeCoverChallenge PrimeCoverSolution} \NormalTok{lake build PrimeStar.Main PrimeStar.Audit} \end{Highlighting} \end{Shaded} Palomar's registered mechanical verification reports only \path{propext}, \path{Classical.choice}, and \path{Quot.sound} for the selected roots. The Lean source is licensed under Apache-2.0; the paper is licensed separately under CC BY 4.0. \subsection*{Declaration of competing interest}\label{declaration-of-competing-interest} \addcontentsline{toc}{subsection}{Declaration of competing interest} The author declares no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. \subsection*{Funding}\label{funding} \addcontentsline{toc}{subsection}{Funding} This research received no external funding. \subsection*{Data availability}\label{data-availability} \addcontentsline{toc}{subsection}{Data availability} No external datasets were used. The finite numerical outputs in Section 9 are reproducible from the scripts in the source repository and are not used as premises of any theorem. \subsection*{Declaration of generative AI and AI-assisted technologies in the manuscript preparation process}\label{declaration-of-generative-ai-and-ai-assisted-technologies-in-the-manuscript-preparation-process} \addcontentsline{toc}{subsection}{Declaration of generative AI and AI-assisted technologies in the manuscript preparation process} During preparation, the author used ChatGPT and OpenAI Codex for proof exploration, manuscript editing, diagnostic code, and Lean 4 formalization. The author reviewed and edited all outputs and takes responsibility for the content of the manuscript. \section*{References}\label{references} \addcontentsline{toc}{section}{References} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item F. Benaych-Georges, C. Bordenave, and A. 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