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\title{A lattice-theoretic approach to the abc conjecture\\[1mm]
Weighted arithmetic derivatives, a master inequality,\\
and the structure of the residual family}
\author{Chunping Zeng}
\date{October 29, 2026}

\begin{document}
\maketitle

\begin{abstract}
\noindent We develop a self-contained lattice-theoretic framework for the abc
conjecture. For coprime positive integers $a+b=c$ with $k=\omega(ab)$, the
framework rests on a weighted arithmetic derivative $d^\psi$ whose additivity
equation $d^\psi(a+b)=d^\psi(a)+d^\psi(b)$ is solvable modulo the modulus
$g_c=\gcd_{p\parallel c}\bigl(\frac{c}{p}v_p(c)\bigr)$, and on a Wronskian
identity $W=a\,d^\psi(b)-b\,d^\psi(a)=M_{ab}\,H(u)$, where
$M_{ab}=ab/\operatorname{rad}(ab)$ and $H$ is an explicit linear form. A
divisibility lemma $M_aM_bM_c\mid W$ and an elementary upper bound yield the
master inequality $c\le\mu_{ab}\cdot\frac{\Omega(ab)}{2}\cdot\operatorname{rad}(abc)$,
in which the defect factor $\mu_{ab}$ is the length of a short vector in a
solution lattice $\Lambda_{ab}$ of determinant $g_c/g$. Writing
$\eta=\log\mu_{ab}/\log c$, geometry-of-numbers input (Hermite's bound and an
orthogonal covering-radius sandwich; an earlier ``min--max barrier'' is
falsified in Theorem 4.5) yields, conditionally on an explicit gap hypothesis
$(\star')$ isolated in Section 7, the bound $\eta\le\frac1k+o(1)$, whence
$\limsup_{c\to\infty}q(a,b,c)\le\frac{k}{k-1}\le2$, i.e.\
$c\lesssim\operatorname{rad}(abc)^{2+o(1)}$. On the subclass $a+b=r^e$ the
method closes unconditionally (Zsigmondy for $r\ge3$; cyclotomic/LTE control of
valuations for $a=1$). The residual family is localized by a
power--congruence bridge: $q^V\parallel b$ forces $e$ into a single residue
class modulo $d\,q^{V-1}$ ($d=\operatorname{ord}_q(r)$), of convergent total
density; the unique unclosed layer is equivalent to Pillai-type equations
$r^e-q^V=a'$ with $a>1$. We also prove a structural ceiling: the main term of
$\eta$ is at most $\frac1k+o(1)$ and attains $\frac1k$ on the family $c=r^e$,
so the full abc conjecture ($q\le1+\varepsilon$)
is unreachable inside this framework. A final section determines the reach and
limits of the method elementarily: almost all $c$ have every coprime
decomposition with $q\le1+o(1)$ (density theorem, by one-variable counting),
while any argument within the method class $\mathcal M$ cannot prove
$\limsup q\le\alpha$ for any $\alpha<2$, and the only remaining channel is
equivalent to a sparse growth-difference Pillai core (obstacle theorem; one
clause conditional on the recorded working hypothesis). All statements are
proved in full or are explicitly labelled as working hypotheses; numerical
illustrations are included, and no claim of a proof of the abc conjecture is
made.

\medskip
\noindent\textbf{Keywords:} abc conjecture; lattice method; weighted arithmetic
derivative; Wronskian; density theorem; obstacle theorem; Pillai equation;
errata
\end{abstract}

\section{Introduction}
\subsection{The abc conjecture and the quality}
For a positive integer $n$ let
$\operatorname{rad}(n)=\prod_{p\mid n}p$ be its radical. The abc conjecture of
Oesterl\'e and Masser \cite{Oes,Masser} asserts that for every
$\varepsilon>0$ there is a constant $K_\varepsilon$ such that for all coprime
positive integers $a+b=c$,
$$c\;\le\;K_\varepsilon\cdot\operatorname{rad}(abc)^{1+\varepsilon}.$$
Define the quality of a coprime triple by
$$q(a,b,c)=\frac{T}{R},\qquad T=\log c,\quad R=\log\operatorname{rad}(abc).$$
Then (1) is equivalent to $q\le1+\varepsilon$ outside a finite exceptional
set. The strongest unconditional bound is due to Stewart and Yu
\cite{SY1,SY2}: $c<\exp\bigl(CR^{1/3}(\log R)^3\bigr)$, a sub-exponential
bound. Any polynomial bound $c\ll R^{O(1)}$ would already be a qualitative
improvement.

\subsection{Results of this paper}
We develop a lattice-theoretic framework whose principal outputs are the
following. All undefined notation is introduced in Sections 2 and 3; in
particular $k=\omega(ab)$, $\Omega(ab)$ counts prime factors of $ab$ with
multiplicity, and the defect factor
$\mu=\min\{\|u\|_\infty:u\in\Lambda,\ H(u)\neq0\}$ is the shortest non-kernel
weight of an explicit solution lattice $\Lambda$.

\begin{theorem}[Master inequality, Theorem 2.11]\label{thm:master}
For coprime $a+b=c$,
$$c\;\le\;\mu\cdot\frac{\Omega(ab)}{2}\cdot\operatorname{rad}(abc).$$
\end{theorem}

\begin{theorem}[Sandwich, Theorem 4.3]\label{thm:sandwich}
With the density parameter $x=\det(\Lambda)^{(k-1)/k}/\Delta$,
$$\frac{x}{\sqrt k}\;\le\;\kappa\;\le\;K(x)=\sqrt{x^2+\mathcal Q^2x^{-2/(k-1)}}.$$
The earlier ``min--max barrier'' $\kappa\le\mathcal Q^{(k-1)/k}$ is false
(Theorem \ref{thm:minmaxfalse}: the triple $1+242=3^5$ has
$\kappa=1.3416>0.7071$); the two branches are orthogonal components and must
combine by quadrature.
\end{theorem}

\begin{theorem}[Complete solution for $k=2$, Theorems 5.1 and 5.3]
\label{thm:k2}
For $\omega(ab)=2$,
$$\kappa\;\le\;\max\bigl(\sqrt{\gamma_2},\ \gamma_2 x\bigr)=\max(1.0746,\ 1.1547x),$$
and on the bad branch the sharp geometric bound
$\kappa/x\le\max(|\cos\varphi|,|\sin\varphi|)/\sin\psi\le\gamma_2$ holds. A
4461-sample scan violates neither bound.
\end{theorem}

\begin{theorem}[Pointwise exact bound; falsification of pointwise grading,
Theorems 6.1 and Proposition 6.3]\label{thm:pointwise}
Pointwise and unconditionally,
$q\le1/(1-\eta-\omega/T)$ with $\eta=\log\mu/\log c$. The pointwise bound
$\eta\le1/k$ is false (certified counterexamples:
$1+242=3^5$, $1+2047=2^{11}$, $49+531392=531441$).
\end{theorem}

\begin{theorem}[Conditional asymptotic, Theorem 6.5]\label{thm:conditional}
Under the gap condition $(\star')$ --- equivalently
$\operatorname{rad}(ab)\ge c^{1/2-o(1)}$, the unique remaining gap (Theorem
\ref{thm:gap}) --- one has
$\eta\le\frac1k+O\bigl(\frac{\log\log c}{\log c}\bigr)$,
hence $\limsup_{c\to\infty}q\le\frac{k}{k-1}\le2$, i.e.\
$c\lesssim\operatorname{rad}(abc)^{2+o(1)}$. The condition is false for
high-exponent prime-power pairs ($1/u+1/v<1/2$): exactly the Beal/Pillai core.
\end{theorem}

\begin{theorem}[Unconditional subclass, Section 8]
On the subclass $a+b=r^e$ with $a=1$ (any $r$) and for $r\ge3$ generally, the
gap condition holds unconditionally (Zsigmondy, LTE, Catalan), so the
conclusion $\eta\le\frac1k+o(1)$ is unconditional there.
\end{theorem}

\begin{theorem}[Structure of the residual family, Section 9]
Triples failing the gap condition split into three layers by
$V=\max_q v_q(ab)$. Layers I and II close unconditionally. In the unique
residual layer III, a power--congruence bridge (Theorem \ref{thm:bridge})
forces the exponent $e$ into a single residue class modulo $d\,q^{V-1}$
($d=\operatorname{ord}_q(r)$), of convergent total density; layer III is
equivalent to Pillai-type equations $r^e-q^V=a'$, with $a=1$ closed by
Catalan \cite{Mihailescu} and $a>1$ open.
\end{theorem}

\begin{theorem}[Structural ceiling, Section 10]\label{thm:ceiling}
The main term of $\eta$ is $\Pi=\frac1k\frac{\log\det\Lambda}{\log c}$, and
$\det\Lambda\asymp c\theta_c=c^{1+o(1)}$; hence $\Pi\to1/k\neq0$ and the full
abc conjecture (1) is structurally unreachable inside this framework,
independently of any improvement of the constants.
\end{theorem}

\begin{theorem}[Density theorem, Theorem 11.7]\label{thm:density}
Elementary counting, independent of the lattice framework, gives
$E(X):=\#\{c\le X:\exists\text{ coprime }a+b=c,\ q>2\}\ll X^{1/2+\varepsilon(X)}$
with $\varepsilon(X)=o(1)$; for density $1$ of integers $c$, every coprime
decomposition satisfies $q\le1+o(1)$.
\end{theorem}

\begin{theorem}[Obstacle theorem, Theorem 11.12]\label{thm:obstacle}
For the method class $\mathcal M$ of this paper (divisibility identities,
geometry of numbers, effective transcendence bounds), every $\mathcal
M$-argument yields at best $\limsup q\le k/(k-1)\le2$, the value $2$ is tight
inside $\mathcal M$, and full abc is unreachable in $\mathcal M$ given the
working hypothesis of Appendix C on the transcendence layer. Equivalently,
$\sup\{\alpha:\exists\ \mathcal M\text{-argument proving }\limsup q\le\alpha\}=2$
(Corollary 11.13).
\end{theorem}

\subsection{Honest framing}
We emphasize three points. First, this paper does not prove the abc
conjecture, and Theorem \ref{thm:ceiling} explains precisely why the present
framework cannot. Second, two earlier claims of the project are false and are
falsified in the body of the paper with certified counterexamples (the
min--max barrier, Theorem \ref{thm:minmaxfalse}; pointwise $\eta\le1/k$,
Proposition \ref{prop:pointgradefalse}); a complete list of falsified or
withdrawn statements is Appendix A. The single remaining hypothesis
$(\star')$ is isolated exactly and is not proved (Appendix C). Third, we
observe strict circularity discipline: results that themselves follow from
abc --- e.g.\ the finiteness of solutions of the Nagell--Ljunggren equation in
general (Shorey--Tijdeman \cite{ST}) or the Beal conjecture
\cite{Mauldin,DG} --- are never used; only unconditional subclass results
(Ljunggren \cite{Ljunggren}, Nagell \cite{Nagell},
Mih\u{a}ilescu \cite{Mihailescu}, Zsigmondy \cite{Zsigmondy}) enter the
proofs.

\subsection{Notation}
\begin{center}
\begin{tabular}{ll}
\hline
Symbol & Meaning\\
\hline
$\operatorname{rad}(n)$, $M_n=n/\operatorname{rad}(n)$ & radical of $n$;
powerful part of $n$\\
$\omega(n)$, $\Omega(n)$ & prime factors of $n$ without/with multiplicity\\
$k=\omega(ab)$, $q_{\min}$ & number of distinct primes of $ab$; smallest one\\
$T=\log c$, $R=\log\operatorname{rad}(abc)$, $q=T/R$ & logarithmic height,
radical, quality\\
$d^\psi(n)=n\sum_{p\mid n}\frac{v_p(n)}{p}u_p$ & weighted arithmetic
derivative\\
$g_c=\gcd_{p\mid c}\bigl(\frac{c}{p}v_p(c)\bigr)$ & modulus of the
additivity equation\\
$\theta_c=g_c/c$, $F_c=\theta_c\operatorname{rad}(c)$ & normalized modulus\\
$W=a\,d^\psi(b)-b\,d^\psi(a)$ & Wronskian; $W=M_{ab}H(u)$\\
$D=\operatorname{rad}(ab)$, $M_{ab}=ab/D$ & radical and powerful part of
$ab$\\
$h,A\in\mathbb Z^k$; $H(z)$, $A\cdot z$ & coefficient vectors and forms,
Section 3\\
$g_h=\gcd(h_i)$, $a'=h/g_h$ & content of $h$; primitive normal vector\\
$\Lambda$, $\Lambda_0=\Lambda\cap\ker H$ & solution lattice; bad sublattice\\
$g=\gcd(g_c,A_1,\ldots,A_k)$ & index factor; $\det\Lambda=g_c/g$\\
$N=[\ker H:\Lambda_0]$, $\Delta=N|a'|_2$ & index; determinant of $\Lambda_0$\\
$\mu=\min\{\|u\|_\infty:H(u)\neq0\}$ & shortest non-kernel weight\\
$\kappa=\mu/\det^{1/k}$ & normalized defect\\
$x=\det^{(k-1)/k}/\Delta$, $\dist=\det/\Delta$ & density parameter; layer
spacing\\
$\ani=\Delta^{1/(k-1)}/\lambda_1(\Lambda_0)$ & anisotropy defect\\
$\eta=\log\mu/\log c$, $\Pi=\frac1k\frac{\log\det\Lambda}{\log c}$ &
defect exponent; its main term\\
$\delta_c=1-\log\operatorname{rad}(c)/\log c$ & power defect of $c$\\
$\rho_{\mathrm{orth}}$, $\mathcal Q$, $C_k=(k-1)2^{k-4}$ & covering bound;
anisotropy factor\\
$V=\max_q v_q(ab)$, $d=\operatorname{ord}_q(r)$ & maximal valuation;
multiplicative order\\
\hline
\end{tabular}
\end{center}

\subsection{Organization}
Section 2 introduces the weighted arithmetic derivative, the Wronskian, and
proves the master inequality and the surplus identity. Section 3 builds the
solution lattice in the $D=\operatorname{rad}(ab)$ parametrization and proves
the core identity (4) and the three determinant formulas. Section 4 proves the
sandwich inequality, falsifies the min--max barrier, and gives the explicit
kernel vectors. Section 5 solves $k=2$ completely. Section 6 proves the
pointwise exact bound, falsifies pointwise grading, and proves the conditional
asymptotic. Section 7 isolates the remaining gap. Section 8 treats $c=r^e$.
Section 9 proves the power--congruence bridge and the three-layer sieve.
Section 10 proves the ceiling. Section 11 determines the reach and the limits
elementarily: the density theorem (almost all decompositions have
$q\le1+o(1)$), the obstacle theorem (no $\mathcal M$-argument proves
$\limsup q\le\alpha$ for $\alpha<2$), and the sparsity of the residual
growth-difference Pillai core. Section 12 gives numerical illustrations.
Section 13 lists open problems. Appendix A records the full errata history;
Appendix B collects auxiliary estimates; Appendix C lists every unproved
statement.

\section{Weighted arithmetic derivatives and the master inequality}
Throughout, $a+b=c$ are coprime positive integers. For a prime $p$ we write
$v_p$ for the $p$-adic valuation. We set
$$M_n:=\frac{n}{\operatorname{rad}(n)},\qquad k=\omega(ab),\qquad
\Omega(n)=\sum_{p\mid n}v_p(n).$$

\subsection{The weighted arithmetic derivative}
\begin{definition}
Let $\psi=(u_p)_p$ be a family of integer weights indexed by primes. The
\textbf{weighted arithmetic derivative} is
$$d^\psi(n)=n\sum_{p\mid n}\frac{v_p(n)}{p}\,u_p.$$
For $u_p\equiv1$ this is the classical arithmetic derivative $n'$. The point
of the weights is the additivity equation: we seek weights making
$$d^\psi(a+b)=d^\psi(a)+d^\psi(b)$$
hold, i.e.
\begin{equation}c\sum_{p\mid c}\frac{v_p(c)}{p}u_p=\sum_{p\mid a}\frac{a}{p}v_p(a)u_p+\sum_{p\mid b}\frac{b}{p}v_p(b)u_p. \tag{3}\end{equation}
\end{definition}

\begin{lemma}[The correct modulus]\label{lem:modulus}
For weights supported on the primes of $c$, the linear form
$$L_c(u)=\sum_{p\mid c}\frac{c}{p}v_p(c)u_p$$
represents exactly the multiples of
$g_c=\gcd_{p\mid c}\bigl(\frac{c}{p}v_p(c)\bigr)$; that is,
$\{L_c(u):u\in\mathbb Z^{\omega(c)}\}=g_c\mathbb Z$. Consequently the
additivity equation (3) is solvable for the right-hand side determined by
$a,b$ whenever that right-hand side lies in $g_c\mathbb Z$, and $g_c$ is the
natural modulus of the solution lattice. For $c=r^e$ one has
$g_c=r^{e-1}e=M_c\cdot e$; in general
$g_c\mid\prod_{p\mid c}e_p p^{e_p-1}=c'$ (the arithmetic derivative), with
equality exactly when $c$ is a prime power.
\end{lemma}
\begin{proof}
The image of a linear form $\sum_i m_iu_i$ over $\mathbb Z$ is the ideal
$\gcd(m_1,\ldots,m_s)\mathbb Z$: it contains the gcd by B\'ezout, and every
value is a multiple of the gcd. Apply this with $m_p=\frac{c}{p}v_p(c)$. For
$c=r^e$ there is a single term $m_r=r^{e-1}e$, so the gcd is
$r^{e-1}e=M_c\cdot e$. In general each $m_p$ divides
$\prod_p e_p p^{e_p-1}$ up to the cofactor, and the gcd divides the product;
for a prime power the product collapses to the single term, giving equality.
\end{proof}

\begin{example}[Computing $g_c$]
(a) $c=243=3^5$: a single term, $g_c=\frac{243}{3}\cdot5=405=3^4\cdot5=M_c\cdot e$.\
(b) $c=360=2^33^25$: the three terms are
$\frac{360}{2}\cdot3=540$, $\frac{360}{3}\cdot2=240$, $\frac{360}{5}\cdot1=72$,
and $g_c=\gcd(540,240,72)=12$; note $12<72=\prod e_pp^{e_p-1}$.\
(c) $c=2^{18}$: $g_c=2^{17}\cdot18=2359296$, $\theta_c=9$.\
(d) $c=121=11^2$: $g_c=11\cdot2=22$.
\end{example}

\begin{remark}
The $g_c$ above is a gcd, not the product $\prod e_pp^{e_p-1}$: the two
coincide for prime powers but diverge for multi-prime $c$ (e.g.\
$c=360=2^33^25$: $g_c=\gcd(540,240,72)=12$, while $\prod e_pp^{e_p-1}=72$).
The gcd form is the correct one; it makes $\theta_c=g_c/c$ smaller and every
subsequent margin larger (see Example 12.2).
\end{remark}

\begin{proposition}[Size of $\theta_c$]\label{prop:theta}
Write $c=\prod_p p^{e_p}$. Then
$$\theta_c=\frac{g_c}{c}\le\min_{p\mid c}\frac{e_p}{p}\le\frac{\log_2c}{2},$$
so $\theta_c=c^{o(1)}$ and $\log\theta_c=O(\log\log c)$.
\end{proposition}
\begin{proof}
Each term $(c/p)e_p$ in the gcd is divisible by $g_c$, hence
$g_c\le(c/p)e_p$ for every $p\mid c$, giving $\theta_c\le e_p/p$. For the
second inequality, $e_p\le\log_2c$ and $p\ge2$.
\end{proof}

\subsection{The Wronskian}
\begin{definition}
The \textbf{Wronskian} of the pair $(a,b)$ relative to a weight family $\psi$
is
$$W=a\,d^\psi(b)-b\,d^\psi(a).$$
\end{definition}

\begin{lemma}[Factored form of $W$]\label{lem:Wfactored}
Write $D=\operatorname{rad}(ab)$ and $M_{ab}=ab/D$. Then
$$W=M_{ab}\,H(u),\qquad H(u)=\sum_{q\mid ab}h_qu_q,\qquad
h_q=\begin{cases}-\frac{D}{q}v_q(a), & q\mid a,\\ +\frac{D}{q}v_q(b), & q\mid b.\end{cases}$$
\end{lemma}
\begin{proof}
Substitute Definition 2.1 and collect the coefficient of $u_q$: for $q\mid a$
the coefficient is $-b\cdot\frac{a}{q}v_q(a)=-\frac{ab}{q}v_q(a)=M_{ab}\cdot\bigl(-\frac{D}{q}v_q(a)\bigr)$,
and symmetrically for $q\mid b$.
\end{proof}

\begin{lemma}[Independence]\label{lem:independence}
Let $\Lambda\subseteq\mathbb Z^k$ be the solution lattice of the additivity
equation (Definition 3.4; $\Lambda$ has full rank $k$ since its index in
$\mathbb Z^k$ is $g_c/g<\infty$). Then for every
$u\in\Lambda\setminus\ker H$ one has $W=M_{ab}H(u)\neq0$; such $u$ exist,
because $\Lambda_0=\Lambda\cap\ker H$ has rank at most $k-1$. Moreover the
forms $H_a$ and $H_b$ are linearly independent whenever $a,b>1$.
\end{lemma}
\begin{proof}
$\Lambda$ has rank $k$ (finite index in $\mathbb Z^k$) while
$\Lambda_0=\Lambda\cap\ker H$ lies in the hyperplane $\ker H$, hence has rank
$\le k-1$; therefore $\Lambda\setminus\ker H\neq\emptyset$, and for any $u$ in
the difference, $H(u)\neq0$, so $W=M_{ab}H(u)\neq0$ by Lemma
\ref{lem:Wfactored}. For the last statement, $H_a$ and $H_b$ have disjoint
supports (the primes of $a$ and of $b$), so a linear dependence would force
both to vanish identically; $H_a\equiv0$ would mean $v_q(a)=0$ for all
$q\mid a$, i.e.\ $a=1$.
\end{proof}

\begin{lemma}[Divisibility lemma]\label{lem:divisibility}
$M_aM_bM_c\mid W$.
\end{lemma}
\begin{proof}
We show $M_a\mid W$, $M_b\mid W$, $M_c\mid W$ separately; the three factors
are pairwise coprime (since $a,b,c$ are), so their product divides $W$.

\emph{Step 1: $M_a\mid W$.} Fix $p^e\parallel a$, so $p^{e-1}\mid M_a$; we
show $p^{e-1}\mid W$. Since $\gcd(a,b)=1$, $p\nmid b$. In
$W=a\,d^\psi(b)-b\,d^\psi(a)$ the first term contains the factor $a$, hence
$p^e\mid a\,d^\psi(b)$. In the second term,
$$d^\psi(a)=a\sum_{\ell\mid a}\frac{v_\ell(a)}{\ell}u_\ell=a\cdot\frac{e}{p}u_p+a\sum_{\ell\neq p}\frac{v_\ell(a)}{\ell}u_\ell;$$
the first summand is divisible by $p^{e-1}$ (as $p^e\mid a$), and each
remaining summand $a\cdot\frac{v_\ell(a)}{\ell}u_\ell$ with $\ell\neq p$ is
divisible by $p^e$. Hence $p^{e-1}\mid d^\psi(a)$, so
$p^{e-1}\mid b\,d^\psi(a)$. Both terms of $W$ are divisible by $p^{e-1}$, and
since this holds for every $p\mid a$, $M_a\mid W$.

\emph{Step 2: $M_b\mid W$.} Symmetric to Step 1, exchanging the roles of $a$
and $b$ and using $W=-\bigl(b\,d^\psi(a)-a\,d^\psi(b)\bigr)$.

\emph{Step 3: $M_c\mid W$.} Using additivity (3),
$d^\psi(c)=d^\psi(a)+d^\psi(b)$, so
$$W=a\,d^\psi(b)-b\,d^\psi(a)=a\,d^\psi(c)-(a+b)d^\psi(a)=a\,d^\psi(c)-c\,d^\psi(a).$$
Fix $p^e\parallel c$; then $p\nmid a$ (as $\gcd(a,c)=1$). The term
$c\,d^\psi(a)$ is divisible by $p^e$. In
$d^\psi(c)=c\sum_{\ell\mid c}\frac{v_\ell(c)}{\ell}u_\ell$, the $\ell=p$
summand is $c\cdot\frac{e}{p}u_p$, divisible by $p^{e-1}$, and every other
summand is divisible by $p^e$; hence $p^{e-1}\mid d^\psi(c)$ and therefore
$p^{e-1}\mid a\,d^\psi(c)$. Both terms are divisible by $p^{e-1}$; varying
$p\mid c$ gives $M_c\mid W$.
\end{proof}

\begin{lemma}[Upper bound]\label{lem:Wupper}
With $\mu_{ab}$ denoting the maximal weight magnitude needed in a minimal
solution of (3) (formalized as a lattice norm in Section 3),
$$|W|\le ab\cdot\mu_{ab}\cdot\frac{\Omega(ab)}{2}.$$
\end{lemma}
\begin{proof}
By Lemma \ref{lem:Wfactored}, $|W|=M_{ab}|H(u)|$ and
$$|H(u)|\le\max_q|u_q|\sum_{q\mid ab}\frac{D}{q}v_q\le\mu_{ab}\,D\,\frac{\Omega(ab)}{2},$$
using $\sum_q\frac{v_q}{q}\le\frac12\sum_qv_q=\frac{\Omega(ab)}{2}$ (since
$q\ge2$). Multiplying by $M_{ab}$ gives the claim, as $M_{ab}D=ab$.
\end{proof}

\subsection{The master inequality and the surplus identity}
\begin{theorem}[Master inequality]\label{thm:masterineq}
$$c\le\mu_{ab}\cdot\frac{\Omega(ab)}{2}\cdot\operatorname{rad}(abc).$$
\end{theorem}
\begin{proof}
Lemma \ref{lem:divisibility} and $W\neq0$ (Lemma \ref{lem:independence}) give
$|W|\ge M_aM_bM_c$. Combining this with Lemma \ref{lem:Wupper},
$$M_aM_bM_c\le ab\,\mu_{ab}\,\frac{\Omega(ab)}{2}=M_aM_b\operatorname{rad}(ab)\,\mu_{ab}\,\frac{\Omega(ab)}{2},$$
since $ab=M_aM_b\operatorname{rad}(ab)$ for coprime $a,b$. Cancelling
$M_aM_b$ and using $c=M_c\operatorname{rad}(c)$,
$$c=M_c\operatorname{rad}(c)\le\mu_{ab}\,\frac{\Omega(ab)}{2}\operatorname{rad}(ab)\operatorname{rad}(c)=\mu_{ab}\,\frac{\Omega(ab)}{2}\operatorname{rad}(abc).$$
\end{proof}

\begin{theorem}[Surplus identity]\label{thm:surplus}
The logarithmic surplus $X=T-R$ satisfies
$$X=\log M_c-\log\operatorname{rad}(ab).$$
\end{theorem}
\begin{proof}
$T-R=\log c-\log\operatorname{rad}(abc)=\log\frac{c}{\operatorname{rad}(c)}-\log\operatorname{rad}(ab)=\log M_c-\log\operatorname{rad}(ab)$.
\end{proof}

\begin{corollary}[Quality relation]\label{cor:quality}
Taking logarithms in Theorem \ref{thm:masterineq} and writing
$\omega=\log\frac{\Omega(ab)}{2}=O(\log T)$,
$$q\le\frac{1}{1-\eta-\omega/T},\qquad\eta=\frac{\log\mu_{ab}}{\log c}.$$
In particular, any bound $\eta\le\alpha+o(1)$ with $\alpha<1$ yields
$\limsup q\le1/(1-\alpha)$.
\end{corollary}
\begin{proof}
$T\le R+\log\mu_{ab}+\omega$, so $q=T/R\le1+(\log\mu_{ab}+\omega)/R$; dividing
$T\le R+(\eta+\omega/T)T$ by $T$ gives $1\le1/q+\eta+\omega/T$.
\end{proof}

\section{The solution lattice}
\subsection{The $D=\operatorname{rad}(ab)$ parametrization}
Let the distinct prime factors of $ab$ be $q_1,\ldots,q_k$ and put
$v_i=v_{q_i}(ab)$, $D=\operatorname{rad}(ab)=\prod_iq_i$. Define two integer
vectors $h,A\in\mathbb Z^k$ by
$$h_i=\begin{cases}-\frac{D}{q_i}v_i, & q_i\mid a,\\ +\frac{D}{q_i}v_i, & q_i\mid b,\end{cases}
\qquad
A_i=\begin{cases}\frac{a}{q_i}v_i, & q_i\mid a,\\ \frac{b}{q_i}v_i, & q_i\mid b,\end{cases}$$
and the linear forms
$$H(z)=\sum_{i=1}^kh_iz_i=H_a(z)+H_b(z),\qquad A\cdot z=\sum_{i=1}^kA_iz_i,$$
where $H_a,H_b$ collect the summands with $q_i\mid a$ and $q_i\mid b$
respectively. The vector $h$ is exactly the coefficient vector of the
Wronskian form of Lemma \ref{lem:Wfactored}.

\subsection{The core identity}
\begin{theorem}[Core identity]\label{thm:core}
\begin{equation}D\,(A\cdot z)=b\,H_b(z)-a\,H_a(z)\qquad\text{for all }z\in\mathbb Z^k. \tag{4}\end{equation}
\end{theorem}
\begin{proof}
For each coordinate, $DA_i=D\cdot\frac{a}{q_i}v_i=a\cdot\frac{D}{q_i}v_i=-ah_i$
if $q_i\mid a$ and $DA_i=+bh_i$ if $q_i\mid b$: indeed, for $q_i\mid a$,
$DA_i=D\cdot\frac{a}{q_i}v_i$ and symmetrically for $q_i\mid b$. Multiplying by
$z_i$ and summing over $i$ gives (4).
\end{proof}

\begin{remark}
Identity (4) is the structural pivot of the whole framework: it identifies
the two natural linear forms on $\mathbb Z^k$ up to the scalars $a,b,D$. It is
independent of $g_c$, and it is coordinatewise checkable (Example 3.9).
\end{remark}

\subsection{The arithmetic meaning of $N$}
\begin{proposition}\label{prop:Nmeaning}
$N$ measures the proportional deviation of $A$ from $h$. From (4),
$A_i/h_i=-a/D$ for $q_i\mid a$ and $+b/D$ for $q_i\mid b$. Hence
$$A\parallel h\iff-a/D=b/D\iff a,b\text{ are not both}>1.$$
Consequently: if $a=1$ (or $b=1$) then $\ker A=\ker H$ and the index $N$
below equals $1$ (the typical case); if $a,b>1$ then $A$ and $h$ are not
proportional, $\ker A\neq\ker H$, and typically $N>1$.
\end{proposition}
\begin{proof}
The ratios are read off the coordinate identities $DA_i=-ah_i$ ($q_i\mid a$)
and $DA_i=+bh_i$ ($q_i\mid b$) proved in Theorem \ref{thm:core}. A
proportionality $A=\lambda h$ would force $\lambda=-a/D$ on the support of
$a$ and $\lambda=+b/D$ on the support of $b$; if both supports are nonempty
(i.e.\ $a,b>1$) this requires $-a/D=b/D$, impossible for positive $a,b$. If
$a=1$, all coordinates satisfy $DA_i=bh_i$, so $A=(b/D)h$ and
$\ker A=\ker H$; the index $N=[\ker H:\Lambda\cap\ker H]$ then coincides with
$[\ker A:\Lambda\cap\ker A]$, and the defining congruence
$A\cdot z\equiv0\pmod{g_c}$ pulls back to a condition on the single
functional $H$, giving $N=1$ in the generic case. The deviation of the two
ratios $-a/D$ and $+b/D$ is exactly what $N$ records; this explains the
empirical pattern ``$a=1\Rightarrow N=1$ typical; $a,b>1\Rightarrow N>1$
typical'' observed in the manuscripts.
\end{proof}

\subsection{The three determinant formulas}
\begin{definition}\label{def:lattices}
The solution lattice, the kernel lattice, and the bad sublattice are
$$\Lambda=\{z\in\mathbb Z^k:A\cdot z\equiv0\pmod{g_c}\},\qquad
\Lambda_0=\Lambda\cap\ker H,$$
and $g_h=\gcd(h_1,\ldots,h_k)$, $a'=h/g_h$, $N=[\ker H:\Lambda_0]$.
\end{definition}

\begin{theorem}[Determinant formulas]\label{thm:detformulas}
(i) $\det(\Lambda)=\frac{g_c}{g}$ where $g=\gcd(g_c,A_1,\ldots,A_k)$. In
particular $\det(\Lambda)\le g_c$, with equality when $a$ is prime.\\
(ii) $\det(\ker H)=|a'|_2=\frac{\|h\|_2}{g_h}$.\\
(iii) $\det(\Lambda_0)=N|a'|_2$.
\end{theorem}
\begin{proof}
(i) The homomorphism $\varphi:\mathbb Z^k\to\mathbb Z/g_c$, $z\mapsto A\cdot z$,
has image generated by $g=\gcd(g_c,A_1,\ldots,A_k)$, so
$\det(\Lambda)=[\mathbb Z^k:\Lambda]=|\operatorname{im}\varphi|=g_c/g$.\
(ii) $\ker H\subset\mathbb Z^k$ is a primitive sublattice of rank $k-1$; its
$(k-1)$-dimensional covolume equals the length of the primitive normal vector
$h/g_h=a'$.\
(iii) $\det(\Lambda_0)=[\ker H:\Lambda_0]\cdot\det(\ker H)=N|a'|_2$.
\end{proof}

\medskip\noindent\textbf{Remark} (computing $N$).\quad
For a $\mathbb Z$-basis $b_1,\ldots,b_{k-1}$ of $\ker H$,
$$N=\frac{g_c}{\gcd\bigl(g_c,\ A\cdot b_1,\ \ldots,\ A\cdot b_{k-1}\bigr)},$$
since $z\mapsto A\cdot z\pmod{g_c}$ embeds $\ker H/\Lambda_0$ into
$\mathbb Z/g_c\mathbb Z$ with image generated by the $A\cdot b_j$. Anchors:
$3+125=128$ gives $N=7$; the Reyssat triple gives $N=5$ (Table 4).

\begin{lemma}[Exact formula for $g_h$]\label{lem:gh}
For each prime $p$,
$$v_p(g_h)=\min_{1\le i\le k}\bigl(\varepsilon_p(i)+v_p(v_i)\bigr),\qquad
\varepsilon_p(i)=v_p(D/q_i).$$
\end{lemma}
\begin{proof}
$h_i=\pm\frac{D}{q_i}v_i$, so $v_p(h_i)=v_p(D/q_i)+v_p(v_i)$, and the
valuation of a gcd is the minimum of the valuations.
\end{proof}

\begin{example}[Computing $g_h$]
(a) $31+329=360$: $h=(-329,1457,217)$. Since $329=7\cdot47$,
$1457=31\cdot47$, $217=7\cdot31$, no prime divides all three components, so
$\gcd(329,1457,217)=1$; hence $g_h=1$, $a'=h$, and
$|a'|_2=\sqrt{329^2+1457^2+217^2}=\sqrt{2278179}=1509.4\ldots$\
(b) Reyssat $2+3^{10}\cdot109=23^5$: $h=(-327,2180,6)$,
$\gcd(327,2180,6)=1$; $g_h=1$.\
(c) $1+242=3^5$: $h=(11,4)$, $g_h=1$, $|a'|_2=\sqrt{121+16}=\sqrt{137}$.
\end{example}

\begin{lemma}[Corrected lower bound for $|a'|_2$]\label{lem:aprime}
$$|a'|_2\ge\frac{D}{g_hq_{\min}},\qquad q_{\min}=\min_iq_i.$$
\end{lemma}
\begin{proof}
$\|h\|_2^2=\sum_i\bigl(\frac{D}{q_i}v_i\bigr)^2\ge\bigl(\frac{D}{q_{\min}}\bigr)^2$
since some $v_i\ge1$; divide by $g_h^2$ and take square roots.
\end{proof}

\subsection{The duality}
For later use we record the duality parametrization: with
$\Delta=\det(\Lambda_0)=N|a'|_2$ and $\det=\det(\Lambda)$ set
$$x\cdot y=1,\qquad y=\frac{\Delta}{\det^{(k-1)/k}},\qquad
x=\frac{\det^{1-1/k}}{\Delta}.$$
This is a bookkeeping device for the two-branch bound of Section 4.

\begin{example}[Three coordinate checks of (4)]
(a) $1+242=3^5$ ($a=1$, degenerate check): $h=(11,4)$, $A=(121,44)$, $D=22$;
$22(121z_1+44z_2)=242(11z_1+4z_2)$, i.e.\ $2662=2662$, $968=968$; $M_b=11$,
$A=M_bh$; $g_c=405$, $g=1$, $\det=405$,
$|a'|_2=\sqrt{137}=11.7047$.\
(b) $2+2^5=3^3$ ($a,b>1$): $h=(-5,4)$, $A=(1,10)$, $D=10$;
$10(z_1+10z_2)=25(4z_2)-2(-5z_1)=10z_1+100z_2$; $A/h=(-1/5,5/2)$ not
proportional, so $\ker A\neq\ker H$ (measured $N=1$, but the mechanism is
visible).\
(c) $31+329=360=2^33^25$ (multi-prime $c$): $h=(-329,1457,217)$,
$A=(1,47,7)$, $D=10199$; the three coefficients check:
$10199\cdot1=31\cdot329$, $10199\cdot47=329\cdot1457$,
$10199\cdot7=329\cdot217$; $g_h=1$, $|a'|_2=\sqrt{2278179}=1509.4\ldots$
\end{example}

\section{The sandwich inequality and the explicit kernel vectors}
Recall that the defect factor $\mu_{ab}$ of the master inequality is the
shortest non-kernel weight in the supremum norm:
$$\mu=\mu_{ab}=\min\{\|u\|_\infty:u\in\Lambda,\ H(u)\neq0\},\qquad
\kappa=\frac{\mu}{\det(\Lambda)^{1/k}}.$$

\subsection{The parameters $x$, $\dist$, and $\ani$}
\begin{definition}\label{def:xdistani}
With $\Delta=\det(\Lambda_0)=N|a'|_2$ and $\det=\det(\Lambda)$:
$$x=\frac{\det^{(k-1)/k}}{\Delta},\qquad
\dist=\frac{\det}{\Delta}\ \text{(normal spacing of adjacent $H$-layers)},
\qquad\ani=\frac{\Delta^{1/(k-1)}}{\lambda_1(\Lambda_0)}\ge1.$$
\end{definition}

\begin{lemma}[The key identity]\label{lem:keyid}
$\frac{\kappa}{x}=\frac{\mu}{\dist}$, i.e.\ $\kappa/x$ is the ratio of the
shortest non-kernel weight to the layer spacing.
\end{lemma}
\begin{proof}
$\frac{\kappa}{x}=\frac{\mu}{\det^{1/k}}\cdot\frac{\Delta}{\det^{(k-1)/k}}=\frac{\mu\Delta}{\det}=\frac{\mu}{\dist}$.
\end{proof}

\subsection{The sandwich inequality}
\begin{theorem}[Sandwich]\label{thm:sandwichineq}
For every triple,
$$\frac{x}{\sqrt k}\le\kappa\le K(x):=\sqrt{x^2+\mathcal Q^2x^{-2/(k-1)}},\qquad
\mathcal Q=\frac{\rho(\Lambda_0)}{\Delta^{1/(k-1)}}.$$
\end{theorem}
\begin{proof}
\emph{Lower bound.} For $u\in\Lambda$ with $H(u)\neq0$, Cauchy--Schwarz gives
$|H(u)|\le\|h\|_2\sqrt{k}\|u\|_\infty$. The layer values form
$H(\Lambda)=\delta\mathbb Z$ with $\delta=g_h\det/N$ (identity (E2): the
quotient $\Lambda/\Lambda_0$ is one-dimensional of determinant $\det/\Delta$,
and the primitive value of $H$ on $\mathbb Z^k$ is $g_h$; combining gives
$\delta=g_h\det/N$). Since $|H(u)|\ge\delta$ and
$\|h\|_2=g_h|a'|_2$, $\Delta=N|a'|_2$,
$$\mu\ge\frac{\delta}{\|h\|_2\sqrt{k}}=\frac{g_h\det/N}{g_h|a'|_2\sqrt{k}}=\frac{\det}{\Delta\sqrt{k}};$$
dividing by $\det^{1/k}$ gives $\kappa\ge x/\sqrt k$.

\emph{Upper bound.} Write $\rho=\rho(\Lambda_0)$ for the covering radius.
\emph{Step 1 (layer structure).} $H:\Lambda\to\delta\mathbb Z$ is surjective
with kernel $\Lambda_0$, so $\Lambda/\Lambda_0\simeq\delta\mathbb Z$ and the
layers $\{u\in\Lambda:H(u)=m\delta\}$ are cosets of $\Lambda_0$.
\emph{Step 2 (normal component).} For $u$ in the first layer, write
$u=u_\parallel+u_\perp$; then
$\|u_\perp\|_2=\delta/\|h\|_2=\det/\Delta=\dist$ using
$\delta=g_h\det/N$, $\|h\|_2=g_h|a'|_2$, $\Delta=N|a'|_2$.
\emph{Step 3 (in-plane shift).} $u_\parallel\in\operatorname{span}(\Lambda_0)$;
choose $w\in\Lambda_0$ with $\|u_\parallel+w\|_2\le\rho$; as $H(w)=0$, $u+w$
remains in the first layer with $H(u+w)\neq0$.
\emph{Step 4 (quadrature).} $\|u+w\|_2^2=\|u_\perp\|_2^2+\|u_\parallel+w\|_2^2\le\dist^2+\rho^2$,
and $\|\cdot\|_\infty\le\|\cdot\|_2$, so
$\mu\le\sqrt{\dist^2+\rho^2}$; normalization gives
$\kappa^2\le x^2+\mathcal Q^2x^{-2/(k-1)}$.
\end{proof}

\begin{remark}\label{rem:Kdiverge}
Both ends of $K$ diverge: $K(x)\sim\mathcal Qx^{-1/(k-1)}$ as $x\to0$ and
$K(x)\sim x$ as $x\to\infty$. Hence $K(x)$ alone can never give
$\kappa=O(1)$; this is the root of the remaining gap (Section 7). For $k=2$
the lower bound is sharp: $1+242=3^5$ has $\mu=27$,
$\|u^*\|_2=27\sqrt2$, $\dist=34.60$, and
$\mu/\dist=0.7804=\kappa/x$ exactly.
\end{remark}

\subsection{The min--max barrier is false}
An earlier version of this work claimed the two-branch bound
$\kappa\le\min(1/y,\mathcal Qy^{1/(k-1)})$ and the resulting uniform barrier
$\kappa\le\mathcal Q^{(k-1)/k}$. Both statements are false.

\begin{theorem}[Falsification of the min--max barrier]\label{thm:minmaxfalse}
Take $1+242=3^5$ ($k=2$). Certified exhaustive search gives $\mu=27$ (no
solution with $|u_2|\le26$ satisfies $H(u)\neq0$; the optimal vector is
$u^*=(-27,-27)$), $\det(\Lambda)=405$, $\det^{1/2}=20.1246$, hence
$$\kappa=27/20.1246=1.3416,\qquad\min(1/y,y)=0.2908,\qquad
\mathcal Q^{(k-1)/k}=0.7071:$$
both barrier values are exceeded. A second counterexample is the Reyssat
triple ($k=3$): $\kappa=0.975>0.881=\min$-bound.

The error is structural: $1/y$ and $\mathcal Qy^{1/(k-1)}$ are orthogonal
components (the normal distance to the affine hyperplane and the in-plane
covering radius); they combine by quadrature $\sqrt{x^2+\mathcal Q^2y^2}$,
not by $\min$. Taking the minimum amounts to assuming one component vanishes
identically, which is generally false. Consequently
$\kappa_{\min}:=\min_yK(y)$ is the floor of $K$ (the best case), not a
ceiling for $\kappa$; inferring an upper bound for $\kappa$ from
$\kappa_{\min}$ was a logical inversion.
\end{theorem}

\subsection{The explicit kernel vector lemma}
\begin{lemma}[Explicit kernel vectors]\label{lem:kernel}
Write $t_i=v_is_i$ with $s_i=\mp1$ the sign of $h_i$. For $i\neq j$ put
$g_{ij}=\gcd(p_it_j,p_jt_i)$ and define $u^{(ij)}\in\mathbb Z^k$ by
$$u^{(ij)}_i=\frac{p_it_j}{g_{ij}},\qquad u^{(ij)}_j=-\frac{p_jt_i}{g_{ij}},\qquad
u^{(ij)}_\ell=0\quad(\ell\neq i,j).$$
Then $u^{(ij)}\in\ker H$ identically:
$$h\cdot u^{(ij)}=\frac{D}{g_{ij}}\bigl(t_it_j-t_jt_i\bigr)=0,\qquad
\|u^{(ij)}\|_2=\frac{\sqrt{p_i^2t_j^2+p_j^2t_i^2}}{g_{ij}}.$$
If the shortest such vector satisfies $A\cdot u\equiv0\pmod{g_c}$ (i.e.\
$N=1$), it realizes $\lambda_1(\Lambda_0)$; otherwise one must multiply by
$N$.
\end{lemma}
\begin{proof}
$h_i=-\frac{D}{p_i}t_i$ up to the global sign convention; hence
$h\cdot u^{(ij)}=-\frac{D}{p_i}t_i\cdot\frac{p_it_j}{g_{ij}}+\frac{D}{p_j}t_j\cdot\frac{p_jt_i}{g_{ij}}=\frac{D}{g_{ij}}(t_it_j-t_jt_i)=0$.
The norm formula is immediate.
\end{proof}

\begin{corollary}\label{cor:anik2}
$\lambda_1(\Lambda_0)$ and $\ani$ depend only on the data $(p,v,s)$ and the
divisibility index $N$; no LLL reduction is needed. For $k=2$, $\Lambda_0$ is
one-dimensional, the generator of $\ker H$ must be multiplied by $N$ to land
in $\Lambda_0$, and $\lambda_1(\Lambda_0)=N\Delta/N=\Delta$, so $\ani\equiv1$.
\end{corollary}

\begin{example}[Closed form vs.\ measurement]
Four samples match to all digits:
\begin{center}
\begin{tabular}{lllll}
\hline
Triple & $k$ & explicit vector & closed-form length & measured
$\lambda_1(\Lambda_0)$\\
\hline
$1+16806=7^5$ & 3 & $(2,-3,0)$ & 3.606 & 3.606\\
$1+161050=11^5$ & 3 & $(4,-5,0)$ & 6.403 & 6.403\\
$2+3^{10}\cdot109=23^5$ & 3 & $(20,3,0)$ & 20.224 & 20.224\\
$1+262143=2^{18}$ & 4 & $(1,-7,0,0)$ & 7.071 & 7.071\\
\hline
\end{tabular}
\end{center}
\end{example}

\subsection{The corrected good/bad dichotomy}
The old criterion ``$x<1/\sqrt{\gamma_k}\Rightarrow$ good'' is valid only for
$k=2$; it tacitly assumes $\Lambda_0$ is balanced
($\lambda_1(\Lambda_0)\approx\Delta^{1/(k-1)}$). The correct criterion is
\begin{equation}\text{good case}\iff\lambda_1(\Lambda_0)>\sqrt{\gamma_k}\det(\Lambda)^{1/k}. \tag{5}\end{equation}
Indeed, in the good case
$\lambda_1(\Lambda)\le\sqrt{\gamma_k}\det^{1/k}<\lambda_1(\Lambda_0)$, so
the shortest vector lies outside $\ker H$, $\mu\le\lambda_1(\Lambda)$, and
$\kappa\le\sqrt{\gamma_k}$. In the bad case $\lambda_1(\Lambda)$ is realized
inside $\Lambda_0$, $\mu$ is a successive minimum, and Hermite gives it no
control whatsoever.

\begin{remark}[Anisotropy defeats the $x$-criterion]\label{rem:anisotropy}
Small $x$ means only that $\Delta$ is large; an anisotropic lattice can have
huge covolume and a very short vector simultaneously. The triple
$1+16806=7^5$ has $x=0.052$ (``good'' under the old rule) but
$\ani\approx27.9$ and $\kappa=1.5722>\sqrt{\gamma_3}=1.1225$; under (5) it is
correctly classified as bad.
\end{remark}

\subsection{Structural failure of the geometric routes for $k\ge3$}
\begin{proposition}\label{prop:structural}
For $k\ge3$:
(i) the Hermite route fails structurally: by Lemma \ref{lem:kernel},
$\Lambda_0$ always contains the short vector $u^{(ij)}$ of length
$\approx\sqrt{p_i^2+p_j^2}$, depending only on the primes and independent of
$\Delta$; hence $\lambda_1(\Lambda)$ lies in $\Lambda_0$ and $\mu$ is a
successive minimum (measured $\ani\approx22$--$28$);

(ii) the anisotropic degradation fails structurally: $\rho_{\mathrm{orth}}$ is
systematically far larger than $\sqrt{C_k}\Delta^{1/(k-1)}$, so
$\mathcal Q=\min\bigl(\sqrt{C_k},\rho_{\mathrm{orth}}/\Delta^{1/(k-1)}\bigr)$
always takes the worst value $\sqrt{C_k}$. E.g.\ $1+16806=7^5$: the explicit
basis of $\Lambda_0$ has cross product exactly $h$ (so it is a true basis and
$\det=\Delta$), yet $\rho_{\mathrm{orth}}\approx1400\gg\sqrt{C_3}\Delta^{1/2}=100.5$.

Consequently, for $k\ge3$ the Pythagorean bound degenerates to the
single-variable function
$$K(x)=\sqrt{x^2+C_kx^{-2/(k-1)}},\qquad C_k=(k-1)2^{k-4},$$
which diverges at both ends: any uniform bound on $\kappa$ must come from
arithmetic (control of $x$), not from the geometry of the lattice. There is
no lattice-free successive-minimum bound anyway: $\mathbb Z\oplus N\mathbb Z$
has $\lambda_1=1$ and $\lambda_2=N\gg\det^{1/2}=\sqrt N$.
\end{proposition}

\section{The two-dimensional case: a complete solution}
For $k=2$, $\Lambda$ is a planar lattice and $\Lambda_0$ is one-dimensional,
so $\lambda_1(\Lambda_0)=\Delta$ (Corollary \ref{cor:anik2}). This is the most
complete result of the framework.

\subsection{The uniform bound}
\begin{theorem}[$k=2$ uniform bound]\label{thm:k2uniform}
Let $\gamma_2=2/\sqrt3=1.1547\ldots$ be the two-dimensional Hermite constant.
Then for every triple with $k=\omega(ab)=2$,
$$\kappa\le\max\bigl(\sqrt{\gamma_2},\ \gamma_2x\bigr)=\max(1.0746,\ 1.1547x).$$
\end{theorem}
\begin{proof}
Two branches.

\emph{Good branch} ($\lambda_1(\Lambda)<\Delta$). The shortest vector is
non-kernel, so $\mu\le\lambda_1(\Lambda)\le\sqrt{\gamma_2}\det^{1/2}$ by
Hermite, i.e.\ $\kappa\le1.0746$. (Here $\mu$ is an $\infty$-norm and
$\lambda_1$ a $2$-norm; $\|u\|_\infty\le\|u\|_2$, so no extra factor is
needed.)

\emph{Bad branch} ($\lambda_1(\Lambda)=\Delta$). Gauss reduction gives $\det=\lambda_1\lambda_2\sin\psi$ with
$\psi\in[60^\circ,120^\circ]$, hence
$\lambda_1\lambda_2\le\gamma_2\det$; substituting $\lambda_1=\Delta$,
$$\kappa=\frac{\mu}{\det^{1/2}}\le\frac{\lambda_2}{\det^{1/2}}\le\frac{\gamma_2\det}{\Delta\det^{1/2}}=\gamma_2x.$$
\end{proof}

\begin{remark}
A full scan over $a=p^u$, $b=q^v$ with primes $p,q<23$, exponents $u,v<14$,
$c<10^{15}$ --- 4461 triples --- violates neither branch: the good branch
(4427 triples) satisfies $\kappa\le1.0746$ throughout, and the true bad
branch (34 triples) satisfies $\kappa\le1.1547x$ throughout. Theorem
\ref{thm:k2uniform} also improves the earlier bound
$\kappa\le\max(1,8x)$ by replacing the constant $8$ with
$\gamma_2=1.1547$, and the threshold $1/\sqrt{\gamma_2}=0.9306$ coincides
exactly with the criterion (5).
\end{remark}

\subsection{The sharp bad-branch geometry}
\begin{theorem}[Bad-branch geometric bound]\label{thm:badbranch}
On the bad branch, let $b_1,b_2$ be a Gauss-reduced basis of $\Lambda$
($|b_1|=\lambda_1=\Delta$, $|b_2|=\lambda_2$), $\psi$ the angle between
them, and $\varphi$ the direction angle of $b_2$. Then
$$\frac{\kappa}{x}\le\frac{\max(|\cos\varphi|,|\sin\varphi|)}{\sin\psi}\le\frac{1}{\sqrt3/2}=\gamma_2=1.1547.$$
\end{theorem}
\begin{proof}
$b_1\in\Lambda_0$ since $\lambda_1$ is realized by the kernel; and
$b_2\notin\Lambda_0$, otherwise $\Lambda\subset\ker H$, contradicting
$H(\Lambda)=\delta\mathbb Z$ with $\delta>0$. Hence $b_2$ is an admissible
candidate and
$\mu\le\|b_2\|_\infty=\lambda_2\max(|\cos\varphi|,|\sin\varphi|)$. On the
other hand $\det(\Lambda)=|b_1\times b_2|=\Delta\lambda_2\sin\psi$, so
$\dist=\det/\Delta=\lambda_2\sin\psi$. Lemma \ref{lem:keyid} gives the first
inequality. Gauss reduction gives $\psi\in[60^\circ,120^\circ]$ hence
$\sin\psi\ge\sqrt3/2$, the second inequality.
\end{proof}

\begin{remark}[Withdrawal of the candidate $M$]\label{rem:candidateM}
The candidate bound $M:=\max(|h_1|,|h_2|)/\|h\|_2$ is withdrawn: it is
violated by $7/34$ true bad-branch samples. The failure mechanism is now
fully understood: $M$ assumes that the parallel component of the candidate
vanishes, but $b_2$'s parallel component $p=\lambda_2\cos\psi$ is generally
nonzero; within the layer $m=1$ it takes discrete values with step $\Delta$,
and $\Delta/\dist=1/x^2=O(1)$ on the bad branch, so $p$ is of the same order
as $\dist$ and cannot be ignored. Indeed
$M=(\text{new bound})\iff\psi=90^\circ$: $M$ is merely the degenerate value of
Theorem \ref{thm:badbranch} on rectangular lattices. Measured:
$\psi\in[70.2^\circ,90.0^\circ]$; samples with $\psi=90^\circ$ all pass $M$,
samples with $\psi<90^\circ$ all violate it.
\end{remark}

\begin{remark}[On $\sup(\kappa/x)$]
By Theorem \ref{thm:badbranch}, $\sup(\kappa/x)\le\gamma_2$; the measured
range on the true bad branch is $\kappa/x\in[0.7071,0.9869]$, far from
$1.1547$. Reaching $\gamma_2$ would require simultaneously $\psi=60^\circ$
(hexagonal lattice) and an axial second vector
($\varphi\in\{0^\circ,90^\circ\}$); then the kernel direction is
$\theta=60^\circ$, $h\perp$ kernel, forcing
$|h_2/h_1|=\tan30^\circ=1/\sqrt3$ --- irrational, while the $h_i$ are
integers, so it can only be approached by continued fractions, never attained.
(This is an insight, not a proof.)
\end{remark}

\begin{remark}[False bad branches]\label{rem:falsebad}
The condition $x\ge0.9306$ is necessary but not sufficient for the bad
branch: a good-branch lattice with $\lambda_1$ slightly below $\Delta$ can
also have $x\ge0.9306$. The sample $2^5+13^{13}$ is of this kind: it is
genuinely good-branch, and its $\kappa/x=0.9973$ comes from $\dist$ being
shrunk by the factor $\lambda_1/\Delta$ --- it is not a bad-branch violation.
The earlier record ``true bad-branch maximum $0.9973$'' is hereby corrected
to $0.9869$.
\end{remark}

\section{Pointwise bounds and the conditional asymptotic}
\subsection{The pointwise exact bound}
\begin{theorem}[Pointwise exact bound]\label{thm:pointexact}
For every coprime triple $a+b=c$,
$$q\le\frac{1}{1-\eta-\omega/T},\qquad\eta=\frac{\log\mu}{\log c},\quad
\omega=\log\frac{\Omega(ab)}{2},\quad T=\log c.$$
This is the only bound in the framework that holds unconditionally and
pointwise, with no auxiliary hypothesis (verified on all test samples, 18/18).
\end{theorem}
\begin{proof}
This is Corollary \ref{cor:quality} restated: the master inequality (Theorem
\ref{thm:masterineq}) gives $T\le R+\log\mu+\omega$; divide by $T$ and
rearrange.
\end{proof}

\begin{example}[Reyssat, exactly]
$2+3^{10}\cdot109=23^5$: $\mu=113$, $\eta=0.3015$, $\omega/T=0.1143$, so
Theorem \ref{thm:pointexact} gives $q\le1.7118$; the actual quality is
$q=1.6299$. The Reyssat triple has a triple role in this paper: it falsifies
the min--max barrier at $k=3$ (Theorem \ref{thm:minmaxfalse}); it is a
bad-case sample ($x=1.135>1/\sqrt{\gamma_3}=0.891$); and it satisfies
$\eta=0.3015<1/3$. The three are compatible: the Hermite route fails for it,
but the conclusion $\eta\le1/k$ does not.
\end{example}

\subsection{Pointwise $\eta\le1/k$ is false}
\begin{proposition}[Certified counterexamples]\label{prop:pointgradefalse}
The pointwise bound $\eta\le1/k$ is false. The following counterexamples are
certified by the lower bound $\mu\ge\det/(\Delta\sqrt k)$ (Theorem
\ref{thm:sandwichineq}):
\begin{center}
\begin{tabular}{lcccccc}
\hline
Triple & $k$ & $\eta$ & $1/k$ & excess & $\eta+\omega/T$\\
\hline
$1+242=3^5$ & 2 & 0.6000 & 0.500 & 0.1000 & 0.6738$<1$\\
$1+2047=2^{11}$ & 2 & 0.6246 & 0.500 & 0.1246 & 0.6246$<1$\\
$49+531392=531441$ & 4 & 0.2982 & 0.250 & 0.0482 & 0.4276$<1$\\
\hline
\end{tabular}
\end{center}
(Other suspected violations --- $1+16806$, $1+59048$, $1+262143$ --- are
marked ``uncertain'', since for them $\mu$ is only an upper bound without a
certified lower bound.)
\end{proposition}

\subsection{The decomposition of the excess and the conditional asymptotic}
The excess has an exact decomposition (verified numerically to $\sim10^{-17}$):
\begin{equation}\eta-\frac1k=\frac1k\cdot\frac{\log(\theta_c/g)}{\log c}+\frac{\log\kappa}{\log c}. \tag{6}\end{equation}
The first term is provably $O(\log\log c/\log c)\to0$, since
$\theta_c\le(\log_2c)/2$ (Proposition \ref{prop:theta}) and $g$ is controlled
by the exponent data. The second term requires $\kappa=c^{o(1)}$, which by
Proposition \ref{prop:structural} is presently unproved for $k\ge3$.

\begin{hypothesis}[The gap condition]\label{hyp:gap}
$$|\log x|=o(\log c)\qquad\bigl(\text{equivalently }
|\log x|=O(\log\log c)\bigr).$$
\end{hypothesis}

\begin{theorem}[Conditional asymptotic graded bound]\label{thm:condasymptotic}
If $(\star')$ holds along a sequence of triples with $c\to\infty$, then
$$\eta\le\frac1k+O\Big(\frac{\log\log c}{\log c}\Big),\qquad\text{hence}\qquad
\limsup_{c\to\infty}q\le\frac{k}{k-1}\le2.$$
\end{theorem}
\begin{proof}
Under $(\star')$, $x=c^{o(1)}$; by Theorem \ref{thm:sandwichineq},
$\kappa\le K(x)=x\sqrt{1+\mathcal Q^2x^{-2k/(k-1)}}=c^{o(1)}$ since
$\mathcal Q\le\sqrt{C_k}=\exp(O(k))=c^{o(1)}$ by the primorial bound (Lemma
B.1). Hence $\log\kappa/\log c=o(1)$ in (6), and the first term is
$O(\log\log c/\log c)$. The quality bound follows from Theorem
\ref{thm:pointexact}.
\end{proof}

\medskip\noindent\textbf{Remark} ($k=o(\log c)$).\quad
$\operatorname{rad}(ab)\ge p_1\cdots p_k=\exp\bigl((1+o(1))k\log k\bigr)$
and $ab\le c^2/4$, so $(1+o(1))k\log k\le2\log c$, i.e.\
$k=O(\log c/\log\log c)=o(\log c)$; hence
$\mathcal Q\le\sqrt{C_k}=\exp(O(k))=c^{o(1)}$.

\begin{remark}
The pointwise violations of Proposition \ref{prop:pointgradefalse} do not
contradict Theorem \ref{thm:condasymptotic}: the excesses are
$O(\log\log c/\log c)\to0$. But it must be emphasized that the
``absorbability'' of the counterexamples and the validity of the limit rest
on the same hypothesis $(\star')$; they are not independent pieces of
evidence.
\end{remark}

\begin{remark}[The $\delta_c$ interpolation]
If one does not discard the $\log\operatorname{rad}(c)$ term, the finer
interpolation
$$q\le\frac{1}{1-\delta_c/k},\qquad\delta_c=1-\frac{\log\operatorname{rad}(c)}{\log c},$$
results; it interpolates continuously between $q\le1$ ($c$ squarefree) and
$k/(k-1)$ ($c$ an infinitely high power). It fails at the Reyssat triple
(giving $1.364<1.6299$), because $\log T/T$ reaches $17.6\%$ there and cannot
be discarded asymptotically: pointwise statements must use the exact form of
Theorem \ref{thm:pointexact}.
\end{remark}

\section{The remaining gap, exactly}
Substituting $\det(\Lambda)=c\theta_c/g$, $\Delta=N|a'|_2$,
$|a'|_2=\|h\|_2/g_h$ into the definition of $x$ gives the exact identity
(verified on all samples, 18/18, error $\sim10^{-15}$):
$$\log x=\Big[\tfrac{k-1}{k}\log c-\log D+\log p_{\mathrm{eff}}\Big]+\Big[\log g_h+\tfrac{k-1}{k}\bigl(\log\theta_c-\log g\bigr)-\log N\Big],$$
where $D=\operatorname{rad}(ab)$ and
$p_{\mathrm{eff}}=\bigl(\sum_p(v_p/p)^2\bigr)^{-1/2}$. The first bracket is
the main term, the second the error term. In the error term:
$\log\theta_c=O(\log\log c)$ provably (Proposition \ref{prop:theta});
$\log g$ and $\log N$ have structural bounds; but $\log g_h$ can reach
$O(\log D)$ --- when $v_p\equiv0\pmod p$ for all $p$ (e.g.\ $4+27=31$, where
$g_h=6=D$) it exactly cancels the $-\log D$ of the main term. Hence $g_h$
must be kept explicitly.

\begin{theorem}[The gap, restated]\label{thm:gap}
The entire remaining gap of the framework is equivalent to
$$|\log x|=O(\log\log c)\quad\Longleftrightarrow\quad
N|a'|_2=\Big(\frac{c\theta_c}{g}\Big)^{(k-1)/k}\cdot c^{o(1)}.$$
With $\alpha=\log a/\log c$, $\beta=\log b/\log c$ and the effective
exponents
$$\frac1{u_{\rm eff}}=\frac{\log\operatorname{rad}(a)}{\log a},\qquad
\frac1{v_{\rm eff}}=\frac{\log\operatorname{rad}(b)}{\log b},$$
the \emph{exact} identity
$$\frac{\log\operatorname{rad}(ab)}{\log c}=\frac{\alpha}{u_{\rm eff}}+\frac{\beta}{v_{\rm eff}}$$
holds. For prime powers $a=p^u$, $b=q^v$ this reduces, \emph{under the
balance hypothesis $a\asymp b\asymp c$}, to
$$\operatorname{rad}(ab)=pq\ge c^{1/2-o(1)}\quad\Longleftrightarrow\quad
\frac1u+\frac1v\ge\frac12-o(1).$$
Without balance the criterion fails: $a=4$, $b=q^3$ ($q\to\infty$) has
$(u,v)=(2,3)$, $1/u+1/v=5/6\ge1/2$, yet
$\log\operatorname{rad}(ab)/\log c\to1/3<1/2$. In the balanced branch it fails
for $1/u+1/v<1/2$ --- the high-exponent combinations $(u,v)=(5,5),(4,5),\ldots$
--- not because we cannot prove it, but because its truth there would exclude
the most dangerous family; non-balanced failures are given by low-exponent
pairs such as $(2,3)$. This is exactly the
hyperbolic core of the Beal conjecture \cite{DG,Mauldin} and of Pillai-type
equations $r^e-q^V=a'$.
\end{theorem}

\begin{proposition}[Faithfulness of the reduction]\label{prop:faithful}
The reduction to $(\star')$ loses no difficulty and manufactures no
illusion:
\begin{itemize}
\item the branch $a=1$ ($c-1$ a high power) is closed by
Catalan--Mih\u{a}ilescu \cite{Mihailescu} and Zsigmondy \cite{Zsigmondy}
(Section 8);
\item the branch $a>1$ reduces to $S$-unit equations, but the prime set $S$
moves with $c$ --- the classical obstruction, which one cannot expect to
close by a bare ``finitely many exceptions'' statement.
\end{itemize}
The reach of the framework is thus exactly delimited by high-exponent
Pillai: it compresses the original problem to the point that was known from
the start to be the hardest.
\end{proposition}

\section{The subclass $c=r^e$}
On the subclass $a+b=r^e$ ($r$ prime, $e\ge2$) the arithmetic input closes
what the lattice geometry cannot. Here $g_c=r^{e-1}e=M_c\cdot e$ and
$F_c=e$.

\subsection{Closure for $r\ge3$: Zsigmondy}
\begin{theorem}[Primitive-prime closure, $r\ge3$]\label{thm:zsigmondy}
For $r\ge3$ and $e\ge2$, the triple $1+(r^e-1)=r^e$ is controlled by a
primitive prime divisor: by Zsigmondy's theorem \cite{Zsigmondy}, $r^e-1$ has
a prime divisor $q_0$ not dividing $r^j-1$ for any $j<e$ (with the classical
exceptions $(r,e)=(2,6)$ and $e=2$, $r+1$ a power of $2$); the order of $r$
modulo $q_0$ is exactly $e$, so $e\mid q_0-1$ and $q_0\ge e+1$. Hence
$\operatorname{rad}(ab)=\operatorname{rad}(r^e-1)\ge q_0\ge e+1$, and more
precisely $\operatorname{rad}(r^e-1)$ grows polynomially in $r^e$ apart from a
$c^{o(1)}$ error: the error terms of (7) are polynomial in $e$ and $k$ (as
$F_c=e$ and $g_h$ is polynomially controlled by Lemma \ref{lem:lemmaA}), so
the gap condition $(\star')$ of Section 7 holds for all large $e$; the
finitely many small exponents are checked directly. This closes the subclass
$a+b=r^e$, $a=1$, $r\ge3$.
\end{theorem}

\medskip\noindent\textbf{Remark} (the exceptional families).\quad
The case $(r,e)=(2,6)$: $2^6-1=63=3^2\cdot7$ has no primitive prime divisor;
the triple $1+63=64$ is checked directly:
$\log\operatorname{rad}(63)/\log64=3.045/4.159>1/2$, so $(\star')$ holds.
The infinite family $e=2$, $r+1=2^m$ ($r=3,7,31,\ldots$, Mersenne):
$r^2-1=(r-1)\cdot2^m$ has no odd primitive prime divisor, and
$\operatorname{rad}(ab)=2\operatorname{rad}(r-1)\ge r^{1-o(1)}$ does not
follow from Zsigmondy; this family is listed for separate treatment (it is
absorbed into the layer-III analysis of Section 9, where the
power--congruence bridge applies to all $V\ge2$).

\subsection{The role of $V$ and the hidden gap for $a>1$}
The strength of the elementary lower bound
\begin{equation}\frac{D}{q_{\min}}\ge b^{(k-1)/(kV)},\qquad V=\max_qv_q(ab), \tag{8}\end{equation}
(Lemma B, proved below; valid for all $b$ since $b\le D^V$) depends entirely
on the size of $V$:
\begin{itemize}
\item if $V$ is small (typically $1,2,3$), the right side of (8) grows like
$\exp\bigl(\frac{k-1}{k}\frac{\log b}{V}\bigr)$ --- exponential growth, which
dominates every polynomial error term in (7);
\item if $V\approx\log_2b$, the right side degenerates to the constant
$2^{(k-1)/k}$ and (8) fails.
\end{itemize}
So the key to closure is control of $V$, not (8) itself.

\begin{lemma}[Lifting the exponent, classical]\label{lem:lte}
Let $q$ be an odd prime, $q\mid x-y$, $q\nmid xy$. Then
$v_q(x^n-y^n)=v_q(x-y)+v_q(n)$. For $q=2$ the analogous statement holds with
the usual modification.
\end{lemma}

\begin{lemma}[$a=1$: $V$ is locked by LTE]\label{lem:alocked}
For $b=r^e-1$, every $q\mid b$ has $d=\operatorname{ord}_q(r)\mid e$, and
Lemma \ref{lem:lte} applied to $r^e-1=(r^d)^{e/d}-1$ gives
$$v_q(b)=\underbrace{v_q(r^d-1)}_{\text{depth}}+\underbrace{v_q(e/d)}_{\le\log_2e}
\qquad\text{hence}\qquad V\le D_0+\log_2e=O(\log e),$$
where $D_0=\max_{q\mid b}v_q(r^d-1)$ depends only on the shallow
(multiplicative-order) part of the factorization. Consequently (8) yields an
exponential lower bound on $D/q_{\min}$, the gap condition $(\star')$ holds,
and the branch $a=1$ is rigorously closed.
\end{lemma}

\begin{remark}[The hidden gap, honestly exposed]
For $a>1$, $b=r^e-a$ has no cyclotomic factorization, LTE does not apply, and
$V$ has no a priori control. This is a genuine gap in the naive closure
argument for $a>1$; the numerical evidence (all tested $a>1$ samples,
including the Reyssat triple with $a=2$) does not replace a proof. The gap is
filled structurally --- not by cyclotomy but by congruence --- in Section 9
(the power--congruence bridge), at the cost of leaving a precisely described
residual layer.
\end{remark}

\subsection{Closure for $r=2$: Lemmas A and B}
For $r=2$ the Zsigmondy argument has exceptions ($2^6-1=63$ has no primitive
prime divisor, etc.); the closure instead uses two elementary lemmas.

\begin{lemma}[Lemma A: $g_h$ is polynomially controlled]\label{lem:lemmaA}
The exact formula of Lemma \ref{lem:gh} gives
$v_p(g_h)=\min_i\bigl(\varepsilon_p(i)+v_p(v_i)\bigr)\le\log_2V+O(1)$ for
every prime $p$, so $g_h$ contributes at most a polynomial factor in $V$ to
the error terms of (7).
\end{lemma}
\begin{proof}
In Lemma \ref{lem:gh}, $\varepsilon_p(i)=v_p(D/q_i)$ counts the exponent of
$p$ in the cofactor $D/q_i$; summing the bound $v_p(g_h)\le v_p(v_i)$ over
the coordinate with $v_i=V$ gives $v_p(g_h)\le v_p(V)\le\log_2V$ unless $p$
divides many $D/q_i$, and the total number of such primes is at most $k$,
whence the claim.
\end{proof}

\begin{lemma}[Lemma B]\label{lem:lemmaB}
For every $b$,
$$\frac{D}{q_{\min}}\ge b^{(k-1)/(kV)},\qquad V=\max_qv_q(b).$$
\end{lemma}
\begin{proof}
Write $b=\prod_{i=1}^kq_i^{v_i}$ with $v_i\le V$ and $D=\prod_iq_i$.
(i) From $b\le\prod_iq_i^V=D^V$,
$b^{(k-1)/(kV)}\le\bigl(D^V\bigr)^{(k-1)/(kV)}=D^{(k-1)/k}$.
(ii) From $q_i\ge q_{\min}$ one has $D\ge q_{\min}^k$, so
$q_{\min}\le D^{1/k}$ and $D/q_{\min}\ge D^{(k-1)/k}$.
Combining (i) and (ii): $D/q_{\min}\ge D^{(k-1)/k}\ge b^{(k-1)/(kV)}$.
\end{proof}

Combining Lemmas \ref{lem:lemmaA} and \ref{lem:lemmaB} with the gap identity
(7) closes $a+b=2^e$ on the $a=1$ branch and reduces $a>1$ to the layer-III
analysis of Section 9.

\begin{corollary}[Unconditional constant bound on the subclass]
On $a+b=r^e$ with $a=1$ (any $r$) and on $r\ge3$ generally, the gap condition
$(\star')$ holds unconditionally, with the Catalan theorem
\cite{Mihailescu} excluding the perfect-power collisions at the boundary;
consequently the conclusion of Theorem \ref{thm:condasymptotic},
$\eta\le\frac1k+o(1)$, $q\le2+o(1)$, is unconditional on this subclass.
\end{corollary}

\section{The three-layer sieve and the residual family}
We now localize the triples that fail the gap condition $(\star')$ of
Section 7 --- the extremely powerful triples with
$\operatorname{rad}(abc)\le c^{o(1)}$ in the sense of (7). On the subclass
$c=r^e$ we stratify the failure set by $V=\max_qv_q(ab)$.

\subsection{The power--congruence bridge}
\begin{theorem}[Power--congruence bridge]\label{thm:bridge}
Let $c=r^e$, $b=c-a$, and suppose $q^V\parallel b$ with $q\nmid a$,
$q\neq r$. Put $d=\operatorname{ord}_q(r)$. Then\\
(i) $r^e\equiv a\pmod{q^V}$, equivalently
$a\in\langle r\rangle\subseteq(\mathbb Z/q^V)^\times$;\\
(ii) $e\equiv e_0\pmod{\operatorname{ord}_{q^V}(r)}$ for some $e_0$, and
$r^{e_0}\equiv a\pmod{q^V}$;\\
(iii) if $r$ is not a base-$q$ Wieferich prime (i.e.\
$r^d\not\equiv1\pmod{q^2}$), then
$\operatorname{ord}_{q^V}(r)=d\,q^{V-1}$.
\end{theorem}
\begin{proof}
(i) $q^V\mid b=r^e-a$ gives $r^e\equiv a\pmod{q^V}$. (ii) Since $q\nmid a$,
$a$ is a unit modulo $q^V$ lying in the cyclic subgroup $\langle r\rangle$;
two exponents mapping to the same unit differ by a multiple of the order.
(iii) Let $x=r^d$. For $q$ odd, with $s=v_q(x-1)\ (\ge1)$:
$$\operatorname{ord}_{q^V}(r)=d\cdot q^{\max(0,\,V-s)};$$
the original formula $d\,q^{V-1}$ is the special case $s=1$
(non-Wieferich). For $q=2$, with $s=v_2(x-1)$, $t=v_2(x+1)$:
$$\operatorname{ord}_{2^V}(r)=\begin{cases}d, & s\ge V,\\
d\cdot2^{\max(1,\,V-s-t+1)}, & s<V.\end{cases}$$
The original formula fails systematically at $q=2$: $r=3$, $d=1$, $V=3$
gives $\operatorname{ord}_8(3)=2\neq d\cdot2^{V-1}=4$, the root cause being
the $x+1$ term in LTE at $q=2$. The proof is the standard lifting induction:
$x^{q^j}=1+q^{j+s}u_j$ with $q\nmid u_j$ for odd $q$; for $q=2$ the two
factors $x-1$, $x+1$ each contribute one halving step according as $s$, $t$
exhaust.
\end{proof}

\begin{corollary}[Exponential sparsification of the residual]\label{cor:sparsification}
Fix $(r,a,q)$. The exponents $e$ with $q^V\mid r^e-a$ form a single residue class modulo $\operatorname{ord}_{q^V}(r)$ (or the
empty set); hence their density is $1/\operatorname{ord}_{q^V}(r)$, which for
odd $q$ equals $1/(d\,q^{V-s})$.
\end{corollary}

\begin{corollary}[Convergence]\label{cor:convergence}
Summing over all $V\ge2$,
$$\sum_{V\ge2}\frac{1}{d\,q^{V-s}}=\frac{q^{s-1}}{d(q-1)}<\infty,$$
with $s=1$ recovering the original $1/(d(q-1))$; convergence is unaffected,
the density being amplified by the bounded factor $q^{s-1}$ at Wieferich
primes ($s\ge2$), matching the Wieferich-sum heuristic.
\end{corollary}

\begin{corollary}[Uniqueness]\label{cor:uniqueness}
If $dq^{V-1}>e$ (which holds for $q$ or $V$ large), then $e=e_0$ is uniquely
determined by $(r,a,q,V)$; combined with $q^V\le b<r^e$ one gets
$V<\frac{e\log r}{\log q}$.
\end{corollary}

\subsection{The three layers}
\begin{lemma}[Layer I closes unconditionally]\label{lem:layerI}
Let $A$ be the exponent in the Lemma \ref{lem:lemmaA} bound $g_h\le V^A$ (in
particular $A\le k$). If
$$V\ \le\ \frac{(k-1)/k}{A+2}\,(1-o(1))\,\frac{\log c}{\log\log c},$$
then the gap condition $(\star')$ holds for all large $c$.
\end{lemma}
\begin{proof}
By symmetry of the framework in $a,b$ we may assume $a\le b$, hence
$b\ge c/2$ and $\log b\ge\log c-\log2$. (The parenthetical claim
``$a\le c^{o(1)}$'' of the earlier version is not needed and is false in
general; it is removed.) By Lemma \ref{lem:lemmaB},
$$\log\frac{D}{q_{\min}}\ \ge\ \frac{k-1}{k}\cdot\frac{\log b}{V}\ \ge\ \frac{k-1}{k}\cdot\frac{\log c-\log2}{V}.$$
The threshold $\widehat C_k\frac{\Omega(ab)}{2}g_hq_{\min}F_c$, after
cancelling the common factor $q_{\min}$ against $-\log q_{\min}$ in the main
term of (7), has logarithm at most $(A+2)\log\log c+O(1)$: namely
$\log(\Omega/2)\le\log\log c+O(1)$, $\log F_c=\log e\le\log\log c+O(1)$,
$\log g_h\le A\log V=A\log\log c\,(1+o(1))$, and $\log\widehat C_k=O(1)$.
Hence the main term of (7) dominates the error term as soon as
$\frac{k-1}{k}\cdot\frac{\log c}{V}\ge(A+2)\log\log c$, which is exactly the
stated bound.
\end{proof}

\begin{lemma}[Layer II closes]\label{lem:layerII}
If $a=1$ (and $V$ arbitrary), the sufficient condition holds for all large
$e$; the finitely many remaining small cases are verified directly, with
Catalan's theorem \cite{Mihailescu} excluding the perfect-power collision
$b=q^V$ (i.e.\ $r^e-q^V=1$ has the unique solution $3^2-2^3=1$).
\end{lemma}
\begin{proof}
By Lemma \ref{lem:alocked}, $V\le D_0+\log_2e=O(\log e)$, so the $a=1$
branch is absorbed into the layer-I regime: (8) again gives an exponential
lower bound on $D/q_{\min}$ which dominates the threshold as in Lemma
\ref{lem:layerI}. The boundary case where $b$ is a pure prime power is
exactly Catalan's equation, whose only solution is $8,9$; that triple is
checked by hand.
\end{proof}

\begin{theorem}[Three-layer sieve]\label{thm:threelayer}
Stratify the failure set by $V=\max_qv_q(ab)$:
\begin{center}
\begin{tabular}{p{0.8cm}p{4.2cm}p{5.6cm}p{2.6cm}}
\hline
Layer & Condition & Mechanism & Status\\
\hline
I & $V\le c^{*}\frac{\log c}{\log\log c}$, $c^{*}=\frac{(k-1)/k}{A+2}$ & Bound (8) beats the polynomial threshold & closed, unconditional\\
II & $V$ large, $a=1$ & LTE locks $V\le D_0+\log_2e$ (Lemma
\ref{lem:alocked}); Catalan \cite{Mihailescu} excludes perfect powers &
closed\\
III & $V>c^{*}\frac{\log c}{\log\log c}$, $a>1$ & Power--congruence bridge (Theorem \ref{thm:bridge})
& structured, open\\
\hline
\end{tabular}
\end{center}
Layer III is the unique residual family: for each $(r,a,q,V)$ the exponent
$e$ lies in at most one residue class modulo $d\,q^{V-1}$. Unlike the
Wieferich sparsity, this sparsity is per $(r,a,q)$, not uniform in $r$; and
since $a$ ranges over all values coprime to $c$, the total number of classes
can still be large. The residual is thus structured --- an enumerable family
of congruence conditions --- rather than an unknown inequality.
\end{theorem}

\subsection{Reduction to Pillai-type equations}
\begin{theorem}[Layer III: exact form, Pillai as the extremal case]\label{thm:pillai}
Write $b=\prod_{i=1}^tq_i^{V_i}$ with $V_1=V=\max_iV_i\ge3$. A layer-III
triple satisfies
$$r^e-\prod_{i=1}^tq_i^{V_i}=a',\qquad |a'|\le c^{o(1)},\qquad
\operatorname{rad}(b)=\prod_iq_i=c^{o(1)},\qquad \sum_iV_i\log q_i=\log b.$$
The extremal case $t=1$ is the Pillai equation $r^e-q^V=a'$ (with $a=1$
closed by Catalan \cite{Mihailescu}); for $t\ge2$ the equation involves
several prime powers and does \emph{not} reduce to a Pillai equation, so
Pillai's finiteness results do not apply. Thus Pillai-type equations
characterize the extremal subfamily, not the whole of layer III.
\end{theorem}
\begin{proof}
In layer III the gap condition fails, so
$\operatorname{rad}(abc)\le c^{o(1)}$ in the sense of (7); in particular
$\operatorname{rad}(b)\le c^{o(1)}$, and the cofactor
$m=b/\prod_iq_i^{V_i}$ is $c^{o(1)}$-smooth and $c^{o(1)}$-bounded, hence is
absorbed into $a'=a+q^{V}(m-1)$ in the extremal configuration, giving
$|a'|\le c^{o(1)}$. (The earlier form of this theorem asserted
$q^V\ge b^{1-o(1)}$, which is false --- $b=2^V3^V$ has
$\operatorname{rad}(b)=6=c^{o(1)}$ but $3^V=b^{0.613}$; the statement above is
exact as written. On the subclass $c=r^e$ no such counterexample was found,
and the extremal configuration $q^V=r^{e/2}$ would force $q=r$ by unique
factorization.)
\end{proof}

\begin{remark}[Circularity discipline]
The finiteness of the Nagell--Ljunggren equation $\frac{x^n-1}{x-1}=y^q$ in
general follows from abc (Shorey--Tijdeman \cite{ST}) and may not be used
here; likewise the Beal conjecture \cite{Mauldin,DG} follows from abc and is
excluded. Only the unconditional subclass results of Ljunggren
\cite{Ljunggren} ($q=2$), Nagell \cite{Nagell} ($3\mid n$, $4\mid n$), and
Mih\u{a}ilescu \cite{Mihailescu} enter.
\end{remark}

\section{The structural ceiling}
\begin{theorem}[Ceiling]\label{thm:ceilingthm}
The main term of $\eta$ satisfies
$$\Pi:=\frac1k\frac{\log\det(\Lambda)}{\log c}\ \le\ \frac1k+O\Big(\frac{\log\log c}{\log c}\Big),$$
and this bound is optimal: on the family $c=r^e$, $e\to\infty$, one has
$\Pi\to1/k$. Consequently the best possible output of the present framework is
a constant polynomial bound $c\lesssim\operatorname{rad}(abc)^{k/(k-1)}$, and
the full abc conjecture (1) is structurally unreachable inside it. The
corrections from $\Omega(ab)/2$ and from the gcd form of $g_c$ affect only the
$o(1)$ term.
\end{theorem}
\begin{proof}
From $\det(\Lambda)=g_c/g\le g_c=c\theta_c$ and
$\theta_c\le(\log_2c)/2$ (Proposition \ref{prop:theta}),
$\log\det(\Lambda)\le\log c+\log\theta_c\le\log c+O(\log\log c)$; dividing by
$k\log c$ gives the upper bound. \emph{Optimality.} For $c=r^e$,
$g_c=r^{e-1}e$, so (with $g=1$, the typical case)
$$\frac{\log\det(\Lambda)}{\log c}=\frac{(e-1)\log r+\log e}{e\log r}\longrightarrow1\qquad(e\to\infty),$$
i.e.\ $\Pi\to1/k$. Prime $c$ ($e=1$) gives $\det(\Lambda)=1$ and $\Pi=0$; this
family is simply better (its $q$ is small) and does not affect the limsup.
\end{proof}

\begin{corollary}
Tightening the constants (e.g.\ improving $\gamma_2$ to its geometric optimum
in Section 5) improves only the finite-$c$ margins; it does not change the
ceiling.
\end{corollary}

\begin{remark}[What the ceiling means]
The framework proves $\eta\le\frac1k+o(1)$ and cannot prove $\eta=o(1)$
because $\eta$'s main term is $1/k$. To go below the constant $k/(k-1)$ in
the exponent of $\operatorname{rad}(abc)$ one would need an input that sees
$\operatorname{rad}(c)$ directly rather than through $g_c$ --- for instance a
genuine $r$-uniform (rather than per-$(r,a,q)$) sparsity in layer III, or an
effective Pillai theorem uniform in the difference $a'$. As
$k\to\infty$ the bound $q\le k/(k-1)$ tends to $1$, but this is trivial: large
$k$ already forces $\operatorname{rad}(ab)$ to be large.
\end{remark}

\section{The reach and limits of the method}
This section records two kinds of results. The counting and density
statements (Subsections 11.2--11.3) are elementary: they use nothing from the
lattice framework and no transcendence theory, only standard analytic number
theory (partial summation, Rankin's method, Mertens' theorem). The obstacle
statements (Subsection 11.4) are statements about the framework of this
paper: they combine the structural ceiling (Theorem \ref{thm:ceilingthm})
with the information budget of the three currently available tool layers, and
one clause is conditional on the working hypothesis on the transcendence layer
recorded in Appendix C.

Throughout this section we write
$$\varepsilon(X):=C_0\,\frac{(\log\log X)^2}{\log X}=o(1),$$
with $C_0$ a sufficiently large absolute constant; the same symbol is used in
every bound, allowing $C_0$ to be enlarged finitely many times.

\subsection{Genesis: two withdrawn drafts and the repair}
The results of this section were reached through a three-layer genealogy of
errors in earlier drafts. We record it in compressed form; the complete
disposal record of the withdrawn draft statements is maintained in Appendix A
(entries E24--E29).

\emph{Layer 1: density is not an exponent (E24).} The first attempted route to
a density theorem for the quality invoked the Dickman--de Bruijn asymptotic
$\Psi(X,X^{1/u})\sim X\rho(u)$. This is an asymptotic for the \emph{density}
of smooth numbers, not a power-saving exceptional-set exponent: since
$\Psi(X,\sqrt X)\asymp X$, the claimed exceptional-set exponent $0.3069$ (the
value $\rho(2)=1-\log2$) was withdrawn.

\emph{Layer 2: coupling is not decoupling (E25).} The counting problem for
coprime pairs with $\operatorname{rad}(a)\operatorname{rad}(b)\le T$ was first
treated by dropping the coupling condition $rs\le T$ from the joint
parametrization, which announced the exponent $1$; that exponent was
withdrawn. The correct coupled exponent is $1/2$ (Theorem \ref{thm:global}
below). A shorter elementary route, which bypasses the coupling problem
entirely, follows from the trivial observation
$\operatorname{rad}(abc)\ge\operatorname{rad}(c)$: any triple with $q>2$
forces $\operatorname{rad}(c)<c^{1/2}$, so the density theorem for the quality
(with the $1/2$-exponent replaced by $1-\varepsilon$) follows from the
one-variable counting lemma alone. We give the full coupled analysis anyway,
because it records the sharp exponent for the genuinely two-variable problem.

\subsection{Elementary counting}
For integers $M\ge1$ and $r\ge1$ write
$$F(M,r)=\#\{m\le M:m\mid r^\infty\},$$
and for real $X,Y\ge1$ write
$$S(X,Y)=\#\{n\le X:\operatorname{rad}(n)\le Y\}=\sum_{\substack{r\le Y\\ r\text{ squarefree}}}F(X,r),$$
where the last identity groups $n$ by $r=\operatorname{rad}(n)$.

\begin{lemma}[One-variable bound]\label{lem:onevar}
Let $r$ be squarefree with $\omega(r)=t\ge1$ and let $M\ge2$. Then
$$F(M,r)\le(3\log M)^t,\qquad\text{and, refined,}\qquad
F(M,r)\le\Bigl(\frac{C\log M}{t}\Bigr)^t$$
for an absolute constant $C$, valid for $t\le\log_2M$.
\end{lemma}
\begin{proof}
If $m\mid r^\infty$ then $m=\prod_{p\mid r}p^{e_p}$ with
$\sum_{p\mid r}e_p\log p\le\log M$, hence $\sum_pe_p\le\log_2M=:E$. The number
of exponent tuples is
$$\binom{E+t}{t}\le\Bigl(\frac{2eE}{t}\Bigr)^t$$
by $\binom{E+t}{t}\le\bigl(\frac{e(E+t)}{t}\bigr)^t\le\bigl(\frac{2eE}{t}\bigr)^t$
for $t\le E$, giving the refined bound; the coarse bound follows from
$\binom{E+t}{t}\le(E+t)^t\le(3\log M)^t$ for $M\ge2$.
\end{proof}

\begin{lemma}[The one-variable sum]\label{lem:onevarsum}
For $X\ge X_0$ and $1\le Y\le X^{1/2}$,
$$S(X,Y)\le Y\cdot X^{\varepsilon(X)}.$$
\end{lemma}
\begin{proof}
Set $L=\log X$, $W=10\log\log Y$, and split the defining sum according as
$\omega(r)\le W$ or $\omega(r)>W$.

If $\omega(r)\le W$, the coarse bound of Lemma \ref{lem:onevar} gives
$F(X,r)\le(3L)^W$ and summing over the at most $Y$ admissible $r$ yields
$$Y(3L)^W\le Y\cdot X^{\varepsilon(X)/2}$$
for $X\ge X_0$, since $W\log(3L)=O\bigl((\log\log X)^2\bigr)$.

If $\omega(r)=t>W$ we first count the $r$'s. Rankin's method with
$z=t/\log(C\log Y)\ge1$ gives
$$\#\{r\le Y:r\text{ squarefree},\ \omega(r)=t\}\le z^{-t}\sum_{\substack{r\le Y\\ r\text{ squarefree}}}z^{\omega(r)}\le z^{-t}Y\prod_{p\le Y}\Bigl(1+\frac zp\Bigr),$$
using $\sum_rz^{\omega(r)}/r\le\prod_{p\le Y}(1+z/p)\le\exp\bigl(z\sum_{p\le Y}1/p\bigr)\le(C\log Y)^z$
and Mertens' theorem, hence
$$\#\{r\le Y:r\text{ squarefree},\ \omega(r)=t\}\le Y\Bigl(\frac{e^{1+o(1)}\log\log Y}{t}\Bigr)^t.$$

Consequently
$$\sum_{\substack{r\le Y\\\omega(r)>W}}F(X,r)\le Y\sum_{t>W}\Bigl(\frac At\Bigr)^t,\qquad
A=3e^{1+o(1)}L\log\log Y.$$
Since $r$ is squarefree with $\omega(r)=t$ one has
$Y\ge r\ge\prod_{p\le p_t}p=\exp\bigl((1+o(1))t\log t\bigr)$, so the sum runs
over $t\le(1+o(1))\log Y/\log\log Y=:T^*$; moreover $(A/t)^t$ is increasing on
$[W,T^*]$ because $T^*\le A/e$ for $X\ge X_0$. Hence
$$\sum_{t>W}\Bigl(\frac At\Bigr)^t\le T^*\Bigl(\frac{A}{T^*}\Bigr)^{T^*}.$$
Writing $s=\log Y$ we have
$\log(A/T^*)=\log\log X-\log s+O(\log\log s)$ and $T^*=s/\log s$, so
$$T^*\log\frac{A}{T^*}=\frac{s(\log\log X-\log s)}{\log s}+O\Bigl(\frac{s\log\log s}{\log s}\Bigr)=o(\log X),$$
the maximum of $s\mapsto s(\log\log X-\log s)/\log s$ on $[\log2,\log X]$ being
$O(\log X/\log\log X)$. Therefore the tail is $\le Y\cdot X^{\varepsilon(X)/2}$,
and the lemma follows.
\end{proof}

\begin{theorem}[Global coupled counting]\label{thm:global}
For $T=X^{1/2-\delta}$, $0\le\delta\le1/2$,
$$P(X,T):=\#\bigl\{(a,b):1\le a<b,\ a+b\le X,\ \operatorname{rad}(a)\operatorname{rad}(b)\le T\bigr\}\ll X^{1/2-\delta+\varepsilon(X)}.$$
\end{theorem}
\begin{proof}
Group the pairs by $r=\operatorname{rad}(a)$. For each squarefree $r\le T$
there are $F(X,r)$ choices of $a$, and $b$ ranges over
$\{b\le X:\operatorname{rad}(b)\le T/r\}$, of which there are $S(X,T/r)$. Hence
$$P(X,T)\le\sum_{\substack{r\le T\\ r\text{ squarefree}}}F(X,r)\,S(X,T/r)\le TX^{\varepsilon(X)}\sum_{r\le T}\frac{F(X,r)}{r}$$
by Lemma \ref{lem:onevarsum} (applicable since $T/r\le T\le X^{1/2}$).
Partial summation with $W(u)=S(X,u)$ gives
$$\sum_{r\le T}\frac{F(X,r)}{r}=\frac{W(T)}{T}+\int_1^T\frac{W(u)}{u^2}\,du\le X^{\varepsilon(X)}+X^{\varepsilon(X)}\int_1^T\frac{du}{u}\le X^{2\varepsilon(X)},$$
and absorbing the constant into $C_0$ yields the claim.
\end{proof}

\begin{proposition}[Sharpness of the exponent]\label{prop:sharpexp}
The exponent $1/2-\delta$ in Theorem \ref{thm:global} is optimal for
elementary counting: taking distinct primes $p,q\le X^{1/4-\delta/2}$ and
$k,\ell\in\{1,2\}$, the pairs $(a,b)=(p^k,q^\ell)$ satisfy $a+b\le X$ and
$\operatorname{rad}(a)\operatorname{rad}(b)=pq\le T$, and there are
$\asymp\frac{X^{1/2-\delta}}{(\log X)^2}$ such pairs.
\end{proposition}

\begin{theorem}[Per-$c$ version]\label{thm:perc}
Fix $c$ and $T=c^\beta$ with $0\le\beta\le1/2$. The number of all
representations
$$\#\bigl\{(a,b):a+b=c,\ \operatorname{rad}(a)\operatorname{rad}(b)\le T\bigr\}\ll c^{\beta+\varepsilon(c)}.$$
The coprime subfamily $(a,b)=1$ therefore also satisfies this bound.
\end{theorem}
\begin{proof}
By symmetry assume $b\ge c/2$; this restriction at most doubles the count.
Since $\operatorname{rad}(b)\le\operatorname{rad}(a)\operatorname{rad}(b)\le T$,
the parameter $b$ satisfies $\operatorname{rad}(b)\le T$, and for each such
$b$ the value $a=c-b$ satisfies
$\operatorname{rad}(a)\le T/\operatorname{rad}(b)$. Hence the count is at most
$$\sum_{\substack{c/2\le b\le c\\\operatorname{rad}(b)\le T}}\#\{a\le X:\operatorname{rad}(a)\le T/\operatorname{rad}(b)\}\le X^{\varepsilon(X)}T\sum_{b\le X}\frac{1}{\operatorname{rad}(b)},$$
by Lemma \ref{lem:onevarsum} applied with $X$ in place of the ambient height
(and $T/\operatorname{rad}(b)\le X^{1/2}$). Finally, grouping $b$ by its
radical,
$$\sum_{b\le X}\frac{1}{\operatorname{rad}(b)}\le\sum_{\substack{r\le X\\ r\text{ squarefree}}}\frac{F(X,r)}{r}\le X^{2\varepsilon(X)}$$
by the same partial-summation step as in Theorem \ref{thm:global}. Since
$c\le X$, this is $\le c^{\beta+\varepsilon(c)}$ after enlarging $C_0$.
\end{proof}

\begin{remark}[$(\star')$-failure counting]\label{rem:starfailcount}
The framework identity expresses the main term of the quality as
$\frac{k-1}{k}\log c-\log\operatorname{rad}(ab)+(\text{effective-prime
correction})$, so $(\star')$-failure forces
$\operatorname{rad}(ab)\le c^{(k-1)/k-\eta}$. Theorem \ref{thm:perc} with
$\beta=\frac{k-1}{k}-\eta$ bounds the number of such exceptional $c\le X$ by
$$\ll_kX^{(k-1)/k-\eta+o(1)}=o(X)\qquad\text{for each fixed }k.$$
In the alternative branch $\operatorname{rad}(abc)\ge\operatorname{rad}(ab)$
one has $R\ge\log\operatorname{rad}(ab)+o(\log c)$ and
$q\le\frac{k}{k-1}+o(1)$ trivially, so the nontrivial exceptional sets are
exactly the sparse ones counted above.
\end{remark}

\subsection{Density consequences}
For $\eta\ge0$ let
$$E_\eta(X):=\#\{c\le X:\exists\text{ coprime }a+b=c\text{ with }q(a,b,c)>2+\eta\},\qquad E(X)=E_0(X).$$

\begin{theorem}[Density theorem]\label{thm:densitythm}
With $\varepsilon(X)$ as above:\\
(i) $\#\{c\le X:\operatorname{rad}(c)\le c^{1-\varepsilon}\}\ll X^{1-\varepsilon/2+\varepsilon(X)}=o(X)$
for every fixed $\varepsilon>0$, hence for density $1$ of integers $c$, every
coprime decomposition $a+b=c$ has $q\le1+o(1)$.\\
(ii) $E_\eta(X)\ll X^{\frac12-\frac{\eta}{2(2+\eta)}+\varepsilon(X)}$, in
particular $E(X)\ll X^{1/2+\varepsilon(X)}$.\\
(iii) The superset $\{c\le X:\operatorname{rad}(c)\le\sqrt c\}$ has size
$X^{1/2\pm o(1)}$, and the exponent $1/2$ is sharp.\\
(iv) For random coprime decompositions of a random $c$, $q\to1$ in
probability.
\end{theorem}
\begin{proof}
(i) By dyadic decomposition and Lemma \ref{lem:onevarsum} with
$Y=(2^j)^{1-\varepsilon}\le2^j$,
$$\#\{c\le X:\operatorname{rad}(c)\le c^{1-\varepsilon}\}\le\sum_{j\le\log_2X}S(2^j,2^{j(1-\varepsilon)})\le\sum_{j\le\log_2X}2^{j(1-\varepsilon+\varepsilon(2^j))},$$
and $\varepsilon(2^j)<\varepsilon/2$ for $j\ge j_0(\varepsilon)$, so the sum
is $\ll_\varepsilon X^{1-\varepsilon/2+\varepsilon(X)}$. The second clause of
(i) uses $\operatorname{rad}(abc)\ge\operatorname{rad}(c)$: a decomposition
with $q>1+\varepsilon'$ forces
$\operatorname{rad}(c)\le c^{1-\varepsilon'}$.

(ii) If $q(a,b,c)>2+\eta$ then
$\operatorname{rad}(c)\le\operatorname{rad}(abc)<c^{1/(2+\eta)}$, so
$E_\eta(X)\le S(X,X^{1/(2+\eta)})\le X^{1/(2+\eta)+\varepsilon(X)}$ by Lemma
\ref{lem:onevarsum}, and
$\frac1{2+\eta}=\frac12-\frac{\eta}{2(2+\eta)}$.

(iii) The upper bound is the case $\eta=0$ of (ii) (or the lemma with
$Y=\sqrt X$); the lower bound comes from the $\pi(\sqrt X)$ squares
$p^2\le X$, all of which satisfy
$\operatorname{rad}(p^2)=p=\sqrt{p^2}$.

(iv) The elementary estimate
$$\sum_{n\le X}\log\frac{n}{\operatorname{rad}(n)}=\sum_p\log p\sum_{k\ge2}\Bigl\lfloor\frac{X}{p^k}\Bigr\rfloor\le X\sum_p\frac{\log p}{p(p-1)}=C\,X,\qquad C=0.755366\ldots,$$
shows that $\log\operatorname{rad}(c)\ge\log c-O(1)$ for density $1$ of $c$;
combined with (i) this gives $q\to1$ in probability over random coprime
decompositions.
\end{proof}

\begin{remark}[Lattice-free]\label{rem:latticefree}
Theorems \ref{thm:global}--\ref{thm:densitythm} are completely independent of
the lattice framework: no solution lattice, defect factor, or $(\star')$
enters their statements or proofs.
\end{remark}

\begin{remark}[The pointwise weight $\omega$ is negligible (E26)]\label{rem:omega}
The $\omega=\log(\Omega(ab)/2)$ in Theorem \ref{thm:pointexact} satisfies
$\Omega(ab)\le\log_2(ab)<2\log_2c$, hence $\omega=O(\log\log c)$, whence
$\omega/T\to0$ pointwise since $T=\log c$. A draft sentence claiming a uniform
$\omega/T$ bound on a parameter range is withdrawn; the pointwise statement is
what Theorem \ref{thm:pointexact} uses, and it is correct.
\end{remark}

\begin{remark}[The exceptional-set exponent]\label{rem:theta0}
Define
$$\theta_0:=\inf\{\theta:E(X)\ll X^\theta\}.$$
Theorem \ref{thm:densitythm}(ii) gives $0\le\theta_0\le1/2$; no lower bound
beyond $\theta_0\ge0$ is currently known, and the exact value is open
(Problem G8).
\end{remark}

\subsection{The obstacle}
\begin{definition}[The method class $\mathcal M$]\label{def:methodclass}
An $\mathcal M$-argument is any argument deriving abc-type estimates through
the three layers used in this paper:\\
(M1) \emph{divisibility identity layer}: the Wronskian factorization and the
master inequality $c\le\mu\,\frac{\Omega(ab)}{2}\operatorname{rad}(abc)$;\\
(M2) \emph{geometric control layer}: Minkowski/Hermite-type bounds
$\mu\le\sqrt{\gamma_k}\det(\Lambda)^{1/k}$ with
$\det(\Lambda)=g_c\theta_c/g$;\\
(M3) \emph{transcendence layer}: effective Baker-type lower bounds for
logarithmic forms $|\Lambda_{\mathrm{LL}}|$ with
$|\Lambda_{\mathrm{LL}}|\le\exp(c^{O(1)})$ (cf.\ the bridge, Section 9).
\end{definition}

\begin{theorem}[Obstacle theorem]\label{thm:obstaclethm}
Let $\mathcal M$ be the method class of Definition \ref{def:methodclass}.\\
(a) \emph{Ceiling.} Every $\mathcal M$-argument yields at best
$\limsup q\le k/(k-1)\le2$; for $k=2$ the value $2$ is the optimal reachable
ceiling inside $\mathcal M$ (this is Theorem \ref{thm:ceilingthm}).\\
(b) \emph{Tightness.} Even with the best-case geometry
($\mu=\det^{1/k}(1+o(1))$ achieved) and the maximal information
($\kappa$ upper-controlled), the provable exponent remains
$\eta\le1/k+o(1)$, i.e.\ $q\le2+o(1)$: the ceiling is not an artifact of poor
constant control.\\
(c) \emph{Full abc unreachable in $\mathcal M$.} Full abc would require
$\eta=o(1)$, pressing $\mu$ below $\det^{1/k}\approx c^{1/k}$ by a full power
of $c$; pure geometry of numbers cannot do this. The only remaining channel is
(M3), which by the working hypothesis of Appendix C on the shape of explicit
Baker bounds in the double-fat window is equivalent to a growth-difference
Pillai problem (Theorem \ref{thm:remaining}(D2) below). No other channel is
available.
\end{theorem}

\begin{corollary}\label{cor:sup2}
$\sup\{\alpha:\exists\ \mathcal M\text{-argument proving }\limsup q\le\alpha\}=2$.
The value $2$ is reached under $(\star')$ (Theorem \ref{thm:condasymptotic}),
and is reached unconditionally in the density and probability senses of
Theorem \ref{thm:densitythm}.
\end{corollary}

\subsection{Remaining difficulty}
\begin{theorem}[Remaining-difficulty theorem]\label{thm:remaining}
Let $R_c=\log\operatorname{rad}(c)$ and $T=\log c$.\\
(D1) \emph{The literal claim is false (E27).} ``$R_c\ge(1-o(1))T$ for almost
all triples'' fails already on the closed family $c=2^e$: $R_c=\log2=o(T)$,
and this class was closed unconditionally in Section 8. More sharply, every
$c$ divisible by $4$ (density $1/4$) satisfies
$\operatorname{rad}(c)\le c/2$, so $R_c\le T-\log2$.\\
(D1$'$) \emph{Restricted characterization.} On the $(\star')$-failure set one
has $\operatorname{rad}(c)\ge c^{1-o(1)}$, hence $R_c\ge(1-o(1))T$;
(D1$'$) is correct exactly on this set.\\
(D2) \emph{The growing-difference Pillai core is sparse.} The only remaining
route (channel (M3) of Definition \ref{def:methodclass}) reduces to a
growth-difference Pillai problem, and by Theorem \ref{thm:perc} the relevant
per-$c$ families are $\ll c^{\beta+o(1)}$ with $\beta\le1/2$; in particular
the growing-difference core has counting function $o(X)$.
\end{theorem}
\begin{proof}
(D1) For $c=2^e$, $\operatorname{rad}(c)=2$. For $4\mid c$, write $c=4m$;
then $\operatorname{rad}(c)=2\operatorname{rad}(m)\le2m=c/2$. (D1$'$) Restates
the definition of $(\star')$-failure via
$\operatorname{rad}(c)\ge c^{1-o(1)}$. (D2) This is Theorem \ref{thm:perc}
together with Remark \ref{rem:starfailcount}.
\end{proof}

\begin{remark}
The constants $2$ and $k/(k-1)$ in the ceiling and in the master inequality
can only be improved on thin sets; the barrier is effective, not a density
artifact.
\end{remark}

\subsection{Numerical cross-checks}
All counts below are exhaustive.

\textbf{Table 1}: Exceptional set $E(X)$, powerful set
$C(X)=\{c\le X:\operatorname{rad}(c)\le\sqrt c\}$; exponents
$\log C/\log X$. All entries verified by exhaustive computation.
\begin{center}
\begin{tabular}{lccccccc}
\hline
$X$ & $10^2$ & $10^3$ & $10^4$ & $10^5$ & $10^6$ & $10^7$\\
\hline
$E(X)$ & 0 & 0 & 0 & 0 & 0 & 0\\
$C(X)$ & 17 & 76 & 338 & 1397 & 5781 & 23368\\
$\log C/\log X$ & 0.615 & 0.627 & 0.632 & 0.629 & 0.627 & 0.624\\
\hline
\end{tabular}
\end{center}

Self-check: $1+4374=4375$ with $4374=2\cdot3^7$, $4375=5^4\cdot7$,
$\operatorname{rad}(abc)=210$, $q=\log4375/\log210=1.5679<2$ and
$c-\operatorname{rad}(abc)^2=4375-210^2<0$, confirming that the family is not
detectable by the $q$-threshold in either direction.

\textbf{Table 2}: Coupled-pair counts
$P_\delta(X)=\#\{1\le a<b,\ a+b\le X:\ (a,b)=1,\ \operatorname{rad}(a)\operatorname{rad}(b)\le X^{1/2-\delta}\}$;
parenthesized: $\log P/\log X$. All entries verified by exhaustive
enumeration.
\begin{center}
\begin{tabular}{lccc}
\hline
$\delta$ & $X=10^4$ & $X=10^5$ & $X=10^6$\\
\hline
0.00 & 3772 (0.894) & 28750 (0.892) & 223604 (0.892)\\
0.10 & 1481 (0.793) & 7506 (0.775) & 47604 (0.780)\\
0.25 & 268 (0.607) & 657 (0.564) & 4140 (0.603)\\
\hline
\end{tabular}
\end{center}

A second implementation (independent code path; its counting convention
differs from the one above --- ordered pairs, possibly including non-coprime
pairs --- and is used only as an order-of-magnitude check) reported
$$(X;P)=(4000;10166),\ (16000;37194),\ (64000;154614),$$
consistent with a $\sqrt X(\log X)^{2.5}$ fit to within $\pm35\%$ (best at
$X=16000$: $0.5\%$). Note that the tabulated $P_\delta$ above are coprime,
unordered counts; the conventions are not identical.

\textbf{Honest reading of the numerics.} The $X^{o(1)}$ slack in the theorems
is numerically indistinguishable from a constant shift of exponent: at
$X=10^6$ the quantity $\varepsilon(X)$ is about $0.5$ with the normalization
used above, and the measured deviations from the clean exponents are about
$0.39$. The tables therefore support the qualitative picture (no exceptional
$c$ up to $10^7$; coupled exponents near $1/2$) but cannot certify sharp
exponents at this height.

\textbf{Table 3}: One-variable counts
$S(\alpha)=\#\{n\le10^6:\operatorname{rad}(n)\le10^{6\alpha}\}$; exponents
$\log S/\log X$ versus the withdrawn draft prediction $\rho(1/\alpha)$
(Dickman--de Bruijn, E24).
\begin{center}
\begin{tabular}{lcccc}
\hline
$\alpha$ & 0.10 & 0.25 & 1/3 & 0.50\\
\hline
$\log S/\log X$ & 0.251 & 0.477 & 0.546 & 0.684\\
draft $\rho(1/\alpha)$ & $\approx0$ & 0.005 & 0.049 & 0.307\\
\hline
\end{tabular}
\end{center}

\subsection{Conclusion of the analysis}
Three statements summarize the reach and the limits. First, the reach of the
method ends at $q\le2$ under $(\star')$, and this ceiling is structurally
stable. Second, on the density side the elementary counting of this section
shows that almost all $c$ have every coprime decomposition with
$q\le1+o(1)$. Third, the difficulty beyond $2$ is precisely the effective
sparsity of the growing-difference Pillai core (Theorem
\ref{thm:remaining}(D2)): not a density obstruction, but an information
obstruction.

\section{Numerical illustrations}
All quantities below are computed directly from the definitions; $\mu$ values
are certified (exhaustive verification of minimality), and the examples are
chosen to exercise the falsified barrier, the sandwich, the explicit kernel
vectors, and the pointwise-vs.-limsup distinction.

\begin{example}[Falsification data]\label{ex:falsdata}
$1+242=3^5$: $\mu=27$ (certified: no solution with $|u_2|\le26$ has
$H(u)\neq0$; optimal vector $u^*=(-27,-27)$), $\det=405$, $\kappa=1.3416$,
while the falsified barriers give $0.2908$ (min) and $0.7071$ (min--max).
The sandwich is tight:
$\mu/\dist=27/34.60=0.7804=\kappa/x$ (Remark \ref{rem:Kdiverge}).
\end{example}

\begin{example}[Multi-prime $c$ and the gcd modulus]\label{ex:gcdmodulus}
$31+329=360=2^33^25$: $k=3$, $h=(-329,1457,217)$, $A=(1,47,7)$, $D=10199$,
$g_h=1$, $|a'|_2=\sqrt{2278179}=1509.4\ldots$,
$\operatorname{rad}(abc)=305970$. With the correct modulus
$g_c=\gcd(540,240,72)=12$ one has $\theta_c=1/30$ and $F_c=1$; the erroneous
product modulus would give $g_c=72$ and $\theta_c=0.2$, shrinking every
subsequent margin fourfold.
\end{example}

\begin{example}[Core sample data]\label{ex:coresamp}
Table 4 collects the certified core samples. Note $3+125=2^7$ with $N=7$ (the
index is visible), and $4+27=31$ with $g_h=6=D$ --- the case where
$\log g_h$ cancels $-\log D$ in (7), forcing $g_h$ to be kept explicitly.
\end{example}

\begin{example}[The $k=2$ full-family scan]\label{ex:k2scan}
Scanning $a=p^u$, $b=q^v$ for primes $p,q<23$, exponents $u,v<14$,
$c<10^{15}$ gives 4461 triples:
\begin{center}
\begin{tabular}{p{6.4cm}p{6.6cm}}
\hline
Item & Result\\
\hline
Good branch ($\lambda_1<\Delta$) & 4427 samples, $\kappa\le1.0746$:
4427/4427 pass\\
True bad branch ($\lambda_1=\Delta$) & 34 samples, $\kappa\le1.1547x$:
34/34 pass\\
False bad branches ($x\ge0.9306$ but good) & 2 (including $2^5+13^{13}$)\\
True bad-branch $\kappa/x$ range & $[0.7071,0.9869]$\\
Geometric bound (Theorem \ref{thm:badbranch}) violations & 0/34\\
Candidate $M$ violations & 7/34 (hence $M$ withdrawn, Remark
\ref{rem:candidateM})\\
$\psi$ range & $[70.2^\circ,90.0^\circ]$ (hexagonal $60^\circ$ not attained)\\
\hline
\end{tabular}
\end{center}
\end{example}

\begin{center}
\textbf{Table 4}: Core sample data; $\mu$ certified throughout.\\[2mm]
\begin{tabular}{lcccccccc}
\hline
Triple & $k$ & $\mu$ & $\det$ & $\Delta$ & $N$ & $\kappa$ & $x$ & $\eta$\\
\hline
$1+78124=5^7$ & 2 & 28 & 109375 & 19531.0 & 1 & 0.0847 & 0.017 & 0.2958\\
$1+2186=3^7$ & 2 & 28 & 5103 & 1093.0 & 1 & 0.3920 & 0.065 & 0.4333\\
$1+242=3^5$ & 2 & 27 & 405 & 11.7 & 1 & 1.3416 & 1.719 & 0.6000\\
$3+125=2^7$ & 2 & 6 & 448 & 72.1 & 7 & 0.2835 & 0.294 & 0.3693\\
$1+15=16$ & 2 & 4 & 32 & 5.8 & 1 & 0.7071 & 0.970 & 0.5000\\
$2+3^{10}\cdot109=23^5$ & 3 & 113 & 1399205 & 11022.0 & 5 & 1.0103 & 1.135 & 0.3015\\
$1+16806=7^5$ & 3 & 36 & 12005 & 10099.2 & 1 & 1.5722 & 0.052 & 0.3683\\
$1+161050=11^5$ & 3 & 29 & 14641 & 20624.5 & 1 & 1.1854 & 0.029 & 0.2809\\
$1+2400=7^4$ & 3 & 5 & 343 & 76.6 & 1 & 0.7143 & 0.640 & 0.2068\\
$1+59048=3^{10}$ & 3 & 45 & 98415 & 2027.9 & 1 & 0.9747 & 1.051 & 0.3465\\
$1+2047=2^{11}$ & 2 & 117 & 11264 & 91.9 & 1 & 1.1024 & 1.155 & 0.6246\\
$1+262143=2^{18}$ & 4 & 28 & 262144 & 9821.8 & 1 & 1.2374 & 1.180 & 0.2671\\
$49+531392=531441$ & 4 & 51 & 1062882 & 18450.2 & 2 & 1.5884 & 1.794 & 0.2982\\
$4+27=31$ & 2 & 1 & 1 & 1.4 & 1 & 0.3780 & 0.435 & 0.0000\\
\hline
\end{tabular}
\end{center}

\begin{example}[The power--congruence bridge, three instances]\label{ex:bridge3}
\begin{center}
\begin{tabular}{cccccc}
\hline
$r$ & $a$ & $q$ & $V$ & residue class of $e$ & modulus $dq^{V-1}$\\
\hline
3 & 1 & 11 & 2 & $e\equiv5$ & 55\\
2 & 1 & 7 & 2 & $e\equiv0$ & 21\\
2 & 5 & 3 & 3 & $e\equiv5$ & 18\\
\hline
\end{tabular}
\end{center}
The first row is $1+242=3^5$: $11^2\parallel242$, $e=5$,
$d=\operatorname{ord}_{11}(3)=5$, $dq^{V-1}=5\cdot11=55>5$, so $e=5$ is the
unique solution (Corollary \ref{cor:uniqueness}) --- the extremal point of the
$a=1$ closure is exactly this mechanism.
\end{example}

\section{Open problems}
\begin{problem}[G1]
Close layer III. Prove an effective Pillai-type theorem for
$r^e-q^V=a'$ uniform for $|a'|$ up to the $c^{o(1)}$ threshold, or an
$r$-uniform version of the power--congruence sparsity (Corollary
\ref{cor:sparsification}). This is the unique residual family and the true
location of the difficulty.
\end{problem}
\begin{problem}[G2]
Pointwise vs.\ limsup. Determine whether $q\le\frac{k}{k-1}$ holds pointwise
outside an explicit finite list; the Reyssat triple shows the list is
nonempty.
\end{problem}
\begin{problem}[G3]
Effective constants. Make the $o(1)$ terms in Theorem
\ref{thm:condasymptotic} effective, with explicit dependence on $k$.
\end{problem}
\begin{problem}[G4]
Joint optimization of $g_h$ and $V$. The exact formula (Lemma \ref{lem:gh})
and the bridge bounds suggest a combined inequality that could shrink the
layer-III family further.
\end{problem}
\begin{problem}[G5]
Beyond $c=r^e$. Extend the sieve of Section 9 to multi-prime $c$, where
$F_c$ replaces $E_c$ and the bridge must be built prime-by-prime.
\end{problem}
\begin{problem}[G6]
Break the ceiling. Any input bounding $\eta$ by $o(1)$ would escape Theorem
\ref{thm:ceilingthm}; within the determinant parametrization this requires
seeing $\operatorname{rad}(c)$ rather than $g_c$.
\end{problem}
\begin{problem}[G7]
Scale of computation. The underlying manuscripts report machine checks
(several hundred triples, including the known high-quality examples); a
systematic computation up to a fixed height, with the scripts and data
published, would sharpen every constant in this paper.
\end{problem}
\begin{problem}[G8]
The exceptional-set exponent. Determine
$\theta_0:=\inf\{\theta:E(X)\ll X^\theta\}$ for the exceptional set $E(X)$
of Theorem \ref{thm:densitythm}; presently $0\le\theta_0\le1/2$
(Remark \ref{rem:theta0}), and even $\theta_0=0$ (the empirical evidence of
Table 1) is consistent with the known bounds.
\end{problem}

\section*{Acknowledgements}
This paper is a synthesis of a series of working manuscripts (Volumes I--X
and their revised second editions) developing the present framework; the
computational verifications appended to those manuscripts were used as
independent checks of the examples in Section 12.

\appendix

\section{Revision history and errata}\label{app:errata}
The framework was developed across ten working volumes, a uniform second
revision, and nineteen further revision batches. For the referee's convenience
we record the errata that shaped the present text; every statement in the body
of this paper already incorporates the corrected form. The later batches
contain two genuine falsifications (E12, E13) and several withdrawals; they
are listed with the same candour.

\subsection*{First generation (volumes I--X and the second edition)}
\begin{itemize}
\item[E1] (danger direction). In the duality $x\cdot y=1$, the dangerous
regime is $x\to\infty$, not $x\to0$.
\item[E2] (kernel bound). $\lambda_1(\ker H)$ is a lower bound for the
geometry of $\Lambda_0$, not an upper bound; the correct input is the
determinant formula $\det(\ker H)=|a'|_2$ (Theorem \ref{thm:detformulas}(ii)).
\item[E3] (good-case criterion). For $a=1$, divisibility of the bad case is
necessary but not sufficient; superseded by (5).
\item[E4] (determinant). $\det\Lambda=g_c/g$ exactly; there is no power law in
$D$ (Theorem \ref{thm:detformulas}(i)).
\item[E5, E6] (bounds). The corrected lower bound
$|a'|_2\ge D/(g_hq_{\min})$ (Lemma \ref{lem:aprime}) replaces two earlier,
weaker estimates.
\item[E7] (literature). An earlier draft misattributed the relevant
regular-prime class to Vandiver; the correct attribution is
Chen--Mih\u{a}ilescu. No mathematical statement was affected.
\item[E8, E9] ($\kappa_{\min}$ formula). An omitted factor
$(k-1)^{1/(2k)}$ was restored. Subsequently the entire $\kappa_{\min}$
concept was reclassified: it is the floor of $K$, not a ceiling for $\kappa$
--- see E12.
\item[E10] ($g_c$ is a gcd). The modulus is
$g_c=\gcd_{p\mid c}\bigl(\frac{c}{p}v_p(c)\bigr)$, not the product
$\prod e_pp^{e_p-1}$ (Lemma \ref{lem:modulus}). The two agree on prime
powers, so the subclass $c=r^e$ was unaffected; for multi-prime $c$ the
correction strengthens every margin and yields the two-line bound
$\theta_c=O(\log c)$ (Proposition \ref{prop:theta}).
\item[E11] ($\Omega/2$ and limsup). The factor $\frac{\Omega(ab)}{2}$ in the
master inequality had been dropped in an intermediate version; restoring it
demotes $q\le\frac{k}{k-1}$ from a pointwise to a limsup bound.
\end{itemize}

\subsection*{Second generation (the nineteen later batches)}
\begin{itemize}
\item[E12] (the min--max barrier is false). The claims
$\kappa\le\min(1/y,\mathcal Qy^{1/(k-1)})$ and $\kappa\le\mathcal Q^{(k-1)/k}$
are falsified by $1+242=3^5$ ($\kappa=1.3416>0.7071$) and by the Reyssat
triple ($\kappa=0.975>0.881$); the two branches are orthogonal components and
combine by quadrature, giving the Pythagorean form $K(x)$ of Theorem
\ref{thm:sandwichineq}. The inference ``$\kappa_{\min}=\min_yK(y)$ bounds
$\kappa$ above'' was a logical inversion and is withdrawn.
\item[E13] (pointwise $\eta\le1/k$ is false). Certified counterexamples in
Proposition \ref{prop:pointgradefalse}; the correct pointwise statement is
Theorem \ref{thm:pointexact}, and the graded bound $\eta\le1/k+o(1)$ is
conditional on $(\star')$ (Theorem \ref{thm:condasymptotic}).
\item[E14] (good/bad criterion). The rule $x<1/\sqrt{\gamma_k}$ tacitly
assumed $\Lambda_0$ balanced; it fails for $k\ge3$
($1+16806=7^5$: $x=0.052$ yet $\kappa=1.5722$). Replaced by (5) together with
the anisotropy defect $\ani$.
\item[E15] ($\gamma_3$). An earlier table filled in $\gamma_2$'s value for
$\gamma_3$; corrected to $\gamma_3=2^{1/3}=1.2599$ (threshold $0.891$).
\item[E16] (the index $N$). The formula $N=g_c/\gcd(A\cdot B_H)$ missed $g_c$
itself in the gcd, truncating $N$ to $0$ at the Reyssat triple and silently
skipping $10/18$ samples; corrected to
$N=g_c/\gcd\bigl(g_c,A\cdot(B_H)_1,\ldots,A\cdot(B_H)_{k-1}\bigr)$.
\item[E17] ($k=2$ anisotropy). The kernel generator had not been multiplied by
$N$ ($3+125=2^7$ reported $\ani=7.00$); corrected: $\ani\equiv1$ for $k=2$
(Corollary \ref{cor:anik2}).
\item[E18] (layer spacing). The quantity $\det^{(k-1)/k}/\Delta$ is $x$
itself; the layer spacing is $\dist=\det/\Delta$ (Definition \ref{def:xdistani}).
\item[E19] ($K^*$ and the in-plane bounds). The candidate
$K^*=\sqrt{\dist^2+\lambda_1(\Lambda_0)^2}$ and the chain
$\lambda_1(\Lambda_0)\le(\text{in-plane})\le\rho(\Lambda_0)$ are both false
($1+78124=5^7$: in-plane $7\ll\lambda_1=19531$); withdrawn.
\item[E20] (the candidate $M$).
$M=\max|h_i|/\|h\|_2$ is violated by $7/34$ true bad-branch samples; $M$ is
merely the degenerate value of Theorem \ref{thm:badbranch} at
$\psi=90^\circ$; withdrawn (Remark \ref{rem:candidateM}).
\item[E21] (old Theorem 19.1). $\kappa\le\max(1,8x)$ superseded by Theorem
\ref{thm:k2uniform} (constant $8\to\gamma_2=1.1547$).
\item[E22] ($\sup(\kappa/x)$). The value $\gamma_2$ would require
$\psi=60^\circ$ and $|h_2/h_1|=1/\sqrt3$ (irrational), hence is not attained;
the empirical supremum is $0.9869$. The earlier record $0.9973$ came from the
false bad branch $2^5+13^{13}$ and is corrected (Remark \ref{rem:falsebad}).
\item[E23] (``Reyssat breaks the graded bound''). Withdrawn: Reyssat has
$\eta=0.3015<1/3$; the pointwise excess is explained by $\omega/T$
(Example 6.2). Failure of the Hermite route is not failure of the conclusion.
\end{itemize}

The ceiling (Theorem \ref{thm:ceilingthm}) is untouched by all errata, and
the falsifications E12--E13 lower the unconditional reach of the framework:
the honest main statement is now the pair (Theorem \ref{thm:pointexact}
pointwise, Theorem \ref{thm:condasymptotic} conditional on the unique gap
$(\star')$).

\subsection*{Third generation (the reach-and-limits chapter, Section 11)}
\begin{itemize}
\item[E24] (density is not an exponent). A draft claimed the exceptional-set
exponent $0.3069$ from the Dickman--de Bruijn asymptotic
$\Psi(X,X^{1/u})\sim X\rho(u)$. That asymptotic controls the \emph{density}
of smooth numbers, not a power-saving exceptional-set exponent; since
$\Psi(X,\sqrt X)\asymp X$, the claimed exponent is withdrawn (it was the value
$\rho(2)=1-\log2$). Table 3 records the measured discrepancy. The corrected
density statements are Theorem \ref{thm:densitythm}(i)--(iii).
\item[E25] (coupling is not decoupling). A draft treated the coupled counting
problem for $\operatorname{rad}(a)\operatorname{rad}(b)\le T$ by dropping the
coupling condition and announced the exponent $1$; withdrawn. The correct
coupled exponent is $1/2$ (Theorem \ref{thm:global}, sharp by Proposition
\ref{prop:sharpexp}); the shorter route via
$\operatorname{rad}(abc)\ge\operatorname{rad}(c)$ bypasses coupling entirely
(Subsection 11.1).
\item[E26] ($\omega/T$ sentence). A draft sentence claimed a uniform
$\omega/T$ bound over a parameter range; withdrawn. The pointwise statement
used by Theorem \ref{thm:pointexact} is correct: $\omega=O(\log\log c)$,
hence $\omega/T\to0$ (Remark \ref{rem:omega}).
\item[E27] (literal remaining-difficulty claim (D1)). The draft claim ``$R_c\ge(1-o(1))T$ for almost all triples'' is false: $c=2^e$ has $R_c=\log2$,
and every $c$ divisible by $4$ has $\operatorname{rad}(c)\le c/2$. The
restricted characterization (D1$'$) on the $(\star')$-failure set is correct
(Theorem \ref{thm:remaining}).
\item[E28] (typographical, Rankin display). In the pre-merge version of the
one-variable counting lemma, the Rankin step was displayed without its factor
$Y$:
$\#\{r\le Y:\omega(r)=t\}\le Y\bigl(e^{1+o(1)}\log\log Y/t\bigr)^t$. The
factor is restored in the proof of Lemma \ref{lem:onevarsum}; the statement
and the final bounds were unaffected.
\item[E29] (``all even $c$''). A draft sentence claimed that every even $c$
satisfies $\operatorname{rad}(c)\le2\sqrt c$; this is false already for
$c=2p$ ($\operatorname{rad}(c)=c$). The corrected density statement --- every
$c$ divisible by $4$ satisfies $\operatorname{rad}(c)\le c/2$ --- is what
Theorem \ref{thm:remaining}(D1) uses.

\item[E30] (Layer I boundary overestimate). Lemma \ref{lem:layerI} as
originally stated claimed closure up to $V\le\log c/\log\log c$; the correct
constant is $c^{*}=\frac{(k-1)/k}{A+2}\le\frac{k-1}{k(k+2)}$ ($\le1/8$ for
$k=2$, $A=2$): the threshold's logarithm carries the coefficient $A+2$, not
$1$. The false parenthetical ``failing triples have $a\le c^{o(1)}$'' is
removed (WLOG $a\le b$ suffices and strengthens the bound). The Layer I/III
dividing line moves down to $c^{*}\log c/\log\log c$; the uncovered band is
absorbed into layer III, which the power--congruence bridge governs for all
$V\ge2$.

\end{itemize}
The complete draft disposal table underlying E24--E30 is not reproduced in
this volume; the entries above record the mathematical content of every
disposal decision.

\section{Auxiliary estimates}
\subsection{Hermite constants}
Hermite's constant $\gamma_k$ is the supremum of
$\lambda_1(L)^2/\det(L)^{2/k}$ over rank-$k$ lattices. The values used in this
paper are $\gamma_2=2/\sqrt3=1.1547\ldots$,
$\gamma_3=2^{1/3}=1.2599\ldots$, and the general bound
$\gamma_k\le k$ (in fact
$\gamma_k\le\frac{2}{\pi}\Gamma\bigl(2+\frac{k}{2}\bigr)^{2/k}\sim k/(\pi e)$
by Minkowski's convex-body theorem; see \cite{Cassels}). All occurrences of
$\sqrt{\gamma_k}$ in the good-case bound are therefore $\le\sqrt k$; for
$k\ge3$ the good case is rare (Proposition \ref{prop:structural}), which is
why the $k=2$ solution of Section 5 does not extend.

\subsection{The Wieferich analogy}
Corollary \ref{cor:convergence} shows that the layer-III residual has
convergent natural density $\sum_{V\ge2}q^{-(V-1)}=1/(d(q-1))$. This is
structurally identical to the heuristic for Wieferich primes \cite{Wieferich}:
the condition $r^{q-1}\equiv1\pmod{q^2}$ has ``probability'' $1/q$ per prime,
and $\sum_q1/q^2$ converges, so the dangerous locus is expected to be sparse
--- here, provably sparse at the level of residue classes. The difference is
that our sparsity is per $(r,a,q)$ rather than uniform in $r$; upgrading it
to an $r$-uniform statement is precisely open problem G1 (Section 13).

\subsection{Comparison with Stewart--Yu}
The best unconditional bound toward abc is
$c<\exp\bigl(CR^{1/3}(\log R)^3\bigr)$ \cite{SY1,SY2} obtained from
Diophantine approximation (linear forms in logarithms). Our conditional
consequence (Theorem \ref{thm:condasymptotic}),
$c\lesssim R^{2+o(1)}$ under $(\star')$, would be polynomial; three caveats
are essential: (i) it is a limsup statement --- the $o(1)$ absorbs constants
and finitely many exceptional regimes, and the pointwise version fails already
at $1+242=3^5$ (Proposition \ref{prop:pointgradefalse}); (ii) it is
conditional on the gap $(\star')$, which is equivalent to
$\operatorname{rad}(ab)\ge c^{1/2-o(1)}$ and false for high-exponent
prime-power pairs (Theorem \ref{thm:gap}); (iii) the residual family of
Section 9 (layer III, Pillai-type) is exactly where $(\star')$ fails. The
comparison is therefore between an unconditional sub-exponential bound for all
triples and a polynomial bound whose hypothesis pinpoints the hard core. The
two methods are complementary: linear forms in logarithms see the archimedean
size, the lattice method sees the $p$-adic valuations.

\subsection{A primorial bound for $C_k$}
\begin{lemma}\label{lem:primorial}
The covering-radius constant
$C_k=\bigl(\rho(\Lambda_0)/\Delta^{1/(k-1)}\bigr)^2$ satisfies
$\sqrt{C_k}=\exp(O(k))$, hence $\sqrt{C_k}=c^{o(1)}$ whenever
$k=o(\log c)$.
\end{lemma}
\begin{proof}
$\Lambda_0$ is a sublattice of $\mathbb Z^{k-1}$ in $k$ coordinates whose
covering radius is bounded by $\frac{k-1}{2}$ times the largest coordinate
vector needed to span it; the explicit kernel vectors of Lemma
\ref{lem:kernel} have sup-norm $\le\max_{i,j}\sqrt{p_i^2+p_j^2}$, and
$\max_ip_i\le p_k\le2k\log k$ by the Chebyshev (primorial) bound. Hence
$\rho(\Lambda_0)\le\frac{k}{2}\cdot2k\log k\cdot\Delta^0$, and after
normalizing by $\Delta^{1/(k-1)}\ge1$ one gets, absorbing the $\log k$ into
the constant, $\sqrt{C_k}=\exp(O(k))$.
\end{proof}

\subsection{The $\Omega(ab)/2$ factor}
The factor $\omega_0=\frac{\Omega(ab)}{2}$ enters the master inequality
through the upper bound for the Wronskian (Lemma \ref{lem:Wupper}): the
coefficient vector $h$ satisfies
$\sum_i|h_i|/(D/q_i)=\sum_iv_i=\Omega(ab)$, and each
$|h_i|=(D/q_i)v_i\le(D/2)v_i$. Since
$\Omega(ab)\le\log_2(ab)\le2\log_2c$, one has $\omega_0=O(\log c)$, so the
factor costs only an $o(1)$ in every exponent --- but it is exactly what turns
the pointwise bound $q\le\frac{k}{k-1}$ into a limsup bound, and it cannot be
removed without improving Lemma \ref{lem:Wupper} itself.

\section{Statements not proved in this paper}
The following items are used as hypotheses or discussed heuristically; none is
proved here, and we list them explicitly to prevent misunderstanding.
\begin{itemize}
\item The gap $(\star')$: $|\log x|=O(\log\log c)$. This is the unique
remaining gap, unproved; it is equivalent to
$\operatorname{rad}(ab)\ge c^{1/2-o(1)}$ and is at risk of being false in
high-exponent regimes (Theorem \ref{thm:gap}).
\item The Pillai reduction for $a>1$: $S$-unit finiteness does not apply with
a moving prime set $S$; no closure is claimed (Proposition
\ref{prop:faithful}).
\item Bad-case $\kappa=O(1)$ for $k\ge3$; empirically true (measured
$\kappa\le1.59$ on all certified samples), unproved.
\item The integrality obstruction to $\sup(\kappa/x)=\gamma_2$: an insight,
not a proof.
\item The abc conjecture itself: unproved; by Theorem \ref{thm:ceilingthm} it
is structurally unreachable within the present framework.
\item The working hypothesis on the transcendence layer (used in Theorem
\ref{thm:obstaclethm}(c)): explicit Baker-type lower bounds for logarithmic
forms in the double-fat window (both variables carrying full radical weight)
have the standard shape with no sub-multiplicative gain, so that channel (M3)
of Definition \ref{def:methodclass} is equivalent to a growth-difference
Pillai problem rather than a shortcut past it. This is a hypothesis about the
shape of known transcendence estimates, not a theorem.
\end{itemize}

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\end{thebibliography}

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