% Options for packages loaded elsewhere \PassOptionsToPackage{unicode}{hyperref} \PassOptionsToPackage{hyphens}{url} \documentclass[ english, a4paper, ]{article} \usepackage{xcolor} \usepackage[margin=28mm]{geometry} \usepackage{amsmath,amssymb} \setcounter{secnumdepth}{1} \usepackage{iftex} \ifPDFTeX \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{textcomp} % provide euro and other symbols \else % if luatex or xetex \usepackage{unicode-math} % this also loads fontspec \defaultfontfeatures{Scale=MatchLowercase} \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1} \fi \usepackage{lmodern} \ifPDFTeX\else % xetex/luatex font selection \fi % Use upquote if available, for straight quotes in verbatim environments \IfFileExists{upquote.sty}{\usepackage{upquote}}{} \IfFileExists{microtype.sty}{% use microtype if available \usepackage[]{microtype} \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts }{} \makeatletter \@ifundefined{KOMAClassName}{% if non-KOMA class \IfFileExists{parskip.sty}{% \usepackage{parskip} }{% else \setlength{\parindent}{0pt} \setlength{\parskip}{6pt plus 2pt minus 1pt}} }{% if KOMA class \KOMAoptions{parskip=half}} \makeatother \usepackage{longtable,booktabs,array} \newcounter{none} % for unnumbered tables \usepackage{calc} % for calculating minipage widths % Correct order of tables after \paragraph or \subparagraph \usepackage{etoolbox} \makeatletter \patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{} \makeatother % Allow footnotes in longtable head/foot \IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}} \makesavenoteenv{longtable} \usepackage{graphicx} \makeatletter \newsavebox\pandoc@box \newcommand*\pandocbounded[1]{% scales image to fit in text height/width \sbox\pandoc@box{#1}% \Gscale@div\@tempa{\textheight}{\dimexpr\ht\pandoc@box+\dp\pandoc@box\relax}% \Gscale@div\@tempb{\linewidth}{\wd\pandoc@box}% \ifdim\@tempb\p@<\@tempa\p@\let\@tempa\@tempb\fi% select the smaller of both \ifdim\@tempa\p@<\p@\scalebox{\@tempa}{\usebox\pandoc@box}% \else\usebox{\pandoc@box}% \fi% } % Set default figure placement to htbp \def\fps@figure{htbp} \makeatother \ifLuaTeX \usepackage[bidi=basic,shorthands=off]{babel} \else \usepackage[bidi=default,shorthands=off]{babel} \fi \ifLuaTeX \usepackage{selnolig} % disable illegal ligatures \fi \setlength{\emergencystretch}{3em} % prevent overfull lines \providecommand{\tightlist}{% \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \usepackage{mathrsfs} \usepackage{bookmark} \IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available \urlstyle{same} % fallback for those not using the hyperref driver hyperxmp: \makeatletter \@ifundefined{xmpquote}{\newcommand{\xmpquote}[1]{#1}}{} \makeatother \hypersetup{ pdftitle={Prescribed Eigenvalues in Power Bands of Finite Prime-Cover Graphs}, pdfauthor={Idris Ali Shaik}, pdflang={en}, pdfkeywords={\xmpquote{prime-cover graph}, \xmpquote{ordered eigenvalues}, \xmpquote{spectral graph theory}, \xmpquote{prime number theorem}, \xmpquote{density-one estimates}}, hidelinks, pdfcreator={LaTeX via pandoc}} \title{Prescribed Eigenvalues in Power Bands of Finite Prime-Cover Graphs} \author{Idris Ali Shaik\thanks{Independent researcher, Dubai, United Arab Emirates. Email: \href{mailto:shaikidris@apache.org}{shaikidris@apache.org}. ORCID: \href{https://orcid.org/0009-0009-9699-9712}{0009-0009-9699-9712}.}} \date{} \begin{document} \maketitle \section*{Abstract}\label{abstract} \addcontentsline{toc}{section}{Abstract} Let \(G_X\) be the graph on the positive integers at most \(X\), with an edge between \(n\) and \(np\) whenever \(p\) is prime and \(np\le X\). Write \(\lambda_1(X)\ge\lambda_2(X)\ge\cdots>0\) for its positive adjacency eigenvalues, counted with multiplicity. We prove that, for every fixed \(21/61<\theta<1/2\) and every integer \(X^\theta/\log X\le a\le X^\theta\), \[ \lambda_a(X)^2=\pi(X/a)+O_\theta\!\left(\frac a{\log X}\right), \] where \(\pi\) is the prime-counting function. This identifies the eigenvalue at the prescribed rank \(a\) throughout a polynomially growing band of centres. For every \(0<\theta<1/2\), ordered mean-square bounds give a weaker estimate outside \(o(X^\theta)\) centres below \(X^\theta\). When \(1/3<\theta<1/2\), the same \(O_\theta(a/\log X)\) error holds outside \(o(X^\theta)\) centres using only global prime estimates. The results extend to deletion of any fixed finite set of prime directions. Edges labelled by primes exceeding \(\sqrt X\) form disjoint stars. Retaining their first-exit interactions gives local comparison operators and a family of approximate eigenvectors. Bounds on their overlaps and residuals control eigenvalue counts across clusters, allowing ranks to be recovered where the global perturbation bound no longer separates neighbouring star levels. The Baker--Harman--Pintz lower bound and the Brun--Titchmarsh upper bound supply the short-interval prime counts used in the uniform every-centre theorem. \section{Introduction}\label{introduction} For \(X\ge1\), we study the graph on the positive integers at most \(X\) in which two vertices are joined when their ratio is prime. Its edges are \[ n\longleftrightarrow np, \qquad p\text{ prime},\quad np\le X. \tag{1.1} \] These are the cover relations of divisibility, so the graph is the Hasse diagram of the divisibility poset on \(\{1,\ldots,\lfloor X\rfloor\}\). Splitting the edges at \(Y=\lfloor\sqrt X\rfloor\) gives an explicit reference spectrum. An integer at most \(X\) cannot contain two distinct prime factors larger than \(Y\), so the edges labelled by \(p>Y\) form disjoint stars. The star centred at \(a\le Y\) has degree \[ d_{a,X}=\pi(X/a)-\pi(Y) \tag{1.2} \] and positive eigenvalue \(\mu_a=\sqrt{d_{a,X}}\). Writing \(j(a)=a\) for the arithmetic rank in the undeleted graph, this suggests the ordered profile \[ \lambda_{j(a)}(X)^2\approx \pi(X/a). \tag{1.3} \] The companion paper {[}15{]} treats fixed centres. The first-order interaction of a star mode with the small-prime adjacency vanishes, allowing the remaining eigenvalue correction to be evaluated through constant order. That work also proves localization of the corresponding eigenvectors. Here the centre grows as a power of \(X\), and we seek the eigenvalue at its prescribed global rank. Comparing the smooth adjacent-star gap \(\mu_a/a\) with the global perturbation norm gives the scale \(a\asymp X^{1/6}\). Beyond this scale, that comparison no longer establishes separation of neighbouring levels. We recover the ranks by comparing families of approximate eigenvectors and counting eigenvalues across spectral clusters. We state the result with a fixed finite set \(S\) of deleted primes. Put \[ V_S(X)=\{n\le X:p\nmid n\text{ for every }p\in S\}, \] and let \(A_{S,X}\) be the adjacency matrix of (1.1) on \(V_S(X)\), with only prime labels outside \(S\). Write \[ \pi_S(t)=\#\{p\le t:p\text{ prime},\ p\notin S\}, \] and enumerate the allowed positive integers as \(a_10\). For \(Y=\lfloor\sqrt X\rfloor\), define \[ d_{S,X}(c)=\pi_S(X/c)-\pi_S(Y) \tag{1.5} \] when \(c\le Y\), and set it equal to zero when the corresponding star is inactive. For an active centre \(a\), put \[ r_{q,X}(a)=\frac{d_{S,X}(aq)}{d_{S,X}(a)}, \qquad s_{q,X}(a)=\frac{d_{S,X}(a/q)}{d_{S,X}(a)}, \tag{1.6} \] where \(q\le Y\) is an allowed prime in the first expression and \(q\mid a\) is prime in the second. Inactive up-stars have \(r_{q,X}(a)=0\). For the ranges below, all nonzero denominators are separated from zero once \(X\) is sufficiently large. Define the first-exit correction \[ \boxed{ M_{S,a,X} =4\sum_{\substack{q\le Y\\q\notin S}} \frac{r_{q,X}(a)}{1-r_{q,X}(a)} +\sum_{\substack{q\mid a\\q\ {\rm prime}}} \frac{5-s_{q,X}(a)}{1-s_{q,X}(a)}.} \tag{1.7} \] Our main theorem identifies every prescribed ordered eigenvalue in a terminal power band below the square-root boundary. \subsection{Theorem 1.1 -- prescribed eigenvalues in a terminal power band}\label{theorem-1.1-prescribed-eigenvalues-in-a-terminal-power-band} Fix a finite set \(S\) of primes and \[ \frac{21}{61}<\theta<\frac12. \] There is a constant \(C_{S,\theta}\) such that, for all sufficiently large \(X\), every allowed centre \[ \frac{X^\theta}{\log X}\le a\le X^\theta \] obeys \[ \boxed{ \left|\lambda_{j_S(a)}(A_{S,X})^2 -\pi_S(X/a)-M_{S,a,X}\right| \le C_{S,\theta}\,\frac a{\log X}.} \tag{1.8} \] Since \(M_{S,a,X}\ll_{S,\theta}\log X=o(a/\log X)\) uniformly on the terminal band, the correction is smaller than the stated error. The same estimate therefore has the following equivalent form. \subsection{Corollary 1.2}\label{corollary-1.2} Under the hypotheses of Theorem 1.1, every allowed centre in the terminal band satisfies \[ \boxed{ \lambda_{j_S(a)}(A_{S,X})^2 =\pi_S(X/a) +O_{S,\theta}\!\left(\frac a{\log X}\right).} \tag{1.9} \] Among all allowed centres \(a\le X^\theta\), the only centres not covered by (1.8)--(1.9) lie in the explicit initial segment \(a0\), \(D\) may be chosen depending on \(S,\theta,R\) so that the terminal-band exceptions number \(O_{S,\theta,R}(X^\theta/(\log X)^R)\). Counting all centres below \(X^\theta\) additionally omits the initial segment \(a{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Estimate \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Centres covered \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Squared-error scale on \(a\asymp X^\theta\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot Raw Weyl, Proposition 2.3 & Every centre, \(0<\theta<1/2\) & \(X^{3/4-\theta/2}/\log X\) \\ Positive-star block, Proposition 2.4 & Every terminal-band centre, \(0<\theta<1/2\) & \(\sqrt X/\log X\) \\ Theorem 1.1 & Every terminal-band centre, \(21/61<\theta<1/2\) & \(X^\theta/\log X\) \\ Theorem 1.3 & Density one, \(0<\theta<1/2\) & \(X^\theta/\log X+(\log X)^D X^{(1-\theta)/2}/\sqrt{\log X}\) \\ Corollary 10.3 & Every terminal-band centre, \(0<\theta<1/2\) & \(X^\theta/\log X+(\log X)^{C'}[X^{(1-\theta)/2}+X^{1/4+\theta/2}]/\sqrt{\log X}\) \\ \end{longtable} } Theorem 1.3 improves the positive-star block scale by a positive power for every fixed \(0<\theta<1/2\), as quantified below. Corollary 10.3 gives a weaker every-centre improvement on the same full range. \subsection{Corollary 1.4 -- density-one improvement over the block baseline}\label{corollary-1.4-density-one-improvement-over-the-block-baseline} Put \[ \delta_0=\min\!\left\{ \frac12-\theta,\quad \frac\theta2 \right\}>0. \] Then, for every \(0<\delta<\delta_0\), outside \(o_{S,\theta}(X^\theta)\) allowed centres \(a\le X^\theta\), \[ \boxed{ \left|\lambda_{j_S(a)}(A_{S,X})^2 -\pi_S(X/a)-M_{S,a,X}\right| \ll_{S,\theta,\delta} X^{1/2-\delta}.} \tag{1.12} \] The exponent comparison follows from Theorem 1.3 and is proved in Section 10. Equation (1.8) identifies the eigenvalue at the specific position \(j_S(a)\). We recover this rank by comparing eigenvalue counts across clusters with the ordered energies of the local comparison operators, called molecules below. Two-sided short-interval prime counts control the arithmetic ordering at the required width. The target width also explains why this collective step is necessary. On a dyadic band \(a\asymp B\), the smooth PNT profile has mean spacing between its squared targets \[ \asymp_{S,\theta}\frac{X}{B^2\log X}, \] whereas the error in (1.8) is \(B/\log X\). Thus one theorem-sized window contains approximately \[ K_B=\frac{B^3}{X} \] model levels. For \(B=X^{\theta+o(1)}\) with \(\theta>1/3\), this number tends to infinity. The error windows overlap, so we use threshold counts across clusters of \(K_B\) labels to recover the prescribed ranks. The mean spacing here is the derivative scale of the smooth PNT profile; consecutive discrete prime-counting targets may coincide. The same argument has a counting form. Corollary 9.4 shows that a buffered interval of terminal-band centres and its corresponding arithmetic energy window contain the same number of modes up to \(O(B^3/X)\) boundary labels. The proof begins with a two-stage Schur comparison between the full graph and the one-exit compression (Section 3). This comparison holds throughout \(a\le X^{1/2-\varepsilon}\). Section 4 constructs the local molecules and evaluates their energies. Sections 5--6 control the Gram matrix and residuals of the molecule family, allowing the vectors to be used together. The proof then divides into two parts. Sections 7--9 establish the sharp every-centre theorem by eliminating spectrally distant modes and bounding the number of internal eigenvalues that can cross a cluster threshold. Short-interval prime counts supply enough arithmetic labels on either side to determine the rank. Section 10 proves the density-one theorem directly from an ordered mean-square comparison above a separated complement. The arithmetic input determines the lower threshold in Theorem 1.1. At the target width, the relevant prime interval has base \(x\asymp X^{1-\theta}\) and length \(h\asymp X^\theta\). The Brun--Titchmarsh upper bound {[}12{]} and the Baker--Harman--Pintz lower bound \(\pi(x)-\pi(x-x^{21/40})\gg x^{21/40}/\log x\) {[}16{]} give the two-sided count used in the proof. The strict inequality \(\theta/(1-\theta)>21/40\) gives \(\theta>21/61\). Using the Guth--Maynard asymptotic {[}11{]} at exponent \(17/30\) would instead require \(\theta>17/47\). More generally, a positive-proportion prime lower bound in intervals of length \(x^{\kappa+\varepsilon}\) gives the arithmetic condition \(\theta>\kappa/(1+\kappa)\). The cluster count also requires \(\theta>1/3\), since \(K_B=B^3/X\) must tend to infinity. The density-one theorem uses global prime estimates and sorting contraction, which accommodates repeated prime-counting targets. Cover-matrix spectra of structured product posets were studied by Anđelić and da Fonseca {[}1{]}. Related divisibility graphs usually retain every comparable pair or restrict to the divisor lattice of one integer {[}2,3{]}; the present graph is the cover graph of the radial truncation \(([1,X],\mid)\). Hub-controlled extreme spectra occur in sparse random graphs {[}4,5{]}. Alt, Ducatez, and Knowles prove Poisson statistics and exponential localization at the spectral edge of sparse Erdős--Rényi graphs {[}6{]}, while Frieze and Tsourakakis obtain a star-forest decomposition in a random Apollonian model {[}7{]}. The present result is deterministic and labels a growing family of ordered modes. Helfgott and Radziwiłł {[}8{]} study a different prime-divisibility operator with additive moves and a global expansion objective. The Schur complement, min--max principle, and inertia arguments used below are standard finite-dimensional tools {[}9,10{]}. Kline's sparse Redheffer descendant is an arithmetic matrix with \(O(X)\) nonzero entries. Its determinant represents the Mertens function, and its two dominant eigenvalues have leading scale \(\pm\sqrt{\pi(X)}\) with a logarithmic correction {[}13,14{]}. That operator retains a distinguished Redheffer interaction. The present paper treats the symmetric zero-diagonal cover adjacency and identifies a family of order \(X^\theta\) ordered positive eigenvalues. A companion Lean 4 formalization of Theorem 1.3 and Corollary 1.4 is registered in Palomar {[}17, version 1{]}. Its scope is described in the formal-verification section. \section{The large-prime forest and the small-prime norm}\label{the-large-prime-forest-and-the-small-prime-norm} Let \(L_{S,X}\) contain the edges labelled by primes \(p>Y\), and put \[ A_{S,X}=L_{S,X}+H_{S,X}. \tag{2.1} \] Figure 1 shows the split for \(X=16\): solid large-prime edges form the disjoint stars, while dashed small-prime edges comprise the remainder. \begin{figure} \centering \includegraphics[width=0.82\linewidth,height=\textheight,keepaspectratio,alt={The prime-cover graph at X=16, split at Y=4. Solid edges belong to L\_\{S,X\}; dashed edges belong to H\_\{S,X\}.}]{figures/fig-cover-split.pdf} \caption{The prime-cover graph at \(X=16\), split at \(Y=4\). Solid edges belong to \(L_{S,X}\); dashed edges belong to \(H_{S,X}\).} \end{figure} \subsection{Proposition 2.1 -- exact star decomposition}\label{proposition-2.1-exact-star-decomposition} The graph with adjacency \(L_{S,X}\) is a disjoint union of stars. The nonempty star centred at \(a\le Y\) has degree \(d_{S,X}(a)\), and its two nonzero normalized eigenvectors are \[ u_a^\pm =\frac1{\sqrt2}\left( \delta_a\pm\frac1{\sqrt{d_{S,X}(a)}} \sum_{\substack{p>Y,\ p\notin S\\ap\le X}}\delta_{ap} \right), \tag{2.2} \] with eigenvalues \(\pm\mu_a\), where \(\mu_a=\sqrt{d_{S,X}(a)}\). \textbf{Proof.} An edge labelled by \(p>Y\) has lower endpoint at most \(Y\) and upper endpoint above \(Y\). An integer at most \(X\) cannot have two distinct prime divisors larger than \(Y\). Hence every large-prime leaf has a unique centre and cannot itself be a centre. The components are stars, and (2.2) is the standard spectrum of \(K_{1,d}\). \(\square\) The remaining graph has the sharp norm scale needed throughout the paper. \subsection{Proposition 2.2 -- small-prime norm}\label{proposition-2.2-small-prime-norm} For fixed \(S\), \[ \|H_{S,X}\|\asymp_S\frac{X^{1/4}}{\sqrt{\log X}}. \tag{2.3} \] \textbf{Proof.} The principal subgraph on the root and the allowed primes at most \(Y\) is \(K_{1,\pi_S(Y)}\), which gives the lower bound, using the prime number theorem {[}12{]}. For the upper bound use the positive Schur weight \(w(n)=n^{-\alpha}\). Uniformly for \(\alpha\) near \(1/2\), partial summation and Chebyshev's estimate {[}12{]} give \[ \sum_{p\le Y}p^{-\alpha}\ll\frac{Y^{1-\alpha}}{\log Y}. \tag{2.4} \] For \(n\le X<(Y+1)^2\), the incoming row sum is \(\sum_{p\mid n,\,p\le Y}p^\alpha\). Prime divisors at most \(Y^{1/2}\) contribute \(o(Y^\alpha)\). For \(Y\ge4\), at most four distinct prime divisors of \(n\) exceed \(Y^{1/2}\): five would have product greater than \(Y^{5/2}\ge(Y+1)^2>X\). Thus the incoming sum is \(O(Y^\alpha)\). With \[ \alpha=\frac12-\frac{\log\log Y}{2\log Y}, \] both row contributions are \(O(Y^{1/2}/\sqrt{\log Y})\). The weighted Schur test proves the upper bound. \(\square\) We shall repeatedly use, uniformly for \(a\le X^{1/2-\varepsilon}\), \[ \mu_a^2\asymp_{S,\varepsilon} \frac{X}{a\log(X/a)}, \qquad \|H_{S,X}\|=o_{S,\varepsilon}(\mu_a). \tag{2.5} \] These follow from the prime number theorem and (2.3). In particular, the min--max principle applied to the positive-star subspace shows that all indices \(j_S(a)\) used below correspond to positive eigenvalues once \(X\) is sufficiently large. \subsection{Proposition 2.3 -- Weyl baseline}\label{proposition-2.3-weyl-baseline} Fix a finite set \(S\) of primes and \(0<\theta<1/2\). Uniformly for allowed centres \(a\le X^\theta\), \[ \left|\lambda_{j_S(a)}(A_{S,X})^2-\pi_S(X/a)\right| \ll_{S,\theta} \frac{X^{3/4}}{\sqrt a\,\log X} +\frac{X^{1/2}}{\log X}. \tag{2.6} \] Consequently the relative estimate (1.11) holds. \textbf{Proof.} Since \(a\mapsto d_{S,X}(a)\) is nonincreasing, the positive large-prime eigenvalue at arithmetic rank \(j_S(a)\) is \(\mu_a\). Weyl's inequality and the exact identity \[ d_{S,X}(a)=\pi_S(X/a)-\pi_S(Y) \] give \[ \begin{aligned} |\lambda_{j_S(a)}(A_{S,X})^2-\pi_S(X/a)| &\le \|H_{S,X}\|(2\mu_a+\|H_{S,X}\|)+\pi_S(Y). \end{aligned} \tag{2.7} \] Combining (2.3), (2.5), and the PNT proves (2.6). After division by \(\pi_S(X/a)\), the dominant term is \(\|H_{S,X}\|/\mu_a\ll_{S,\theta}X^{-(1-2\theta)/4}\); the squared-norm and boundary terms are \(O_{S,\theta}(X^{-1/2+\theta})\). This proves (1.11). \(\square\) \subsection{Proposition 2.4 -- positive-star block baseline}\label{proposition-2.4-positive-star-block-baseline} Fix a finite set \(S\) of primes and \(0<\theta<1/2\). Uniformly for allowed centres \(X^\theta/\log X\le a\le X^\theta\), \[ \boxed{ |\lambda_{j_S(a)}(A_{S,X})^2-\pi_S(X/a)| \ll_{S,\theta}\frac{\sqrt X}{\log X}.} \tag{2.8} \] \textbf{Proof.} Choose a sufficiently large fixed \(C_\star\), put \(Z=\lceil C_\star a\rceil\), and let \(P_Z\) project onto the positive large-prime modes with allowed centres \(c\le Z\). All these stars are active for large \(X\). The compression is \(P_ZA_{S,X}P_Z=\Lambda_Z+W_Z\), with \(\Lambda_Z=\operatorname{diag}(\mu_c)\). Unique factorization gives \[ (W_Z)_{c,cq}=\frac{1+\sqrt{d_{S,X}(cq)/d_{S,X}(c)}}2\le1 \qquad(cq\le Z), \tag{2.9} \] for allowed primes \(q\), and every other off-diagonal entry is zero. The diagonal is zero. Entrywise domination by the prime-cover adjacency on \(V_S(Z)\), followed by the large/small-prime split at size \(Z\), therefore gives \[ \|W_Z\|\le\|A_{S,Z}\| \ll_S\sqrt{\frac Z{\log Z}} \ll_{S,\theta}\sqrt{\frac a{\log X}}. \tag{2.10} \] Here nonnegative symmetric matrices satisfy this norm domination by the Rayleigh principle applied to coordinatewise absolute values. The bound on \(A_{S,Z}\) uses \(\|L_{S,Z}\|\le\sqrt{\pi_S(Z)}\), Proposition 2.2 at size \(Z\), and the PNT; it uses no molecule estimate. Let \(\tau_j\) be the ordered eigenvalues of \(\Lambda_Z+W_Z\). Monotonicity of the star degrees and Weyl give \(|\tau_{j_S(a)}-\mu_a|\le\|W_Z\|=o(\mu_a)\). On \(P_Z^\perp\), the forest has positive top at most \(\mu_b\), where \(b\) is the first allowed centre above \(Z\); all other sectors are nonpositive. Uniform PNT gives \(\mu_b^2/\mu_a^2\le C_\star^{-1}+o(1)\). Consequently the full adjacency complement has top below \(\tau_{j_S(a)}-c\mu_a\), after increasing \(C_\star\), since \(\|H_{S,X}\|=o(\mu_a)\). Its coupling to \(P_Z\) has norm at most \(\|H_{S,X}\|\). The finite-dimensional Schur comparison in Lemma 5.2 now yields \[ |\lambda_{j_S(a)}(A_{S,X})-\mu_a| \le\|W_Z\|+O_{S,\theta}(\|H_{S,X}\|^2/\mu_a) \ll_{S,\theta}\sqrt{\frac a{\log X}}. \tag{2.11} \] Squaring costs \(O_{S,\theta}(\sqrt X/\log X+a/\log X)\). Finally \(\mu_a^2=\pi_S(X/a)-\pi_S(Y)\) and \(\pi_S(Y)\ll_S\sqrt X/\log X\), proving (2.8). \(\square\) \section{The exact one-exit compression}\label{the-exact-one-exit-compression} Let \(P=P_+\) be the orthogonal projection onto the span of all positive star modes \(u_a^+\), and put \(Q=I-P\). Define \[ C_0=QL_{S,X}Q, \qquad K=QH_{S,X}P. \tag{3.1} \] Let \(\mathcal E\) be the smallest reducing subspace of the self-adjoint operator \(C_0\) containing \({\rm ran}\,K\). The one-exit space and its compressed adjacency are \[ \mathcal R=P\ell^2(V_S(X))\oplus\mathcal E, \qquad G_{S,X}=P_{\mathcal R}A_{S,X}|_{\mathcal R}. \tag{3.2} \] This space keeps every positive star mode and exactly the nonpositive large-prime spectral components reached in one small-prime step. It is a proper compression of the full graph: most mean-zero leaf modes of a large star are not charged by \(K\). We first record the linear-algebra comparison that transfers results for \(G_{S,X}\) back to \(A_{S,X}\). \subsection{Lemma 3.1 -- two-stage Schur comparison}\label{lemma-3.1-two-stage-schur-comparison} Let \[ \mathcal A= \begin{pmatrix}T&K^*&0\\K&D&R^*\\0&R&C\end{pmatrix}, \qquad \mathcal G=\begin{pmatrix}T&K^*\\K&D\end{pmatrix} \tag{3.3} \] be Hermitian, and suppose \[ \lambda_{\max}(D),\lambda_{\max}(C),\|K\|,\|R\|\le B. \] If the \(j\)-th eigenvalue \(\rho_j\) of \(\mathcal G\) satisfies \(\rho_j\ge4B\), then \[ 0\le\lambda_j(\mathcal A)^2-\rho_j^2 \le 7\frac{B^4}{\rho_j^2}. \tag{3.4} \] \textbf{Proof.} Interlacing gives the lower bound. Put \[ \delta=3B^4/\rho_j^3, \qquad z=\rho_j+\delta, \qquad s=B^2/(z-B). \] Eliminating the last block at energy \(z\) adds to the \(D\)-block a positive operator bounded above by \(sI\). Eliminate the \(D\)-block in this upper comparison and compare it with the Schur form of \(\mathcal G-\rho_j\). If \(s\le\delta\), the change of self-energy is nonpositive. If \(s>\delta\), the resolvent identity gives \[ \left\|K^*\bigl((z-s-D)^{-1}-(\rho_j-D)^{-1}\bigr)K\right\| \le\frac{8B^4}{3\rho_j^3}<\delta. \] Sylvester inertia therefore shows that \(\mathcal A\) has at most \(j-1\) eigenvalues above \(z\), so \(0\le\lambda_j(\mathcal A)-\rho_j\le\delta\). Since \(\rho_j\ge4B\), squaring gives (3.4). \(\square\) \subsection{Theorem 3.2 -- transfer from one exit to the full graph}\label{theorem-3.2-transfer-from-one-exit-to-the-full-graph} Let \(\rho_1\ge\rho_2\ge\cdots>0\) be the positive eigenvalues of \(G_{S,X}\). For fixed \(\varepsilon>0\), uniformly whenever \(a_j\le X^{1/2-\varepsilon}\), \[ \boxed{ 0\le\lambda_j(A_{S,X})^2-\rho_j^2 \ll_{S,\varepsilon} \frac{a_j\log(X/a_j)}{(\log X)^2} \ll_{S,\varepsilon}\frac{a_j}{\log X}.} \tag{3.5} \] \textbf{Proof.} With \(\mathcal F=Q\ell^2\ominus\mathcal E\), the decomposition \(P\oplus\mathcal E\oplus\mathcal F\) has the form (3.3). The \(P\)-to-\(\mathcal F\) block vanishes by the definition of \(\mathcal E\), and all remaining off-diagonal blocks have norm at most \(B_X=\|H_{S,X}\|\). The \(D\)- and \(C\)-blocks have top eigenvalue at most \(B_X\), since \(C_0\le0\). Interlacing with the positive-star compression gives \(\rho_j\ge\mu_{a_j}-B_X\). Equations (2.3)--(2.5) imply \(\rho_j\ge4B_X\) uniformly in the stated range. Lemma 3.1 yields \[ \lambda_j^2-\rho_j^2 \ll\frac{B_X^4}{\mu_{a_j}^2} \ll_{S,\varepsilon} \frac{a_j\log(X/a_j)}{(\log X)^2}, \] which is (3.5). \(\square\) \subsection{Corollary 3.3 -- uniform relative one-exit comparison}\label{corollary-3.3-uniform-relative-one-exit-comparison} Fix \(\varepsilon>0\). Uniformly over the indices whose arithmetic centres satisfy \(a_j\le X^{1/2-\varepsilon}\), \[ \boxed{ \sup_{a_j\le X^{1/2-\varepsilon}} \frac{\left|\lambda_j(A_{S,X})^2-\rho_j(G_{S,X})^2\right|} {\pi_S(X/a_j)}\longrightarrow0.} \tag{3.6} \] In particular, (3.6) holds simultaneously for every \(a_j\le X^\theta\) when \(\theta<1/2\) is fixed. \textbf{Proof.} Theorem 3.2 and the prime number theorem give, uniformly in the stated range, \[ \frac{\left|\lambda_j^2-\rho_j^2\right|}{\pi_S(X/a_j)} \ll_S \frac{a_j^2\log^2(X/a_j)}{X(\log X)^2} \ll_{S,\varepsilon}X^{-2\varepsilon}=o(1). \] This proves (3.6). \(\square\) \section{The first-exit correction and principal molecules}\label{the-first-exit-correction-and-principal-molecules} Fix an active centre \(a\). Put \(u=u_a^+\), \(\mu=\mu_a\), \[ h_a=P H_{S,X}u, \qquad k_a=Q H_{S,X}u. \tag{4.1} \] Every component of \(h_a\oplus k_a\) lies in a large-prime star reached from the \(a\)-star by multiplying or dividing the centre by one allowed small prime. Distinct targets are orthogonal. If \(q\le Y\) is an allowed prime, the up-target is \(aq\); if \(q\mid a\), the down-target is \(a/q\). \begin{figure} \centering \includegraphics[width=0.7\linewidth,height=\textheight,keepaspectratio,alt={The two first-exit families from the star at a. Multiplication by q reaches the up-star at aq; division by a prime q\textbackslash mid a reaches the down-star at a/q.}]{figures/fig-first-exit.pdf} \caption{The two first-exit families from the star at \(a\). Multiplication by \(q\) reaches the up-star at \(aq\); division by a prime \(q\mid a\) reaches the down-star at \(a/q\).} \end{figure} \subsection{Lemma 4.1 -- exact local self-energy}\label{lemma-4.1-exact-local-self-energy} Let \(D_+\) be the restriction of the positive-star energy operator to the target modes reached from the \(a\)-star, excluding \(u_a^+\). The vector \(h_a\) belongs to this target space. All inverses involving \(\mu-D_+\) below are taken on that space, not on the full positive sector, where \(\mu\) is an eigenvalue. Retain \(C_0\) on the nonpositive sector. For the sub-square-root ranges used below the target energies are separated from \(\mu\), and \[ \boxed{ 2\mu\left( \langle h_a,(\mu-D_+)^{-1}h_a\rangle +\langle k_a,(\mu-C_0)^{-1}k_a\rangle \right)-\pi_S(Y) =M_{S,a,X}.} \tag{4.2} \] \textbf{Proof.} For an active up-target put \(r=d_{S,X}(aq)/d_{S,X}(a)\). Its positive and negative coefficients are \[ \frac{1+\sqrt r}{2}, \qquad \frac{1-\sqrt r}{2}, \] at energies \(+\mu\sqrt r\) and \(-\mu\sqrt r\). Their contribution to twice the normalized self-energy is \[ \frac{(1+\sqrt r)^2}{2(1-\sqrt r)} +\frac{(1-\sqrt r)^2}{2(1+\sqrt r)} =1+\frac{4r}{1-r}. \tag{4.3} \] An inactive up-target contributes \(1\). Summing these leading terms gives \(\pi_S(Y)\). For a down-target \(c=a/q\), put \(s=d_{S,X}(c)/d_{S,X}(a)>1\). The positive, negative, and mean-zero leaf components on the \(c\)-star give \[ \frac{5-s}{1-s}. \tag{4.4} \] This follows by inverting \(\mu\begin{smallmatrix}1&-\sqrt s\\-\sqrt s&1\end{smallmatrix}\) on the centre--constant-leaf plane and adding the kernel mass \((s-1)/(2s\mu)\). Unique factorization makes the target spaces orthogonal, so (4.3)--(4.4) add to (4.2). \(\square\) Let \(\mathcal B_a\) be the coordinate space of the entire boundary star at \(a\), including both modes \(u_a^\pm\) and its mean-zero leaf kernel. Let \(\mathcal U_a\) be the orthogonal sum of the entire target stars at \(aq\), for allowed primes \(q\le Y\), and at \(a/q\), for prime divisors \(q\mid a\). An inactive up-target is the singleton \(\{aq\}\), not an omitted coordinate. On a fixed sub-square-root range all these singleton vertices are at most \(X\). Put \(K_a=P_{\mathcal U_a}H_{S,X}|_{\mathcal B_a}\). The raw principal molecule is \[ \mathcal M_a= \begin{pmatrix} L_{S,X}|_{\mathcal B_a}&K_a^*\\ K_a&L_{S,X}|_{\mathcal U_a} \end{pmatrix} \tag{4.5} \] on \(\mathcal B_a\oplus\mathcal U_a\). This is a principal submatrix of \(A_{S,X}\). Indeed, small-prime edges preserve the retained large-prime factor and reverse the parity of the remaining number of prime factors; all first-exit targets have the same such parity, so no small-prime edge joins two of them. The stars themselves are disjoint. Let \(\nu_a>0\) be the eigenvalue continuing \(\mu_a\), and extend its unit eigenvector \(\widetilde f_a\) by zero to the full graph, with \(\langle u_a^+,\widetilde f_a\rangle>0\). Write its boundary part as \(\alpha_a^+u_a^++\alpha_a^-u_a^-+\zeta_a\), where \(\zeta_a\in\ker L_{S,X}|_{\mathcal B_a}\). The full boundary, rather than only \(u_a^+\), is needed for the cancellation below. An uncharged zero-mode component may lie outside \(\mathcal R\). We therefore use the projected, normalized vector \[ z_a=(I-P_{\mathcal R})\widetilde f_a, \qquad f_a=\frac{P_{\mathcal R}\widetilde f_a} {\|P_{\mathcal R}\widetilde f_a\|}, \qquad g_a=G_{S,X}f_a-\nu_af_a. \tag{4.6} \] Put \[ F_{S,a,X}=\pi_S(X/a)+M_{S,a,X}. \tag{4.7} \] \subsection{Proposition 4.2 -- molecule energy, projection, and residual}\label{proposition-4.2-molecule-energy-projection-and-residual} Fix \(0<\vartheta<1/2\). Uniformly for allowed \(a\le X^\vartheta\), \[ \nu_a^2=F_{S,a,X}+e_{a,X}, \tag{4.8} \] where \[ |e_{a,X}| \ll_{S,\vartheta} (\log\log X+\omega(a))\frac a{\sqrt X} +\frac{\log X}{\sqrt X}. \tag{4.9} \] The discarded projection satisfies \[ \begin{aligned} \|f_a-u_a^+\|^2&\ll_{S,\vartheta}\frac a{\sqrt X}, &L_{S,X}z_a&=0,\\ \|z_a\|^2&\ll_{S,\vartheta} \frac{a(\log X)^2}{X^{3/2}} +\frac{a^3\omega(a)\log X}{X^2},\\ \|A_{S,X}z_a\|^2&\ll_{S,\vartheta} \frac{a\log X}{X}+\frac{a^3\omega(a)}{X^{3/2}}. \end{aligned} \tag{4.10} \] Apart from the projection term in (4.10), \(g_a\) comes only from small-prime edges from a target space to the next unretained layer. Moreover, for every fixed \(C_\star\), every \(B\le X^\vartheta\), and some fixed exponent \(C_S\), \[ \sum_{a\le C_\star B}\|g_a\|^2\ll_{S,\vartheta}B(\log X)^{C_S}. \tag{4.11} \] \textbf{Proof.} All active up-target ratios are at most \(1/2+o(1)\), and all down-target ratios are at least \(2+o(1)\); the latter follows from the fixed-ratio argument in (9.7b). Thus the diagonal forest in (4.5) has a simple eigenvalue \(\mu_a\) separated from its other eigenvalues by a fixed multiple of \(\mu_a\). Since \(\|K_a\|\le\|H_{S,X}\|=o(\mu_a)\), Weyl's inequality gives a unique molecule eigenvalue in a fixed neighborhood of \(\mu_a\). Eliminating the target stars gives the exact boundary equation \[ \bigl(\nu_a-L_{\mathcal B_a}-\Sigma_a(\nu_a)\bigr)v_a=0, \qquad \Sigma_a(\lambda)=K_a^*(\lambda-L_{\mathcal U_a})^{-1}K_a, \quad v_a=P_{\mathcal B_a}\widetilde f_a. \tag{4.12} \] First compress this equation to \(\operatorname{span}\{u_a^+,u_a^-\}\); the boundary-kernel contribution will be bounded below. Write \(z=\lambda/\mu_a\), \(P=\pi_S(Y)\), and \(d=\mu_a^2\). Let \(\mathbf F(z)\) have entries \(\mu_a\langle K_au_a^\sigma,(z\mu_a-L_{\mathcal U_a})^{-1} K_au_a^\tau\rangle\). The two-mode equation is \(\det(\operatorname{diag}(z-1,z+1)-\mathbf F(z)/d)=0\). For an up-target of ratio \(r\), its deviation from \((2z)^{-1}\begin{pmatrix}1&1\\1&1\end{pmatrix}\) is \[ \mathbf R_r(z)= \frac{r}{2z(z^2-r)} \begin{pmatrix} (z+1)^2&-(z^2-1)\\ -(z^2-1)&(z-1)^2 \end{pmatrix}. \tag{4.13} \] After the boundary self-energy is divided by \(d\) in the dimensionless two-mode determinant, the universal terms sum to \(\gamma(z)\begin{smallmatrix}1&1\\1&1\end{smallmatrix}\), where \(\gamma(z)=P/(2zd)\), and the corresponding determinant is exactly \[ (z-1-\gamma)(z+1-\gamma)-\gamma^2 =z^2-1-P/d. \tag{4.14} \] Thus no pure \(P^2/d^2\) term remains. The matrices (4.13), their first derivatives, and the analogous down-target rational functions have total norm \(O_{S,\vartheta}(\log\log X+\omega(a))\) on a fixed neighborhood of \(z=1\); all denominators there are separated from zero. The first local identities also give \[ 2R_{++}(1)=M_{S,a,X}, \qquad R_{+-}(1)=-\omega(a)/2. \] Let \(\nu_a^{(2)}\) denote the positive root of this two-mode equation. Expanding its determinant at the unique positive root of (4.14), the pure \(P\)-terms cancel and every remaining term contains at least one \(R_{\sigma\tau}\)-factor. The mean-value theorem and the preceding derivative bounds therefore give \[ |z-1|\ll_{S,\vartheta}\frac a{\sqrt X}, \qquad |(\nu_a^{(2)})^2-F_{S,a,X}| \ll_{S,\vartheta} (\log\log X+\omega(a))\frac a{\sqrt X} +\frac{\log X}{\sqrt X}. \] This proves (4.8)--(4.9) for the two-mode compression, apart from the boundary-kernel response estimated below. The cancellation between the positive and negative entries of an up-star is essential; estimating them separately would leave a spurious quadratic boundary term. Before treating the uncharged kernel, the direct-sum target resolvent gives \[ \|\widetilde f_a-u_a^+\|^2 \ll_{S,\vartheta} \frac{\pi_S(Y)+\omega(a)}{\mu_a^2} +O_{S,\vartheta}\!\left(\frac{a^2}{X}\right) \ll_{S,\vartheta}\frac a{\sqrt X}. \] Projection and normalization change this by the smaller quantity estimated below, proving the first assertion of (4.10). We first bound the boundary kernel \(\zeta_a\), separately from the projection \(z_a\). For a boundary leaf \(ap\), put \[ n_a(p)=\#\{q\le Y:q\notin S,\ q\text{ prime},\ apq\le X\}. \] Let \(Q_0\) project onto the boundary leaf kernel, and define \(c_{a,\pm}=Q_0\Sigma_a(z\mu_a)u_a^\pm\). Put \[ H_a(p;z)= \sum_{\substack{q\le Y,\ q\notin S\\q\ {\rm prime},\ apq\le X}} \frac1{z^2-r_{q,X}(a)},\qquad \overline H_a(z)=\frac1{d_a}\sum_{YY\) is unique. These facts bound multiplicity before squaring. Put \(\mathcal P_Y=1+\sum_{q\le Y}q^{-1}\). Chebyshev summation gives \[ \#\{q,r\text{ prime}:qr\le T\} \ll \frac{T\log\log(3T)}{\log(2T)}. \] For leaf labels sum over the retained prime first; its logarithm is \(\asymp\log X\). The coefficient table yields {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}} >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Residual family \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Squared-mass bound, up to a fixed power of \(1+\omega(a)\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot Centre, two up-steps & \(C\mathcal P_Y\) \\ Centre, mixed or two down-steps & \(C(\pi_S(Y)+1)/d_a\) \\ Leaf, two up-steps & \(C\mathcal P_Y^2/d_a\) \\ Leaf, mixed or two down-steps & \(C(1+a)\mathcal P_Y/d_a\) \\ \end{longtable} } For the last row, an inner down-leaf segment has at most \(O(d_a q\mathcal P_Y)\) subsequent prime pairs and coefficient \(O(d_a^{-1})\); use \(q\le a\). On the outer segment the same count is multiplied by \(q^{-2}\), retaining the down-star weight. For two up-steps the leaf count is at most \(O(d_a\mathcal P_Y^2)\). All four rows are polylogarithmic uniformly on \(a\le X^\vartheta\), because \(\pi_S(Y)/d_a\ll a/\sqrt X\) and \(a/d_a\ll a^2\log X/X=o(1)\). The signed raw residual therefore has squared norm \(O((\log X)^C)\). The kernel-driven residual has squared norm at most \(C\|H_{S,X}\|^4\|\zeta_a\|^2/\mu_a^2\), which is smaller. Projection contributes \(P_{\mathcal R}H_{S,X}z_a\), followed by the bounded normalization in (4.6). Summing its squared norm gives \[ \sum_{a\le C_\star B}\|P_{\mathcal R}H_{S,X}z_a\|^2 \le \|H_{S,X}\|^2 \sum_{a\le C_\star B}\|z_a\|^2 \ll_{S,\vartheta} \frac{B^2L}{X}+\frac{B^4L^C}{X^{3/2}} \ll_{S,\vartheta}B L^C. \] Here \(L=\log X\), and the fixed exponent absorbs the bounded word multiplicities. The raw-residual bound and this projection estimate prove (4.11). \(\square\) \subsection{Lemma 4.3 -- size of the first-exit correction}\label{lemma-4.3-size-of-the-first-exit-correction} For every fixed \(S\) and \(0<\vartheta<1/2\), uniformly for allowed \(a\le X^\vartheta\), \[ |M_{S,a,X}|\ll_{S,\vartheta}\log X. \tag{4.18} \] \textbf{Proof.} Uniform PNT bounds on this sub-square-root range give \(r_{q,X}(a)\ll_{S,\vartheta}q^{-1}\), with all nonzero up-ratios bounded away from one. Thus the up-sum is \(O_S(\log\log X)\). They also give \(s_{q,X}(a)\ge1+c_{S,\vartheta}\) for every prime divisor \(q\mid a\), so every down summand is \(O_{S,\vartheta}(1)\). The down-sum is \(O_{S,\vartheta}(\omega(a))=O(\log X)\). \(\square\) \section{Finite-dimensional tools}\label{finite-dimensional-tools} We now pass from individual molecules to their combined span. The following lemmas control reorthonormalization, the complementary spectrum, and ordered eigenvalue errors. Section 6 applies them to the molecule family. \subsection{Lemma 5.1 -- frame commutator}\label{lemma-5.1-frame-commutator} Let \(T=T^*\), let \(V\) have full column rank, and suppose \[ TV=VD+J,\qquad D=D^*,\qquad \Gamma=V^*V. \tag{5.1} \] Then \[ D\Gamma-\Gamma D=V^*J-J^*V. \tag{5.2} \] If \(\|\Gamma-I\|\le1/2\), put \[ Q=V\Gamma^{-1/2},\qquad E=TQ-QD. \tag{5.3} \] For either the operator norm or the Hilbert--Schmidt norm, denoted by \(\|\cdot\|_*\), \[ \|E\|_*\ll\|J\|_*. \tag{5.4} \] Moreover, with \[ M=Q^*TQ,\qquad R=(I-QQ^*)TQ, \] one has \[ \|M-D\|_*\le\|E\|_*, \qquad \|R\|_*\le\|E\|_*, \qquad *\in\{2,\infty\}. \tag{5.5} \] \textbf{Proof.} Self-adjointness gives \[ V^*TV=\Gamma D+V^*J=D\Gamma+J^*V, \] which proves (5.2). On \([1/2,3/2]\), the divided difference of \(x\mapsto x^{-1/2}\) is bounded. The integral formula for negative fractional powers (equivalently, the divided-difference formula in an eigenbasis of \(\Gamma\)) gives, for \(*\in\{2,\infty\}\), \[ \|[D,\Gamma^{-1/2}]\|_* \ll\|[D,\Gamma]\|_* \ll\|J\|_*. \] Since \[ E=V[D,\Gamma^{-1/2}]+J\Gamma^{-1/2}, \] the frame bounds prove (5.4). Finally, \(M-D=Q^*E\) and \(R=(I-QQ^*)E\). \(\square\) \textbf{Hilbert--Schmidt refinement.} Under the same hypotheses, write \(\operatorname{sym}B=(B+B^*)/2\). Then \[ \boxed{ \|M-D-\operatorname{sym}(V^*J)\|_{\rm HS} \ll\|\Gamma-I\|\,\|J\|_{\rm HS}.} \tag{5.5a} \] Indeed, self-adjointness gives \(V^*TV=(\Gamma D+D\Gamma)/2+\operatorname{sym}(V^*J)\). Conjugating the symmetrized residual by \(\Gamma^{-1/2}\) changes it in Hilbert--Schmidt norm by at most \(O(\|\Gamma-I\|\,\|J\|_{\rm HS})\). In an eigenbasis of \(\Gamma\), the remaining difference from \(D\) has entries \[ \frac{(\sqrt{\gamma_i}-\sqrt{\gamma_j})^2} {2\sqrt{\gamma_i\gamma_j}}D_{ij}. \] They vanish when \(\gamma_i=\gamma_j\). Otherwise divide their coefficient by \(\gamma_j-\gamma_i\); the resulting multiplier is \(O(\|\Gamma-I\|)\), uniformly for \(\gamma_i,\gamma_j\in[1/2,3/2]\). Equation (5.2) and \(\|V\|\le\sqrt{3/2}\) give (5.5a). In particular, under the global estimates (6.3) and (6.4), the squared cost of this Gram correction is bounded by \[ (\log X)^C\left(\frac{B^2}{X}+\frac{B^5}{X^2}\right) =o\!\left(\frac{B^3}{X\log X}\right) \tag{5.5b} \] on a fixed positive sub-square-root power band. This is a bound for the complete matrix before taking its local compression. The matrix that remains to be estimated is \(\operatorname{sym}(V^*J)\), whose entries involve the complete output vectors; coefficients of \(U^*J\) for an auxiliary star-mode frame \(U\) are not interchangeable with them. \subsection{Lemma 5.2 -- ordered eigenvalues above a separated complement}\label{lemma-5.2-ordered-eigenvalues-above-a-separated-complement} Let \[ T=\begin{pmatrix}M&R^*\\R&C\end{pmatrix}=T^* \] on \(\mathbb C^m\oplus\mathcal K\), and let \(\tau_1\ge\cdots\ge\tau_m\) be the eigenvalues of \(M\). If \[ \sup\sigma(C)\le\beta<\tau_k, \] then \[ \boxed{ 0\le\lambda_k(T)-\tau_k \le \frac{\|R\|^2}{\tau_k-\beta}.} \tag{5.6} \] \textbf{Proof.} Compression interlacing gives \(\lambda_k(T)\ge\tau_k\). Put \[ \delta=\frac{\|R\|^2}{\tau_k-\beta}, \qquad t=\tau_k+\delta. \] The complement block \(C-tI\) is negative definite. Sylvester inertia and Schur elimination give \[ \#\{j:\lambda_j(T)>t\} =n_+\!\left(M-tI+R^*(t-C)^{-1}R\right). \] Since \[ 0\le R^*(t-C)^{-1}R \le \frac{\|R\|^2}{t-\beta}I \le\delta I, \] the Schur form is bounded above by \(M-\tau_kI\), which has at most \(k-1\) positive eigenvalues. Hence \(\lambda_k(T)\le t\), proving (5.6). \(\square\) \subsection{Lemma 5.3 -- complement of an energy-weighted graph}\label{lemma-5.3-complement-of-an-energy-weighted-graph} Let \(L=L^*\) reduce \(\mathcal P\oplus\mathcal P^\perp\), suppose \(L|_{\mathcal P}=\Lambda\ge0\), and write \(\beta=\sup\sigma(L|_{\mathcal P^\perp})\ge0\). Let \(\mathcal Q=\{p+Tp:p\in\mathcal P\}\) be the graph of \(T:\mathcal P\to\mathcal P^\perp\). For every self-adjoint \(H\), the compression \(C=P_{\mathcal Q^\perp}(L+H)|_{\mathcal Q^\perp}\) satisfies \[ \boxed{\sup\sigma(C)\le \beta+\|T\Lambda^{1/2}\|^2+\|H\|.} \tag{5.7} \] \textbf{Proof.} Every \(x\in\mathcal Q^\perp\) has the unique form \(x=-T^*y+y\), with \(y\in\mathcal P^\perp\). The two summands are orthogonal, so \(\|y\|\le\|x\|\), and the reducing decomposition gives \[ \langle x,Lx\rangle =\langle y,L|_{\mathcal P^\perp}y\rangle +\langle T^*y,\Lambda T^*y\rangle \le(\beta+\|T\Lambda^{1/2}\|^2)\|y\|^2. \] For unit \(x\), add \(|\langle x,Hx\rangle|\le\|H\|\). The coefficient \(\beta+\|T\Lambda^{1/2}\|^2\) is nonnegative, so replacing \(\|y\|^2\) by \(\|x\|^2=1\) preserves the upper bound. The variational principle proves (5.7). \(\square\) \subsection{Lemma 5.4 -- ordered residual comparison}\label{lemma-5.4-ordered-residual-comparison} Let \(T=T^*\) on a finite-dimensional Hilbert space \(\mathcal H\), let \(Q:\mathbb C^m\to\mathcal H\) be an isometry, and let \(D=\operatorname{diag}(\nu_1,\ldots,\nu_m)\) be real. Write \(\widehat\nu_1\ge\cdots\ge\widehat\nu_m\) for the decreasing rearrangement of its entries and put \[ E=TQ-QD,\quad e=\|E\|,\quad h=\|E\|_{\rm HS},\quad M=Q^*TQ,\quad C=T|_{{\rm ran}(Q)^\perp}, \] where the last expression denotes compression. Let \(\xi_j\) and \(\zeta_j\) be the decreasing eigenvalues of \(M\) and \(T\), respectively. Suppose \(\mu>0\), \(04\) is fixed. Let \(V_Be_c=f_c\), \(D_B=\operatorname{diag}(\nu_c)\), and \[ J_B=G_{S,X}V_B-V_BD_B,\qquad \Gamma_B=V_B^*V_B. \tag{6.2} \] The following estimates bound the Gram matrix and the residual of this molecule family. All occurrences of \(X^{o(1)}\) in Sections 6--9 arise from bounded-word divisor multiplicities. After \(S\) and \(\theta\) are fixed, only finitely many such factors are composed. Consequently, for every fixed \(\xi>0\), their complete product is \(O_{S,\theta,\xi}(X^\xi)\). Only the later target-width threshold-count argument requires \(\theta>1/3\) and chooses \(\xi<3\theta-1\). The global estimates in this section hold throughout \(0<\theta<1/2\). \subsection{Proposition 6.1 -- collision budget}\label{proposition-6.1-collision-budget} For some fixed exponent \(C_S\), uniformly in (6.1), \[ \|\Gamma_B-I\| \ll_{S,\theta}(\log X)^{C_S} \left(\sqrt{\frac BX}+\frac{B^2}{X}\right)=o(1), \tag{6.3} \] \[ \|J_B\|_{\rm HS}^2\ll_{S,\theta}B(\log X)^{C_S}, \qquad \boxed{ \|J_B\|^2\ll_{S,\theta}(\log X)^{C_S} \left(1+\frac{B^2}{\sqrt X}\right).} \tag{6.4} \] \textbf{Proof.} We first work with the raw full-star vectors \(\widetilde f_c\) and raw residuals \(\widetilde g_c=(A_{S,X}-\nu_c)\widetilde f_c\); projection and normalization will be estimated at the end. Their support dictionary is \[ c,\quad cp,\quad cq,\quad cqp,\quad c/q,\quad(c/q)p, \qquad p>Y,\quad q\le Y, \tag{6.4a} \] with \(q\mid c\) for a down-step. A signed raw residual has one further small-prime edge, and no coordinate on its own full principal molecule. We use the coefficients in (4.17b), keeping the outer down-leaf factor \(q^{-1}\). First consider the Gram matrix. If two source centres are prime-adjacent, each boundary star belongs to the other molecule. Self-adjointness gives \[ (\nu_c-\nu_d)\langle\widetilde f_d,\widetilde f_c\rangle =\langle\widetilde g_d,\widetilde f_c\rangle -\langle\widetilde f_d,\widetilde g_c\rangle. \] Only first-exit interiors occur in these pairings. Their norms are \(O(\sqrt{c/\sqrt X})\) and \(O(\sqrt{d/\sqrt X})\), while the raw residuals are polylogarithmically bounded by Proposition 4.2. The two roots are separated by \(\gg\max(\mu_c,\mu_d)\). If the centres are not prime-adjacent, their boundary stars do not intersect the other molecule. Their interiors can share at most two target stars: solve \(cq^{\pm1}=dr^{\pm1}\). On any such target the complete response has norm \(O(1/\mu_c)\), respectively \(O(1/\mu_d)\), since the one-target coupling is a partial coordinate isometry and the target resolvent gap is \(\gg\mu_c\). Consequently \[ |\langle\widetilde f_c,\widetilde f_d\rangle| \ll L^C \begin{cases} \sqrt{cd}/X^{3/4},&c,d\text{ prime-adjacent},\\ 1/(\mu_c\mu_d),&c,d\text{ share a target but are not adjacent},\\ 0,&\text{otherwise}, \end{cases} \qquad c\ne d. \tag{6.5} \] For the first matrix use the Schur weight \(w_c=c^{-1/2}\). An up-neighbour \(d=cq\le C_\star B\) contributes \(c/X^{3/4}\) to the weighted row; there are at most \(C_\star B/c\) such neighbours. Down-neighbours contribute the same weight and have at most \(\omega(c)\) choices. The other matrix has ordinary row sum at most \(L^C B^2/X\), using \(\mu_c^{-1}\ll\sqrt{cL/X}\). Thus \[ \|\widetilde V_B^*\widetilde V_B-I\| \ll L^C\left(\frac{B}{X^{3/4}}+\frac{B^2}{X}\right) \ll L^C\left(\sqrt{\frac BX}+\frac{B^2}{X}\right). \tag{6.6} \] This controls the complete raw vectors, including their boundary kernels. For the residual operator, take two distinct sources \(c,d\). A centre-type equality has the form \[ cu_1/u_2=dv_1/v_2,\qquad \frac dc=\frac{u_1v_2}{u_2v_1}, \tag{6.7} \] where each side uses at most two prime steps. Denominator slots divide the source at the step where they are removed. There are a bounded number of orientations, and at most \(L^4\) source-dividing choices for each. Since \(c\ne d\), at least two of the four prime slots are fixed by the reduced quotient; a free prime must occupy one slot on each side. There is therefore at most one free common small prime, of range at most \(Y\). This gives \(O(YL^C)\) representations, including the common-descendant multiplicity. For leaf-type equalities the same large prime occurs on both sides. On a source piece indexed by \(\ell_c\), the leaf range is \(p\le X\ell_c/c\) and the coefficient is \(O((\ell_c d_c)^{-1})\): take \(\ell_c=1\) for up-leaves and inner down-leaves, and \(\ell_c=q\) for an outer down-leaf segment. When no free common small prime remains, summing \(p\) gives at most \(XL^C\min(\ell_c/c,\ell_d/d)\) choices. Their weighted contribution is bounded using the exact inequality \[ \frac{\min(\ell_c/c,\ell_d/d)}{\ell_c\ell_d} \le\frac1{\max(c,d)}. \] Thus the extended outer down-leaf range retains its reciprocal down-prime factor. If a common small prime also remains, an inner down-leaf segment can have a larger product cutoff. We use the safe bound \(XL^C\) for the number of prime pairs, with coefficients \(O(d_c^{-1})\) and \(O(d_d^{-1})\), instead of claiming the preceding smaller range. Centre and leaf residuals have disjoint large-prime signatures. Together with the diagonal bound from Proposition 4.2, the two off-diagonal residual-Gram row bounds are \[ \sum_{d\le C_\star B} O\!\left(\frac{YL^C}{\mu_c\mu_d}\right) \ll L^C\frac{B^2}{\sqrt X}, \tag{6.8a} \] and \[ \sum_{d\le C_\star B} O\!\left(\frac{XL^C}{d_cd_d}\right) \ll L^C\frac{B^3}{X} \le L^C\frac{B^2}{\sqrt X}. \tag{6.8b} \] The Schur test proves the signed raw-residual operator bound in (6.4). Its Hilbert--Schmidt bound is the sum of the one-source estimates. It remains to pass to the actual projected columns. Let \(Z_B\) have columns \(z_c\), and let \(N_B\) be diagonal with entries \((1-\|z_c\|^2)^{-1/2}\). Exactly, \[ V_B=P_{\mathcal R}\widetilde V_BN_B,\qquad J_B=(P_{\mathcal R}\widetilde J_B-P_{\mathcal R}H_{S,X}Z_B)N_B. \tag{6.8c} \] By (4.10), \[ \|Z_B\|_{\rm HS}^2 \ll L^C\left(\frac{B^2}{X^{3/2}}+\frac{B^4}{X^2}\right), \qquad \|H_{S,X}Z_B\|_{\rm HS}^2 \ll L^C\left(\frac{B^2}{X}+\frac{B^4}{X^{3/2}}\right). \tag{6.8d} \] The first is \(o(1)\) and is \(O(L^C B^2/X)\). Consequently (6.6) survives the Gram subtraction \(Z_B^*Z_B\) and the diagonal normalization, proving (6.3). The kernel-driven raw residual has squared Hilbert--Schmidt norm at most \(L^C B^3/X^{3/2}\), by (4.17a) and the extra \(\|H_{S,X}\|/\mu_c\) factor in its target response. This and the second bound in (6.8d) are \(O(L^C(1+B^2/\sqrt X))\) in operator-norm squared and \(O(BL^C)\) in Hilbert--Schmidt norm squared, because \(B/\sqrt X\to0\). Normalization has the same or a smaller cost. These estimates complete (6.4) for the full projected residual. The \(B^2/\sqrt X\) term retains the coherent prime-source rectangle. \(\square\) Because of (6.3), the symmetric reorthonormalization \[ Q_B=V_B\Gamma_B^{-1/2} \tag{6.9} \] is defined for large \(X\). Put \[ E_B=G_{S,X}Q_B-Q_BD_B. \tag{6.10} \] \subsection{Corollary 6.2 -- signed prefix budget}\label{corollary-6.2-signed-prefix-budget} Uniformly in (6.1), \[ \boxed{ \|E_B\|_{\rm HS}^2\ll_{S,\theta}B(\log X)^{C_S}, \qquad \|E_B\|^2\ll_{S,\theta}(\log X)^{C_S} \left(1+\frac{B^2}{\sqrt X}\right).} \tag{6.11} \] \textbf{Proof.} Apply Lemma 5.1 to Proposition 6.1. \(\square\) The Hilbert--Schmidt bound in (6.11) controls the number of bad cluster boundaries. The operator bound controls the self-energy after elimination of the separated complement, where its coherent term is divided by a root-scale gap. The factor \(1+B^2/\sqrt X\) is retained in that estimate. \subsection{Lemma 6.3 -- the prefix as an energy-weighted graph}\label{lemma-6.3-the-prefix-as-an-energy-weighted-graph} Let \(\mathcal P_B\) be the span of the positive star modes with allowed centres \(c\le C_\star B\), let \(\Pi_B\) be its orthogonal projection, and write \(\Lambda_B=L_{S,X}|_{\mathcal P_B}\). For all sufficiently large \(X\), the map \(W_B=\Pi_BV_B\) is invertible, and \({\rm ran}\,Q_B={\rm ran}\,V_B\) is the graph of \[ T_B=(I-\Pi_B)V_BW_B^{-1}: \mathcal P_B\longrightarrow\mathcal P_B^\perp. \tag{6.12} \] Moreover \[ \boxed{ \|T_B\|^2\ll_{S,\theta}L^{C_S}\frac B{\sqrt X}=o(1), \qquad \|T_B\Lambda_B^{1/2}\|^2 \ll_{S,\theta}L^{C_S}\sqrt{\frac BL}=o(\mu_B).} \tag{6.13} \] The largest forest eigenvalue on \(\mathcal P_B^\perp\) is \(\sqrt{d_{S,X}(c_B)}\), where \(c_B\) is the first allowed centre above \(C_\star B\). \textbf{Proof.} Let \(U_Be_c=u_c^+\), and identify \(\Lambda_B\) on the source space with \(\widehat\Lambda_B=\operatorname{diag}(\mu_c)\). The coefficient table following (4.17b) gives both \[ \|V_B-U_B\|^2\ll L^C\frac B{\sqrt X}, \qquad \|(V_B-U_B)\widehat\Lambda_B^{1/2}\|^2 \ll L^C\sqrt{\frac BL}. \tag{6.14} \] Here are the row and column counts for the weighted estimate; the unweighted estimate follows from the same counts without the column factor \(\sqrt{\mu_c}\). \begin{itemize} \tightlist \item An up-centre row has at most \(\omega(n)\le L\) source representations. Its weighted column squared mass is \(O(\pi_S(Y)/\mu_c)\ll\sqrt{c/L}\). \item A down-centre coefficient is \(O((q\sqrt{\mu_c})^{-1})\). The matrix of weights \(q^{-1}\) has logarithmic row and column sums. Replacing \(\mu_c\) by its prefix minimum bounds its squared norm by \(L^C/\min\mu_c\). \item An up-leaf row has at most \(L\) source representations after its unique large prime is removed. Its weighted column squared mass is \(O(\mathcal P_Y/\mu_c)\). \item A down-leaf row has at most \(C_\star B\) sources. The inner and outer segments each have weighted column squared mass \(O(L^C/\mu_c)\); on the outer segment use \(d_{c/q}\ll qd_c\) with its coefficient \(q^{-1}d_c^{-1}\). Their squared norms are at most \(L^C B/\min\mu_c\). \end{itemize} The last bound is at most \(L^{C'}\sqrt{B/L}\), since \(B/\sqrt X\to0\). The signed boundary coefficients differ from \(u_c^+\) by \(O(c/\sqrt X)\), and their weighted squared mass is at most \(L^C c^{3/2}/\sqrt X\). The boundary-kernel contribution has weighted Hilbert--Schmidt mass at most \(L^C B^{3/2}/X\), by (4.17a); its target response has the extra factor \(\|H\|/\mu_c=o(1)\). Finally projection is a contraction fixing \(U_B\), and normalization adds only the diagonal error \(O(\|z_c\|^2)\). These estimates prove (6.14). It follows that \(U_B^*W_B=I+o(1)\), proving invertibility and the graph representation. Put \(Z_B'=(I-\Pi_B)V_B\). In particular \[ \mu_c\|Z_B'e_c\|^2\ll L^C\sqrt{c/L}, \tag{6.15} \] and the operator estimate in (6.14), rather than only this column bound, gives \[ \|Z_B'\widehat\Lambda_B^{1/2}\|^2 \ll L^C\sqrt{B/L}. \tag{6.16} \] To move the energy weight through \(W_B^{-1}\), set \(A_B=U_B^*W_B-I\). Its signed off-diagonal entries are nonzero only for prime-adjacent centres. After conjugation by \(\widehat\Lambda_B^{1/2}\), their bounds, with source \(a\) and output \(b\), are \[ C\sqrt{\frac LX} \begin{cases} (ab)^{1/4},&b=aq,\\ b^{3/4}a^{-1/4},&b=a/q. \end{cases} \] The Schur weight \(a^{-1/2}\) gives row and column sums \(L^C B^{3/4}/\sqrt X\). The diagonal normalization and source-amplitude terms have size \(O(B/\sqrt X)\); these terms must also be retained. The kernel-driven positive response has Hilbert--Schmidt norm at most \(L^C B^{3/2}/X\), and conjugation costs at most \(O(B^{1/4})\). Thus the sufficient conjugated bound is \[ \|\widehat\Lambda_B^{-1/2}A_B\widehat\Lambda_B^{1/2}\| \ll L^C\frac B{\sqrt X}=o(1). \tag{6.17} \] Indeed \(B^{7/4}/X\le B/\sqrt X\) on this range. The inverse conjugation is therefore uniformly bounded, and \[ \begin{aligned} \widehat\Lambda_B^{-1/2}(I+A_B)^{-1}\widehat\Lambda_B^{1/2} &=(I+\widehat\Lambda_B^{-1/2}A_B\widehat\Lambda_B^{1/2})^{-1},\\ \|T_B\Lambda_B^{1/2}\| &\le\|Z_B'\widehat\Lambda_B^{1/2}\| \,\|\widehat\Lambda_B^{-1/2}(I+A_B)^{-1} \widehat\Lambda_B^{1/2}\|. \end{aligned} \tag{6.18} \] Equations (6.14), (6.16), and (6.18) prove (6.13). Removing the positive modes with centres at most \(C_\star B\) leaves forest top \(\sqrt{d_{S,X}(c_B)}\); the other sectors are nonpositive. \(\square\) \section{Arithmetic clusters and exact complement elimination}\label{arithmetic-clusters-and-exact-complement-elimination} Let \[ A=X^\theta,\qquad L=\log X,\qquad K_B=\left\lfloor\frac{B^3}{X}\right\rfloor, \tag{7.1} \] and use the root scale \[ \mu_B^2\asymp_{S,\theta}\frac{X}{BL}, \qquad \Delta_B=\frac{B}{\mu_BL}. \tag{7.2} \] Since \[ \Delta_B^2\asymp\frac{B^3}{XL}=\frac{K_B}{L}, \] the quantity \(\Delta_B\) is the root width of a cluster of \(K_B\) labels; the derivative-scale spacing between adjacent roots is \(\asymp\Delta_B/K_B\). \subsection{Proposition 7.1 -- target-width prime clusters}\label{proposition-7.1-target-width-prime-clusters} Suppose \(21/61<\theta<1/2\). Uniformly for \(B\) in (6.1), if allowed centres \(a\frac{21}{40}. \tag{7.5} \] Two classical inputs give a two-sided short-interval count. Baker, Harman, and Pintz {[}16{]} prove, in an equivalent forward-interval form, \[ \pi(z+z^{21/40})-\pi(z)\gg\frac{z^{21/40}}{\log z} \] for all large \(z\). Greedily partition \([x,x+h]\) into consecutive blocks \([z,z+z^{21/40}]\), discarding only the final incomplete block. Because \(h/x\to0\), all block lengths and logarithms are uniformly comparable to \(x^{21/40}\) and \(\log x\). Summing the block bounds gives the lower half of \[ \pi(x+h)-\pi(x)\asymp_\theta\frac{h}{\log x}, \tag{7.6} \] uniformly whenever \(h/x^{21/40}\to\infty\) and \(h=o(x)\); the upper half is the Brun--Titchmarsh inequality {[}12{]}, since by (7.5) the ratio \(\log h/\log x\) is bounded below by a fixed \(\theta\)-constant. Integer parts cost \(O(1)\), and deleting the fixed set \(S\) does not change the interval count for large \(X\). Thus the prime-count difference is \(\asymp_{S,\theta}B/L\). The square-root boundary cancels between the two star degrees. Lemma 4.3 gives \(|M_{S,a,X}-M_{S,b,X}|\ll_{S,\theta}L=o(B/L)\), proving (7.4). If \(b-a\ge K_B/L^4\), the interval length \(h_{\mathrm{sort}}\asymp B/L^4\) still satisfies \[ \frac{h_{\mathrm{sort}}}{x^{21/40}} \asymp\frac{X^{(61/40)(\theta-21/61)}}{(\log X)^4}\to\infty, \] so the lower half of (7.6) applies and \[ F_{S,a,X}-F_{S,b,X}\gg_{S,\theta}B/L^5>0. \] Thus an inversion in the labelled target list can involve only centres in an interval of length \(K_B/L^4\). A fixed finite prime deletion changes the number of labels in such an interval by at most a fixed factor, so sorting moves each label by \(O_S(K_B/L^4)=o(K_B)\) positions. The upper oscillation bound follows from the upper half of (7.6), applied across a fixed number of \(K_B\)-blocks, together with Lemma 4.3. The sorting statement holds throughout the full retained prefix \(c\le C_\star B\). Uniform PNT at fixed multiplicative scales separates labels outside an overlapping dyadic neighborhood by \(\gg X/(BL)\), while the preceding short-interval argument handles the overlapping neighborhoods. Thus no distant label can cross a target in \([B/2,B]\). Finally Proposition 4.2 gives, uniformly for \(c\asymp B\), \[ |\nu_c^2-F_{S,c,X}|=o_{S,\theta}(B/L^5). \] This error is smaller than the separation just used, so sorting the \(\nu_c\) has the same \(O_S(K_B/L^4)\) label displacement. The target oscillation bound then proves (7.4a). \(\square\) \subsection{Lemma 7.2 -- quantitative cluster-boundary transport}\label{lemma-7.2-quantitative-cluster-boundary-transport} Let \(x_1\ge\cdots\ge x_m\) and \(y_1\ge\cdots\ge y_m\), and let \(K\to\infty\) with \(K=o(m)\). Suppose that at all but \(N\) grid boundaries \(b=rK\), \[ |N_x(y_b)-b|\le C_0K, \tag{7.6a} \] where \(N_x(t)=\#\{i:x_i\ge t\}\). If the target oscillation across any fixed number of adjacent grid blocks is at most \(\eta\), then \[ \#\{i:|x_i-y_i|>C\eta\} \ll_{C_0}K(N+1). \tag{7.6b} \] \textbf{Proof.} Put \(R=\lceil C_0\rceil+2\), and discard every block within \(R\) grid steps of a bad boundary. These blocks contain \(O_{C_0}(KN)\) indices. For an index \(i\) in a remaining block, choose good grid boundaries \(b_-21/61>1/3\); the other limits use \(\theta<1/2\). Write \[ M_B=Q_B^*G_{S,X}Q_B,\qquad C_B=(I-Q_BQ_B^*)G_{S,X}(I-Q_BQ_B^*), \] \[ R_B=(I-Q_BQ_B^*)G_{S,X}Q_B. \tag{7.9} \] For a target centre \(a_b\in[B/2,B]\), set \[ z_b=\sqrt{F_{S,a_b,X}}. \tag{7.10} \] \subsection{Proposition 7.3 -- separated complement}\label{proposition-7.3-separated-complement} For \(C_\star\) sufficiently large and all large \(X\), \[ \sup\sigma(C_B)\le z_b-c_{S,\theta}\mu_B \tag{7.11} \] uniformly in \(B\) and \(b\). \textbf{Proof.} Let \(\mathcal Q_B={\rm ran}\,Q_B\). By Lemma 6.3 this is the graph of \(T_B:\mathcal P_B\to\mathcal P_B^\perp\). The large-prime forest reduces \(\mathcal P_B\oplus\mathcal P_B^\perp\), and its top on the second summand is \(\sqrt{d_{S,X}(c_B)}\ge0\), where \(c_B\) is the first allowed centre above \(C_\star B\). Thus the sign hypothesis in Lemma 5.3 holds, and that lemma gives the direct Rayleigh bound \[ \sup\sigma(C_B) \le \sqrt{d_{S,X}(c_B)} +\|T_B\Lambda_B^{1/2}\|^2+\|H_{S,X}\|. \tag{7.14} \] Here Lemma 5.3 is first applied to \(L_{S,X}+H_{S,X}=A_{S,X}\) on the full graph of \(T_B\). The operator \(C_B\) is the further compression of that orthogonal-complement form to \(\mathcal R\cap\mathcal Q_B^\perp\); restricting the form can only lower its spectral supremum. Thus (7.14) applies to the one-exit complement used in (7.9). Uniform PNT at the two fixed multiplicative scales gives \(d_{S,X}(c_B)\le(2/C_\star+o(1))\mu_B^2\), whereas every target root in \([B/2,B]\) is bounded below by a fixed multiple of \(\mu_B\). By (6.13) and (7.7), the last two terms in (7.14) are \(o(\mu_B)\). Choose \(C_\star\) large. This proves (7.11). \(\square\) \section{Quantitative cluster-boundary estimate}\label{quantitative-cluster-boundary-estimate} The every-centre proof below uses a threshold count. We first record the Hilbert--Schmidt cluster estimate from which that refinement starts. \subsection{Proposition 8.1 -- quantitative almost-all comparison}\label{proposition-8.1-quantitative-almost-all-comparison} Fix \(21/61<\theta<1/2\), put \(A=X^\theta\), and let \[ \mathcal E_B=\left\{a\in V_S(X): B/2\le a\le B,\ \left|\lambda_{j_S(a)}(A_{S,X})^2-F_{S,a,X}\right| >C_{S,\theta}B/\log X\right\}. \] For a sufficiently large fixed exponent \(D=D(S,\theta)\), \[ |\mathcal E_B| \ll_{S,\theta}(\log X)^D \left(\frac X{B^2}+\frac{B^3}{X}\right). \tag{8.9} \] Consequently the union of these exceptional sets over \(A/\log X\le B\le A\) has the same form with \(B=A\), up to a further power of \(\log X\). \textbf{Proof.} We first prove the result for \(G_{S,X}\). Restrict to \[ A/L\le a\le A,\qquad A=X^\theta, \tag{8.1} \] and decompose this interval into dyadic centre bands \([B/2,B]\). Fix one such band. Let \(\xi_{B,1}\ge\cdots\ge\xi_{B,m_B}\) be the eigenvalues of \(M_B\), and retain the notation \(\widehat\nu_{B,k}\) from Proposition 7.1. Corollary 6.2 and (5.5) give \[ \sum_{k=1}^{m_B}|\xi_{B,k}-\widehat\nu_{B,k}|^2 \le\|M_B-D_B\|_{\rm HS}^2 \ll_{S,\theta}BL^{C_S}. \tag{8.2} \] Put \(K=K_B\), and list the allowed labels in the band in increasing order. At a grid boundary \(b=rK\), compare the tail count of the \(\xi_{B,k}\)'s at the target root \(z_b=\sqrt{F_{S,a_b,X}}\) with its arithmetic index. If this count is wrong by more than a fixed multiple of \(K\), then Proposition 7.1 forces at least \(K\) ordered pairs in (8.2) to move by \(\gg_{S,\theta}\Delta_B\). Choosing a disjoint subfamily of bad boundaries gives \[ N_B \ll_{S,\theta} \frac{BL^{C_S}}{K\Delta_B^2} \ll_{S,\theta}\frac{BL^{C_S+1}}{K^2}, \tag{8.3} \] where \(N_B\) denotes the number of bad grid boundaries and \(\Delta_B^2\asymp K/L\). This is the step at which an exceptional set enters: (8.2) controls the total squared displacement, not its maximum. It remains to transfer the good boundary counts from the retained prefix to the full signed one-exit operator. Proposition 7.3 separates the complement from every target root by \(\gg_{S,\theta}\mu_B\). For \(z\) in the target window, exact Schur elimination and (6.11) give \[ \left\|R_B^*(z-C_B)^{-1}R_B\right\| \ll_{S,\theta} \frac{L^{C_S}}{\mu_B} \left(1+\frac{B^2}{\sqrt X}\right). \tag{8.4} \] Relative to the root width this is \[ \frac{\|R_B^*(z-C_B)^{-1}R_B\|}{\Delta_B} \ll_{S,\theta}L^{C_S+1} \left(\frac1B+\frac B{\sqrt X}\right)=o(1), \tag{8.5} \] uniformly for \(A/L\le B\le A\). Sylvester inertia therefore changes the prefix count only across a root interval of width \(o(\Delta_B)\). Proposition 7.1 shows that such an interval contains \(O_{S,\theta}(K)\) target roots. Hence, outside the \(N_B\) boundaries counted in (8.3), \[ |N_G(F_{S,a_{rK},X})-rK|\ll_{S,\theta}K, \tag{8.6} \] where \(N_G(t)=\#\{j:\rho_j(G_{S,X})^2\ge t\}\), with the fixed offset of the dyadic band understood. Lemma 7.2 and the oscillation estimate in Proposition 7.1 now give \[ \rho_{j_S(a)}(G_{S,X})^2 =F_{S,a,X}+O_{S,\theta}(a/L) \tag{8.7} \] outside a set of size \[ \ll_{S,\theta}K(N_B+1) \ll_{S,\theta} \frac{XL^{C_S+1}}{B^2}+\frac{B^3}{X}. \tag{8.8} \] Finally choose a fixed \(\varepsilon\) with \(0<\varepsilon<1/2-\theta\). Theorem 3.2 transfers (8.7) from \(G_{S,X}\) to the full adjacency \(A_{S,X}\) at an additional cost \(O_{S,\theta}(a/L)\). Summing (8.8) over the dyadic bands \(A/L\le B\le A\) gives \[ \ll_{S,\theta} L^{D_{S,\theta}} \left(\frac X{A^2}+\frac{A^3}{X}\right) \] exceptions. This proves (8.9). \(\square\) \section{Buffered threshold counts and proof of Theorem 1.1}\label{buffered-threshold-counts-and-proof-of-theorem-1.1} To compare eigenvalues at the cluster root width \(\Delta_B\), we separate the coherent common-descendant interactions into couplings with spectrally distant molecules and interactions within the target band. Put \[ L=\log X,\qquad B/2\le a\le B,\qquad \mu_B^2\asymp\frac{X}{BL}, \] and define the target label and root widths \[ K_B=\frac{B^3}{X},\qquad \Delta_B=\frac{B}{\mu_BL},\qquad \Delta_B^2\asymp\frac{B^3}{XL}. \tag{9.1} \] In the range \(21/61<\theta<1/2\), \(K_B\) is a fixed positive power of \(X\). \subsection{Lemma 9.1 -- buffered exterior elimination}\label{lemma-9.1-buffered-exterior-elimination} Choose fixed \(0C_+B\) below it, with a gap \(\gg\mu_B\). Thus \(\operatorname{diag}(\nu_c:c\notin\mathcal A_B)\oplus C_B\) is gapped on both sides of every target threshold. The actual exterior differs from this reference by \(O(\|E_B\|)\). By (6.11), \[ \frac{\|E_B\|^2}{\mu_B^2} \ll L^C\left(\frac BX+\frac{B^3}{X^{3/2}}\right)=o(1). \] The homotopy multiplying this difference by a parameter in \([0,1]\) therefore preserves the gap and its inertia. At parameter zero the above-threshold exterior count is exactly the number of allowed labels \(c0\). \textbf{Proof.} Use the raw splitting \(\widetilde f_a=s_a+i_a+k_a^{\rm ker}\) from Proposition 4.2, and put \(g_a^s=(I-P_a)H_{S,X}i_a\). On the full prefix define the complete signed-response pairing \[ (\mathscr S_B)_{ba}=\langle s_b+i_b,g_a^s\rangle. \] This matrix, not the selected coefficients \(\langle u_b^+,g_a\rangle\), is the starting comparison object. \textbf{Coherent entries.} A prime-sibling pair has the form \[ a=gr,\qquad b=gq,\qquad r,q\text{ prime},\quad r\ne q. \tag{9.6} \] For a fixed retained large-prime signature, \(i_b\) has parity \(\Omega(b)+1\), while \(g_a^s\) has parity \(\Omega(a)\). These are opposite for a sibling pair, so \(\langle i_b,g_a^s\rangle=0\) exactly. An edge from the boundary at \(b\) to a target at \(a\) reduces to \(q t^\sigma=r u^\tau\). Unique factorization leaves only the common down-target \(g\) and common up-target \(grq\). Both active and inactive up-targets are included. For example, if \(gr\le gq\) and \(grq>Y\), write \(s_a=\alpha\delta_{gr}+\beta\psi_{gr}\) and put \(z=\nu_{gr}/\mu_{gr}\), \(s=d_g/d_{gr}\), and \(t=d_{gq}/d_{gr}\). Pairing the common-core response with \(u_{gq}^+\) gives \[ \frac1{\sqrt2\,\mu_{gr}}\left[ \frac{\alpha}{z} +\frac{z\alpha+\beta}{z^2-s} +\sqrt t\left\{ \frac{\alpha+z\beta/s}{z^2-s} +\frac{\beta(1-1/s)}z \right\}\right]. \tag{9.7a} \] For a denominator bound uniform in \(r\), put \(x=X/(gr)\) and \(P_Y=\pi_S(Y)\). Monotonicity gives \[ s=\frac{\pi_S(rx)-P_Y}{\pi_S(x)-P_Y} \ge\frac{\pi_S(2x)-P_Y}{\pi_S(x)-P_Y} \ge\frac{\pi_S(2x)}{\pi_S(x)} =2+o_{S,\theta}(1). \tag{9.7b} \] The second inequality uses \(\pi_S(2x)\ge\pi_S(x)>P_Y\). Since \(z^2=1+o(1)\), \[ |z^2-s|\ge\tfrac12 \tag{9.7c} \] eventually. The output ratio \(t\) is not inverted; \(t=1\) is allowed. The coefficient table (4.17b) also covers the reverse orientation and active common up-targets. A centre contribution is \(O(1/\mu_a)\). For an up-target there are at most \(d_b\) output leaves, each with coefficient \(O(1/d_a)\). For the down-target split the output leaves at \(X/a\); the inner coefficient is \(O(1/d_a)\), and the outer coefficient is \(O(1/(r d_a))\). Pairing with either output-star sign therefore gives, in both orientations, \[ |(\mathscr S_B)_{ba}| \ll_{S,\theta} \frac1{\mu_a}+\frac{\mu_b}{\mu_a^2}. \tag{9.7} \] This estimates the full signed output \(s_b\), since its two coefficients are bounded, and its interior contribution has already vanished. For a given pair \(a,b\), the core is uniquely \(g=\gcd(a,b)\); its two common targets are added within that entry. There are at most \(O(B/g)\) possible sources and outputs, and \(\mu_a,\mu_b\asymp\mu_B\) in the buffer. Hence \[ \sum_g\ \sum_{\substack{a=gr,\ b=gq\\a,b\in\mathcal A_B}} |(\mathscr S_B)_{ba}|^2 \ll\sum_g\frac{B^2}{g^2}\frac{BL}{X} \ll\frac{B^3L}{X} \asymp L^2\Delta_B^2. \tag{9.8} \] Thus the contributions of all cores are aggregated at the actual matrix entries before taking the norm. \textbf{Noncoherent entries.} If the source and output have the same parity, only the boundary pairing can occur. Outside the sibling mask, its nonzero entries have \(a/b=qr\) or \(b/a=qr\). Since both labels lie in the fixed buffer, \(qr\le C_+/c_-\); each row has only boundedly many such partners. The coefficient bound (4.17b) makes their operator norm \(O(1/\mu_B)\). For opposite parities only \(\langle i_b,g_a^s\rangle\) occurs. Equality of its labels uses one prime step at \(b\) and two at \(a\). A free common small prime uses one slot on each side, leaving \(a/b\) or \(b/a\) equal to one prime. That prime is bounded by \(C_+/c_-\), so there are boundedly many such partners in each row. Their centre-coordinate sum is \(O(L^C Y/(\mu_a\mu_b))\). For leaf coordinates, sum a possible common small prime together with the unique large prime under their product cutoff. If the common step is upward, its prime \(q\) and the retained prime \(p\) satisfy \(pq\ll X/B\), giving \(O(XL^C/B)\) choices. If the common step is downward, \(q\) divides the source and has only \(O(L)\) choices. The inner segment has \(p\ll X/B\); the outer segment has \(p\ll Xq/B\) but retains a factor \(q^{-1}\) from each outer response. Splitting inner and outer segments therefore gives the same \(O(XL^C/B)\) weighted count. The remaining source/output ratio is a bounded prime, so changing its orientation changes only the fixed constant. If no free common small prime remains, all three small-prime slots are fixed by the reduced quotient, up to \(L^C\) bounded-word choices. Centre terms then cost \(O(L^C/(\mu_a\mu_b))\); leaf terms satisfy the weighted-range inequality preceding (6.8a). Summing these bounds over the at most \(O(B)\) partners gives, in both row and column orientation, \[ \|\operatorname{sym}(\mathscr S_B)_{\rm noncoh}\| \ll L^C\left(\mu_B^{-1}+\frac B{\sqrt X}+\frac{B^2}{X}\right) \le L^{C'}. \] The diagonal is zero, since \(g_a^s\) is exterior to its own molecule. \textbf{Complete vectors and the fixed local matrix.} Let \(\widetilde J_B\) be the raw full residual synthesis matrix. Equation (4.17a) gives \(\sum_a\|k_a^{\rm ker}\|^2\ll L^C B^2/X^{3/2}\). The kernel-driven residual has squared Hilbert--Schmidt norm \(O(L^C B^3/X^{3/2})\). Since \(\|\widetilde V_B\|=O(1)\) and \(\|\widetilde J_B\|_{\rm HS}^2\ll BL^C\), it follows that \[ \|\widetilde V_B^*\widetilde J_B-\mathscr S_B\|_{\rm HS}^2 \ll L^C B^3/X^{3/2}. \] Next use the exact projection identities (6.8c), including \(Z_B\), \(H Z_B\), and \(N_B\). Equations (6.8d) bound their total paired-matrix cost. Finally (5.5a), on the full prefix before local compression, prices the Gram correction. Combining the three costs gives \[ \left\|Q_B^*G_{S,X}Q_B-D_{\rm prefix} -\operatorname{sym}(\mathscr S_B)\right\|_{\rm HS}^2 \ll L^C\left(\frac{B^2}{X}+\frac{B^4}{X^{3/2}}\right) =o(\Delta_B^2). \tag{9.9} \] For projection, the terms \(Z_B^*\widetilde J_B\) and \((P_{\mathcal R}\widetilde V_B)^*H Z_B\) cost, respectively, \(L^C(B^3/X^{3/2}+B^5/X^2)\) and \(L^C(B^2/X+B^4/X^{3/2})\); normalization is smaller. For the Gram term use (5.5b) and \(B/\sqrt X=o(1)\). The ratio of the displayed majorant to \(\Delta_B^2\) is \(O(L^{C+1}(1/B+B/\sqrt X))\), which tends to zero. Restrict (9.9) to the buffer. Define \(\mathcal R_{B,\mathrm{noncoh}}\) to be the noncoherent part of \(\operatorname{sym}(\mathscr S_B)\), and \(\mathcal K_{B,\mathrm{int}}\) to be its coherent part plus the exact matrix difference in (9.9). This defines (9.4) on the same coordinate space and proves (9.5). The matrix difference in (9.9) accounts for the additional entries introduced by Gram conjugation. \(\square\) \subsection{Proposition 9.3 -- threshold-count comparison}\label{proposition-9.3-threshold-count-comparison} Let \(N_G(t)\) count positive one-exit eigenvalues at least \(t\), and let \(N_D(t)\) count entries of \(D_B\) at least \(t\). There is an integer \(r_B\), independent of \(t\) in the target window, such that \[ r_B+N_D(t+2\Delta_B)-X^{o(1)} \le N_G(t) \le r_B+N_D(t-2\Delta_B)+X^{o(1)}. \tag{9.10} \] \textbf{Proof.} Lemma 9.1 and Sylvester inertia reduce the full count at each fixed threshold to the local matrix (9.4), with the constant higher-energy exterior count \(r_B\), at an operator cost \(o(\Delta_B)\). Apply Hoffman--Wielandt only to the fixed matrix \(\mathcal K_{B,\mathrm{int}}\). A root that crosses a threshold by more than \(\Delta_B\) spends at least \(\Delta_B^2\) of (9.5), so at most \(X^{o(1)}\) roots can do so. The noncoherent term and the exterior Schur term consume the other root-width unit. \(\square\) \textbf{Proof of Theorem 1.1.} Proposition 7.1 implies, uniformly for every allowed \(a\in[B/2,B]\), \[ \begin{aligned} r_B+N_D(\nu_a-c\Delta_B)-j_S(a)&\gg_{S,\theta}K_B,\\ j_S(a)-r_B-N_D(\nu_a+c\Delta_B)&\gg_{S,\theta}K_B \end{aligned} \tag{9.11} \] for a sufficiently large fixed \(c\). To see the margin, first fix a positive integer \(m\), then take \(mK_B\) allowed labels on either side of \(a\). A fixed finite deletion set has bounded gaps between allowed integers, so these labels lie within \(O_S(mK_B)\) in centre distance. The upper target-oscillation estimate in Proposition 7.1 places all their squared targets within \(C_{S,\theta,m}B/L\) of the target at \(a\), and hence all their molecule roots within \(C'_{S,\theta,m}\Delta_B\) of \(\nu_a\). Choose \(c>C'_{S,\theta,m}+2\) after \(m\). Both label blocks are then contained in the corresponding root window. Since \(mK_B/B\to0\), the buffered band contains these blocks even at a dyadic endpoint. Sorting moves only \(o(K_B)\) labels, leaving a fixed positive multiple of \(K_B\) on each side. The offset \(r_B\) accounts for the higher-energy labels outside the local ordering. Since \(\theta>21/61>1/3\), \[ X^{o(1)}=o(K_B). \] Combining (9.10)--(9.11), with a larger fixed root-width constant, gives \[ N_G(\nu_a-C\Delta_B)\ge j_S(a),\qquad N_G(\nu_a+C\Delta_B)\le j_S(a)-1. \tag{9.12} \] Thus \[ |\rho_{j_S(a)}(G_{S,X})-\nu_a|\ll_{S,\theta}\Delta_B \] for every allowed centre in the terminal band. Squaring costs \[ O(\mu_B\Delta_B+\Delta_B^2)=O(B/L), \] because \(\Delta_B^2/(B/L)=B^2/X=o(1)\). Proposition 4.2 replaces \(\nu_a^2\) by \(F_{S,a,X}\), and Theorem 3.2 transfers the result to \(A_{S,X}\) at the same squared scale. Summing over the dyadic bands \(X^\theta/L\le B\le X^\theta\) proves (1.8). Lemma 4.3 gives \(M_{S,a,X}\ll_{S,\theta}\log X=o(a/\log X)\) on the terminal band, proving Corollary 1.2. \(\square\) \subsection{Corollary 9.4 -- terminal-band counting rigidity}\label{corollary-9.4-terminal-band-counting-rigidity} Under the hypotheses of Theorem 1.1, let \([B/2,B]\) be a consumed dyadic band and put \(K_B=\lfloor B^3/X\rfloor\). There are constants \(C_1,C_2>0\), depending only on \(S\) and \(\theta\), with the following property. Let \(I=[u,v]\) be an interval for which \[ [u-C_1K_B,v+C_1K_B]\subset[B/2,B], \] and suppose that \(I_S=I\cap V_S(\infty)\) is nonempty. For \[ J_I=\left[ \min_{a\in I_S}F_{S,a,X}-C_2\frac B{\log X}, \max_{a\in I_S}F_{S,a,X}+C_2\frac B{\log X} \right], \] one has \[ \left| \#\{k:\lambda_k(A_{S,X})^2\in J_I\}-\#I_S \right| \ll_{S,\theta}K_B. \tag{9.13} \] \textbf{Proof.} Theorem 1.1 places the eigenvalue at every rank \(j_S(a)\), \(a\in I_S\), inside \(J_I\) once \(C_2\) is sufficiently large. Proposition 7.1 shows that moving a fixed multiple of \(K_B\) allowed labels past either endpoint changes the target by a fixed multiple of \(B/\log X\), while sorting the labelled targets moves each label by only \(o(K_B)\) places. Choose \(C_1\) after \(C_2\). Theorem 1.1 then puts the corresponding exterior boundary eigenvalues strictly above and below \(J_I\). Since the positive eigenvalues are ordered, only \(O_{S,\theta}(K_B)\) ranks adjacent to the two endpoints can enter or leave the window. This proves (9.13). \(\square\) \section{Ordered comparison and proof of Theorem 1.3}\label{ordered-comparison-and-proof-of-theorem-1.3} The density-one argument uses the full prefix and the maximum-norm contraction of sorted values. Throughout this section fix \(0<\theta<1/2\), \(L=\log X\), \(A=X^\theta\), and a dyadic scale \(A/L\le B\le A\). Put \[ \mathcal A_B=[B/2,B]\cap V_S(\infty),\qquad \mu_B=\sqrt{\frac{X}{BL}},\qquad \mathfrak d_{a,X} =|\lambda_{j_S(a)}(A_{S,X})^2-F_{S,a,X}|. \] \subsection{Theorem 10.1 -- ordered mean-square and tail estimates}\label{theorem-10.1-ordered-mean-square-and-tail-estimates} There are constants \(C,C_0,C_1,C_2>0\), depending only on \(S,\theta\), such that, uniformly in \(B\) for all sufficiently large \(X\), \[ \boxed{ \sum_{a\in\mathcal A_B}\mathfrak d_{a,X}^2 \ll_{S,\theta}XL^C+\frac{B^3}{L^2}.} \tag{10.1} \] More precisely, for every \(t>0\), \[ \boxed{ \#\{a\in\mathcal A_B: \mathfrak d_{a,X}>C_0B/L+C_1t\mu_B\} \le C_2\frac{BL^C}{t^2}.} \tag{10.2} \] \textbf{Proof.} Retain all allowed centres \(c\le C_\star B\), with \(C_\star\) a sufficiently large fixed constant, and use \(Q_B,D_B,E_B\) from Section 6. Choose \(\theta<\vartheta<1/2\); then \(C_\star B\le X^\vartheta\) eventually. Proposition 4.2 and Lemma 4.3 give, uniformly on this complete prefix, \[ \nu_c>0,\qquad |\nu_c^2-p_c|\ll_{S,\theta}L,\qquad p_c=\pi_S(X/c). \tag{10.3} \] Indeed the error in (4.9) is \(o(1)\), using \(\omega(c)\ll\log c\) and \(C_\star B/\sqrt X\to0\). The list \(p_c\), in arithmetic order of \(c\), is nonincreasing. Apply (5.12) to the squared molecule energies. Since all molecule roots are positive, sorting their squares also sorts the roots. Thus \[ |\widehat\nu_{j_S(a)}^2-p_a|\ll L,\qquad |\widehat\nu_{j_S(a)}^2-F_{S,a,X}|\ll L \quad(a\in\mathcal A_B). \tag{10.4} \] The index here is the global arithmetic rank: the prefix includes every allowed centre smaller than \(a\). No band-local reindexing or strict separation of the \(p_c\)'s is used. Global PNT gives \(\widehat\nu_{j_S(a)}\asymp_{S,\theta}\mu_B\). Write \(e_B=\|E_B\|\) and \(h_B=\|E_B\|_{\rm HS}\). Corollary 6.2 gives \[ h_B^2\ll BL^C,\qquad e_B^2\ll L^C(1+B^2/\sqrt X),\qquad \frac{e_B^2}{\mu_B^2} \ll L^{C+1}\left(\frac BX+\frac{B^3}{X^{3/2}}\right)=o(1). \tag{10.5} \] The complementary compression is below the entire target band. To see this without an arithmetic cluster, apply Lemmas 5.3 and 6.3 directly: \[ \sup\sigma(C_B) \le \sqrt{d_{S,X}(c_B)} +L^C\sqrt{B/L}+\|H_{S,X}\|, \tag{10.6} \] where \(c_B\) is the first allowed centre above \(C_\star B\). Relative to \(\mu_B\), the last two terms are \(O(L^CB/\sqrt X)\) and \(O(\sqrt B/X^{1/4})\), respectively, hence are \(o(1)\). By PNT the first term is \(O_{S,\theta}(C_\star^{-1/2}\mu_B)\). Choose \(C_\star\) large enough for the complement condition in (5.8). The constants in the lower bound for the target roots are independent of this prefix factor. Lemma 5.4, applied to \(T=G_{S,X}\), now supplies errors \(\epsilon_j\ge0\) satisfying \[ \sum_j\epsilon_j^2\ll BL^C,\qquad \max_j\epsilon_j\le e_B, \qquad |\rho_{j_S(a)}(G_{S,X})^2-\widehat\nu_{j_S(a)}^2| \ll\mu_B\epsilon_{j_S(a)}+e_B^2. \tag{10.7} \] Its ordered indices are unchanged by the complement elimination. In particular the relevant eigenvalues are positive, so they are the positive eigenvalues used in Theorem 3.2. The deterministic Schur cost is smaller than \(B/L\): \[ \frac{e_B^2}{B/L} \ll L^{C+1}\left(\frac1B+\frac B{\sqrt X}\right)=o(1). \tag{10.8} \] Both limits are uniform on \(A/L\le B\le A\), since \(0<\theta<1/2\) is fixed. Also \(L=o(B/L)\). Combining (10.4), (10.7), and the full-graph transfer (3.5) gives \[ \mathfrak d_{a,X} \le C_0B/L+C_1\mu_B\epsilon_{j_S(a)} \quad(a\in\mathcal A_B). \tag{10.9} \] Squaring and summing proves (10.1), because \(\mu_B^2BL^C=XL^{C-1}\), which is absorbed after increasing \(C\). Markov's inequality applied to the squared errors \(\epsilon_j^2\) proves (10.2). \(\square\) Global prime estimates suffice for this proof. Sorting contraction in (10.4) accommodates repeated targets, and the coherent term retained in (10.5) is bounded using the complement gap. \textbf{Proof of Theorem 1.3.} Fix \(R>0\) and take \(t=L^D\) in (10.2), with \(2D>C+R\). Outside \(O_{S,\theta,R}(B/L^R)\) centres of each dyadic band, \[ \mathfrak d_{a,X} \ll_{S,\theta,R}\frac aL+ L^D\sqrt{\frac{X}{aL}}. \tag{10.10} \] Here \(a\) and \(B\) differ by at most a factor of two. Cover \([A/L,A]\) by the bands with upper endpoints \(A, A/2,\ldots\), stopping when the upper endpoint first lies in \([A/L,2A/L)\). Their lengths have geometric sum \(O(A)\). All estimates use the same constants, so their exceptional counts sum to \(O(A/L^R)\). The omitted initial segment has \(O(A/L)\) centres. This proves the stated density-one conclusion below \(A\), as well as the sharper terminal-band exceptional count. Below \(A\), the two contributions to the exceptional set have sizes \(O(A/L^R)\) and \(O(A/L)\), respectively. For the relative conclusion, division of (10.10) by \(\pi_S(X/a)\asymp X/(aL)\) leaves \[ O_{S,\theta,R}\left( \frac{a^2}{X}+L^{D+1/2}\sqrt{\frac aX}\right)=o(1). \] Thus (1.10) holds throughout the asserted range. \(\square\) \textbf{Proof of Corollary 1.4.} On the retained terminal bands, the two error exponents in (1.10), apart from fixed logarithmic factors, are \(\theta\) and \((1-\theta)/2\). Their margins below the block-baseline exponent \(1/2\) are \(1/2-\theta\) and \(\theta/2\). Any smaller positive margin absorbs the logarithmic factors. \(\square\) \subsection{Corollary 10.2 -- quantitative sharp density-one law}\label{corollary-10.2-quantitative-sharp-density-one-law} If \(1/3<\theta<1/2\), then on each retained dyadic band the number of centres for which \[ |\lambda_{j_S(a)}^2-F_{S,a,X}|>C_3B/L \] is at most \[ O_{S,\theta}\left(\frac{XL^{C+1}}{B^2}\right)=o(B). \tag{10.11} \] Consequently the error is \(O_{S,\theta}(a/L)\) outside \(o(X^\theta)\) centres below \(X^\theta\). \textbf{Proof.} Set \(t=B/(\mu_BL)\) in (10.2) and take \(C_3=C_0+C_1\). Then \(BL^C/t^2=XL^{C+1}/B^2\), and its ratio to \(B\) tends to zero uniformly because \(B\ge X^\theta/L\) and \(3\theta>1\). The dyadic summation and initial segment are handled as above. \(\square\) \subsection{Corollary 10.3 -- every-centre bound on a sub-square-root band}\label{corollary-10.3-every-centre-bound-on-a-sub-square-root-band} For every fixed \(0<\theta<1/2\), every allowed \(X^\theta/L\le a\le X^\theta\) satisfies \[ \boxed{ |\lambda_{j_S(a)}^2-F_{S,a,X}| \ll_{S,\theta}\frac aL+ L^{C'}\left[ \sqrt{\frac{X}{aL}}+ X^{1/4}\sqrt{\frac aL}\right].} \tag{10.12} \] This is smaller by a positive power than the block baseline \(\sqrt X/L\). \textbf{Proof.} Use \(\epsilon_j\le e_B\) in (10.9), and take the square root of the corrected operator bound in (10.5): \[ \mu_Be_B \ll L^{C'}\left[ \sqrt{\frac{X}{BL}}+ X^{1/4}\sqrt{\frac BL}\right]. \] Replace \(B\) by \(a\) within its dyadic band. The three error exponents are \(\theta\), \((1-\theta)/2\), and \(1/4+\theta/2\). Their minimum margin below \(1/2\) is \[ \min\{\theta/2,(1-2\theta)/4\}>0. \] This proves the comparison after absorbing logarithmic factors. Theorem 1.1 gives the sharper error on its stated range. \(\square\) \section{Scope of the argument}\label{scope-of-the-argument} Theorem 1.1 identifies every prescribed rank in \([X^\theta/\log X,X^\theta]\) for \(21/61<\theta<1/2\). The proof eliminates couplings with spectrally distant modes and bounds the number of internal eigenvalues that can cross a cluster threshold. The prime-counting comparison supplies more arithmetic labels in each cluster than this bound allows to cross. The initial segment below \(X^\theta/\log X\) is the theorem's only omission among centres up to \(X^\theta\). The ordered mean-square argument gives Theorem 1.3 throughout \(0<\theta<1/2\), using global prime estimates. Its error improves the positive-star block benchmark by a positive power of \(X\). For \(\theta>1/3\), Corollary 10.2 gives the sharp \(O(a/\log X)\) error outside a quantitative exceptional set. Corollary 10.3 gives a weaker every-centre improvement throughout the full sub-square-root range. At \(B\asymp\sqrt X\), the frame and squaring factor \(B^2/X\) ceases to tend to zero, the retained prefix reaches the large-prime cutoff, and \(\|H_{S,X}\|/\mu_B\) is no longer small. The target-star denominator in (9.7b) remains separated from zero. Extending the results to the endpoint requires control of the full boundary block beyond the smallness estimates used here. \section*{Formal verification}\label{formal-verification} \addcontentsline{toc}{section}{Formal verification} The Palomar registration {[}17, version 1{]} covers the Lean 4 formalization of Theorem 1.3 and Corollary 1.4. The selected declarations were checked by the Lean kernel and NanoDa. These two density-one statements constitute the registered verification scope. Formalization of the sharp every-centre Theorem 1.1 and Corollary 1.2 is work in progress. \section*{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}{section}{Declaration of generative AI and AI-assisted technologies in the manuscript preparation process} During the preparation of this work, the author used OpenAI Codex, including AI-assisted research and coding tools, to support mathematical exploration, proof-development assistance, manuscript revision, finite diagnostic design, and Lean 4 formalization. The author reviewed and edited the output as needed and takes full responsibility for the content of this manuscript. \section*{References}\label{references} \addcontentsline{toc}{section}{References} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item M. Anđelić and C. 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