% Formal English paper synthesized from the dissertation % "Power-kernel fluctuations, Lambert-W subpolynomial source, and local CRT conditional law". % Fix applied relative to the dissertation: the weight w(d) is DEFINED as prod_{p|d} p/(p+1) % (the dissertation's notation section printed the reciprocal; the discrete law, its proof, % the normalization identity, and the dissertation's own numerics all require prod p/(p+1)). \documentclass[11pt,a4paper]{amsart} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs} \usepackage[margin=1.1in]{geometry} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \numberwithin{equation}{section} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem*{convention}{Status convention} \theoremstyle{plain} \newtheorem{discovery}[theorem]{Discovery} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\Var}{Var} \DeclareMathOperator{\Corr}{Corr} \DeclareMathOperator{\Ei}{Ei} \newcommand{\e}{\mathrm{e}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Cov}{Cov} \DeclareMathOperator{\Li}{Li} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\Prb}{\mathbb{P}} \newcommand{\bfp}{\mathfrak{p}} \title[Power-Kernel Fluctuations and the Lambert $W$ Subpolynomial Source]{Power-Kernel Fluctuations, the Lambert $W$ Subpolynomial Source,\\ and a Local CRT Conditional Law} \author{Chunping Zeng} \date{September 2026} \begin{document} \begin{abstract} For a positive integer $n$ we define the \emph{power kernel} $d_n=n/\rad(n)$ and the \emph{fluctuation} $X_n=\log d_n$, where $\rad(n)=\prod_{p\mid n}p$. This paper develops three mutually supporting lines around this ``relative fluctuation''. \emph{First, the microscopic theory.} From the kernel decomposition $n=v_d\,s$ ($s$ squarefree, $(s,d)=1$, $v_d=d\,\rad(d)$) we obtain the exact discrete law \[ \Prb(X_n=\log d)=\frac{6}{\pi^{2}}\,\frac{w(d)}{d\,\rad(d)},\qquad w(d):=\prod_{p\mid d}\frac{p}{p+1}, \] together with a three-branch parity-conditioned refinement. Parity is a first-order effect: the measured gap $\E[X_n\mid n\text{ even}]-\E[X_n\mid n\text{ odd}]=0.69008$ matches $\log 2=0.69315$. \emph{Second, the conditional law.} For fixed $c$, the distribution of $X_{c-a}$ is completely determined by $\mathrm{mask}(a)$ through a three-case local CRT convolution: the local contribution vanishes identically when $p\mid c$ or $p\mid a$, and otherwise follows a \emph{corrected} law rather than the usual one, differing by a factor $\frac{p}{p-1}$ (a factor $2$ at $p=2$). The corrected law agrees with $220$ masks to $0.993$, while the usual law underestimates the tail by $47\%$--$78\%$. This replaces an earlier erroneous assertion of equidistribution. \emph{Third, the subpolynomial source.} For radical-type Euler products $\prod_p\bigl(1+\frac{\eta(p)}{p^\sigma-1}\bigr)$ with $\eta(p)\to a$, one has as $\sigma\to0^+$ \[ \Phi(\sigma)=\frac{a}{\sigma W(a/\sigma)}\Bigl[1+\frac{\kappa_1}{W}+\frac{\kappa_2}{W^2}+\cdots\Bigr], \] where $W$ is the Lambert function. The main term comes not from small primes but from the prime range $[e^{v^*},e^{1/\sigma}]$ --- which explains why no finite prime table can reproduce it. The coefficients satisfy $\kappa_n=(n-1)!+\sum_j c_{n,j}\zeta(2j)$, a sum over \emph{even} zetas only, because the two-sided integral coefficients satisfy $\iota(r)=(-1)^{r+1}\rho(r)$, forcing $\zeta(\mathrm{odd})$-terms to cancel exactly at every order. This unifies $T(x)=\sum_{n\le x}1/\rad(n)$, $\Sigma(D)$ and $\Sigma_\perp(D;Q)$, and explains at one stroke why all previously fitted ``constants'' are transient. Applications include: a quantitative formula for Mazur parameters in the open range $12$, while Mazur parameters always have $\alpha,\beta<1$. Our weak abc bound is weaker than de Bruijn's 1962 trivial bound; this is recorded as well. \end{abstract} \maketitle \section{Introduction and notation} \subsection{Background} Let $a+b=c$ with $(a,b)=1$, and write $\rad(n)=\prod_{p\mid n}p$ and $q(a,b,c)=\frac{\log c}{\log\rad(abc)}$. The abc conjecture (Oesterl\'e--Masser) asserts that for every $\varepsilon>0$ there are only finitely many primitive triples with $q>1+\varepsilon$. The starting point of this paper is the observation that $(a,b)=1$ implies that $a,b,c$ are pairwise coprime, so $\rad(abc)=\rad(a)\rad(b)\rad(c)$, and therefore \[ q>1\iff\rad(a)\rad(b)0$: $C(\beta):=\sum_p\frac{\log p}{p^\beta(p-1)}$, $P(s):=\sum_p p^{-s}$. Numerically $C(1)=0.755368$, $C(2)=0.262276$, $P(2)=0.452247514$. \emph{Counting function.} $\mathcal N_\varepsilon(N):=\#\{(a,b,c):a+b=c,\ (a,b)=1,\ c\le N,\ q>1+\varepsilon\}$. \emph{Mazur parameters.} For $0<\alpha,\beta,\gamma<1$, $s:=\alpha+\beta+\gamma$, $S_{\alpha,\beta,\gamma}(N):=\#\{a+b=c\le N:(a,b)=1,\ \rad(a)\le a^\alpha,\ \rad(b)\le b^\beta,\ \rad(c)\le c^\gamma\}$. \emph{Tail functions.} $\Sigma(D):=\sum_{d>D}u(d)$; with coprimality conditions, $\Sigma_\perp(D;Q)$. \section{The exact discrete law of the power kernel}\label{sec:2} \begin{lemma}[Kernel decomposition]\label{lem:2.1} $X_n=\log d$ if and only if $n=v_d\,s$ with $s$ squarefree and $(s,d)=1$. \end{lemma} \begin{proof} Write $n=\prod_p p^{e_p}$. Then $d_n=\prod_{e_p\ge2}p^{e_p-1}$ and $\rad(d_n)=\prod_{e_p\ge2}p$. Hence $d_n=d$ forces $s=n/v_d$ to be squarefree and coprime to $d$; conversely, if $s$ is squarefree with $(s,d)=1$, then the $p$-adic orders in $n=v_d s$ are $v_p(d)+1\ge2$ for $p\mid d$ and $1$ for $p\mid s$, so the support and exponents match exactly. \end{proof} \begin{theorem}[Exact discrete law]\label{thm:2.2} For every $d\ge1$, \[ \Prb(X_n=\log d)=K_6\,u(d)=\frac{6}{\pi^{2}}\cdot\frac{w(d)}{d\,\rad(d)}. \] \end{theorem} \begin{proof} By Lemma~\ref{lem:2.1}, $\Prb(X_n=\log d)=\rho(d)/v_d$ where $\rho(d)$ is the local density of admissible $s$. For a prime $p$: if $p\mid d$ then $p\nmid s$ (factor $1-\frac1p$); if $p\nmid d$ then $v_p(s)\le1$ (since $s$ is squarefree, factor $1-\frac1{p^2}$). By CRT the primes are independent, so \[ \rho(d)=\frac{1}{\zeta(2)}\prod_{p\mid d}\Bigl(1-\frac1p\Bigr) \prod_{p\nmid d}\Bigl(1-\frac1{p^2}\Bigr) =K_6\prod_{p\mid d}\frac{1-\frac1p}{1-\frac1{p^2}} =K_6\prod_{p\mid d}\frac{p}{p+1}=K_6\,w(d). \qedhere \] \end{proof} \begin{proposition}[Normalization]\label{prop:2.3} $\sum_{d\ge1}u(d)=\frac{\pi^2}{6}=\zeta(2)$, via the Euler factor $1+\frac{1}{p^2-1}=\frac{p^2}{p^2-1}$. \end{proposition} \begin{proof}[Verification] The partial sum over $d\le 2\times10^{6}$ is $1.64492235$, relative difference $7.1\times10^{-6}$ from truncation alone. \end{proof} \begin{proposition}[Moments]\label{prop:2.4} With $Y_p(n):=(v_p(n)-1)_+\log p$ one has $X_n=\sum_p Y_p(n)$ and \begin{align*} \E[X_n]&=C(1),\\ \E\Bigl[\sum_{p\mid n}\log p\Bigr]&=C(1)-C(2)=\sum_p\frac{\log p}{p^2(p-1)},\\ \E\Bigl[\sum_{p\mid n}(\log p)\,\mathbf 1_{p\parallel n}\Bigr]&=C(2). \end{align*} \end{proposition} \begin{proof} $\Prb(v_p=k)=(1-\frac1p)p^{-k}$ for $k\ge0$ gives $\E[(v_p-1)_+]\log p=\frac{\log p}{p(p-1)}$; similarly $\E[(v_p-1)_+\mathbf 1_{v_p\ge2}] \log p=\frac{\log p}{p^2(p-1)}$. Summing over $p$ gives $C(1)$, and the difference of the two sums is exactly the contribution of primes dividing $n$ exactly once. \end{proof} \emph{Numerical check.} Direct summation against Theorem~\ref{thm:2.2} gives $\E[X_n]=0.755255$ versus $C(1)=0.755368$ (relative difference $1.5\times10^{-4}$; two independent paths are self-consistent). Also $\E[X_n^2]=2.194778$, $\Var(X_n)=1.624368$. \begin{proposition}[Tail function]\label{prop:2.5} $\Prb(X>y)=K_6\,\Sigma(e^y)$, and numerically $\Prb(X>y)\approx 0.67\,y\,e^{-y}$ in the computable range. \end{proposition} \begin{corollary}[Finite-size ceiling]\label{cor:2.6} $\Xi_n\le(\lfloor\log_2 n\rfloor-1)\log 2$; for $n\le10^5$ this bound is $10.397$. \end{corollary} \begin{proof} $d_n=d$ forces $v_d\le n$, and $v_d=d\,\rad(d)$ grows slowest for $d=2^k$: then $v_{2^k}=2^{k+1}\le n$ gives $k\le\log_2 n-1$ and $X_n=(k-1)\log 2$. \end{proof} \emph{Important remark.} $\Sigma(D)\approx\kappa' D\log D$ with $\kappa'\approx1.1$ is a \emph{transient} law, not an asymptotic one; the true asymptotic is the Lambert source of Section~\ref{sec:7}. This is the common root of all ``constant drift'' phenomena below. \subsection{Parity is a first-order effect} \begin{discovery}[Numerical]\label{disc:2.7} $\E[X\mid n\text{ even}]=1.09696$, $\E[X\mid n\text{ odd}]=0.40687$; the difference $0.69008$ matches $\log 2=0.69315$. \end{discovery} \emph{Mechanism.} Writing $n=2^k m$ with $m$ odd, $X_n=(k-1)\log 2+X_m$, and $\E[k-1\mid n\text{ even}]=1$ (geometric distribution). The parity gap is exactly $\log 2$. \begin{theorem}[Parity-refined discrete law]\label{thm:2.8} Let $(d,Q)=1$ be implicit in the support. Then \[ \Prb\bigl(X_n=\log d\ \big|\ (n,Q)=1,\ n\equiv r\ (2)\bigr)=G\,u(d)\times \begin{cases} 1, & 2\mid Q,\ r=1,\\[2pt] 2\,\mathbf 1_{r=0}, & 2\nmid Q,\ 2\mid d,\\[2pt] \frac23\,\mathbf 1_{r=0}+\frac43\,\mathbf 1_{r=1}, & 2\nmid Q,\ 2\nmid d. \end{cases} \] (In the first branch the hypotheses force $n$ odd, hence $d$ odd, so $2\mid d$ does not arise.) \end{theorem} \begin{proof} By Lemma~\ref{lem:2.1}, $n=v_d s$ with $s$ squarefree, $(s,dQ)=1$; the parity of $n$ is the parity of $v_d s$. If $2\mid d$ then $v_d$ is even so $n$ is even: the $r=1$ probability is $0$ and the $r=0$ weight is doubled by normalization. If $2\nmid d$, then $n\equiv s\ (2)$; among squarefree $s$ coprime to $d$, odd $s$ carry density $\frac23$ and even $s$ (necessarily of the form $2s'$) density $\frac13$, giving weights $\frac43$ and $\frac23$; the parity-conditioned total normalizes with $G=K_6\Pi_Q$. \end{proof} \emph{Verification.} Normalization holds to $3\times10^{-5}$; the sharpest prediction --- $\Prb(d\text{ even}\mid r=1)\equiv0$ --- is confirmed exactly on $n\le4\times10^5$; both parity classes agree to $10^{-4}$ for $y\lesssim4$; $\Sigma/\Sigma_\perp\asymp e^{0.0722}$ over $D:4\to10^6$. \section{A combinatorial restatement of abc}\label{sec:3} \begin{theorem}[Master identity]\label{thm:3.1} Let $(a,b)=1$, $a+b=c$. Then \[ q>1\iff\rad(a)\rad(b)1\iff c>\rad(a)\rad(b)\rad(c)\iff c/\rad(c)>\rad(a)\rad(b)$. \end{proof} \emph{Equivalent form.} Since $a/\rad(a)=d_a$ and $c/\rad(c)=d_c$, the identity reads $\rad(b)1$ then \[ a\cdot\rad(d_b)\le d_ad_c,\qquad b\cdot\rad(d_a)\le d_bd_c,\qquad \frac{ab}{c}\cdot\rad(d_c)\le d_ad_b. \] \end{theorem} \begin{proof} From Theorem~\ref{thm:3.1}, $\rad(b)1\Rightarrow d_b>1$: $b$ is never squarefree. \end{corollary} \begin{proof} From $d_a\le a$, $d_c\le c<2b$ one gets $\frac{d_ad_c}{ab}<\frac{a\cdot2b}{ab}=2$, so $\rad(b)\le b/2$, i.e.\ $d_b\ge2$; the critical case $d_b=1$ would force $\rad(b)=b\le d_ad_c/ab>c/2$, contradicting the strict inequality of Theorem~\ref{thm:3.1}. \end{proof} \begin{theorem}[Prime exclusion]\label{thm:3.5} If $(a,b)=1$ then $p\mid a\Rightarrow p\nmid b$; hence the events $\{p^k\parallel a\}$ and $\{p^j\parallel b\}$ are mutually exclusive for every prime $p$. \end{theorem} \begin{proposition}[Covariance in closed form]\label{prop:3.6} With $Y_p(n):=(v_p(n)-1)_+\log p$, \[ \Cov(X_a,X_b)=-\sum_{\substack{p\\ p\nmid c}}\E[Y_p(a)]\E[Y_p(b)] \longrightarrow-\sum_{p}\frac{(\log p)^2}{p^2(p-1)^2}=-0.163290. \] \end{proposition} \begin{proof} By Theorem~\ref{thm:3.5}, for $p\nmid c$ the events $\{Y_p(a)>0\}$ and $\{Y_p(b)>0\}$ are mutually exclusive, so $\E[Y_p(a)Y_p(b)]=0$ while an independent model gives $\E[Y_p(a)]\E[Y_p(b)]$; the difference is the negative contribution. For $p\mid c$ there is no constraint and the contribution vanishes asymptotically. By Proposition~\ref{prop:2.4}, $\E[Y_p]=\frac{\log p}{p(p-1)}$, and the sum over $p$ converges to the stated constant. \end{proof} \emph{Numerical.} $N=2\times10^4$: $\Cov_{\mathrm{emp}}=-0.076072$, $\Cov_{\mathrm{th}}=-0.079157$ ($4\%$ agreement), correlation $\rho=-0.0633$; prime-by-prime, $\E[Y_p(a)Y_p(b)]\equiv0$ on $p\le130$ as measured, versus a $10^3$-scale prediction from the independent model. \begin{theorem}[Hard data]\label{thm:3.7} Exact counts: $\mathcal N_0(N)=419,734,1269,1752,2054$ for $N=10^5,3\times10^5,10^6,2\times10^6,3\times10^6$. \end{theorem} \begin{proposition}[Growth law, numerical]\label{prop:3.8} $\mathcal N_0(N)\approx1.965\,N^{0.4676}$ with residual $\pm2.5\%$ ($5$ points, $30\times$ range). The ratios $\mathcal N_0/\sqrt N=1.325,1.340,1.269,1.239,1.186$ decrease slowly. Leave-one-out refitting keeps the exponent in $[0.4504,0.4753]$ (span $0.025$): a pure power law is robustly supported by the data. \end{proposition} \begin{remark}[Identifiability of the logarithmic correction]\label{rem:3.1} The same data were once written as $\mathcal N_0(N)\approx3.77\sqrt N(\log N)^{-0.419}$ ($\dagger$), which fits equally well (residual $\pm2.8\%$), but the exponent $-0.419$ is \emph{not} independently measured: (i) it is a conditional parameter obtained with $\alpha=\frac12$ pre-fixed (we reproduce $-0.4189$); freeing $\alpha$ gives $\alpha=0.4676$ and the logarithmic coefficient becomes unidentifiable; (ii) $\log\log N$ and $\log N$ are collinear on the computable range ($\log\log N\in[2.44,2.70]$ while $\log N\in[11.5,14.9]$); a free three-term fit is pathologically ill-conditioned (leave-one-out spans $2.808$ to $27.6$, including meaningless values); (iii) by our own criterion of \S\ref{sec:15.5} --- slowly varying functions manufacture stable-looking false constants --- the exponent in $(\dagger)$ is exactly such an artifact. \emph{Conclusion:} the main result is the pure power law; $(\dagger)$ may be exhibited only as compatibility with a $\sqrt N$ power, never as an independently measured exponent. \end{remark} \begin{remark}[Compatibility with the square-root barrier] An independent lattice program proves unconditionally $\#\{q>2\}\le X^{1/2+o(1)}$ with $o(1)=O(\log\log X/\log X)$. Our exponent $0.467$ satisfies $0.5-0.467=0.033$, while $(\log\log X)/\log X\approx0.49$ at $X=3\times10^6$: the barrier's slack accommodates the measured exponent an order of magnitude over. The mechanistic attribution of the exponent is in Section~\ref{sec:14}. \end{remark} \begin{proposition}[The signature $\beta(\varepsilon)$, numerical]\label{prop:3.9} Fitting $\mathcal N_\varepsilon(N)\asymp N^{\beta}$ on $10^5\to3\times10^6$: \[ \begin{array}{c|ccccccc} 1+\varepsilon&1.00&1.05&1.10&1.20&1.30&1.40&1.50\\ \hline \beta&0.467&0.433&0.388&0.313&0.315&0.204&0.000\\ \mathcal N(3\times10^6)&2054&1047&569&148&38&12&2 \end{array} \] The fitted exponents decrease to $0$ (the single wobble $0.313\to0.315$ is within fit noise); no new solutions appear beyond $q>1.5$ in a $30\times$ range. The median $q$ is $1.051$ ($99\%$ quantile $1.34$); the maximum is $1.56789$ ($2+3^{10}\cdot109=23^{5}$, i.e.\ $4374+1=4375$). \end{proposition} \section{The local CRT conditional law}\label{sec:4} \begin{theorem}\label{thm:4.1} Given $(a,c)$, the hit probability is $\Prb(q>1\mid a,c)=\Prb(X_b>y)$ with $y:=\log(ab)-X_a-X_c$. \end{theorem} \emph{Correction.} Early versions wrote the hit probability as the tail of the three-sum $S=X_a+X_b+X_c$. Given $(a,c)$, $X_a$ and $X_c$ are constants, so the correct object is a \emph{single-variable} tail. This correction is the premise of every subsequent quantitative result. \begin{theorem}[Local CRT conditional law]\label{thm:4.2} Let $c$ be fixed and $a$ uniform on $A:=\{ay\mid\mathrm{mask}(a)=m)=\mathcal S_{m,c}(y)$, a sum over $p\nmid c$, $p\in m$ of the case-C laws. \emph{Key point:} case C is \emph{not} the usual law. The usual law $\Prb(v_p=k)=(1-\frac1p)p^{-k}$ has mean $\E[Y_p]=\frac{\log p}{p(p-1)}$, while the corrected law gives \[ \E[Y_p\mid p\nmid a]=\frac{\log p}{(p-1)^2}, \] a factor $\frac{p}{p-1}$ larger (exactly $2$ at $p=2$). Equivalently $\Prb(Y_p=0)=1-\frac{1}{p(p-1)}$, $\Prb(Y_p=(k-1)\log p)=p^{-k}$ ($k\ge2$). \end{theorem} \begin{proof} Case A: $(b,c)=(c-a,c)=1$, so $p\mid c\Rightarrow v_p(b)=0$. Case B: $p\mid a\Rightarrow p\nmid b$ by Theorem~\ref{thm:3.5}. Case C: let $p\nmid c$; $a$ is uniform on the $\varphi(p^K)=p^{K-1}(p-1)$ reduced residue classes. For $k\ge1$, $v_p(b)=k$ iff $a\equiv c\pmod{p^k}$ and $a\not\equiv c\pmod{p^{k+1}}$: among the $p^{K-k-1}$ lifts of the $p-1$ admissible classes, so $\Prb(v_p(b)=k)=\frac{(p-1)p^{K-k-1}}{p^{K-1}(p-1)}=p^{-k}$. Independence follows from CRT applied to $\prod_p p^{K_p}\mid a$-conditions. Finally $\E[Y_p\mid\text{case C}]=\sum_{k\ge2}(k-1)p^{-k}\log p=\frac{\log p}{(p-1)^2}$. \end{proof} \begin{proposition}[Consistency]\label{prop:4.3} Averaging over the unconditioned prime with $\Prb(p\mid a)=\frac1p$, cases B and C mix to recover the usual law, hence unconditionally $\E[X_b]=C(1)$. \end{proposition} \begin{proof} For $k\ge1$, $\Prb(v_p(b)=k)=\Prb(p\nmid a)p^{-k}=(1-\frac1p)p^{-k}$, and $\Prb(p\nmid b)=\frac1p+\frac{p-1}{p}\cdot\frac{p-2}{p-1}=\frac{p-1}{p}$, matching the usual law. \end{proof} \subsection{Numerical verification, five items} (i) \emph{Prime-by-prime} (case C): with $c\approx8\times10^5$ fixed and $2.4\times10^6$ pairs $(a,c)$: \[ \begin{array}{c|ccccc} p&2&3&5&13&47\\ \hline \text{measured }\E[Y_p]/\log p&0.999513&0.249999&0.062500&0.006944&0.000473\\ \text{corrected }(p-1)^{-2}&1.000000&0.250000&0.062500&0.006944&0.000473\\ \text{usual }(p(p-1))^{-1}&0.500000&0.166667&0.050000&0.006410&0.000463 \end{array} \] Case B ($p\mid a$) is identically $0$ in measurement, zero violations. (ii) \emph{CRT independence}: $\Prb(v_p=k,v_q=j)/((1-\frac1p)(1-\frac1q)p^{-k}q^{-j})= 1.000$ for $(p,q)=(2,3)$, $n=8\times10^5$. (iii) \emph{Mask convolution}: $c=800083$, $3222$ masks ($P\subseteq\{2,3,5,7,11,13\}$, $P\ne\emptyset$), $y=2$: measured/theory $0.9929\pm0.0086$. (iv) \emph{End-to-end decisive comparison} ($c=800083$): the usual law underestimates the tail by $47\%$--$78\%$ with errors accumulating in the tail; the corrected law agrees to $0.4\%$--$2\%$. Mask-by-mask at $y=2$: corrected $0.9929\pm0.0086$ versus usual $1.4417\pm0.2521$ --- the usual law is biased by $44\%$ with $30\times$ the variance. (v) \emph{Provable range}: stratified by $\lambda=\log(D/\sqrt c)$ ($c=800083$, $\log\sqrt c=6.796$): the ratio stays $\approx1$ for $\lambda\lesssim0$ and degrades monotonically across $\lambda\ge0.7$, consistent with the finite-size ceiling $X_b\le\log(b/2)=12.21$. \subsection{Range of validity and failure mechanism} Inside $D\lesssim\sqrt c$ the ratio is $0.96$--$1$; beyond $\sqrt c$ it degrades monotonically ($0.96\to0.72$). This range is exactly the boundary provable by Theorem~\ref{thm:6.3} (main term) plus the trivial tail bound. Theorem~\ref{thm:4.2} replaces the earlier ``equidistribution'' assertion: its error is \emph{structural}, not statistical --- the unconditional law neither knows which primes $c$ and $a$ have excluded (cases A, B) nor uses the case-C distribution (off by the factor $\frac{p}{p-1}$). \section{The analytic theory of \texorpdfstring{$C(\beta)$}{C(beta)}}\label{sec:5} \begin{proposition}[Complete monotonicity]\label{prop:5.1} $C(\beta)$ is analytic on $(0,\infty)$ with $C^{(k)}(\beta)=(-1)^k\sum_p\frac{(\log p)^{k+1}}{p^\beta(p-1)}$, so $(-1)^kC^{(k)}(\beta)>0$. The unique singularity is at $\beta=0$; there is \emph{no phase transition} at $\beta=1$ or $\beta=2$. (An earlier conjecture of a phase transition at $\beta=1,2$ was wrong; see Section~\ref{sec:15.4}.) \end{proposition} \begin{proposition}[Closed-form integral]\label{prop:5.2} $\displaystyle\int_1^2 C(\beta)\,d\beta=\sum_p\frac{p^{-1}-p^{-2}}{p-1}=\sum_p\frac1{p^2}=P(2)=0.452247514$. \end{proposition} \begin{proof} $\int_1^2 p^{-\beta}d\beta=\frac{p^{-1}-p^{-2}}{\log p}$; multiply by $\frac{\log p}{p-1}$ and sum. \end{proof} \emph{Numerical:} numerical integration agrees with the closed form to $9.4\times10^{-8}$. An earlier erroneous value $P(2)-P(3)=0.277$ came from a typo $\frac1{p-1}\to\frac1p$; excluded. \begin{corollary}\label{cor:5.3} $\Delta_*:=\frac{C(1)+C(2)}{2}-P(2)=0.056574$, i.e.\ $11.12\%$ of the chord area. An earlier value $0.23134$ ($45.47\%$) is void. \end{corollary} \begin{theorem}[Laurent expansion]\label{thm:5.4} As $\beta\to0^+$, $C(\beta)=\frac1\beta-\gamma+1.5732\,\beta+O(\beta^2)$. \end{theorem} \begin{proof}[Four steps] \emph{Step 1 (functional recursion).} From $\frac1{p-1}=\sum_{k\ge1}p^{-k}$, $C(\beta)=C(\beta+1)+\sum_p(\log p)\,p^{-(\beta+1)}$. \emph{Step 2 (prime zeta).} $-\frac{\zeta'}{\zeta}(s)=\sum_p\frac{\log p}{p^s-1}=\sum_p\frac{\log p}{p^s}+D(s)$ with $D(s)=\sum_p\frac{\log p}{p^s(p^s-1)}$. \emph{Step 3.} With the Stieltjes expansion $\zeta(1+\beta)=\frac1\beta+\gamma-\gamma_1\beta+\cdots$ one gets $-\frac{\zeta'}{\zeta}(1+\beta)=\frac1\beta-\gamma+(\gamma^2+2\gamma_1)\beta+\cdots$, and $D(1+\beta)=C(1)-E_1\beta+\cdots$ with $E_1=\sum_p\frac{(\log p)^2(2p-1)}{p(p-1)^2}$. \emph{Step 4 (assembly).} Substituting into Step 1, the constant $C(1)$ cancels exactly, leaving $C(\beta)=\frac1\beta-\gamma+[(\gamma^2+2\gamma_1)+E_1+C'(1)]\beta+\cdots$. Numerically $C'(1)=-0.981957$, $\gamma^2=0.333178$, $2\gamma_1=-0.145632$, $E_1=2.367561$, total $1.573151$; independent extrapolation gives $1.57317$. \end{proof} \begin{theorem}[Pole structure and natural boundary]\label{thm:5.5} \[ C(\beta)=\sum_{m\ge1}\mu(m)\sum_{k\ge1}\Bigl(-\frac{\zeta'}{\zeta}\Bigr)\bigl(m(\beta+k)\bigr), \qquad \operatorname*{Res}_{\beta=\frac{\rho}{m}-k}C=\frac{\mu(m)}{m}\operatorname*{Res}_{s=\rho}\Bigl(-\frac{\zeta'}{\zeta}\Bigr). \] The families sit at $\beta=\frac1m-1$ (residue $\mu(m)/m$), $\beta=\frac{\rho}{m}-1$ (accumulating at $\Re\beta=-1$), and $\beta=-\frac{2n}{m}-1\le-1$. Hence $\Re\beta=-1$ is a natural boundary (poles accumulate). Numerically the residue at $\beta=-\frac12$ is $-0.0909087$ ($m=1$ prediction $-0.090909$) and $+0.500189$ at $\beta=0$ ($m=2$, $\rho=\frac12$). \emph{RH formulation:} the RH subfamily ($\beta=\frac{\rho}{m}-1$, $m=1$) lies on $\Re\beta=-\frac12$; but $\beta=-\frac12$ is itself a pole ($m=2$, $\rho=\frac12$), so no functional equation about $\Re\beta=-\frac12$ exists. \end{theorem} \begin{theorem}[$S$--$E$ decomposition and telescoping]\label{thm:5.6} With $S(\beta):=\sum_{k\ge1}\bigl(-\frac{\zeta'}{\zeta}\bigr)(\beta+k)$ and $E(\beta):=\sum_p\log p\sum_{m\ge2}\frac1{p^{m\beta}(p^m-1)}$, $C(\beta)=S(\beta)-E(\beta)$ and $S(1)-S(2)=-\frac{\zeta'(2)}{\zeta(2)}=0.569961$, $C(1)-C(2)=-\frac{\zeta'(2)}{\zeta(2)}-[E(1)-E(2)]=0.493091$. Numerically $S(1)=0.850312$, $E(1)=0.094946$, $S(2)=0.280351$, $E(2)=0.018076$; $S-E$ agrees with $C$ to $10^{-15}$. \end{theorem} \begin{theorem}[The subexponential term and the $2\log2$ cancellation]\label{thm:5.7} For the unconditional three-sum with $\Lambda_{\mathrm H}'(0)=1.741549$, $\Lambda_{\mathrm H}''(0)=3.502899$, with $w=W(6T)$, $c=\frac14$, \[ I_{\mathrm H}(T)=T-\sqrt{\frac{6T}{w}}\Bigl(2+\frac2w+\frac{4(2c-1)}{w^2}+O(w^{-3})\Bigr). \] The source of the constant: the reduced integral $J(A):=\int_{\log2}^\infty\log\bigl(1+\frac{Ae^{-u}}{u}\bigr)\frac{e^u}{u}du$ splits at $u_0$ ($u_0e^{u_0}=A$); the outer region loses $\frac{2\log2-1}{u_0}e^{u_0}$ and the inner region contributes $\frac{2\log2}{u_0}e^{u_0}$; the $2\log2$ cancels, leaving $\frac{e^{u_0}}{u_0}$. Numerically at $u_0=160$, $(J/e^{u_0}-1)u_0=1.027\to1$. The term is subpolynomial and does not change the critical threshold; the same cancellation recurs at higher orders in Section~\ref{sec:7}. \end{theorem} \section{Sieve upper bounds: an honest failure}\label{sec:6} \begin{theorem}[Rigorous tail bound for $\Sigma_\perp$]\label{thm:6.1} For $\sigma\in(0,1)$, \[ \Sigma_\perp(D;Q)\le D^{-\sigma}\Phi(\sigma)\prod_{p\mid Q}\frac{p^2-1}{p^2-p^{-\sigma}}, \qquad \Phi(\sigma)=\zeta(1+\sigma)\prod_p\Bigl(1+\frac{p^{-\sigma}}{p^2-1}\Bigr). \] \end{theorem} \begin{proof}[Sketch] Markov: $\Sigma_\perp(D;Q)\le D^{-\sigma}F_Q(-\sigma)$ with $F_Q(-\sigma)=\sum_{(d,Q)=1}u(d)d^\sigma$. Splitting inside/outside $Q$, the error factors satisfy $R_p\le Cp^{-1-\sigma}$, and $\prod_p(1+R_p)\le\zeta(1+\sigma)\prod_{p\mid Q}(1-\frac1{p^{1+\sigma}})^{-1}$; the identity $\frac{1-p^{-1-\sigma}}{1+\frac{p^{-\sigma}}{p^2-1}}=\frac{p^2-1}{p^2-p^{-\sigma}}$ finishes. \end{proof} \begin{corollary}\label{cor:6.2} With $\sigma=1-\frac1{\log D}$: $\Sigma_\perp\le C_1\frac{\log D}{D}\prod_{p\mid Q}\frac{p^2-1}{p^2-p^{-\sigma}}$, $C_1\approx1.1$. \end{corollary} \begin{theorem}[Sieve theorem with remainder]\label{thm:6.3} Let $Q$ be squarefree, $|I|\ge3$. Then \[ \mathcal N(I;D,Q)\le|I|K_6\Pi_Q\,\Sigma_\perp(D;Q)+C\,2^{\omega(Q)}\sqrt{|I|}(\log|I|)^2. \] \end{theorem} \begin{proof}[Sketch] By Lemma~\ref{lem:2.1}, $\mathcal N=\sum_{d>D,(d,Q)=1}S_d$ with $S_d=\#\{s\le|I|/v_d:\mu^2(s)=1,(s,dQ)=1\} =\frac{|I|}{v_d}K_6w(dQ)+O(2^{\omega(dQ)}\sqrt{|I|/v_d})$; the main terms assemble to $|I|K_6\Pi_Q\Sigma_\perp(D;Q)$ and the remainders are controlled by $v_d\ge2d$. \end{proof} \emph{Effective range (the key limitation).} The main term must dominate the remainder: \[ D\ll\frac{\sqrt{|I|}}{2^{\omega(Q)}(\log|I|)^2}\approx\frac{\sqrt c}{(\log c)^3}, \] while abc requires $D\sim10^5$--$10^7$: a gap of about five orders of magnitude. \begin{proposition}[Moment/Chernoff strict insufficiency]\label{prop:6.4} The moment method gives $\mathcal N\le D^{-\sigma}\sum_n d_n^\sigma$, which is strictly weaker than the target inequality. \end{proposition} \begin{proposition}[Union bound loss]\label{prop:6.5} Interchanging the sums loses $\log D$; for $\varepsilon=0$ the resulting series diverges. \end{proposition} \begin{proposition}[Joint constraints do not decouple, numerical]\label{prop:6.6} At $\varepsilon=0.1$, $c\le3\times10^4$: the joint three-radical constraint gives $94$ solutions; the single $\rad(b)$ constraint gives $36362$: an inflation by a factor $387$. Any decoupling suffices to turn a convergent series into a divergent one. \end{proposition} \begin{theorem}[Weak abc, unconditional --- and weaker than the trivial bound]\label{thm:6.7} \[ \small \mathcal N_\varepsilon(N)\le\sum_{c\le N}\min(A_c,B_c),\quad A_c=\#\{a<\tfrac c2:\rad(a)0$ satisfy $\eta(p)\to a\in(0,\infty)$ (error $O(p^{-\delta})$). Define \[ \Phi(\sigma):=\sum_p\log\Bigl(1+\frac{\eta(p)}{p^\sigma-1}\Bigr),\qquad v^*:=W\Bigl(\frac a\sigma\Bigr)\quad(\text{Lambert }W). \] \begin{theorem}[Main theorem]\label{thm:7.1} As $\sigma\to0^+$, \[ \Phi(\sigma)=e^{v^*}\Bigl[1+\frac{\kappa_1}{v^*}+\frac{\kappa_2}{v^{*2}}+\frac{\kappa_3}{v^{*3}}+\cdots\Bigr], \] where $\frac{a}{\sigma v^*}=e^{v^*}$, so the leading term is $\frac{a}{\sigma W(a/\sigma)}$. The coefficients are universal: independent of the shape of $a$ and $\eta$. \end{theorem} \begin{proof}[Skeleton] \emph{Step 1 (PNT to integral).} By Abel summation and the prime number theorem, $\Phi(\sigma)=\int_{\log2}^\infty\frac{e^v}{v}\log\bigl(1+\frac{a}{e^v(e^{\sigma v}-1)}\bigr)dv+\mathcal E$, with $\mathcal E$ controlled by the standard PNT remainder (Section~\ref{sec:12.4}). \emph{Step 2 (regime change).} With $\psi(v):=\frac{a}{e^v(e^{\sigma v}-1)}$ one has $\psi(v^*)=1+O(e^{-v^*})$ since $\sigma v^*=e^{-v^*}$. \emph{Step 3 (identification of the dominant range).} For $v\ge v^*$, $\psi\le1$ and $\log(1+\psi)=\psi+O(\psi^2)$, so $\int_{v^*}^{\infty}\frac{e^v}{v}\psi(v)dv=\int_{v^*}^{\infty}\frac{dv}{v(e^{\sigma v}-1)} =\int_{e^{-v^*}}^\infty\frac{du}{u(e^u-1)}$ (substituting $u=\sigma v$) $=e^{v^*}-\frac{v^*}{2}+C_0+O(e^{-v^*})$ by the Bernoulli expansion $\frac1{u(e^u-1)}=\frac1{u^2}-\frac1{2u}+\frac1{12}+O(u)$. Hence the leading term is $\frac{a}{\sigma v^*}=e^{v^*}$. \textbf{The dominant primes lie in $[e^{v^*},e^{1/\sigma}]$:} there $\log(1+\psi)\approx\psi\approx\frac{a}{p\sigma\log p}$, summing over $p$ to $\frac{a}{\sigma v^*}$. This explains why no finite prime table can reproduce $\Phi$: $e^{1/\sigma}$ is astronomically large. \emph{Step 4 (exact $1/v^*$ correction).} In the dominant range $\sigma v\le\sigma v^* =e^{-v^*}\ll1$, so $e^{\sigma v}-1=\sigma v(1+O(\sigma v))$ and $\psi(v)=\frac{v^*}{v}e^{v^*-v}(1+O(e^{-v^*}))$. Region I ($\log2\le v\le v^*$, $\psi\ge1$): substituting $v=v^*-t$, $\Phi_{\mathrm I}=\frac{e^{v^*}}{v^*}\underbrace{\int_0^\infty e^{-t}\log(1+e^t)dt}_{=:\,\iota_0} +O\bigl(\frac{e^{v^*}}{v^{*2}}\bigr)$. Region II ($v\ge v^*$, $\psi\le1$): substituting $v=v^*+s$ and subtracting the $\psi-\log(1+\psi)$ deficit, $\Phi_{\mathrm{II}}=e^{v^*}-\frac{e^{v^*}}{v^*}\underbrace{\int_0^\infty e^s\bigl[e^{-s}-\log(1+e^{-s})\bigr]ds}_{=:\,\rho_0} +O\bigl(\tfrac{e^{v^*}}{v^{*2}}\bigr)+O(v^*)$. Both integrals have closed forms: $\iota_0=2\log2$ and $\rho_0=2\log2-1$. Hence the $1/v^*$ correction is $\frac{\Phi_{\mathrm I}+\Phi_{\mathrm{II}}-e^{v^*}}{e^{v^*}/v^*}=\iota_0-\rho_0 =2\log2-(2\log2-1)=\mathbf1$, i.e.\ $\kappa_1=1$ \emph{exactly}: region I contributes $2\log2$, region II contributes $-(2\log2-1)$, and the net is $1$. This cancellation recurs at every higher order (Section~\ref{sec:7.4}). \emph{Caveat.} The $O(e^{v^*}/v^{*2})$ error above gives only the magnitude; the precise values of $\kappa_2,\kappa_3,\dots$ come from the closed-form integrals for $\iota_N$, $\rho_N$ (Sections~\ref{sec:7.3}--\ref{sec:7.5}); the early sloppy statement ``region I is $O(e^{v^*}/v^{*2})$'' does not determine $\kappa_1$ --- the $1/v^*$ main part is the carrier of $\kappa_1$. \end{proof} \begin{lemma}[Consistency of the PNT exchange]\label{lem:7.1} Let $\Phi_{\mathrm{PNT}}(\sigma)$ be the integral main term of Step 1. As $\sigma\to0^+$ (equivalently $v^*\to\infty$), $\frac{|\Phi_{\mathrm{true}}(\sigma)-\Phi_{\mathrm{PNT}}(\sigma)|}{\Phi_{\mathrm{PNT}}(\sigma)} \ll\exp(-c\sqrt{v^*})\to0$. \end{lemma} \begin{proof}[Sketch] Abel summation gives $\Phi_{\mathrm{true}}=-\int_2^\infty\pi(t)f_\sigma'(t)dt$ with $f_\sigma(t)=\log(1+\frac{\eta(t)}{t^\sigma-1})$; writing $\pi(t)=\Li(t)+E(t)$ with $|E(t)|\ll t\exp(-c\sqrt{\log t})$, the main term is $\Phi_{\mathrm{PNT}}$ and the remainder is $\ll\exp(-c\sqrt{v^*})\Phi_{\mathrm{PNT}}$ because the dominant range forces $\log t\ge v^*$. Outside the dominant range both $\Phi$ and its PNT error are $O(e^{v^*}/v^*)=o(\Phi_{\mathrm{PNT}})$. \end{proof} \begin{remark}[The series is asymptotic, not convergent] The right side of Theorem~\ref{thm:7.1} is a Poincar\'e asymptotic: for fixed truncation order $M$ the remainder is $O(e^{v^*}v^{*-M-1})$. For fixed $v^*$ the series diverges: $\kappa_n\sim(n-1)!$ makes the terms grow beyond $n\gtrsim v^*$, so the optimal truncation is at $n\approx v^*$. In the computable range $v^*\sim0.9$--$1.3$, $1/v^*$ is \emph{not} a small parameter, so every application must use the exact $\Phi$ (numerical integration or high-precision prime summation) for the saddle point, never the truncated expansion. This is the unified root of the ``constant drifts'' of $\kappa'\approx1.1$, $\kappa_1\approx1.67$, $\omega\approx0.66$. \end{remark} \subsection{Closed forms for \texorpdfstring{$\kappa_1$--$\kappa_8$}{kappa1-kappa8}}\label{sec:7.3} \[ \begin{array}{c|l} n&\kappa_n\text{ (closed form)}\\ \hline 1&1\\ 2&1+2\zeta(2)=1+\frac{\pi^2}{3}\\ 3&2+\zeta(2)=2+\frac{\pi^2}{6}\\ 4&6+7\zeta(2)+42\zeta(4)=6+\frac{7\pi^2}{6}+\frac{7\pi^4}{15}\\ 5&\frac{119}{24}+23\zeta(2)-\frac52\zeta(4)\\ 6&120+121\zeta(2)+448\zeta(4)+1395\zeta(6) \end{array} \] ($\kappa_7,\kappa_8$ similar; $\kappa_5$ is negative --- the first sign change, with absolute value exceeding $(n-1)!=24$.) Verification by two independent paths (integral coefficients vs.\ closed form): \[ \begin{array}{c|l|l|l} n&\text{numerical}&\text{closed form}&\text{error}\\ \hline 1&1.000000000000000&1&5\times10^{-31}\\ 2&4.28986813369645287&1+\pi^2/3&10^{-28}\\ 3&3.64493406684822644&2+\pi^2/6&3.5\times10^{-18}\\ 4&62.97211428380538909885397&6+\frac{7\pi^2}{6}+\frac{7\pi^4}{15}&1.25\times10^{-19} \end{array} \] $\kappa_5$--$\kappa_8$ confirmed by PSLQ at $1300$ digits with reconstruction error $10^{-19}$. \subsection{The cancellation mechanism \texorpdfstring{$\iota(r)=(-1)^{r+1}\rho(r)$}{iota(r)=(-1)^{r+1}rho(r)}}\label{sec:7.4} At $v^*$ the integral splits into region I ($vv^*$); writing the coefficients of $\zeta(r)\log2$ on the two sides as $\iota(r)$ and $\rho(r)$, one has $\kappa_n=\iota_{n-1}-\rho_{n-1}$; e.g.\ for $n=2$: $\rho_2=-2-\frac{\pi^2}{12}+4\log2+\frac92\zeta(3)$, $\iota_2=\frac{\pi^2}{12}+4\log2+\frac92\zeta(3)$. Hence $\kappa(r)=\iota(r)-\rho(r)=[(-1)^{r+1}-1]\rho(r)$: \[ \begin{array}{c|c|c} r&(-1)^{r+1}-1&\text{result}\\ \hline \text{odd (incl.\ }r=1\text{'s }\log2,\ \zeta(3),\zeta(5),\dots)&1-1=0&\text{exact cancellation}\\ \text{even }(\zeta(2),\zeta(4),\dots)&-1-1=-2&\text{doubled, survives} \end{array} \] This is the complete mechanism behind ``even zetas only'': the two-sided kernels are related by $\log(1+e^t)=t+\log(1+e^{-t})$, i.e.\ $t\leftrightarrow s$ with reversed orientation, producing the factor $(-1)^{r+1}$ at order $r$. It is the higher-order version of the first-order cancellation of Theorem~\ref{thm:5.7}. \subsection{Stirling convolution and the general term \texorpdfstring{$c_{n,j}$}{c\_{n,j}}}\label{sec:7.5} \begin{theorem}[Structure theorem]\label{thm:7.2} $\displaystyle\kappa_n=(n-1)!+\sum_{j=1}^{\lfloor n/2\rfloor}c_{n,j}\,\zeta(2j)$, with $c_{n,1}=(n-1)!+(-1)^n$ (confirmed at $16$ data points); only even zeta values occur, no $\log2$, no odd zetas. \end{theorem} \begin{proof}[Sketch of the $\rho$-side Stirling closed form] Partial fractions for $\frac{P_N(m)}{(m+1)m^{N+1}}=\frac{C}{m+1}+\sum_{r=1}^{N+1}\frac{D_r}{m^r}$ with $P_N(m)=\prod_{i=2}^{N+1}(m+i)$ give $C=N!(-1)^{N+1}$, $D_1=(-1)^NN!$, and for $1\le r\le N+1$ an explicit Stirling-number form for $D_r^{(N)}$; hence $\rho_N[\zeta(r)]=-N!(-1)^{N+1}D_r^{(N)}(1-2^{1-r})$ and finally \[ c_{n,j}=2(-1)^nD_{2j}^{(n-1)}\bigl(1-2^{1-2j}\bigr). \] \emph{New identity:} $(n-1)!=|D_1^{(n-1)}|$, so the constant term is the $r=1$ Stirling coefficient; the $\log2$ term vanishes identically because $(1-2^{1-2j})=0$ has no solution --- wait: precisely, at $r=1$ the factor $(1-2^0)=0$ kills the $\log2$-carrying term, leaving the $(n-1)!$ constant. Same mechanism, second manifestation. Also $D_2^{(n-1)}=(-1)^n(n-1)!+1$, giving $c_{n,1}=(n-1)!+(-1)^n$. \end{proof} \emph{New predictions} from the general term (first $\zeta(2)$-generated constant terms): $\kappa_9=40320+40319\zeta(2)+\frac{279759}{4}\zeta(4)-\frac{25141}{2}\zeta(6)-\frac{2387727}{4}\zeta(8)+\cdots$; $\kappa_{10}=362880+362881\zeta(2)+\frac{254851}{4}\zeta(4)+\frac{766519}{8}\zeta(6)+\frac{3785299}{8}\zeta(8)+7243425\zeta(10)$. \subsection{Generating function} With $z=1/v^*$ ($\sigma=az\,e^{-1/z}$): \[ F(z):=\sum_{n\ge0}\kappa_nz^n=e^{-1/z}\Phi\bigl(\sigma(z)\bigr) =\int_{-\infty}^{\infty}\frac{ze^u}{1+uz}\log\Bigl(1+\frac{e^{-u}}{1+uz}\Bigr)du, \] giving any $\kappa_n$ by termwise expansion --- no PSLQ needed. \subsection{Saddle constant and transient laws}\label{sec:7.7} \begin{theorem}\label{thm:7.3} $\mathfrak c=1$ (asymptotically). \end{theorem} \begin{proof}[Sketch] For $\Psi(s)=\Phi(s)+sL-\log s$ the Edgeworth first non-Gaussian correction decays super-exponentially: \[ \footnotesize \begin{array}{c|cccccc} 2^j&2&4&8&16&32&64\\ \hline \text{correction}&-2.8\times10^{-3}&-5.7\times10^{-5}&-2.0\times10^{-8}&-2.3\times10^{-15}&-3.0\times10^{-29}&\text{(below precision)} \end{array} \] so $\mathfrak c\to1$ far faster than any power; the scaling is $\asymp e^{-v^*}$. \end{proof} \emph{Important correction.} Earlier reports $\mathfrak c\approx0.84$--$0.89$ and $\log\mathfrak c=-0.392/L^{0.421}$ are \textbf{void}: they came from insufficient PNT integration accuracy (a $10^{-4}$ relative error in $\Phi$ amplified through subtraction of $O(4)$-magnitude terms into a fake $O(0.1)$ bias). High-precision recomputation gives $\mathfrak c=1.00$--$1.06\to1$. The theoretical asymptotic bias is below current numerical reach; $\mathfrak c=1$ is to be regarded as confirmed. \emph{Transient law (derived, not fitted):} \[ \log T(x)=\underbrace{\bigl(\Phi(\sigma^*)+\sigma^*L\bigr)}_{>\sqrt L} -\underbrace{\log\bigl(\sigma^*\sqrt{2\pi\Phi''}\bigr)}_{>\log L} +\underbrace{\log\mathfrak c}_{O(0.1)}=1.667\sqrt L-1.246. \] \emph{Core methodology:} at computable scales $v^*\sim1$, $1/v^*$ expansion diverges ($1+\frac1{v^*}+4.29+\cdots$ grows), so the only correct procedure is to saddle with the \emph{exact} $\Phi$. Every ``unstable constant'' is the error of treating a transient as an asymptotic. \subsection{Classification theorem and double criticality}\label{sec:7.8} Let $\eta(p)\asymp a\,p(\log p)^{-\alpha-\beta}$. Then: \[ \begin{array}{c|l|c|l} \text{class}&\Phi(\sigma),\ \sigma\to0^+&\delta_{\mathfrak c}&\text{Tauberian: }\log A(x)\\ \hline \alpha<1&\text{diverges}&1-\alpha&(1-\alpha)\log x\ \text{(polynomial)}\\ \alpha>1&C\sigma^{-1/\alpha}&0&c_a(\log x)^{1/(\alpha+1)}\\ \alpha=1,\ \beta>0&a/\sigma\ (\beta+1)L^{\beta+1}&0&\sim2\sqrt{C\log x}\\ \alpha=1,\ \beta=0&\frac{a}{\sigma W(a/\sigma)}&0&2a\frac{\log x}{\sqrt{2\log\log x}} \end{array} \] Unified by the rescaled Lambert variable $v_g=(\beta+1)W\bigl(\frac{(a/\sigma)^{1/(\beta+1)}}{\beta+1}\bigr)$, $\Phi=\frac{e^{v_g}}{\beta+1}(1+\cdots)$. $\alpha=1,\beta=0$ is the \emph{double critical point}: it carries one fewer $\sqrt{\log\log x}$ than both neighbors --- the signature where $W$-resummation and order-by-order cancellation appear. Numerically the ratio $\frac{\log A}{2\sqrt{2L/\log L}}$ decreases from $3.86$ ($L\le55$) to $1.044$ ($L=10^{21}$), extremely slowly: this quantitatively explains why empirical fits look like $\sqrt L$ rather than $\sqrt{L/\log L}$. \emph{Unification.} $T(x)=\sum_{n\le x}\frac1{\rad(n)}$ ($\eta_p=1$), $\Sigma(D)$ ($\eta_p=\frac1p$... more precisely $\eta(p)=1/(p+1)$ forms) and $\Sigma_\perp(D;Q)$ all satisfy $p\,\eta(p)\to1$ and share the same $\Phi$; measured $\Psi/\Phi=0.9707\to1.000000$ as $\varepsilon:10^{-2}\to10^{-10}$. \emph{Non-member:} $\sum\frac1{\varphi(n)}$ has local factor $\frac{p}{(p-1)(p^{1+s}-1)}$, an effective parameter $s+1$ on the boundary of the $\Psi$-class, giving $A\log x$; smooth numbers have a different structure and are not in this class. \section{Quantitative theory of Mazur parameters}\label{sec:8} \subsection{Two-form formula} From Section~\ref{sec:7}, the one-sided probability $p_\nu(n)=K_6\Sigma(n^{1-\nu})$ has two forms, with $2v^*+\log v^*=\log\log D$: \[ \begin{array}{c|l|l} \text{form}&\text{condition}&\Sigma(D)\\ \hline \text{transient}&v^*\le1&\log D\sim K'D\ (\text{one }\log\text{ per side})\\ \text{asymptotic}&v^*>1&C\exp\bigl(2\sqrt2\sqrt{\tfrac{\log D}{\log\log D}}\bigr)\ (\text{no }\log) \end{array} \] \emph{Computable law} ($12$, while Mazur parameters always have $\alpha,\beta<1$: the circle-method remainder is \emph{never} controllable. This explains why BLT had to switch to geometric number theory, and turns ``Fourier cannot save abc'' from intuition into a criterion with an exponent. \section{Ceiling extremal points}\label{sec:9} \begin{proposition}[Two exact identities]\label{prop:9.1} With $\lambda:=\frac{c}{\rad(abc)}=\frac{\rad(c)}{d_c}\rad(a)\rad(b)$: \[ q=\frac{\log c}{\log c-\log\lambda}, \] so $q$ is a race: the numerator gains from the power-kernelization $d_c$, the denominator from $\rad(a)\rad(b)$ being small. On $234$ solutions, $\max|q-\frac{\log c}{\log c-\log\lambda}|=4.4\times10^{-16}$. \end{proposition} \begin{theorem}[Ceiling theorem, sharp]\label{thm:9.1} For any coprime triple $a+b=c$, $(a,b)=1$, $1\le a\frac12=\frac{\log2}{\log2+\log\rad(c)}$: the inequality is violated. The cause: in the proof, the lower bound $\rad(a)\rad(b)\ge2$ degenerates to $1$ when $a=b=1$. Exhaustive check at $c\le2\times10^4$: under the relaxed convention $a\le b$, $(1,1,2)$ is the \emph{unique} violator; under the paper's domain ($a1$ solutions, while $(1,1,2)$ has $q=1$ and slipped through the gap --- a specimen of \emph{verification-set self-selection bias} (\S\ref{sec:15.5}). \end{remark} \subsection{Extremal reconfiguration and the bearing rate} Since $q$ decreases in $\rad(a)\rad(b)$, $q_{\max}(c)=\frac{\log c}{\log c-\log(d_c/\rho(c))}$ with $\rho(c):=\min_{ay\}}{\sum_m\mathrm{count}[m]\,\mathcal S_{m,c}(y)}, \qquad \lambda:=\log\frac{D}{\sqrt c}=y-\frac12\log c. \] \[ \begin{array}{c|ccc|c} \lambda&c=10^4&c=10^5&c=10^6&\text{between-group spread/stat.\ error}\\ \hline -2.0&1.025&1.022&1.015&1.5\\ -1.0&1.054&1.057&1.041&0.7\\ -0.5&1.084&1.122&1.068&0.9\\ 0.0&0.925&0.943&0.948&0.8\\ 1.0&0.852&0.860&0.870&0.6\\ 1.25&0.808&0.833&0.840&0.9\\ 1.5&0.669&0.753&0.798&1.2\\ 2.0&0.525&0.635&0.781&2.9 \end{array} \] Collapse holds for $\lambda\in[-1.5,1.25]$ (ratio $\approx1$); it breaks for $\lambda\ge1.5$. The curve runs from $1.02$ ($\lambda=-2$) through a peak ($\approx1.09$ at $\lambda\approx-0.5$) down to $0.83$ ($\lambda=1.25$); note $R>1$ means measurement exceeds theory. \subsection{Hard truncation mechanism} In reality $n=v_ds\le c$, so for squarefree $d$: $v_d\le c\iff d\le\sqrt c$. Thus $\lambda=0$ is exactly the switch where ``squarefrees begin to be hard-truncated''. For $\lambda<0$ squarefrees dominate, no truncation, $R\approx1$; for $\lambda>0$ only power-kernelized $d$ (small $\rad$, $v_d\ll d^2$) survive, whose distribution is governed by the Lambert $W$ source with an extra $c$-dependent collapse breakdown. This explains why abc is out of reach: it requires $D$ in $\lambda>1$, a region with no universal law. \subsection{Exact closed form of the lattice surplus \texorpdfstring{$r(L)$}{r(L)}} For even $c$, $\ell=c/2$, $|J|=c/2$, and $L_d:=|J|/v_d$: the $s$-window is $[\lceil L_d\rceil,\lfloor2L_d\rfloor]=[L_d,2L_d)$ (the doubling window). Confirmed: $A_d=\operatorname{nint}(L_d)$ covers $99.89\%$; the squarefree count in $[\lceil L\rceil,\lfloor2L\rfloor]$ covers $99.83\%$ (max deviation $1$). This explains the universality of $r(L)$: the window depends only on $L_d$, not on $c$. \begin{proposition}[Exact closed form]\label{prop:11.1} With $G(L,M):=\#\{s\in[L,2L):\mu^2(s)=1,\ (s,M)=1\}$ and $[j^2,k]:=\operatorname{lcm}(j^2,k)$: \[ G(L,M)=\sum_{j\ge1}\sum_{k\mid M}\mu(j)\mu(k)\Bigl(\Bigl\lfloor\frac{2L}{[j^2,k]}\Bigr\rfloor-\Bigl\lfloor\frac{L}{[j^2,k]}\Bigr\rfloor\Bigr). \] For $M=1$: $\#\{s\in[L,2L):\mu^2(s)=1\}=\sum_{j\ge1}\mu(j)\bigl(\lfloor\frac{2L}{j^2}\rfloor-\lfloor\frac{L}{j^2}\rfloor\bigr)=K_6L+\Delta(L)$, with $\Delta(L)$ the deterministic arithmetic oscillation of the squarefree counting function in the doubling window (not noise): $|\Delta|\asymp L^{1/4}$, relative amplitude $\asymp L^{-3/4}$. Measured: $r(L)=2.05,1.05,1.14,0.82,1.00,0.98$ for $L=1,2.4,4.5,5.5,20,100$ (consistent across $c=10^4\to10^6$). Note $r(1)=2.05$: when the window contains exactly one integer, it is forced to be $1$, $\mu^2(1)=(1,d)=1$ holds, but the density only gives $K_6w(d)<1$: the extreme form of Ehrhart surplus. It resolves an earlier paradox (``the exact ceiling asymptotic density underestimates by a factor $2$--$3$''): the underestimate is the $r(L)>1$ part, and the peak of $R$ at $\lambda\approx-0.5$ corresponds to $L_d\approx e^{-2\lambda}\approx2.7$, exactly the surplus interval of $r$. \end{proposition} \subsection{The second scaling variable} The crossover scale is not $\sqrt c$. Collapse search with $t=\log(D/c^{1/2+\gamma})$: \[ \begin{array}{c|cccc} \text{exponent }1/2+\gamma&0.50\ (\text{no shift})&0.70&0.77&0.80\\ \hline \text{between-}c\text{ spread}&0.0644&0.0224&0.0067&0.0063 \end{array} \] a $10\times$ collapse improvement. Hence \[ \tau=\log D-\kappa\log c,\qquad \kappa=\frac1{1+\theta},\qquad \theta:=\Bigl\langle\frac{\log\rad(d)}{\log d}\Bigr\rangle_{\text{tail }d}; \] squarefrees give $\theta=1$, $\kappa=\frac12$ (a special case); tail measurements ($c=10^6$, $c=10^7$) give $\theta=0.248$, $\kappa=0.800,0.785$, consistent with the fitted $0.77$--$0.80$. The crossover scale is $c^{0.78}$ because the tail-contributing $d$ satisfy $\rad(d)\approx d^{1/4}$, $v_d\approx d^{5/4}\le c\Rightarrow d\lesssim c^{0.8}$. \emph{Ambiguity in the definition of $\theta$ (important).} Direct $u(d)$-weighted averages give $\theta_{\mathrm{glob}}\approx0.72$--$0.76$, while back-solving from $\kappa$ gives $\theta=0.248$: these differ by a factor $3$ and are two different quantities. $\theta_{\mathrm{bound}}=1/\kappa-1$: the upper bound forced by $\rad(d)\le c/d$ (literature back-solved values belong to this class); $\theta_{\mathrm{typ}}=\frac{1-\kappa}{1+\kappa}$: measured typical values (deviation only $0.02$--$0.03$, valid throughout $\kappa\in[0.5,0.9]$); $\theta_{\mathrm{glob}}$: the untruncated global average, irrelevant here. Thus $\kappa$ is not a constant but a function of $\theta$; ``$\kappa\to4/5$'' and ``$\theta\to1/4$'' are two statements of the same fact with no independent predictive power. \section{BGT slow variation and Bahadur--Rao}\label{sec:12} \subsection{\texorpdfstring{$T(x)$}{T(x)} is slowly varying} $\frac{T(2x)}{T(x)}$: $1.2331$, $1.2120$, $1.1828$, $1.1477$ (at $x=10^3,10^4,10^5,10^6$), decreasing monotonically to $1$; similarly ($\lambda=10$) $1.987\to1.632$. Theoretically, $\log\frac{T(\lambda x)}{T(x)}\sim\frac{\sqrt2\log\lambda}{\sqrt{L\log L}}\to0$ for fixed $\lambda$, so $T$ is slowly varying. The paper's earlier homemade term ``subpolynomial source'' is standardly called \emph{slow variation}. \subsection{Second-order slow variation and the de Haan auxiliary function} The BGT second-order theory asks for an auxiliary $g$ with $\frac{A(\lambda x)/A(x)-1}{g(x)}\to\log\lambda$. Here the natural choice is $g(x):=\frac{d\log T}{d\log x}$. Measured (smoothed local regression): \[ \begin{array}{c|cccc} x&10^3&10^5&10^6&10^7\\ \hline g(x)&0.313&0.246&0.2206&0.2055 \end{array} \] $g\to0$ slowly and second-order slow variation holds; $\log\frac{T(\lambda x)}{T(x)} /\log\lambda$ checks agree to $3\%$ (residual: $g$ decreasing on the interval). This is the standard name for the ``common orbit'': all the Section~\ref{sec:7} transient constants drifting along one orbit parametrized by $v^*$ is the statement ``$T$ is second-order slowly varying with de Haan auxiliary given by the Lambert saddle''. \subsection{Envelope theorem} From the Perrinversion saddle \[ \log T=\bigl(\Phi(\sigma^*)+\sigma^*L\bigr)-\log\bigl(\sigma^*\sqrt{2\pi\Phi''}\bigr)+\log\mathfrak c \] with $\Phi'(\sigma^*)=-L+\frac1{\sigma^*}$: differentiating in $\sigma^*$, $\partial_{\sigma^*}[\Phi+\sigma L]=\Phi'+L=\frac1{\sigma^*}\ne0$ --- the term $\frac1{\sigma^*}\frac{dL}{d\sigma^*}$ was initially missed; the complete formula is \[ g=\frac{d\log T}{d\log x}=\sigma^*+\frac{\Phi'''}{2\Phi''\bigl(\Phi''+\sigma^{*-2}\bigr)}. \] Verification: ratio measured/predicted $=1.043,1.031,1.016,1.017$ at $x=3\times10^3,10^5,10^6,3\times10^6$ (agreement $1.5\%$--$4\%$; before adding the missed term the ratio was $0.83$, off by $17\%$). The auxiliary function is determined by the third derivative of $\Phi$ at the saddle. \subsection{Far-tail rigorous version and the PNT transition}\label{sec:12.4} Exact Mellin inversion: $T(x)=\frac1{2\pi}\int_{\R}D(c+it)\frac{x^{c+it}}{c+it}dt$ with $D(s)=\sum_n\frac1{\rad(n)n^s}$; writing $\Psi(s)=\log D(s)+sL-\log s$, the Gaussian expansion gives $\mathfrak c$ exactly as the far-tail relative contribution: \[ \mathfrak c=\frac1{\sqrt{2\pi}}\int_{\R}\exp\Bigl(\sum_{n\ge3}\frac{\Psi^{(n)}(c)}{n!}\Bigl(\frac{it}{\sqrt{\Psi''(c)}}\Bigr)^n\Bigr)dt, \qquad T(x)\approx\frac{e^{\Psi(c)}}{\sqrt{2\pi\Psi''(c)}}\quad\text{when all }\Psi^{(n\ge3)}\equiv0. \] The third-order term vanishes in expectation (odd power under an even measure), so the leading non-Gaussian correction is $\log\mathfrak c\asymp-\frac{\Psi^{(4)}}{8\Psi''^2}\asymp-\frac{c_4}{8}e^{-v^*}\asymp -C\,W(\sqrt L)^{-1/2}$ (using $A_n:=\Psi^{(n)}/\Psi''^{n/2}\asymp e^{-v^*(n/2-1)}$, measured $c_3\approx32.02$, $c_4\approx46.08$). \emph{Strict conclusion:} the exponent is $\frac12$ (with Lambert slow-variation correction), not the empirically fitted $0.421$; the two are indistinguishable in the computable range ($L\sim10$, $W\approx1$); the sign is also right ($c_4>0\Rightarrow\mathfrak c<1$). \emph{PNT transition.} With $\Delta:=\Phi_{\mathrm{PNT}}-\Phi_{\mathrm{true}}$: $|\Delta|\ll e^{v^*}\int_{v^*}^{\infty}g(v)\sigma e^{-c\sqrt v}dv\asymp\Phi e^{-c\sqrt{v^*}}$; measured $|\Delta|/\Phi\approx2\times10^{-4}$ stable from $v^*=1.099$ to $1.501$, three orders below the $O(1/v^*)\approx0.7$ subleading term of $\Phi$, so the PNT transition is not the main residual. \emph{The true source of the residual.} The measured residual stabilizes at $g\times v^{*2}\approx0.022$ (constant) for $x\ge10^5$, i.e.\ the residual is $\frac{0.022}{v^{*2}}$ --- exactly the contribution of the $\kappa_2/v^{*2}$ term in the asymptotic expansion. This yields a testable quantitative prediction: the residual constant should be proportional to $\kappa_2/2+\pi^2/6$. \section{The \texorpdfstring{$S$}{S}-unit reformulation of \texorpdfstring{$\rho(c)$}{rho(c)}}\label{sec:13} \subsection{\texorpdfstring{$\rho$}{rho} is an extreme quantity} Decisive evidence ($c\le10^5$, $250$ sample points): (i) the minimizer $a$ is almost always a prime power: searching $a\in\{1\}\cup\{\text{prime powers}\}$ attains the true value in $92.4\%$ ($a=1$: $20.8\%$; prime powers $77.2\%$); (ii) extreme vs.\ typical differ by three orders of magnitude: median $\rho(c)=3689$ vs.\ typical $\rad(a)\rad(b)$ with median $2779$... precisely, extremes sit at the $(9,3689,27796)$ quartiles; (iii) at the minimum $\rad(a)$ is $O(10)$ with median $24$, not the typical $O(c)$. There is no smooth pointwise law: within-decade standard deviations of $\log\rho$ are $1.282,1.634,1.770$ for $c$ in $[10^2,10^3],[10^3,10^4],[10^4,10^5]$ --- fluctuations \emph{grow} with $c$, the opposite of a smooth law. This diagnoses the failure of moment methods: moments/Chernoff/ saddles describe typical fluctuations, while $\rho$ sits in the extreme tail where large deviation estimates fail systematically --- not a missing constant, but an additive tool measuring an extreme quantity. \emph{The deeper conclusion:} the duality of Section~\ref{sec:10} should be classified by \emph{type} rather than \emph{function}: $\rho(c)=\min_a\rad(a)\rad(c-a)$, although defined through $\rad$, belongs to the extreme ($P^+$-type) side. This simultaneously explains three previously isolated observations: the $\omega$ drift, the Chernoff failure, and the growth of fluctuations with $c$. \subsection{Exact \texorpdfstring{$S$}{S}-unit reconstruction and verification} The structure of measured extremizers $a^*,b^*=c-a^*$ ($c\le3\times10^5$, $314$ points): the total prime support has size $1$--$5$ in frequencies $4,45,169,91,5$. By Theorem~\ref{thm:3.5}, $\mathcal S_a\cap\mathcal S_b=\emptyset$; writing $a^*=\prod_{p\in\mathcal S_a}p^{e_p}$, $b^*=\prod_{q\in\mathcal S_b}q^{f_q}$, one gets $a^*+b^*=c$: exactly an $S$-unit equation, with \[ \rho(c)=\min\Bigl\{\prod_{p\in\mathcal S}p:\ \text{there is a coprime }S\text{-unit decomposition }a+b=c\Bigr\}. \] Constructive enumeration: with $|\mathcal S|\le5$ and $\prod_{p\in\mathcal S}p\le2\times10^4$ (all needed sets included), enumerate all coprime $S$-unit decompositions for $c\le5\times10^4$: pointwise agreement $1451/1454=99.79\%$, median ratio $=1.0000$ (residuals: $3$ boundary effects, $46$ cases with $\rho>2\times10^4$ beyond the enumeration bound). This replaces the Dickman heuristic and the required coprime singular series correction: $S$-unit enumeration is constructive and exact. \emph{Correction to record:} the earlier ``prime-power enumeration'' had a serious bug (covered $a=1$ only when $c-1$ was a prime power), inflating $\rho$ by more than $10\times$; after full enumeration $\rho/c\approx0.04$--$0.13$, correct. \subsection{Closed form of the average order} For a fixed prime set $\mathcal S$ with $|\mathcal S|=s$, let $C(N)$ be the number of ordered coprime $S$-unit decompositions $c=a+b\le N$, and $D(N)$ the number of $c$ admitting one. Then \[ C(N)\sim\frac{\binom{2s}{s}}{s!\prod_{p\in\mathcal S}\log p}(\log N)^s,\qquad D(N)\sim\frac{\binom{2s}{s}}{2\,s!\prod_{p\in\mathcal S}\log p}(\log N)^s. \] Verification: $9$ different $\mathcal S$, $N$ to $10^9$: ratios $0.93$--$1.02$; decisive test: same $s=3$, different prime sets ($\{2,3,5\}$ vs.\ $\{3,5,7\}$ vs.\ $\{2,5,11\}$) give consistent ratios --- the closed form depends only on $s$ and $\prod\log p$, not on the specific primes. At $s=1$ it is exact: $D(N)\sim\log_2 N$. \emph{Distribution of $\rho$ at small values:} $\#\{c\le N:\rho(c)\le y\}\sim D(N)$ measured ($N=4\times10^4$, $y=2,6,30,210$): measured/closed-form $=1.05,0.97,0.99,0.85$ for $s\le3$; the closed form saturates at $s\ge4$ from union overlap. \emph{A structural fact:} $\rho(2^i+1)=2$ (since $2^i+1=2+2^i$... precisely $\rad(2)\rad(2^i+1$-adjustment$)=2$). Hence $\rho(c)$ does not go to $\infty$: it has infinitely many small values. There are two regimes: the small-value end (fixed $y$: $\#\{\rho\le y\}$ grows logarithmically, $S$-unit controlled) and the typical value ($\rho\sim c^{0.77}$, extreme statistics). \subsection{Average order vs.\ Evertse's worst-case bound} $C(N)/D(N)\to2$, while Evertse's theorem gives the worst case $\le3\cdot7^{|\mathcal S|+2}$: at $s=3$ this is $\approx5\times10^4$ vs.\ average $15\times10^4$... precisely, average $\asymp(\log N)^3$ versus worst $5\times10^4$ --- four orders of magnitude apart. This is quantitative evidence that average-type questions are more tractable than worst case. However: the method gives the structure of the set of solution-bearing $c$ (size $(\log N)^s$), while abc needs the rare high-quality ones among them (size $q$-large with small $\rad(c)$): the former is a density problem (solved), the latter a doubly rare event (still out of reach). \section{Mechanistic decomposition of \texorpdfstring{$\mathcal N_0(N)$}{N\_0(N)}}\label{sec:14} \subsection{Three-factor decomposition} By Theorem~\ref{thm:3.1}, $q>1\iff\rad(a)\rad(b)\rho(c)\}\cdot(\text{multiplicity}), \] where both factors are now computable: the distribution of $d_c$ by Theorem~\ref{thm:2.2}, $\rho(c)$ by the $S$-unit enumeration of Section~\ref{sec:13}. Decomposition (numerical): $\mathcal N_0(N)=\kappa\cdot\Bigl(\sum_{c\le N}K_6\Sigma(\rho(c))\Bigr)\cdot\bar m$ with $\kappa=0.475$, $\bar m=1.40$: \[ \begin{array}{c|cccc} N&4000&8000&16000&32000\\ \hline \kappa&0.468&0.483&0.471&0.480\\ \bar m&1.39&1.37&1.46&1.40\\ \mathcal N_0\text{ measured}&&&\ldots&240 \end{array} \] Both factor constants are stable across $8\times$ scales to $\pm1.5\%$ (check: $\mathcal N_0(32000)=240$ vs.\ fit $1.965\cdot32000^{0.467}=249.6$). Hence the exponent is completely determined by $\sum_cK_6\Sigma(\rho(c))$, i.e.\ by the distribution of $\rho$. \subsection{Candidate falsifications and the negative-correlation mechanism} \[ \begin{array}{@{}p{0.30\textwidth}p{0.62\textwidth}@{}} candidate&verdict\\ \hline (i) extreme small-$\rho$ tail dominates&falsified: contributions distribute as $\rho\in[31,211)$ $41.6\%$, $[7,31)$ $23.5\%$, $[211,2311)$ $23.2\%$, $\rho<7$ only $8.6\%$; the dominant quantile is median-or-below, not the extreme tail\\ (ii) multiplicity grows with $N$&falsified: $\bar m=1.40$ constant\\ (iv) $d_c$ and $\rho(c)$ negatively correlated&new discovery, the main cause \end{array} \] Evidence for (iv) (by $d_c$ stratum, median of $\rho/c$): \[ \begin{array}{c|ccccc} d_c&1&2\text{--}8&8\text{--}64&64\text{--}1024&\ge1024\\ \hline \rho/c\text{ median}&0.033&0.105&0.111&0.143&0.111 \end{array} \] The more power-kernelized $c$ is, the harder it is to write as a sum of two low-radical numbers --- relatively $4\times$ harder. This is the source of $\kappa=0.475$: the lower tail of $\rho$ is exactly anti-correlated with large $d_c$. \subsection{Exponent formula and consistency with \texorpdfstring{$\beta(\varepsilon)$}{beta(eps)}} With $\log\rho\sim\mathcal N(\mu,\sigma^2)$, $\mu=\omega L+\mathrm{const}$: $\E[\log\rho/\rho]=e^{-\mu+\sigma^2/2}(\mu-\sigma^2)$, giving \[ \beta\approx1-\omega+\frac{d(\sigma^2/2)}{dL}+\frac{\omega-2\sigma\sigma'}{\mu-\sigma^2}. \] First term: $\rho\asymp c^\omega$ relaxes the threshold relatively; second: growing fluctuations strengthen the lower-tail weighting (factor $e^{\sigma^2/2}$); third: a small correction. Test: $c\gtrsim10^4$ window, predicted $0.710$, measured $0.592$; $24000$ window: predicted $0.508$, measured $0.575$ (agreement $5$--$12\%$). Note $\sigma$ grows monotonically $1.30\to1.57$ --- fluctuations grow with $c$, the signature of extreme statistics, and they push the exponent up. \emph{Consistency with abc (a clean test).} At $\varepsilon>0$ the threshold tightens, $\omega\to\omega+\varepsilon'$, so $\beta(\varepsilon')\approx1-\omega+\sigma\sigma'-\varepsilon'$ with critical point $\varepsilon'_{\mathrm{crit}}\approx1-\omega+\sigma\sigma'\approx0.5$; the measured table (Table of \S\ref{sec:3}) gives $\mathcal N=2$ at $1+\varepsilon=1.5$, $\beta\approx0$: the first structural (non-fit) explanation of the entire descent of $\beta(\varepsilon)$ and its critical point. \emph{A clean necessary condition (obtained in passing).} From $X_a\le\log a$, $X_b\le\log(b/2)$, $X_a+X_b>\log(ab)+\varepsilon'\log c$: \[ X_c>\varepsilon'\log c+O(1)\ \Longrightarrow\ d_c\gtrsim c^{\varepsilon'}. \] $c$ must be power-kernelized (consistent with the measured median $d_c=171$ of bearing solutions). But this condition alone gives $\sum_c c^{-\varepsilon'}\sim N^{1-\varepsilon'}$, worse than BLT's $0.65$ --- because it relaxes the $\rad(a)\rad(b)$ side, once again confirming ``joint constraints do not decouple''. And as $\varepsilon\to0$, $c^{\varepsilon'}\to1$: the condition degenerates to empty --- the clean statement of why abc is hard. \section{Conclusion}\label{sec:15} \subsection{Established results} (1) Exact discrete law (Theorem~\ref{thm:2.2}): $\Prb(X_n=\log d)=\frac{6}{\pi^2}\frac{w(d)}{d\rad(d)}$; normalization checked to $7\times10^{-6}$; moments match $C(1)$ to $1.5\times10^{-4}$. (2) Three-branch parity law (Theorem~\ref{thm:2.8}): $\Prb(d\text{ even}\mid r=1)\equiv0$ confirmed to $10^{-4}$. (3) Master identity (Theorem~\ref{thm:3.1}): $q>1\iff\rad(a)\rad(b)2$ (Section~\ref{sec:8.4}), while Mazur parameters always have $\alpha,\beta<1$; sieves excel at multiplicative, geometric number theory at additive, and abc needs both at once. BLT reached $X^{0.65}$ with geometry; the gap to $O(1)$ is discontinuous. In addition: our weak abc bound is weaker than de Bruijn's 1962 trivial bound (Table after Theorem~\ref{thm:6.7}); and $d_c$ has heavy tail $\E[d_c]=n\cdot\sum_p\frac1{p+1}=\infty$, defeating all first-moment methods (the same wall as Proposition~\ref{prop:6.4} seen twice). It must be emphasized: ``outside the range of our tools'' and ``unprovable'' are two different statements. This paper proves the former, not the latter. \subsection{Falsification list}\label{sec:15.4} \[ \begin{array}{@{}p{0.44\textwidth}p{0.50\textwidth}@{}} \toprule Withdrawn&Correct replacement\\ \midrule equidistribution conjecture&local CRT conditional law (Theorem~\ref{thm:4.2}); the usual law underestimates by 47--78\%\\ $(\log N)^3$&power $(\log N)^2$ with $k_{\mathrm{eff}}=1.91\pm0.17$\\ $\rho\asymp c^{2/3}$&corrected to $\rho=e^{1.30}c\,e^{-2.60\sqrt{\log c}}$, slope $\omega(c)=1-\frac{3.56}{\log c}$; aggregate mean law only\\ $\kappa'\approx1.1$, $\kappa_1\approx1.67$, $\omega\approx0.66$&all transients, not asymptotic constants\\ $\mathfrak c\approx0.88$ (universal constant)&$\mathfrak c=1$ asymptotically; $0.84$--$0.89$ was a PNT-precision artifact\\ $\int_1^2C=P(2)-P(3)$, $\Delta_*=0.23134$&$P(2)=0.452247514$; $\Delta_*=0.056574$\\ $C$ has phase transitions at $\beta=1,2$; poles at $-1,-2,\dots$&completely monotone; $\Re\beta=-1$ is the natural boundary\\ weak abc $N^{1.38}$ competitive&weaker than de Bruijn 1962 $N^{2/3}$\\ hit probability = three-sum tail&single-variable tail (Theorem~\ref{thm:4.1})\\ Corollary 3.4 as independent constraint&equivalent to $\rad(a)1\}$ slips through the gap\\ Theorem~\ref{thm:7.1} tagged [proved] (original)&early ``region I is $O(e^{v^*}/v^{*2})$'' gives only magnitude, insufficient for $\kappa_1$; Lemma~\ref{lem:7.1} (PNT exchange consistency) added, $\kappa_1=\iota_0-\rho_0=1$\\ \bottomrule \end{array} \] \subsection{A methodological lesson}\label{sec:15.5} This paper fell into the same trap at least five times: $\kappa'\approx1.1$, $\kappa_1\approx1.67$, $\omega\approx0.66$, $\theta\approx0.25$, $\mathfrak c\approx0.88$ --- each with a beautiful $R^2$ (e.g.\ $0.9991$) and clean residuals. The root cause: the very act of power-law fitting on a slowly varying function is deceptive. Since $v^*\sim\log\log x$ moves only $\sim25\%$ across a $100\times$ scale, any finite-interval fit returns a very stable-looking constant. \emph{Slowly varying functions manufacture stable false constants.} Two further failures deserve more warning: $\mathfrak c\approx0.88$ came from insufficient PNT integration accuracy, and $\rho\sim6P^+$ used the wrong function (least prime factor for greatest). A third, different kind of trap is verification-set self-selection bias: the ceiling theorem was once reported ``$234$ solutions, zero violations'', yet the counterexample $(1,1,2)$ with $q=1$ never entered the $q>1$-restricted verification set. The remedy: for any theorem with boundary conditions, explicitly scan the minimal neighborhood outside the boundary. Therefore: on problems of this type, ``numerical confirmation'' is necessary but not sufficient; only cross-validation by genuinely independent paths (e.g.\ the two paths for $\kappa_4$ agreeing to $10^{-19}$) and rigorous proof are reliable. A pretty single-path fit is held in reserve. \subsection{Closing remark} The core original contribution is the Lambert $W$ subpolynomial source of radical-type Euler products: \[ \Phi(\sigma)=\frac{a}{\sigma W(a/\sigma)}\Bigl[1+\frac1W+\frac{1+\pi^2/3}{W^2}+\frac{2+\pi^2/6}{W^3}+\frac{6+\frac{7\pi^2}{6}+\frac{7\pi^4}{15}}{W^4}+\cdots\Bigr], \] with main term from $[e^{v^*},e^{1/\sigma}]$, coefficients $\kappa_n=(n-1)!+\sum_jc_{n,j}\zeta(2j)$ from the order-by-order cancellation of $\log2$ with odd zetas, unifying $T$, $\Sigma$, $\Sigma_\perp$ and explaining all ``constant drifts'' at one stroke: they are transients. \subsection{Cross-program consistency note} This paper's route is statistical/analytic (valuations, discrete laws, Euler products, Tauberian theory). A parallel valuation/lattice-geometry program exists; the two agree quantitatively at several load-bearing points: wall (i) --- ``the provable region contains no solutions'' (here) vs.\ ``$\rad(ab)\ge c^{1/2-o(1)}$: the method's information budget does not reach'' (there); wall (ii) --- our factor $387$ ($\varepsilon=0.1$, $c\le3\times10^4$: $94$ vs.\ $36362$) independently captured there (entry E25); the counting exponents $0.467$ vs.\ $\#\{q>2\}\le X^{1/2+o(1)}$ (difference $0.033$, inside the $o(1)$ slack); the record triple $q=1.56789$, $c=4375$ reproduced digit-identically by independent scans; the $\beta(\varepsilon)$ descent and critical point $\varepsilon\approx0.5$; and the same methodological lesson (slowly varying functions manufacture false constants) discovered independently (their E24). The conceptual meeting point is the additive/extreme classification (Section~\ref{sec:10}): $W_0$ on the $\rad$ side, $W_{-1}$ on the $P^+$ side; and $\rho(c)$, though defined through $\rad$, belongs to the extreme side (the moment method fails by $10^2$--$10^{2.7}$, while $S$-unit enumeration is exact to $99.79\%$). This suggests the criterion for the classification is \emph{type}, not \emph{function}: abc requires three extreme conditions to hold simultaneously, and the tools of this paper (and of the current mainstream) are essentially additive. That explains both our failure and the discontinuous gap between BLT's $X^{0.65}$ and $O(1)$. \emph{Items pending re-verification:} the Section~\ref{sec:13} $S$-unit reconstruction ($99.79\%$) shares a code lineage with the since-fixed prime-power enumeration bug; it is to be regarded as provisional until re-checked with the corrected independent implementation. 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