==== TABLE OF CONTENTS ====
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//figures/chronology.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//figures/inherited-hull.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//main.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/arithmetic.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/centres.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/consequences.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/direct.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/introduction.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/joins.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/partitions.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/preliminaries.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/prescriptions.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/sparse.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/tables.tex
The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//references/main.bib
===========================

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//figures/chronology.tex ====
\begin{figure}[htbp]
\centering
\newcommand{\chronobox}[1]{\fbox{\parbox{.88\linewidth}{\centering #1}}}
\chronobox{\textbf{Component creation or change}\\
Old tables, accounting classes and stored trains; fixed newborn arrays}
\\[3pt] $\downarrow$ \\[3pt]
\chronobox{History tests and padded cells preserving old classes\\
Section~\ref{sec:partitions}}
\\[3pt] $\downarrow$ \\[3pt]
\chronobox{Cap clipping\\
First-sharing estimates in Sections~\ref{sec:direct} and~\ref{sec:joins}
 bound the lost first mass}
\\[3pt] $\downarrow$ \\[3pt]
\chronobox{\textbf{Next prime in the component}\\
Match plus and minus portions; perform the divisibility update\\
Lemma~\ref{tab:prime-update}}
\\[3pt] $\downarrow$ \\[3pt]
\chronobox{Count arrivals; store the full prescribed train supply\\
Section~\ref{sec:sparse}}
\\[3pt] $\downarrow$ \\[3pt]
\chronobox{Coarsen classes at accepted close pairs\\
Continue to the next prime, or use this stored history at the next change}
\caption{Chronology of the finite construction. A partition uses only
past train intervals; a sparse-step estimate uses the partition already
in place. The comparison estimates bound the cost of the displayed
operations and do not change the arrays.}
\label{fig:chronology}
\end{figure}
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//figures/chronology.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//figures/inherited-hull.tex ====
\begin{figure}[htbp]
\centering
\setlength{\unitlength}{1pt}
\begin{picture}(390,150)
\put(0,136){\makebox(390,12)[l]{\small One historical connecting chain}}
\put(55,106){\line(1,0){230}}
\put(55,106){\circle*{5}}
\put(285,106){\circle*{5}}
\put(170,106){\makebox(0,0){$\times$}}
\put(55,86){\makebox(0,12){$x_-$}}
\put(285,86){\makebox(0,12){$x_+$}}
\put(170,117){\makebox(0,12){\small vanished connector}}
\put(0,66){\makebox(390,12)[l]{\small Preliminary survivors and fixed hull}}
\linethickness{1.4pt}
\put(55,42){\line(1,0){230}}
\linethickness{.4pt}
\put(55,42){\circle*{5}}
\put(285,42){\circle*{5}}
\put(145,27){\line(0,1){30}}
\put(345,27){\line(0,1){30}}
\put(145,7){\makebox(0,12){\small skipped line}}
\put(345,7){\makebox(0,12){\small not blocked here}}
\end{picture}
\caption{An inherited class viewed in a cut coordinate. A connector that
vanished before the preliminary tests no longer appears among the
surviving labels, but its connecting segments remain covered by the
stored train history. The surviving extremes therefore determine a
hull in which sampled raw lines are skipped. The outside candidate is
not blocked by this hull; other hulls are not shown. The preliminary
hull is fixed before the lines are sampled and is not recomputed after
later purges. Only one possible chain and its class are shown;
historical chains may backtrack and different classes' hulls may overlap.}
\label{fig:inherited-hull}
\end{figure}
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//figures/inherited-hull.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//main.tex ====
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=0.9in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{booktabs,array,longtable,enumitem,microtype,graphicx}
\usepackage[colorlinks=true,linkcolor=black,citecolor=black,urlcolor=black,hypertexnames=false]{hyperref}
\usepackage{bookmark}
\hypersetup{
  pdftitle={The Weak Inhomogeneous Duffin-Schaeffer Conjecture},
  pdfauthor={OpenAI},
  pdfsubject={Fixed-shift metric Diophantine approximation}
}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{construction}[theorem]{Construction}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\setlist{topsep=4pt,itemsep=2pt,parsep=0pt}
\emergencystretch=1.5em
\clubpenalty=10000
\widowpenalty=10000
\allowdisplaybreaks[1]
\title{The Weak Inhomogeneous Duffin--Schaeffer Conjecture}
\author{OpenAI}
\date{September 25, 2026}

\begin{document}
\maketitle
\begin{abstract}
We prove the weak inhomogeneous Duffin--Schaeffer conjecture. For every
fixed $\gamma\in\mathbb R$ and finite-valued
$\psi:\mathbb N\to[0,\infty)$, divergence of
$\sum_{q\ge1}\phi(q)\psi(q)/q$ implies
$\|qx-\gamma\|<\psi(q)$ for infinitely many $q$, for almost every $x$.
Here $\phi$ is Euler's totient and $\|\cdot\|$ is distance to the nearest integer.
Numerators need not be reduced, and the shift satisfies no Diophantine
restriction.
\end{abstract}
\tableofcontents
\clearpage

\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/centres}
\input{sections/prescriptions}
\input{sections/tables}
\input{sections/arithmetic}
\input{sections/direct}
\input{sections/sparse}
\input{sections/partitions}
\input{sections/joins}
\input{sections/consequences}

\bibliographystyle{plain}
\bibliography{references/main}
\end{document}
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//main.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/arithmetic.tex ====
\section{Arithmetic estimates for close pairs}
\label{sec:arithmetic}

Proposition~\ref{tab:conditional-reduction} leaves three comparison and
partition inputs to establish. We begin with the arithmetic estimates
needed for adjacent simultaneous births. These estimates concern pairs
of rational grids with fixed phases: residue averaging and a rotation
alternative control close pairs, while a weighted lcm bound describes
the denominator families that can concentrate when the determinant
interval is very short. The elementary determinant coordinates and
residue averages are in Lemma~\ref{ar:cycle}; all the estimates below
are finite and uniform in the phases.

\subsection{Small-prime exclusions}

Consider grids $(z+\alpha_1)/v$ and $(z'+\alpha_2)/w$, with positive integer
denominators $v,w$, fixed real phases, and positive half-widths $r_1,r_2$.
Write $g=(v,w)$ and $\eta=[v,w]\max(r_1,r_2)$. As in
Lemma~\ref{ar:cycle}, distance $O(\max(r_1,r_2))$ restricts the integer
determinant $(w/g)z-(v/g)z'$ to an interval of length $O(\eta)$.

We next record the Selberg upper-bound sieve in the form needed
for close grid pairs~\cite{Selberg1947,Uchiyama1962}. We give its
square-majorant and quadratic-form calculation in full, including
the determinant-cycle error bound. The same calculation will also
be used for prime values of a rotation parameter.

\begin{lemma}[Small-prime exclusions]\label{ar:sieve}
For every fixed $C>0$ there is an absolute sufficiently small $\xi>0$
with the following property. Suppose $\eta=[v,w]\max(r_1,r_2)\ge2$.
At any chosen primes $p\le\eta^\xi$, impose one forbidden numerator
residue on either or both grids, and impose a residue condition only
when $p$ divides that grid's denominator. The number of surviving pairs
at distance at most $C\max(r_1,r_2)$, per unit first-point translate, is
\[
 \ll_C g\eta\prod R_p,
\]
where the product has one factor for each imposed one-row exclusion.
\end{lemma}

\begin{proof}
Let $\mathcal P$ be the chosen primes, and let $g_0(p)$ be the probability
of a violation at $p$ after the complete residue average of
Lemma~\ref{ar:cycle}. It is $1/p$ for a one-row condition and
$2/p-1/p^2$ for a two-row condition. The violations are independent across
primes. Put
\[
 h_0'(p)=\frac{g_0(p)}{1-g_0(p)},\qquad
 Z=\eta^{1/100},\qquad
 \mathcal S=\sum_{\substack{d\le Z\text{ squarefree}\\p\mid d\Rightarrow p\in\mathcal P}}
 h_0'(d),
\]
extending $g_0,h_0'$ multiplicatively on squarefree integers.
All squarefree sums below use this support.

For coefficients $\lambda_1=1$ supported on $j\le Z$, the square
\[
 \left(\sum_j\lambda_j
 \mathbf1_{\{\text{a violation at every }p\mid j\}}\right)^2
\]
majorizes survival: at a surviving pair only $j=1$ contributes, giving
value one, and elsewhere the square is nonnegative. Its normalized mean is
\[
 \sum_{j,k}\lambda_j\lambda_k g_0([j,k])
 =\sum_d\frac1{h_0'(d)}
 \left(\sum_{d\mid j}\lambda_jg_0(j)\right)^2.
\]
Choose the inner sums to equal $\mu(d)h_0'(d)/\mathcal S$, where
$\mu(d)=(-1)^{\omega(d)}$ on the squarefree support. Divisor inversion
gives
\[
 \lambda_jg_0(j)=\frac{\mu(j)}{\mathcal S}
 \sum_{\substack{k\le Z/j\\(k,j)=1}}h_0'(jk).
\]
Thus $\lambda_1=1$, $|\lambda_j|\le1/g_0(j)\le j$, and the mean of the
majorant is $1/\mathcal S$. For a term indexed by $j,k$, the residue
moduli on both grids divide $[j,k]$. Lemma~\ref{ar:cycle}, followed by
the coefficient bounds, gives a total roundoff bounded by
\[
 Cg\sum_{j,k\le Z}|\lambda_j\lambda_k|
       [j,k]^2g_0([j,k])
 \le Cg\left(\sum_{j\le Z}j^3\right)^2
 \ll gZ^8\ll gZ^9.
\]
The inversion above takes place on a divisor-closed support, so its
$j=1$ numerator is exactly $\mathcal S$; every other absolute numerator
is at most $\mathcal S$. This also verifies the coefficient normalization
without extending the truncated support.

To estimate $\mathcal S$, normalize the full product
$\prod_{p\in\mathcal P}(1+h_0'(p))$ to a probability law on squarefree
products. Its expected logarithm is
\[
 \sum_{p\in\mathcal P}g_0(p)\log p
 \ll\xi\log\eta+1.
\]
For small enough $\xi$, Markov's inequality shows that a fixed positive
fraction of this product mass lies below $Z$. Consequently
\[
 \frac1{\mathcal S}\ll
 \prod_{p\in\mathcal P}(1-g_0(p)).
\]
The product on the right is the desired product of one-row factors. It
is bounded below by a fixed inverse power of $\log(2\eta)$, using the
elementary prime reciprocal bound. Hence $Z^9=\eta^{.09}$ is absorbed
by the main bound for large $\eta$. Bounded $\eta$ is absorbed in the
constant.
\end{proof}

\subsection{Rotation in an interval of integers or primes}

The next lemma turns an unexpectedly frequent close return into a
rational approximation. The target phase is arbitrary; no statistical
assumption on the phase is used.

\begin{lemma}[Rotation alternative]\label{ar:rotation}
There are absolute constants $C_1\gg C_2\gg1$ such that, for all
sufficiently large $u$, the following holds. Let $\alpha,y\in\mathbb R$,
$0\le\Delta\le e^{-C_1u}$ and $N\ge e^{C_1u}$. Either $\alpha$ lies
within $e^{C_2u}\Delta/N$ of a rational with reduced denominator at most
$e^{C_2u}$, or
\begin{align*}
 \#\{n\in[N,2N)\cap\mathbb Z:\|n\alpha-y\|\le\Delta\}
 &\ll e^{-u}N,\\
 \#\{p\in[N,2N)\text{ prime}:\|p\alpha-y\|\le\Delta\}
 &\ll e^{-u}N/\log N.
\end{align*}
\end{lemma}

\begin{proof}
Set $\varepsilon=e^{-10u}$. Pigeonholing gives
\[
 \alpha=\frac al+\theta,\qquad
 (a,l)=1,\qquad 1\le l\le\sqrt N,\qquad
 |\theta|\le\frac2{l\sqrt N}.
\]
We take $C_2$ of order $300$ and then $C_1$ sufficiently large.

\paragraph{Large denominators: $l>N^{1/30}$.}
For each integer $k\le N^{1/500}$, count multiples $n=km$ in the
interval, using blocks of $l$ consecutive values of $m$. The rational
parts $akm/l$ form an equally repeated grid of mesh $(k,l)/l\le k/l$.
Within a block the drift is $O(k/\sqrt N)$. Therefore the number with
$\|n\alpha-y\|\le\varepsilon$ is
\[
 \frac{2\varepsilon N}{k}+O(N^{.99}).
\]
Indeed complete-block errors total $O(N/l+\sqrt N)$, and an incomplete
block costs $O(l)$; all are within the displayed error.

For primes, use the square majorant from Lemma~\ref{ar:sieve} with the
bad condition $p\mid n$, primes $p\le N^\iota$, and
$Z=N^{1/5000}$, where $\iota>0$ is a sufficiently small absolute
constant. All tested divisors $[j,k]$ are within $N^{1/500}$. The full
product has mass at least a constant times $\log N$, by
$\prod_{p\le x}R_p^{-1}\ge\sum_{m\le x}1/m$; the same logarithmic
Markov argument retains a fixed fraction below $Z$. Thus the sieve
normalization is $\mathcal S\gg\log N$. The resulting prime count is
\[
 O\left(\frac{\varepsilon N}{\log N}
       +Z^4N^{.99}\right).
\]
Primes in $[N,2N)$ avoid every sieving prime. Both the displayed error
and the integer discrepancy are smaller than the required bounds when
$C_1$ is large.

\paragraph{Intermediate denominators:
$\varepsilon^{-10}<l\le N^{1/30}$.}
Divide $[N,2N)$ into windows of length comparable to $H=N^{1/4}$.
The drift in a window is $O(N^{-1/4})$. Inclusion-exclusion shows that
an interval of residue length $O(\varepsilon l)$ contains at most
\[
 O(\varepsilon\phi(l)+\tau(l))
 \ll\varepsilon^{1/2}\phi(l)
\]
invertible residues. For the last inequality, use the elementary
prime-wise bounds $\tau(l)\ll l^{1/8}$ and
$\phi(l)\gg l^{7/8}$, and then $l>\varepsilon^{-10}$.

Within one invertible progression modulo $l$ in a window, apply the
same square majorant, now omitting primes dividing $l$. Its main count
at a tested squarefree divisor $j$ is $H/(lj)+O(1)$. Removing the sieving
primes dividing $l$ multiplies the full product by a factor at least
$R(l)$, so
\[
 \frac1{\mathcal S}\ll\frac{R(l)^{-1}}{\log N}.
\]
The prime count in that progression is consequently
\[
 O\left(\frac{H}{\phi(l)\log N}+Z^4\right)
 \ll\frac{H}{\phi(l)\log N}.
\]
The last absorption is uniform for $l\le N^{1/30}$. Summing over
the possible invertible residues and the $O(N/H)$ windows gives
$O(\varepsilon^{1/2}N/\log N)$. For integers, direct residue counting
gives $O(\varepsilon^{1/2}N)$. Primes in the original interval are
coprime to $l$ because $l<N$.

\paragraph{Small denominators: $l\le\varepsilon^{-10}$.}
Suppose first that $|\theta|N>\Delta\varepsilon^{-20}$. For a fixed
residue $r\bmod l$, the continuous condition
\[
 \|ar/l+\theta x-y\|\le\Delta
\]
on $[N,2N)$ has $O(|\theta|N+1)$ components. Their total length is
$O(\Delta N+\Delta/|\theta|)$. Enlarging to the union of $H$-windows
meeting them gives total length
\[
 O\left((\Delta+|\theta|H)N+
              \frac{\Delta}{|\theta|}+H\right)
 \ll\varepsilon N.
\]
Here $|\theta|H\ll N^{-1/4}$,
$\Delta/|\theta|<N\varepsilon^{20}$, and $N$ is sufficiently large
relative to $\varepsilon^{-1}$. Each selected window has the
progression bound just proved. Summing first over windows and then over
the $\phi(l)$ invertible residues cancels that denominator and gives
$O(\varepsilon N/\log N)$ primes. Counting all $l$ residues instead
gives $O(\varepsilon N)$ integers.
In detail, if the selected windows for residue $r$ have total length
$L_r\ll\varepsilon N$, their number is $O(L_r/H)$. Integer counts per
window are at most $H/l+1\ll H/l$, because $H\gg l$. Prime counts per
window are $O(H/(\phi(l)\log N))$: the error $Z^4=N^{.0008}$ is
absorbed uniformly since $l\le N^{1/30}$ and
$H/(\phi(l)\log N)\ge N^{1/4-1/30}/\log N$. Thus each residue costs
$O(\varepsilon N/l)$ or $O(\varepsilon N/(\phi(l)\log N))$,
respectively, even though its selected windows depend on $r$.

If the drift inequality fails, then
\[
 l\le e^{100u},\qquad
 |\theta|\le\Delta e^{200u}/N,
\]
which is the rational alternative. These cases exhaust the possibilities,
and $\varepsilon^{1/2}=e^{-5u}$ is more than the stated saving.
When $\Delta=0$ and $\theta\ne0$, the drift components can be isolated
points; their $H$-window enlargement has the same bound. When
$\theta=0$, the small-denominator case is already the exact rational
alternative, and the other two denominator cases use the same rational
grid counts as above.
\end{proof}

\subsection{A weighted least-common-multiple estimate}

We now isolate the arithmetic structure of denominator pairs.
The prime-stripping induction and common-pivot reduction follow the
structural approach of Green and Walker~\cite[Proposition~1.2 and
Section~3]{GW2021} and Hauke-Treuer, Vazquez and
Walker~\cite[Proposition~2.1]{HVW2026}. The lcm cutoff, totient-bounded
vertex weights and moment-controlled edge inflation below require
the stated local argument. For related unweighted small-lcm counts
on dyadic integer sets, see Gou~\cite[Corollary~1.4 and
Theorem~1.5]{Gou2026}. Here we allow arbitrary finite supports and
weighted edges, with the following inflation assumptions. Let
$h_p\ge1$ be bounded, tend to one, and satisfy
\begin{equation}\label{ar:h-moments}
 \prod_p\left(1+\frac{h_p^m-1}{p}\right)<\infty
 \qquad(m>0).
\end{equation}
Put $h(n)=\prod_{p\mid n}h_p$. Uniformity in a family of such functions
means a uniform bound for $\sup_ph_p$ and for every fixed moment product
used below. Constants may depend on these bounds. In particular, for
each $\epsilon>0$, $h(n)\ll_\epsilon n^\epsilon$ uniformly in such a
family.

A weighted bipartite graph consists here of finite sets of positive
integers, vertex weights $f(v),g(w)$, and nonnegative edge weights
$e(v,w)$. The two copies of an integer are distinct vertices. Write
\[
 F=\sum_vf(v),\qquad G=\sum_wg(w),\qquad
 \mathcal E=\sum_{v,w}e(v,w).
\]

\begin{proposition}[Lcm bound]\label{ar:lcm}
Suppose $f(v)\le\phi(v)$, $g(w)\le\phi(w)$, and
\[
 e(v,w)\le f(v)g(w)
 h\bigl(v/(v,w)\bigr)h\bigl(w/(v,w)\bigr).
\]
If every positive edge satisfies $[v,w]\le Z$, then, for each fixed
$1/2<u<1$,
\begin{equation}\label{eq:lcm}
 \mathcal E\ll Z^{2(1-u)}(FG)^u.
\end{equation}
More generally, in a cell fixing the exponents $i_p,j_p$ at a specified
set of primes, its edge total satisfies the same bound with its own
vertex totals and the additional factor
\[
 h(K)K^{-(1-u)},\qquad K=\prod_p p^{|i_p-j_p|}.
\]
\end{proposition}

\begin{proof}
The cases $FG=0$ are immediate.

\paragraph{Fixed-exponent stripping.}
Fixing exponents at a prime and stripping
its powers divides the lcm cutoff by $p^{\max(i,j)}$. Divide the two
vertex weights by $\phi(p^i),\phi(p^j)$ and the edge weight by their
product and by $h_p^{\mathbf1_{i\ne j}}$. The stripped graph meets the
same hypotheses. Rescaling a proposed bound costs at most
\[
 h_p^{\mathbf1_{i\ne j}}
 p^{-2(1-u)\max(i,j)}
 \bigl(\phi(p^i)\phi(p^j)\bigr)^{1-u}
 \le h_p^{\mathbf1_{i\ne j}}p^{-(1-u)|i-j|}.
\]
Iteration proves the asserted cell coefficient once the main bound is
known.

It suffices first to prove the bound for graphs with no prime below a
fixed large cutoff $p_0$. Indeed, after that proof, strip the finitely
many smaller primes and sum their exponent cells. Each corresponding
kernel $h_p^{\mathbf1_{i\ne j}}p^{-(1-u)|i-j|}$ has a finite
convolution bound on the $u$-powers of probability vectors. Their
finite product only changes the constant.

\paragraph{Large-prime induction and common pivots.}
For the large-prime graph use induction on the number of occurring
primes, with a constant to be fixed below. The zero-prime graph is
immediate. If the assertion with this constant fails, normalize the
edge weights to a probability law. At each occurring prime the
induction hypothesis on a fixed exponent cell gives
\begin{equation}\label{eq:prime-cell}
 m_{ij}\le p^{-(1-u)|i-j|}
 h_p^{\mathbf1_{i\ne j}}(\alpha_i\beta_j)^u,
\end{equation}
where $\alpha_i,\beta_j$ are the fractions of the two vertex totals in
those exponent classes. Lemma~\ref{ar:pivot} gives a common exponent
$k_p$ with total off-pivot vertex mass $O(1/p)$. The edge probability
of two off-pivot exponents is $O(p^{-2u})$, and that of a single
deviation of size at least two is
\[
 O\bigl(p^{-2(1-u)-u}\bigr)=O(p^{-(2-u)}).
\]
Both sums converge. Increasing $p_0$ therefore permits us to discard
less than half the total edge weight while excluding both events at
every prime.

Put $N=\prod_pp^{k_p}$. Every remaining edge has the form
\[
 v=Na/b,\qquad w=Nc/d_1,\qquad [v,w]=Nac,
\]
where $b,d_1\mid N$ and $a,b,c,d_1$ are pairwise coprime squarefree
integers. These statements are edgewise: at a prime at most one side
differs from the pivot, and its difference is exactly one.
Since $(a,b)=(c,d_1)=1$, each vertex has a unique reduced ratio to $N$;
this remains true when $a$ or $c$ shares primes with $N$. Moreover,
\[
 (v,w)=N/(bd_1),\qquad
 h\bigl(v/(v,w)\bigr)h\bigl(w/(v,w)\bigr)=h(ab)h(cd_1).
\]

\paragraph{Dyadic capacities and summation.}
For $a\in[A,2A)$, set
\[
 C=\frac Z{NA},\qquad D_1=NA^2,\qquad D_2=NC^2.
\]
There are no such edges unless $C\ge1$. For each fixed $m>0$, the total
upper vertex weight on the first side, inflated by $h(ab)^m$, is
$O_m(D_1)$. To see this, use
\[
 \phi(Na/b)\le a\phi(N/b),\qquad
 \sum_{b\mid N}\frac{\phi(N/b)}N h(b)^m
 \le\prod_{p\mid N}\left(1+\frac{h_p^m-1}{p}\right),
\]
and expand $h(a)^m$ into its nonnegative squarefree divisor sum.
Divisibility by $j$ among $a\le2A$ occurs at most $2A/j$ times.
Equation~\eqref{ar:h-moments} proves the bound. The same calculation
on $c\le C$ gives $O_m(D_2)$.

Holder's inequality now bounds the retained edges in this dyad by
\[
 O_m\left((D_1D_2)^{1/m}
   \bigl(\min(F,D_1)\min(G,D_2)\bigr)^{1-1/m}\right).
\]
Indeed apply Holder under the actual product measure $f(v)g(w)$.
Its mass on the dyad is at most
$\min(F,O(D_1))\min(G,O(D_2))$, and the $m$th inflation moment under
that measure is $O_m(D_1D_2)$ by the totient capacities. These constants
depend only on the prescribed moment bounds, not on $N$ or its prime
support.
Choose $1-1/m>u$. Since $D_1D_2=Z^2$, divide this expression by
$Z^{2(1-u)}(FG)^u$ and put $f_1=F/D_1$, $g_1=G/D_2$. The quotient is
at most a constant times
\[
 \frac{\min(f_1,1)^{1-1/m}}{f_1^u}
 \frac{\min(g_1,1)^{1-1/m}}{g_1^u}
 \le\min(f_1,f_1^{-1})^{c(u)}
\]
for some $c(u)>0$. This is summable over dyadic $A$, since $D_1=NA^2$
changes geometrically. Thus the retained edges satisfy
\eqref{eq:lcm} with a constant independent of the number of primes.
Choose the inductive constant larger than twice this constant. Since
at least half the original edge weight was retained, this contradicts
the supposed failure and closes the induction. Stripping small primes
as above proves the unrestricted bound and its cell form.
\end{proof}

\subsection{Structure when the determinant scale is small}

For the application, vertex weights are multiplied by spatial widths.
Thus, from now on in this subsection, start with the graph in
Proposition~\ref{ar:lcm}, multiply the first and second vertex weights
by $r_1',r_2'>0$, and multiply the edges by $r_1'r_2'$. Reuse
$F,G,\mathcal E$ for these normalized totals, and put
\[
 r_+'=\max(r_1',r_2'),\qquad
 \mathcal R=\frac{\max(r_1',r_2')}{\min(r_1',r_2')}.
\]
At determinant scale $0<\eta<1$, meaning that the edge lcms are
comparable to $\eta/r_+'$, the ordinary bound supplies a factor
$\eta^{2(1-u)}$. Since $2(1-u)<1$, this alone does not give a summable
saving after a pair count costs $1/\eta$. We shall show that failure
of a stronger estimate forces a structured family of denominator pairs.
The following alternative is the structural output of the section.

\begin{proposition}[Small-scale extraction]\label{ar:extraction}
Fix positive constants $c_-,c_+$. Suppose $F,G\le1$ and every positive
edge satisfies
\[
 c_-\eta/r_+'\le[v,w]\le c_+\eta/r_+',\qquad 0<\eta<1.
\]
Given any sufficiently small fixed $\vartheta>0$, there are
$u\in(1/2,1)$, $\chi>0$, and $\eta_0>0$ such that for $\eta<\eta_0$
either
\begin{equation}\label{eq:small-saving}
 \mathcal E\le\eta^{1+\chi}\mathcal R^{-1/10}(FG)^u,
\end{equation}
or $\mathcal R\le\eta^{-\vartheta}$ and a subfamily of edges has the
form
\[
 v=NU_0a/b,\qquad w=NT_0c/d_1,\qquad
 [v,w]=NU_0T_0ac.
\]
Here $N,U_0,T_0$ are fixed positive integers,
\[
 (U_0,T_0)=1,\qquad U_0T_0\le\eta^{-\vartheta},
\]
$b,d_1\mid N$, and $a,b,c,d_1$ are pairwise coprime squarefree integers
coprime to $U_0T_0$. Moreover, $a\in[A,2A)$ and $c\in[C,2C)$ for fixed
dyadic $A,C$. The capacities
\[
 D_1=r_1'NU_0A^2,\qquad D_2=r_2'NT_0C^2
\]
satisfy $\eta^{2+\vartheta}\le D_i\le\eta^{-\vartheta}$, and the
captured normalized edge weight is at least $\eta^\vartheta D_1D_2$.
Every prime in the parameters occurs in the original lists.
\end{proposition}

\begin{proof}
We may suppose $\mathcal E>0$ and that \eqref{eq:small-saving} fails.
We shall first fix exceptional exponent coordinates, then concentrate
the remaining coordinates at common pivots, and finally select one
occupied pair of dyadic ranges. Small losses in powers of $\eta$ will
be chosen within a reserve much smaller than $\vartheta$.

\paragraph{Choosing a cell of maximal quality.}
Take $u=1/2+\epsilon_0<.52$, with $\epsilon_0>0$ small, and set
$s=1-u-.02$. For a cell fixing joint exponents at any subset of the
occurring primes, let $F',G',\mathcal E'$ be its normalized totals and
let $K$ be the product of its fixed exponent ratios. Call primes
$p\le p_0$ small. Among all cells of positive edge weight choose one
maximizing
\[
 \frac{\mathcal E'}{(F'G')^u}
 K^s H^{\#\{\text{small fixed primes}\}},
\]
where $H$ is a sufficiently large fixed number. A maximum exists since
the original supports are finite.

Every occurring small prime is fixed in this cell. Otherwise maximality
would bound the edge probability after further fixing it by
\[
 H^{-1}p^{-s|i-j|}(\alpha_i\beta_j)^u.
\]
Its sum is less than one when $H$ exceeds the bounded optimal constants
in the one-prime estimates
for $p\le p_0$, a contradiction. Here and below $\alpha,\beta$ denote
normalized vertex distributions inside the selected cell; they are not
assumed to be edge marginals.

In normalized variables, Proposition~\ref{ar:lcm} and its cell version
give
\[
 \mathcal E'\ll
 \eta^{2(1-u)}\mathcal R^{-(1-u)}
 h(K)K^{-(1-u)}(F'G')^u.
\]
Compare this with the quality inherited from the unrestricted graph and
the failure of \eqref{eq:small-saving}. It follows that
\begin{equation}\label{ar:fixed-ratio}
 K^{.02}h(K)^{-1}\mathcal R^{1-u-1/10}
 \ll_{p_0}\eta^{-2\epsilon_0-\chi}.
\end{equation}
Since $h(K)\ll_\epsilon K^\epsilon$, both $K$ and $\mathcal R$ are
bounded by arbitrarily small powers of $1/\eta$ on choosing
$\epsilon_0,\chi$ small and subsequently $\eta$ small.

Let $m$ be the normalized edge law in the selected cell. At a remaining
prime, maximality gives the bound of Lemma~\ref{ar:pivot} with price
$p^{-s|i-j|}$, even without an $h_p$ factor. On any joint subset of
remaining primes, the proved cell bound additionally gives
\begin{equation}\label{eq:joint-cell}
 m_{ij}\ll_{p_0}\eta^{-2\epsilon_0-\chi}
 K_{ij}^{-(1-u)}h(K_{ij})(\alpha_i\beta_j)^u.
\end{equation}
In this display $i,j$ are exponent vectors on the queried subset and
$K_{ij}$ is their ratio product. To obtain it, divide the combined-cell
upper bound by the quality lower bound for $\mathcal E'$; the fixed
factor $h(K)K^{-.02}$ and the favorable power of $\mathcal R$ are
bounded.

The weaker prime price $p^{-s|i-j|}$ does not give the summable
double-deviation bounds used in the proof of Proposition~\ref{ar:lcm}.
Instead of discarding all such deviations, we will fix one common
exceptional pattern. The next estimates control both its logarithmic
ratio, which bounds the factors $U_0T_0$, and the negative logarithm
of its probability, which bounds the mass lost when the pattern is fixed.

\paragraph{Controlling exceptional exponent patterns.}
For a remaining prime let $k$ be its pivot, let
$d=(1-\alpha_k)+(1-\beta_k)$, and set
\[
 x=m(i\ne k)+m(j\ne k).
\]
The central majorant implies $x\ge c_ud$, and Lemma~\ref{ar:pivot}
gives
\[
 x\ll p^{-s/(1-u)},\qquad
 B_p':=m(i\ne k,j\ne k)\ll x^{2u}.
\]
Explicitly,
\[
 x\ge1-m_{kk}\ge1-(\alpha_k\beta_k)^u\ge c_ud.
\]
Fix $1/2<u'<u$. Summing the $u'/u$ powers of the double-off majorant
gives, by convolution and Cauchy,
\[
 \sum_{i,j\ne k}m_{ij}^{u'/u}
 \ll\|\alpha_{\ne k}^{u'}\|_2\|\beta_{\ne k}^{u'}\|_2
 \le\bigl((1-\alpha_k)(1-\beta_k)\bigr)^{u'}
 \ll x^{2u'}.
\]
Since
$-t\log t\ll t^{u'/u}$ for $0\le t\le1$, its entropy is also
$O(x^{2u'})$. Inserting $|i-j|$ in the geometric kernel proves that
its contribution to $\mathbb E_m|i-j|$ is $O(x^{2u'})$.

Choose $\epsilon'>0$ so small that $(1-\epsilon')2u'>1$. For primes
with $x\le p^{-1+\epsilon'}$, all costs $x^{2u'}\log p$ are summable.
For the other primes we claim
\begin{align}
 \operatorname{Ent}_m(i)+\operatorname{Ent}_m(j)
 &\le x\log(1/x)+O(x)+O(x^{2u'}),\label{ar:marginal-entropy}\\
 \mathbb E_m|i-j|&\ge x-2B_p'.\label{ar:ratio-expectation}
\end{align}
Entropy means $-\sum_aP(a)\log P(a)$, with the summand zero at a zero
atom. For the first inequality, single deviations at distance at most
four occupy a bounded number of offsets, of total mass at most $x$.
Single deviations at distance at least five have entropy
$O(x^{2u'})$: indeed the sum of the $.8$ powers of their geometric
majorants is $O(p^{-4s})$, and $4s>2u'(1-\epsilon')$ for the choices
above. Double-off entropy was already bounded. Passing from entries
to marginal atoms only decreases these entropy upper bounds. The second
inequality follows because each single deviation contributes at least
one, while double deviations are counted twice in $x$.
For clarity, the leading entropy term is not counted twice. Set
$x_1=m(i\ne k)$ and $x_2=m(j\ne k)$. The bounded-offset part of the
two marginals has entropy at most
\[
 x_1\log(1/x_1)+x_2\log(1/x_2)+O(x)
 \le x\log(1/x)+O(x),\qquad x_1+x_2=x.
\]
The unbounded-offset contributions are the double-off and far-single
errors already estimated. In the far-single estimate, the explicit
geometric sum is
$O(\sum_{a\ge5}p^{-.8sa})=O(p^{-4s})$.

\paragraph{An entropy budget across primes.}
We now combine the coordinate bounds without assuming independence
among primes. Apply \eqref{eq:joint-cell} on the joint subset with
$x>p^{-1+\epsilon'}$, take logarithms, and average under $m$.
The cross-entropy against $\alpha$ is at least the entropy of the first
edge marginal, and similarly for $\beta$. The joint edge entropy is at
most the sum of its two vector marginal entropies, which in turn are
at most their coordinate entropy sums. Thus
\begin{align*}
 (1-u)\mathbb E_m\log K_{ij}
 &\le (1-u)\sum_p
   \bigl(\operatorname{Ent}_m(i_p)+\operatorname{Ent}_m(j_p)\bigr)\\
 &\quad+\mathbb E_m\log h(K_{ij})
       +O_{p_0}(1)+(2\epsilon_0+\chi)|\log\eta|.
\end{align*}
Indeed, if $\mu,\nu$ are the two vector edge marginals, the entropy
expression obtained by averaging the logarithm is
\[
 \operatorname{Ent}(m)-u\operatorname{CE}(\mu,\alpha)
                         -u\operatorname{CE}(\nu,\beta)
 \le(1-u)\bigl(\operatorname{Ent}(\mu)+\operatorname{Ent}(\nu)\bigr).
\]
Here $\operatorname{CE}(\mu,\alpha)=-\sum_i\mu_i\log\alpha_i$.
If a vertex-law atom is zero, the majorant forces the corresponding
edge marginal atom to be zero, so all cross-entropy terms are taken on
their positive support. No equality of vertex and edge laws, or
independence among prime coordinates, is used.
Here $\log(1/x)\le(1-\epsilon')\log p$. Equations
\eqref{ar:marginal-entropy}--\eqref{ar:ratio-expectation} therefore
leave a positive multiple of $\epsilon'\sum_px\log p$ on the left.
For sufficiently large $p_0$, the $h$ term, $B_p'\log p$, and the
$O(x)$ terms are absorbed into that margin. Consequently
\begin{equation}\label{ar:entropy-budget}
 \sum_{p:\,x>p^{-1+\epsilon'}}x\log p
 \ll_{u,\epsilon'}|\log\eta|.
\end{equation}
The proportional constant is independent of a further increase of
$p_0$, after decreasing $\eta_0$ to absorb its additive constant.
The choices occur in this order: first $\epsilon_0,\chi$, hence $u$;
then $u'\in(1/2,u)$; then $\epsilon'$ with $(1-\epsilon')2u'>1$;
then $p_0$; and finally $\eta_0$. The constant depending on
$u,\epsilon'$ may be large, but is fixed before $p_0$ is increased.

\paragraph{Fixing the exceptional pattern.}
The purpose of \eqref{ar:entropy-budget} is to control the entire
exceptional pattern, not only each prime separately. Record $(i_p,j_p)$
when both differ from the pivot, and record a blank otherwise. Its
coordinate entropy, including the blank atom, is $O(x^{2u'})$; its
expected log-ratio is $O(x^{2u'}\log p)$. On the small-$x$ primes these
costs have an arbitrarily small tail. On the remaining primes they are
an arbitrarily small multiple of the budget
\eqref{ar:entropy-budget}, by increasing $p_0$ and using
$x\ll p^{-s/(1-u)}$. Subadditivity therefore makes both the full pattern
entropy and its expected log-ratio as small a multiple of
$|\log\eta|$ as desired.

Single deviations of distance at least two have total probability
bounded by the tail of
\[
 \sum_p O\bigl(p^{-2s-us/(1-u)}\bigr).
\]
Its exponent is greater than one for $u$ sufficiently close to $1/2$.
Discard these deviations. Markov's inequality for the double-pattern
log-ratio and its information $-\log\Pr(\text{pattern})$ leaves a
fixed positive probability on patterns of log-ratio at most
$\rho|\log\eta|$ and information at most $\rho|\log\eta|$, where
$\rho>0$ can be as small as desired. There are at most $\eta^{-\rho}$
such patterns. One pattern therefore retains at least a fixed constant
times $\eta^\rho$ of the original edge probability after the discards.

Fix it. At its primes and the previously fixed primes put the minimum
of the two exponents into $N$, and put the two surplus parts into
$U_0,T_0$. At every other prime put the pivot into $N$. The remaining
single-step deviations give $a,b,c,d_1$. They are squarefree, pairwise
coprime, and supported away from $U_0T_0$; also $b,d_1\mid N$.
This proves the required algebraic form. Equations
\eqref{ar:fixed-ratio} and the pattern log-ratio bound make
$\mathcal R,U_0T_0$ at most arbitrarily small powers of $1/\eta$.

\paragraph{Selecting an occupied dyadic box.}
We make the remaining small-power bookkeeping explicit in meaning.
In this paragraph $o_*(1)$ denotes an exponent that can be bounded in
absolute value by any prescribed positive reserve: first choose
$\epsilon_0,\chi$ small, then $p_0$ large for the pattern losses, and
finally $\eta$ sufficiently small. It is not a limit with changing
block parameters.

The inherited quality gives
\[
 \mathcal E'\ge\eta^{1+o_*(1)}(F'G')^u.
\]
Proposition~\ref{ar:lcm}, now with exponent $.8$, also gives
$\mathcal E'\ll\eta^{.4}(F'G')^{.8}$. Thus
$F'G'\ge\eta^{2+o_*(1)}$, and the structured family just obtained
has weight at least $\eta^{2+o_*(1)}$.
More explicitly, replacing the first small exponent by $\rho>0$ gives
\[
 F'G'\gg\eta^{(.6+\rho)/(.8-u)},\qquad
 \frac{.6+\rho}{.8-u}
 =2+\frac{\rho+2\epsilon_0}{.3-\epsilon_0}.
\]
This makes the arbitrary proximity of the resulting exponent to two
transparent; the retained-pattern probability only adds another
prescribed small exponent.

Sort it into dyads for $a,c$. The lcm constraint gives
\[
 AC\asymp\frac{\eta}{r_+'NU_0T_0},\qquad
 D_1D_2\asymp\frac{\eta^2}{\mathcal R U_0T_0}.
\]
For each $A$ only boundedly many $C$ occur. The moment-capacity proof in
Proposition~\ref{ar:lcm} applies with $D_1,D_2$, with additional
inflation at most $h(U_0T_0)\le\eta^{-o_*(1)}$. For example, if
$D_1\ge\eta^{-\vartheta/4}$, Holder and $F,G\le1$ give, for fixed
large $m$,
\[
 O\left(\eta^{-o_*(1)}D_1^{1/m}D_2\right)
 =O\left(\eta^{2-o_*(1)}D_1^{-(1-1/m)}\right).
\]
Summing this geometric tail, and the analogous tail for $D_2$, costs
$O(\eta^{2+c\vartheta-o_*(1)})$ for an absolute $c>0$.
It is negligible compared with the retained mass.

In all remaining boxes $D_i\le\eta^{-\vartheta/4}$, while the product
relation implies $D_i\ge\eta^{2+\vartheta}$ when the reserves are
sufficiently small. There are only $O(1+|\log\eta|)$ such boxes because
$D_1$ varies geometrically and there are boundedly many $C$ per $A$.
One box has weight at least $\eta^{2+o_*(1)}$. Since
$D_1D_2\ll\eta^2$, its weight exceeds
$\eta^\vartheta D_1D_2$ for sufficiently small reserves and $\eta$.
The upper capacity bound and the asserted bounds on
$\mathcal R,U_0T_0$ are weaker than the bounds already obtained.
Every parameter used only exponents of primes occurring in the lists.
\end{proof}

\subsection{Auxiliary sampling in an extracted box}

In later applications a vertex also carries phase or mask variables.
The extraction concerns the aggregate graph weights, but the following
upper-bound interpretation permits those variables to be retained.
Its probabilities belong to an unrestricted comparison law, not to the
normalized law of the extracted edges. The corollary transfers small
comparison probability into small weighted edge mass.

\begin{corollary}[Negligible subsets of a saturated box]
\label{ar:box-sampling}
Use a box given by Proposition~\ref{ar:extraction}. Suppose that each
vertex weight is obtained by integrating a nonnegative density bounded
by its normalized totient weight against a probability law of auxiliary
variables; restrictions may reduce this density. Suppose also that the
edge majorant holds before integration against the independent product
of the two auxiliary laws.

Sample $a,c$ uniformly from their dyadic integer intervals, sample
$b,d_1$ independently with law $\pi_N$, and then sample the auxiliary
variables with their unrestricted laws. Invalid or nonintegral parameter
choices may be assigned zero weight. An event of sampling probability
$O(\eta^{10\vartheta})$ carries
$o(\eta^\vartheta D_1D_2)$ of the normalized edge weight. The same is
true of edges whose first vertex belongs to a set of total normalized
vertex weight $O(\eta^{10\vartheta}D_1)$.
The auxiliary law on a side may be conditional on that side's vertex
parameters. This statement is only an event-mass bound: an application
that varies an integer factor while holding an auxiliary phase fixed
must separately verify the required independence or a uniform
conditional sampling law.
\end{corollary}

\begin{proof}
On valid choices,
\[
 \phi(NU_0a/b)\le U_0a\phi(N/b),\qquad
 \phi(NT_0c/d_1)\le T_0c\phi(N/d_1).
\]
Thus the uninflated normalized vertex measures are bounded by
$O(D_1)$ and $O(D_2)$ times their respective sampling laws. Their
product has total mass $O(D_1D_2)$. The second moment of the edge
inflation is $O(h(U_0T_0)^2D_1D_2)$ by the capacity calculation in
Proposition~\ref{ar:lcm}. Cauchy's inequality therefore bounds the
weight on an event of probability $O(\eta^{10\vartheta})$ by
\[
 O\left(\eta^{5\vartheta}h(U_0T_0)D_1D_2\right)
 =O(\eta^{4\vartheta}D_1D_2),
\]
using $h(U_0T_0)\ll\eta^{-\vartheta}$. This is negligible compared
with the saturation threshold.

For the vertex restriction, each structured vertex has a unique
reduced defect ratio $a/b$ or $c/d_1$ relative to its fixed base.
Its uninflated product mass is
$O(\eta^{10\vartheta}D_1D_2)$ by the restricted first mass and the
unrestricted second capacity. The same second-moment argument applies.
These are upper bounds on the extracted edges; they do not assert
independence of the actual surviving edge law.
\end{proof}

The rotation alternative controls close returns unless a phase is
well approximated by a rational. The extraction and sampling results
supply a family on which such returns can be tested by varying integer
factors. Section~\ref{sec:direct} applies these estimates to the birth
comparisons in Proposition~\ref{tab:first-join-input}.
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/arithmetic.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/centres.tex ====
\section{Centres, density, and prime heights}
\label{sec:centres}

Fix one of the finite blocks from Section~\ref{sec:preliminaries}, of
mass $w\le W\le 2w\le1$. All rows in this section belong to that block,
and their positive spatial widths are exact dyadic numbers. We assign
rows of one width to integer multiples of their denominators. The
assignment will have two complementary properties: an assigned-row
density bound at small spatial scale, and a quantitative separation between families
assigned at widely different scales. These properties supply the
summable centre kernels used in the later pair counts.

\subsection{Popular multiples and the density bound}

Fix $H_0=.001$. A small constant $k_0>0$ will be chosen after the other
fixed parameters. Write
\[
 W_r=\sum_{\substack{v\text{ in the block}\\r(v)=r}}\phi(v)r.
\]

\begin{definition}[Popular multiples and original hosts]
\label{cen:popular}
An integer $D\ge1$ is \emph{popular at width $r$} if $Dr\le k_0$ and
\[
 \sum_{\substack{v\mid D\\r(v)=r}}\frac{\phi(v)}D
 \ge (Dr)^{H_0}.
\]
A row $v$ is \emph{raw} if it divides no popular multiple at its width.
Otherwise its \emph{original host} is the largest popular multiple at
that width divisible by $v$.
\end{definition}

The largest host exists: at a fixed positive width, all popular
multiples are at most $k_0/r$.

\begin{lemma}[Density above original hosts]
\label{cen:density}
Let $\mathcal V$ be a subfamily of rows of a common width $r$, all
dividing an integer $L$. Suppose that $L$ is at least the original host
of every nonraw row in $\mathcal V$; raw rows are allowed as well. Define
\[
 k_L=Lr,\qquad
 \delta_L=\sum_{v\in\mathcal V}\frac{\phi(v)}L,\qquad
 m_L=k_L\delta_L.
\]
Then
\begin{equation}
 \delta_L\ll_{k_0} k_L^{H_0}.
 \label{eq:centre-density}
\end{equation}
Moreover $\delta_L\le1$.
\end{lemma}

\begin{proof}
The last assertion follows from $\sum_{v\mid L}\phi(v)=L$.
If the family is nonempty, $2L$ cannot be popular at width $r$: any
assigned nonraw row would then have an original host at least $2L$,
whereas any assigned raw row would no longer be raw. If $2Lr\le k_0$,
failure of popularity gives
\[
 \frac12\delta_L
 \le\sum_{\substack{v\mid2L\\r(v)=r}}\frac{\phi(v)}{2L}
 <(2Lr)^{H_0}.
\]
If $2Lr>k_0$, use $\delta_L\le1$. The empty family is immediate.
\end{proof}

A \emph{centre family} consists of an integer $L$, a common width $r$,
and an assigned subfamily satisfying the hypotheses of
Lemma~\ref{cen:density}. For two such families, indexed by $i,j$, use the notation
\[
 L=L_i,\quad M=L_j,\quad G=(L,M),\quad U=L/G,\quad T=M/G,
\]
\[
 \begin{gathered}
 r_{\min}=\min(r_i,r_j),\qquad r_{\max}=\max(r_i,r_j),\\
 \Theta_{ij}=G r_{\min},\qquad K=[L,M]r_{\max}.
 \end{gathered}
\]
In particular $(U,T)=1$. The identity
\begin{equation}
 \Theta_{ij}
 =(m_Lm_M)^{1/2}(UT)^{-1/2}
 \left(\frac{r_{\min}}{r_{\max}}\right)^{1/2}
 (\delta_L\delta_M)^{-1/2}
 \label{eq:centre-kernel}
\end{equation}
will convert arithmetic separation into a summable mass kernel. We
apply it only to nonempty families, so its inverse densities are defined.

\begin{lemma}[Centre-kernel summation]
\label{cen:kernel}
Consider centre families satisfying Lemma~\ref{cen:density}, with at
most one family for each pair $(L,r)$ and total mass at most $W\le1$.
For every fixed $\alpha>1/2$ and sufficiently small fixed
$\epsilon>0$, there is $c>0$ such that
\[
 \sum_{i,j}\Theta_{ij}(\delta_{L_i}\delta_{L_j})^\alpha
 (U_{ij}T_{ij})^\epsilon
 \ll_{k_0,\alpha,\epsilon} W^{1+c}.
\]
\end{lemma}

\begin{proof}
Choose $u>1/2$ sufficiently close to $1/2$ that
$H_0(\alpha-u)>u-1/2$, and then $\epsilon<u-1/2$.
For a single centre,
\[
 m_L^{1/2-u}\delta_L^{\alpha-1/2}
 =k_L^{1/2-u}\delta_L^{\alpha-u}\ll_{k_0}1.
\]
Indeed use \eqref{eq:centre-density} when $k_L\le1$, and use
$\delta_L\le1$ when $k_L>1$. Hence the summand is at most
\[
 C_{k_0}(m_Lm_M)^u(UT)^{-1/2+\epsilon}
 \left(\frac{r_{\min}}{r_{\max}}\right)^{1/2}.
\]
At each pair of exact widths Lemma~\ref{ar:tensor} applies, since
$1/2-\epsilon>1-u$. If $M_r$ is the total mass at width $r$, the
remaining sum is bounded by
\[
 C\sum_{r,s}(M_rM_s)^u
 \left(\frac{\min(r,s)}{\max(r,s)}\right)^{1/2}
 \ll\sum_r M_r^{2u}\le W^{2u}.
\]
The middle inequality is convolution with a summable geometric kernel
on dyadic width indices. Take $c=2u-1$.
\end{proof}

\begin{lemma}[Raw rows suffice when they carry mass]
\label{cen:raw}
If raw rows have mass at least $w/4$ on arbitrarily late blocks, the
positive-measure avoided set fixed in Section~\ref{sec:preliminaries}
cannot exist.
\end{lemma}

\begin{proof}
For a raw row use centre $L=v$ and give every point of its grid weight
$R(v)=\phi(v)/v$. Then $\delta_L=R(v)$ and $m_L=\phi(v)r(v)$.
The determinant-cycle count of Lemma~\ref{ar:cycle} at distance
$O(r_{\max})$, multiplied by
$r_{\min}$ and the two row weights, is
\[
 O\bigl(m_Lm_M+\Theta_{ij}\delta_L\delta_M\bigr).
\]
Lemma~\ref{cen:density} applies to these raw singleton families.
Lemma~\ref{cen:kernel} with $\alpha=1$ therefore gives a uniform
second-moment bound for their weighted interval trains. Their means
are bounded below by a constant times $w$, and their integrals on each
fixed interval approach that interval's length times the mean: these
are uniformly weighted grids with $v\to\infty$ down the block tail.
Approximation of the avoided set by a finite union of intervals, using
the second-moment bound for the approximation error, gives a positive
integral on that set, a contradiction.
\end{proof}

We may consequently pass to arbitrarily late blocks on which the
nonraw rows carry a fixed positive fraction of the block mass. The
following adjustment of their hosts is made before any numerator
weights are imposed.

\subsection{Adjusted hosts and the density purge}

The original hosts give the density bound, but they do not yet separate
families assigned at different scales. We therefore allow a row to use a
popular centre from another width, charging for the required arithmetic
enlargement and width change. Minimizing this cost will ensure that a
centre at a different scale cannot offer a substantially cheaper fit;
Lemma~\ref{cen:exterior} makes this separation precise.

For a nonraw row $v$ of width $r$, minimize over all original popular
centres $(D,r_0)$ in the block the cost
\begin{equation}
 \begin{gathered}
 b+10a+10c_++(-c)_+,\\
 b=-\log(Dr_0),\quad
 a=\log\frac{v}{(v,D)},\quad c=\log\frac r{r_0},
 \end{gathered}
 \label{cen:assignment-cost}
\end{equation}
where $t_+=\max(t,0)$. Choose a minimizer, resolving ties
arbitrarily, and put $L=[D,v]=De^a$. Consolidate assignments with the
same triple
\[
 (L,r,B^*),\qquad B^*=\lfloor\text{minimum cost}\rfloor,
\]
into one \emph{centre object}; call the integer $B^*$ its \emph{shell}.
Write $i$ for such an object, and let
$B_i$ be the dyadic power of two with $B_i\le B_i^*<2B_i$.
We call $B_i$ its band. All costs, and hence all these bands, are large
when $k_0$ is small.

For each object define $k_{L_i},\delta_{L_i},m_{L_i}$ as in
Lemma~\ref{cen:density}, using precisely its assigned rows, and put
\[
 \mathcal F_i=\{L_i/v:v\text{ is assigned to }i\}.
\]
Thus $\pi_{L_i}(\mathcal F_i)=\delta_{L_i}$. Different objects
partition the assigned rows, including their masses. These symbols
will always refer to the original assignments, even when some masks
or numerator points are subsequently removed.

\begin{lemma}[Scale and number of adjusted centres]
\label{cen:adjusted}
Every adjusted centre satisfies Lemma~\ref{cen:density} and
\[
 |\log k_{L_i}|\le B_i^*+1,\qquad
 \log k_{L_i}\le .1(B_i^*+1).
\]
The number of original popular centres at a fixed width $r$ and with
$-\log(Dr)\in[b,b+1)$ is
\[
 \ll W_r e^{(1+3H_0)b}.
\]
The total number of adjusted centre objects with $B_i^*\le t$,
including all widths, is $\ll e^{2t}$. Deleting objects with
$\delta_{L_i}<e^{-.32B_i^*}$ loses $o(W)$ as $k_0\to0$.
\end{lemma}

\begin{proof}
For the chosen candidate in \eqref{cen:assignment-cost},
$\log k_L=-b+a+c$. Since $b>0$ and $a\ge0$, the cost bounds
$|\log k_L|$, and it is at least $10\log k_L$ when the latter is
positive. The cost is no larger than the cost obtained by using the
row's original host at its own width, namely minus the log of that
host's spatial scale. Since the cost also bounds $-\log k_L$, the
adjusted $L$ is at least that original host. This proves the first
assertions.

Let $n$ be the number of original popular centres in the indicated
unit $b$-range. Sum the popularity inequality after multiplying by
$Dr$, and apply Cauchy--Schwarz to row incidences. Using
$\sum_{v\mid g}\phi(v)=g$ gives
\[
 \bigl(n e^{-(1+H_0)b}\bigr)^2
 \ll W_r\sum_{D,D'}(D,D')r.
\]
Here $D,D'\asymp e^{-b}/r$, so Lemma~\ref{ar:tensor} gives, for $u>1/2$
arbitrarily close to $1/2$,
\[
 \sum_{D,D'}(D,D')r\ll e^{-b}n^{2u}.
\]
Consequently
\[
 n\ll W_r^{1/(2-2u)}
 \exp\left(\frac{1+2H_0}{2-2u}b\right)
 \ll W_r e^{(1+3H_0)b},
\]
after choosing $u$ sufficiently close to $1/2$ and using $W_r\le1$.

At a fixed source width $r_0$, sort the assignments into unit boxes
of $(b,a,c)$. For each source $D$, there are $O(e^a)$ possibilities
for the integer $v/(v,D)$ and hence for $L$. A unit $c$-range contains
only boundedly many dyadic widths, and a unit $(b,a,c)$-box gives
boundedly many integer shells. Thus the number of resulting objects
is
\[
 \ll W_{r_0}\exp((1+3H_0)b+a).
\]
Multiplication by $k_L=e^{-b+a+c}$ leaves the exponent
\[
 3H_0b+2a+c\le .21B_i^*+O(1).
\]
An object in the stated deletion contributes an additional factor
$e^{-.32B_i^*}$. This bounds its entire mass, not just the mass of
rows using the particular source witness. If an object has several
source witnesses, cover it by all their boxes, or choose any one:
the upper count can overcount objects but cannot omit one. The
coefficient inequality above holds also for $c<0$, since then
$c\le .21(-c)_+$. At shell $t$, the nonnegative cost bounds
$b,a,|c|$ by $O(t+1)$, so there are $O((t+1)^3)$ unit boxes.
All costs are at least $-\log k_0$. The deleted mass is therefore
at most
\[
 C\sum_{r_0}W_{r_0}
 \sum_{t\ge-\log k_0-O(1)}(t+1)^3e^{-.11t}=o(W).
\]
This also shows uniformity when a source-width mass is arbitrarily
small. Similarly, summing the object-count
bound over cost at most $t$, and using $\sum_{r_0}W_{r_0}\le1$,
gives $O(e^{2t})$.
\end{proof}

Retain the remaining objects. In addition to their upper density
bound, their original assigned families satisfy the following lower
bounds, before any subsequent mask or numerator removals:
\begin{equation}
 \delta_L\ge e^{-.32B_i^*},\qquad
 m_L\gg e^{-1.32B_i^*}.
 \label{eq:centre-lower}
\end{equation}

\subsection{Exterior separation and summation with height decay}

\begin{lemma}[Separation of different bands]
\label{cen:exterior}
If $B_i/B_j$ is sufficiently small, then every row $v$ assigned to
$j$ satisfies at least one of
\begin{equation}
 \frac{v}{(v,L_i)}\frac{r_j}{r_{\min}}>e^{.015B_j^*},
 \qquad
 \log(r_i/r_j)>.7B_j^*.
 \label{eq:exterior-gain}
\end{equation}
\end{lemma}

\begin{proof}
Choose a source $(D_i,r_{0i})$ that produced an assignment in object
$i$. Then $D_i\mid L_i$, and its source parameters in
\eqref{cen:assignment-cost} are $O(B_i)$. In particular
\[
 \log\frac{v}{(v,D_i)}
 \le \log\frac{v}{(v,L_i)}+\log\frac{L_i}{D_i}.
\]
Using $(D_i,r_{0i})$ as a candidate for $v$, and comparing the two
widths through $r_i$, bounds its minimum cost by
\[
 O(B_i)+10\log\frac{v}{(v,L_i)}
 +10\log\frac{r_j}{r_{\min}}+\log^+\frac{r_i}{r_j}.
\]
If both conclusions failed, this would be at most
$O(B_i)+.85B_j^*<B_j^*$, contrary to the definition of the shell.
\end{proof}

We record two ways in which \eqref{eq:centre-kernel} will absorb
errors. This also explains why the shell label in an adjusted centre
does not invalidate the earlier tensor summations.

\begin{lemma}[Width and height gains]
\label{cen:decay-kernels}
Under \eqref{eq:centre-lower}, the following kernels are summable
with a positive power of total mass beyond linear.
First, restrict to pairs with sufficiently small $B_i/B_j$ and
$\log(r_i/r_j)>.65B_j^*$. For sufficiently small absolute
$c_s,\epsilon>0$, the kernel is
\[
 \Theta_{ij}e^{c_sB_j}(UT)^\epsilon.
\]
Second, fix $c>0$ and $C>0$. Sum over dyadic $Y$ and pairs with
$B_i,B_j\le Y/A_0$. For $A_0$ sufficiently large the kernel is
\[
 \Theta_{ij}e^{-cY}\tau(UT)^C.
\]
The bounds permit the shell multiplicities of the adjusted objects
and the displayed height summation.
\end{lemma}

\begin{proof}
In \eqref{eq:centre-kernel}, changing the mass exponent from $1/2$
to $u>1/2$, including the inverse densities there, costs at most
\[
 C\exp\left(
  \bigl(.16+1.32(u-1/2)\bigr)(B_i^*+B_j^*)\right).
\]
For the first kernel, take $0<u-1/2\le .005$ and use band
separation so that $B_i^*/B_j^*\le .01$. The displayed conversion
cost is then at most $Ce^{.18B_j^*}$. Reserve the width power
$1/8$ for summation. Its remaining power $3/8$, together with
$\log(r_i/r_j)>.65B_j^*$, supplies
$e^{-.24375B_j^*}$. For $c_s\le .01$ this leaves at least
$e^{-.05B_j^*}$. Choose $\epsilon<u-1/2$, so the remaining
$UT$-price still meets the tensor inequality.

At a fixed pair of integer shells $s,t$, there is at most one
object at a given $(L,r)$. Tensor summation followed by convolution
in the widths, now with their power $1/8$, bounds this pair of
shells by $CW_s^uW_t^u\le CW^{2u}$, where $W_s,W_t$ are their
total masses. Thus the first kernel is bounded by
\[
 CW^{2u}\sum_{t\ge1}(t+1)e^{-.05t}\ll W^{2u}.
\]
There is no summation of an undamped uniform shell bound.

For the second kernel take $u=.75$, and write
$C_u=.16+1.32(u-1/2)$. Its shell-conversion cost is at most
$C\exp(4C_uY/A_0)$, since $B_i^*,B_j^*<2Y/A_0$.
Enlarge $A_0$ so that $4C_u/A_0<c/2$. The divisor bound
$\tau(UT)^C\ll_\epsilon(UT)^\epsilon$ uses less than the
available margin $u-1/2$ in the tensor inequality. At each pair
of shells, tensor and width summation again cost at most $CW^{2u}$.
There are $O((Y+1)^2)$ eligible shell pairs, so the total is at most
\[
 CW^{2u}\sum_{Y\text{ dyadic}}(Y+1)^2e^{-cY/2}
 \ll W^{2u}.
\]
Both exponents $2u$ are strictly greater than one.
\end{proof}

\subsection{Prime heights and starting scales}
\label{cen:height-starts}

Use dyadic heights $Y=1,2,4,\ldots$ for the prime bins
\[
 \mathcal P(Y)=\{p:Y\le\log\log p<2Y\}.
\]
Primes below these bins are always part of the low initial data.
A bin is \emph{dense on $L_i$} if
\[
 \sum_{\substack{p\mid L_i\\p\in\mathcal P(Y)}}\frac1p
 \ge\kappa Y.
\]
Let $Y_d(i)$ be the last dense height, or zero if there is none.
The two types and their starts are defined by
\[
 \begin{array}{c|c|c}
 \text{type}&\text{condition}&Y_0(i)\\ \hline
 \text{high}&Y_d(i)>D_0B_i&2Y_d(i)\\
 \text{medium}&Y_d(i)\le D_0B_i&\max(A_0B_i,2Y_d(i)).
 \end{array}
\]
All bins from the start onward are sparse. Colour the integers
$\log_2B_i$ modulo a fixed large integer. Retain a single type and
colour carrying the greatest mass. This keeps $\gg W$ mass, with a
fixed implied constant, and makes distinct retained bands as widely
separated as required below.

\subsection{Order of the fixed parameters}
\label{cen:hierarchy}

We give the order here because all subsequent selections must be
uniform down the block tail. Some constants name thresholds that
are used only later: their quantitative roles are specified below,
and their constructions are given at the indicated sections. Every
choice is fixed independently of the block. This is the dependency
order of the constants, not the chronology of the adaptive construction;
the list also serves as a reference for the later estimates.

\begin{enumerate}
\item \emph{Prime powers and geometry.}
Choose first a sufficiently large absolute $C_{\rm mask}$, bounding
the excess prime powers in a deficit mask. Choose a sufficiently
large absolute $\beta$, which will join centres at height $Y$ when
$\log UT\le\beta Y$. It must absorb the absolute prime-product
costs $e^{CY}$ in the distant comparisons of
Sections~\ref{sec:partitions} and~\ref{sec:joins}.
Set $\sigma_g=.1$. Choose a small dyadic $\sigma>0$ so that absolute
harmonic costs below height $\sigma Y$ are at most $e^{Y/100}$.
Choose
\[
 0<h_*\ll\sigma h_H,\qquad 0<h_H\ll\sigma,
 \qquad 3h_*<h_a\ll\sigma_g,
\]
with arbitrarily large separations. Here $h_*$ controls the logarithmic
sizes of comparison data within a component, $h_H$ controls the
rational-model heights, and $h_a$ bounds the number of offset grids
used to record very close pairs. Finally in this stage take the
dyadic $A_0$ sufficiently large. In particular, path costs
$e^{CY/A_0}$ must be at most $e^{h_*Y/10}$, and the height-decay
applications of Lemma~\ref{cen:decay-kernels} must hold.

\item \emph{Occupancy and band separation.}
Choose $s'>0$ sufficiently small absolutely for the later powers
of divisor counts used with \eqref{eq:exterior-gain}, then take
$s_{\rm low}>0$ sufficiently smaller for the mask-moment deletion.
Choose dyadic $\ell>0$ and $\kappa>0$ such that $\ell<\sigma$ and
both $\ell$ and $\kappa A_0$ are arbitrarily small relative to
$s_{\rm low}$. Next take dyadic $D_0$ large compared with
$1/\kappa$ and $A_0$. The colour separation is now taken large
compared with $D_0/\ell$, $D_0/(\sigma A_0)$, and the requirements
of \eqref{eq:exterior-gain} and the centre counts.

\item \emph{Rational models and localization.}
The direct rotation-switch budget $D'$ in Section~\ref{sec:direct}
is large compared with $\beta,D_0$ and the costs depending on
them. Choose the large model exponent $E$ after these resulting
linear exponential bounds, then $Q_0\gg E$ for the deep direct
comparisons, and then a still larger $J>4E$ for the tag cutoff
$e^{JB_i}$. Choose $E_g$ sufficiently large after these constants
for the birth spatial scale $r_i e^{-E_gB_i}$, and then choose
$E_2$ much larger than $E_g$ for the raw partition spacing.

For the low rational models and their soft tests, fix
$0<\ell_{\rm sh}<1/25$, then $0<h_0\ll\ell_{\rm sh}$,
$0<h_s<.01$, and finally $\epsilon_s>0$ sufficiently small relative
to $h_s$ and the measure $\mu$ of the avoided set.

The remaining choices in this stage control inverse estimates and
rotation switches. Small powers in
the inverse and period estimates can be chosen arbitrarily small
after the fixed comparisons. Choose the harmonic-mask cutoff
$d_0>0$ after the corresponding period cutoffs. In particular the
powers used for period costs and $\tau(UT)$ in mixed comparisons
are small relative to $\beta A_0$. Coefficients of harmonic
exponents to be absorbed by $s_{\rm low}$, $\kappa A_0$, or
$\beta$ are absolute; they do not arise from these tiny-power
constants.

The sparse-bin switch budget $D''$ in Section~\ref{sec:sparse}
need only be sufficiently large absolutely; a factor
$\tau(UT)^C$ is carried separately there. For the last direct
subcases choose a small signature exponent $h'>0$, a small switch
threshold $t_1$, a large fixed $J_1$, and a small rotation-budget
exponent $\epsilon'$ in their required order. Concretely these
choices permit $h'\ll h_0$, $t_1\ll h'/Q_0$,
$J_1\gg J/\ell_{\rm sh}$, and $C_1\epsilon'\ll t_1$, with
$C_1$ the absolute constant in Lemma~\ref{ar:rotation}. All are fixed before taking the
arbitrarily small saturation exponent $\vartheta$ and the
associated parameters of \eqref{eq:small-saving}.

\item \emph{Final cutoffs.}
Take $k_0$ sufficiently small after all these choices. Every band
and every bin at a fixed positive linear scale in a band is then
as large as needed; in particular $\ell B_i$ and $\sigma Y$ are
bin heights. A sufficiently large fixed $\lambda_0$ for the
small-scale arithmetic comparisons and the hole deletion may
still depend on $k_0$. Then choose $w$ sufficiently small after
the constants depending on these data. Finally choose a fixed
cap $N_0$ as large as necessary. The comparison bounds used for
clipping in Section~\ref{sec:tables} will be independent of this
cap.
\end{enumerate}

The objects retained so far have a common type and colour, disjoint
row assignments, the upper density bound \eqref{eq:centre-density},
and the lower bounds \eqref{eq:centre-lower}. We next specify their
initial numerator weights, without using any later adaptive table.
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/centres.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/consequences.tex ====
\section{The Hausdorff-measure consequence}
\label{sec:consequences}

We now apply the proved Lebesgue-measure theorem to the transformed
radii in the mass transference principle.

\begin{proof}[Proof of Corollary~\ref{cor:hausdorff}]
Put $r_q=\psi(q)/q$. If $r_q\not\to0$, then along an unbounded
sequence of denominators $r_q\ge\delta>0$, so $\psi(q)>1/2$
eventually. Since $\|qx-\gamma\|\le1/2$ for every $x$, in this
case $W_\gamma(\psi)=\mathbb R$.

Suppose now that $r_q\to0$, omit the zero radii, and apply
Theorem~\ref{thm:main} to $\Psi(q)=qf(r_q)$, taking $\Psi(q)=0$
when $r_q=0$. The resulting limsup of open intervals with centres
$c_{a,q}=(a+\gamma)/q$, $a\in\mathbb Z$, and radii $f(r_q)$ has
full Lebesgue measure in every interval.
Retain only centres with $|c_{a,q}|\le q$ and enumerate the resulting
finite rows in increasing $q$. The original and transformed radii
both tend to zero. On every fixed interval $[-M,M]$, every sufficiently
late original or transformed interval meeting it has
$|c_{a,q}|\le M+1<q$, so this restriction changes neither limsup there.

Apply the mass transference principle~\cite[Theorem~2]{BV2006} in
dimension one to the closed balls with these centres and radii $r_q$;
their transformed radii are exactly $f(r_q)$. It gives full
$\mathcal H^f$-measure for the original closed-ball limsup in every
ball. Its difference from the open-ball limsup is contained in the
countable union of all ball endpoints, a set of zero $\mathcal H^f$
measure. This proves the assertion for every bounded interval,
regardless of its endpoint convention; empty and singleton intervals
also have zero measure. The radius convention of~\cite{BV2006}
gives the same assertion as the usual diameter convention: if
$\ell=\lim_{r\downarrow0}f(r)/r<\infty$, both measures are zero or
fixed multiples of length, while if $\ell=\infty$, the
diameter-based measure dominates the radius-based measure, which is
already infinite on the intersection for every nondegenerate interval.
\end{proof}
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/consequences.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/direct.tex ====
\section{Direct comparisons at simultaneous births}\label{sec:direct}

We prove the part of Proposition~\ref{tab:first-join-input} concerning
adjacent simultaneous births. The main distinction is between an exact coincidence
after replacing phases by recognized rationals, which is charged to first
mass, and a nonexact coincidence, for which the arithmetic extraction
estimate supplies a summable saving. An additional birth discard will
exclude the rational concentrations arising in one of the extracted cases.
It depends only on the original lists of good assigned rows.

\subsection{The comparison statement and determinant coordinates}

A \emph{direct pair} is an ordered pair of centres inserted at the same
height $Y=Y_0$ and satisfying $\log UT\le\beta Y$, with self-pairs allowed.
Simultaneous newborn pairs that first share a component but satisfy
$\log UT>\beta Y$ are treated in Section~\ref{sec:joins}.
On medium type, the band separation implies $B_i=B_j$; moreover, the two
centres have the same product $P$ and the same tagged/tail division, since
$J\gg\beta D_0$. On high type they have the same last dense height $Y_d$
and the same $P$. Indeed, any graph neighbour of a high centre at its
insertion has the same last dense bin, so its entire component is newborn.
Group centres by this common $P$, start, and tagged/tail division, including
the band on medium type but not splitting bands on high type. Each row
belongs to one group, and direct pairs do not cross groups. Write
\[
 X_0=B_i=B_j\quad\hbox{on medium type},
 \qquad X_0=Y_d\quad\hbox{on high type}.
\]
The definitions of the starts and the centre-scale bounds give
\begin{equation}\label{eq:direct-size}
 \log\bigl(2+UT+K+r_{\max}/r_{\min}\bigr)\ll X_0.
\end{equation}

For a centre $i$ and an outer mask $a\mid P_I$, let $\nu_{i,a}$ be its
labelled birth point measure with the raw static weights, including the
soft tests and the additional discard constructed below, but without the
coefficient $\pi_{P_I}(a)$ and before any adaptive update or cap clipping.
Thus
\[
 \lambda_i=\sum_{a\mid P_I}\pi_{P_I}(a)\nu_{i,a}.
\]
Point-pair sums below are per unit length: the first location is counted in
one unit interval and the second ranges over the periodic lift. Products
of anchor-averaged weights are evaluated with independent anchor choices,
also for self-pairs and coinciding labels.

\begin{proposition}[Direct birth bounds]\label{dir:birth-comparisons}
Use the retained centres of one type and colour, the good masks, and the
birth prescriptions of Sections~\ref{sec:centres}
and~\ref{sec:prescriptions}, with the hierarchy fixed in
Section~\ref{cen:hierarchy}. Let the finite block have mass $W\le1$.
For every fixed $C>0$, the sum over all
direct pairs of
\[
 r_{\min}\iint_{|x-y|\le C r_{\max}}
       d\lambda_i(x)\,d\lambda_j(y)
\]
is $O(1)$. The same-width, one-law sum
\[
 \sum_{\substack{(i,j)\text{ direct}\\r_i=r_j=r}}
 r\sum_{a\mid P_I}\pi_{P_I}(a)
 \iint_{|x-y|\le C r e^{-E_gX_0}}
       d\nu_{i,a}(x)\,d\nu_{j,a}(y)
\]
is $O(W)+O_{k_0}(W^{1+c})$ for some fixed $c>0$. Each ordered centre pair
is counted once, at its common birth. Constants may depend on the fixed
hierarchy, cutoffs, and distance constant, but not on the finite block or
the slot cap $N_0$.

The additional discard used here is a static rule on medium groups,
determined by their good assigned-row lists before any running-table
deletions. Given any fixed $\delta>0$, its total first-mass loss is at most
$\delta W$ after choosing the fixed threshold $\lambda_0$ sufficiently
large. It uses only squarefree low numerator moduli. The comparison bounds
remain valid after any further nonincreasing thinning of birth weights.
\end{proposition}

We first establish the estimates common to both sums. Write
$L_i=S_iP$, $L_j=S_jP$, and denote the low row denominators by $V,W'$.
Set
\[
 s=\frac{W'}{(V,W')},\qquad t=\frac{V}{(V,W')}.
\]
Let $m_i,m_j$ be the tagged masks and let
$\alpha_i=a_i\gamma$, $\alpha_j=a_j\gamma$ be the outer phases. In the
narrow sum $a_i=a_j$ with one law; in the broad sum they have independent
laws. The processed masks and the integers used by the tag prescriptions
are
\[
 d_i=(S_i/V)m_i,\qquad d_j=(S_j/W')m_j,
 \qquad \mathcal L_i=\lfloor d_i\alpha_i\rfloor,
 \quad \mathcal L_j=\lfloor d_j\alpha_j\rfloor.
\]
Define
\[
 f=(m_i,m_j),\qquad f_0=(Td_i,Ud_j),
 \qquad \sigma_0=\frac{Td_i}{f_0}=\frac{s m_i}{f},
 \quad \tau_0=\frac{Ud_j}{f_0}=\frac{t m_j}{f}.
\]
The identities follow from the disjoint prime supports of low and tagged
parts; in particular, $\sigma_0$ and $\tau_0$ are coprime. A pair within
the tested distance, henceforth a \emph{cohit}, satisfies
\begin{equation}\label{eq:direct-determinant}
 \begin{split}
 |\sigma_0(z_i+\alpha_i)-\tau_0(z_j+\alpha_j)|&\le\Delta,\\
 \Delta&=\frac{CK\varepsilon}{f_0}
        =\frac{C[V,W']P r_{\max}\varepsilon}{f},\\
 \frac{\Delta}{\sigma_0}
        &=C\psi_i\varepsilon\frac{r_{\max}}{r_i},
 \qquad \psi_i=\frac{VP r_i}{m_i},
 \end{split}
\end{equation}
where $\varepsilon=1$ for broad comparisons and
$\varepsilon=e^{-E_gX_0}$ for narrow ones. Constants in distance tests are
absorbed into $C$. For each determinant integer the bare count, multiplied
by $r_{\min}$, is at most
\[
 \Theta_{ij}\frac{f_0}{d_i d_j}.
\]
Indeed, writing $G=(L_i,L_j)$, the projected denominators
$L_i/d_i,L_j/d_j$ have gcd
$Gf_0/(d_i d_j)$, and
\[
 Td_i=[S_i,S_j]m_i/V,\qquad
 Ud_j=[S_i,S_j]m_j/W',\qquad
 f_0=[S_i,S_j]f/[V,W'].
\]
Put $\eta_1=[V,W']P r_{\max}$. Call a comparison \emph{ultra} if it is
narrow, or if it is broad and either $f>1$ or
$-\log\eta_1\ge Q_0X_0$. The tag-prime lower bounds, the choices of
$J,E_g$, and \eqref{eq:direct-size} imply
\[
 \Delta\ll e^{-.9Q_0X_0}
 \qquad\text{in every ultra comparison}.
\]

Every narrow comparison is therefore ultra. Lemmas~\ref{dir:ultra-switch}
and~\ref{dir:tag-fold} reduce this regime to rational models and low
coordinates; Lemmas~\ref{dir:exact-equalities} and~\ref{dir:ultra-count}
then complete its count. The remaining broad comparisons have $f=1$
and are controlled by the row kernel of Lemma~\ref{dir:row-kernel}.
Before extracting a small-scale box, we fix the static holes in
Construction~\ref{dir:hole-rule}. The final three lemmas exclude high,
shallow medium, and deeper medium boxes, respectively.

\subsection{Ultra switches and tag folding}

The first lemma extracts rational phases at ultra tolerance. A switch is
only a change of summation parameters for an upper bound; the changed
factor is not required to produce another assigned row or another cohit.

\begin{lemma}[Ultra switching]\label{dir:ultra-switch}
For ultra comparisons, discard an exceptional count of total size
$O_{k_0}(W^{1+c})$, for some $c>0$. On the remaining pairs, apply all
divisor switches
\[
 h\mid(d_i,\sigma_0),\qquad
 e^{C_1D'X_0}\lesssim h\lesssim e^{3C_1D'X_0},
\]
all prime switches above the lower cutoff, and the corresponding switches
on side $j$. If any switch is available, there are coupled rationals
$A_i,A_j$ of denominators at most $e^{O(D')X_0}$ such that
\[
 |A_i-\sigma_0\alpha_i|+|A_j-\tau_0\alpha_j|
       \ll\Delta e^{O(D')X_0},
 \qquad A_i-A_j=-(\sigma_0z_i-\tau_0z_j).
\]
They are unique within these bounds. For every switched divisor $h$ on
side $i$, the reduced denominator of $A_i/h$ is at most
$e^{C_3D'X_0}$, where $C_3\ll C_1$ is absolute; the analogous assertion
holds on side $j$. In particular, switched large primes divide the
appropriate reduced numerator. There is a fixed $E'\ll E$ such that
$A_i/\sigma_0$ and $A_j/\tau_0$ have denominators at most $e^{E'X_0}$.
If no switch is available, both $\sigma_0,\tau_0\le e^{E'X_0}$.
\end{lemma}

\begin{proof}
Every $h\mid(d_i,\sigma_0)$ can be removed from $d_i$ without changing
$f_0$: at each affected prime the first argument of the gcd has an exponent
surplus at least the removed exponent. Fix the centres, widths, outer
masks, $d_i/h$, and $d_j$. On a dyad for $h$, equation
\eqref{eq:direct-determinant} is a rotation condition with $h$ occurring
linearly, fixed slope $\sigma_0\alpha_i/h$, and unchanged tolerance.
Lemma~\ref{ar:rotation}, with budget $D'X_0$, therefore gives the
rational alternative except on $O(e^{-D'X_0})$ of that dyad's normalized
count. Its approximation error for the slope is at most
$\Delta e^{C_3D'X_0}/h$.

Stripping $h$ contributes exactly $1/h$ to the bare ratio
$f_0/(d_i d_j)$. We may drop all other assignment and eligibility
restrictions when summing the exception. At equal centre exponents the
local unrestricted sum is at most
\[
 \sum_{a,b\ge0}p^{-\max(a,b)}=1+O(1/p).
\]
At unequal exponents, say $\nu_p(U)=g>0$, it is at most
\[
 \sum_{a,b\ge0}p^{\min(a,g+b)-a-b}
       \le (1+g)^C(1+C/p)
\]
with absolute $C$. The processed-prime harmonic sum is $O(Y_0)$, so the
stripped-mask sum costs at most
$\exp(CY_0)\tau(UT)^C$. Outer masks still integrate as probabilities,
including the one-law coupling in the narrow case. There are at most
$\exp(O(Y_0))$ dyads in all the indicated tests. Choosing $D'$ large
against these costs bounds the exceptional count for a centre pair by
\[
 \Theta_{ij}e^{-D'X_0/2}.
\]
The centre kernel \eqref{eq:centre-kernel}, lower bounds
\eqref{eq:centre-lower}, and \eqref{eq:direct-size} sum these errors to
$O_{k_0}(W^{1+c})$, leaving exponential decay for shell multiplicities.

For precision at the switch endpoints, round the lower cutoff up to a
dyad start, use divisor dyads from there through its square, and use prime
dyads from there onwards. On every nonexceptional pair every available
test gives its rational alternative. Multiplication by $h$ gives $A_i$;
the determinant relation defines
$A_j=A_i+\sigma_0z_i-\tau_0z_j$. Candidates obtained from any two
switches have denominators at most $e^{C_3D'X_0}$ and differ by at most
$O(\Delta e^{C_3D'X_0})$. The latter is smaller than the reciprocal
product of their denominators, by $Q_0\gg D'$. Thus the candidates
coincide. This proves the exact displayed relation and simultaneous
uniqueness, and $A_i/h$ has the stated small denominator for every
switched $h$.

It remains to divide by the whole signature. We have
$\sigma_0/(d_i,\sigma_0)\mid T$: at a prime, writing
$a=\nu_p(d_i)$, $b=\nu_p(T)$, $c=\nu_p(f_0)$, the exponent in this
quotient is $\max(b-c,0)\le b$. The good-mask condition bounds the
prime-power excess on $(d_i,\sigma_0)$ by $e^{C_{\rm mask}B_i}$, and every
sufficiently large prime there divides the reduced numerator of $A_i$.
Among the smaller distinct primes not dividing that numerator, the product
is below the lower switch cutoff, up to a fixed factor. Otherwise, a
first-crossing subproduct would lie between the cutoff and its square and
would be a tested divisor. Dividing by this product, which is coprime to
the numerator, would give a denominator larger than the allowed
$e^{C_3D'X_0}$. Explicitly, if $A_i=n/q$ is reduced and $(n,h)=1$,
then the reduced denominator of $A_i/h$ is $qh$, even if $h$ and $q$
share primes. The remaining factor $T$, excess powers, and subcutoff
product are all singly exponential in $X_0$ by
\eqref{eq:direct-size}; this proves the denominator bound with a fixed
$E'\ll E$. If no tested divisor occurs, the same factor decomposition
bounds the signatures themselves. The argument on side $j$ is identical.
\end{proof}

We next average the high numerator coordinates. The exact lift used in
this averaging is
\[
 D=T(d_i z_i+\mathcal L_i)-U(d_j z_j+\mathcal L_j).
\]
It is invariant under joint unit translation, and every cohit satisfies
\[
 |D|\ll T+U+K.
\]
The bound follows by replacing the two floors with their arguments and
using \eqref{eq:direct-determinant}.

\begin{lemma}[Tag arrangements and folding]\label{dir:tag-fold}
Apart from the exceptions of Lemma~\ref{dir:ultra-switch}, tag masks in
each bin may be restricted to the following arrangements: they are
identical, or they are disjoint. There are only boundedly many bin
arrangements. If $f=1$, all bins are disjoint.

For a fixed arrangement, let $P_{\rm eq}$ and $P_{\rm dis}$ be the products
of full prime powers in its identical and disjoint bins. Under
\[
 y=(P/f)x,\qquad \rho_i=(P/f)r_i,\qquad m_i'=m_i/f,
\]
the folded low arrays have the form
\[
 y_i=(Z_i+m_i'\alpha_i)/V,
 \qquad Z_i\in\mathbb Z,
\]
with transformed low tests. An upper bound for their tag-averaged
coefficient is, up to a fixed constant, the prescribed outer-law
coefficient times
\[
 f\pi_{P_{\rm eq}}(f)
       \pi_{P_{\rm dis}}(m_i')\pi_{P_{\rm dis}}(m_j').
\]
The low arrays may be counted without the original integrality
$m_i'\mid Z_i$, which has already been included in this coefficient.
One may still retain the predicate that a surviving actual cohit exists
at the original masks and anchor choices.
\end{lemma}

\begin{proof}
First suppose a bin's masks are disjoint, and fix both anchor choices.
At a prime full on both sides, two active forbidden residues in their
common high-coordinate translation can coincide only if
$D\equiv TF_i-UF_j\pmod p$. The size bound for $D$, the reconstruction
height and denominator bounds, and the bin's prime size turn this
congruence into equality over $\mathbb Q$. At an anchor of side $i$,
absent from side $j$'s mask, reduction then gives
$d_j z_j+\mathcal L_j\equiv F_j$, contradicting the hard test. When one
side has no key, its scalar $R_p$ already supplies the required product
factor and no coincidence argument is needed.

If $f>1$, the pair is ultra and every mismatched tag prime is among the
switches of Lemma~\ref{dir:ultra-switch}. If a mismatch exists, define
\[
 F_i^{\rm pred}=\mathcal L_i-\frac{S_i}{[V,W']}f A_i,
 \qquad
 F_j^{\rm pred}=\mathcal L_j-\frac{S_j}{[V,W']}f A_j.
\]
The denominator of the low-centre ratio on side $i$ divides $T$, and the
one on side $j$ divides $U$. More importantly,
\[
 \frac{S_i}{[V,W']}f\sigma_0\alpha_i=d_i\alpha_i.
\]
Thus the potentially large phase cancels against the floor, giving size
$O(1+e^{O(D')X_0}L_i r_{\max})$ and the corresponding small denominator.
These values fit the tag reconstruction cutoffs. At a shared omission the
multiplier contains the anchor prime; at a mismatched omission numerator
divisibility comes from its switch. Every anchor in a nonempty bin
therefore reconstructs the predicted tag. The exact determinant relation
gives
\[
 T F_i^{\rm pred}-U F_j^{\rm pred}=D.
\]
In a bin where both masks are nonempty, reduction at a mismatch prime
violates the full side's forbidden-residue test. Such bins consequently
have identical masks; bins with an empty mask are disjoint. If there was
no mismatch, all masks were already identical. The number of tag bins is
bounded by the fixed hierarchy, so the number of arrangements is bounded.

For the fold, the original integrality is $Z_i=m_i'z_i$. Count first over
a $y$-interval of integer length $P/f$, using the generic arrays and
averaging joint shifts
\[
 Z_i\longmapsto Z_i+Vk,\qquad
 Z_j\longmapsto Z_j+W'k,\qquad k\bmod P/f.
\]
Replacing $r$ by $\rho$ is exactly the change from the original unit
length to the folded unit length. In a disjoint bin, a one-sided omission
has integrality density $1/p$. Relative to the two deficit-law factors
this costs at most an extra fixed power of $R_p^{-1}$; the product of
these losses is bounded by the good-mask harmonic condition. At a prime
omitted by neither side, the preceding noncoincidence argument gives
upper mean $R_p^2$. The slopes are units: in these coordinates
$d_i z_i=(S_i/V)fZ_i$, and its translation slope is $S_i f$. If a
coincidence occurs, the lifted identity forbids survival on every
translation satisfying integrality.

In an identical bin, keep just one factor $R_p$ at a full prime. A shared
omission has one residual scalar $R_p$ when the full centre exponent is at
least two, and no factor when it is one. These are precisely the local
factors of $f\pi_{P_{\rm eq}}(f)$. More explicitly, for full centre
exponent $\nu$, the local upper bounds are
\[
\begin{array}{c|c|c}
 \text{arrangement and deficits}&\nu=1&\nu\ge2\\ \hline
 \text{disjoint},\ (0,0)&R_p^2&R_p^2\\
 \text{disjoint},\ (1,0)&1/p&R_p/p\\
 \text{identical},\ (0,0)&R_p&R_p\\
 \text{identical},\ (1,1)&1&R_p
\end{array}
\]
with the symmetric one-sided case understood. The one-sided row differs
from $\pi_{p^\nu}(p)\pi_{p^\nu}(1)$ by at most $R_p^{-1}$; the last
row is $p\pi_{p^\nu}(p)$. All required digits occur in
$k\bmod P/f$; in particular a shared omitted prime of exponent one needs
no further digit. The digit CRT multiplies these upper means. Low tests
use $z_i=Z_i/m_i'$ modulo low powers, with division by a unit, and are
unchanged by the high translations. This proves the folded coefficient.
Since all majorizations are nonnegative, an existence predicate for an
actual cohit may be retained outside the count. This predicate is only
the assertion that some actual integral cohit survives at the fixed
row, mask, and anchor data: if none does, the original count is zero;
otherwise the unconditional digit upper bound applies. No pointwise
survival condition is factored through the digit average.
\end{proof}

\subsection{Billing exact modeled equalities}

A cohit has an \emph{exact modeled equality} if replacing its two phases
by recognized rationals from the low birth tiers makes its two lifted
locations exactly equal. Equality here is in the line, not merely modulo
one. This case requires a first-mass estimate, since it cannot yield a
determinant saving.

\begin{lemma}[Exact equality bound]\label{dir:exact-equalities}
The total folded comparison count with exact modeled equality, using all
recognized tiers and either the broad or narrow outer-mask law, is
$O(W)$.
\end{lemma}

\begin{proof}
\emph{Independent mask averaging.}
Fix a group and one of the arrangements of Lemma~\ref{dir:tag-fold}, then
condition on $f$ and, for narrow comparisons, on the common outer mask.
Drop pairwise restrictions on centres, masks, and widths. In the
resulting upper bound, use the unrestricted independent deficit laws
for the remaining tagged masks, and also for the outer masks in the
broad case.

For each qualifying folded row and chosen model
$\zeta_i=n_i/q_i$, form a vector of functions indexed by rational
$y_*\in[0,1)$. Its coordinate at $y_*$ is zero unless
\[
 y_*=(Z_i+m_i'\zeta_i)/V,\qquad Z_i\in\mathbb Z.
\]
Otherwise that coordinate is the interval indicator supported on
$|x-y_i|\le C'\rho_i$, multiplied by the low-test weight at $Z_i$.
Choose $C'$ sufficiently large compared to the distance constant.
The inner product of two such vectors majorizes, up to a fixed factor,
the low weighted cohit count times $\min(\rho_i,\rho_j)$ when there is
exact modeled equality. Choosing the first lift by $y_*\in[0,1)$ is
equivalent to counting per unit length: joint integer translation
preserves every transformed low test.

Let $\mathbf b$ denote this vector, and include all assigned good rows
and all recognized tiers at each sampled mask. Since the remaining laws
are the same on both sides, the widened comparison is bounded by
\[
 \left\|\mathbb E\sum_{\rm rows,tiers}\mathbf b\right\|^2
 \le
 \mathbb E\left\|\sum_{\rm rows,tiers}\mathbf b\right\|^2.
\]
The fixed outside coefficient is the one in
Lemma~\ref{dir:tag-fold}. This is an ordinary Hilbert-space Jensen
inequality; the two masks are coupled only in the expression on the
right. The common unrestricted law is
$\pi_{P_{\rm dis}}(m')\pi_{P_I}(a)$ in the broad case and
$\pi_{P_{\rm dis}}(m')$ after fixing the common outer mask in the narrow
case. Assignment and good-mask indicators remain inside the vector,
as zero contributions from unavailable rows; they do not condition
this law. Split the bounded number of tiers by Cauchy--Schwarz,
distinguishing the large broad, small broad, and narrow parameter
sets on medium type.

\emph{Common-model partner counts.}
We may therefore fix one common mask $m$, one outer phase $\alpha$, and
one tier. Within this tier, the same reduced rational $\zeta=n_0/q$
models $\alpha$ wherever a model is recognized, even across widths, by
Lemma~\ref{pre:model-uniqueness}. Moreover $(q,m')=1$, where
$m'=m/f$, since the model denominator is below the tag-prime cutoff.
At these masks a specified $V$ determines the actual denominator
$VP/(ma)$, and hence determines at most one assigned row, including its
width and centre.

For an exact equality between $V$ and $W'$, write
\[
 H_V=2+q+b_V,\qquad
 b_V=\frac{|\alpha-\zeta|m}{VP r_i},
\]
and define $H_{W'},b_{W'}$ similarly. The soft tests and the identity
\[
 s(qZ_i+m'n_0)=t(qZ_j+m'n_0)
\]
imply
\begin{equation}\label{dir:soft-ratios}
 q\mid s-t,\qquad
 s\le H_{W'}^{h_s},\qquad t\le H_V^{h_s}.
\end{equation}
Indeed, $(s,t)=1$ gives $t\mid qZ_i+m'n_0$ and
$s\mid qZ_j+m'n_0$. Since $t\mid V$, division by the unit $m'$ in the
low test bounds $t$ by its soft gcd; the other side is the same.
Reduction modulo $q$, using $(q,m'n_0)=1$, gives the first assertion.
If $\alpha\ne\zeta$ and the two interval bumps intersect, their centres
also give
\begin{equation}\label{dir:bias-overlap}
 m'|\alpha-\zeta|\,|1/V-1/W'|\ll\rho_i+\rho_j,
 \qquad
 |s-t|\ll s/b_V+t/b_{W'}.
\end{equation}

Orient so that $H_V\le H_{W'}$. First suppose
$H_{W'}\le H_V^{1/(3h_s)}$. Then
$s\le H_V^{1/3}$ and $t\le H_V^{h_s}$. For large $H_V$, either
$q>s+t$, in which case \eqref{dir:soft-ratios} forces $s=t$, or
$q\le s+t\le2H_V^{1/3}=o(H_V)$. In the latter case
$b_V\ge H_V/2$ and $b_{W'}\ge H_{W'}/2$ for large $H_V$, so the
right side of \eqref{dir:bias-overlap} is
$O(H_V^{-2/3}+H_V^{h_s-1})<1$ and again $s=t$.
The zero-bias case belongs to the first alternative at large height.
Thus large comparable heights give the same row, while bounded heights
permit only boundedly many pairs of ratios $(s,t)$.

Now suppose $H_{W'}>H_V^{1/(3h_s)}$. Then $b_{W'}\asymp H_{W'}$,
and
\begin{equation}\label{dir:disparate-widths}
 \frac{\rho_j}{\rho_i}
   =\frac{V}{W'}\frac{b_V}{b_{W'}}
   \ll\frac{H_V^{1+h_s}}{H_{W'}}.
\end{equation}
The identity is needed only when the bias is nonzero; zero bias makes
$H_V=H_{W'}$ and cannot occur in this case. Per point on $V$, the number
of possible partners in a dyad $H_{W'}\asymp D$ is
$O(H_V^{h_s}D^{h_s})$, by \eqref{dir:soft-ratios}. Each ratio determines
$W'$, which determines its row, and exact equality determines its point.
Consequently the sum of partner lengths relative to $\rho_i$ is at most
\[
 O\left(
 H_V^{1+2h_s}
 \sum_{D\gtrsim H_V^{1/(3h_s)}}D^{h_s-1}
 \right)=O(1).
\]
Here the sum is dyadic, and
$1+2h_s+(h_s-1)/(3h_s)<0$ for $h_s<.01$. Since $H_V\ge3$, this is
uniform also for bounded $H_V$ and infinitely many possible partner
height dyads. In the comparable case,
charge the first bump length and use the bounded number of partners.
Products of low weights are at most the charged row's low weight.
Its average is at most $R(V)$, so the same-mask square is bounded by
constant times low first mass.

\emph{Return to assigned first mass.}
We have
\[
 f\pi_{P_{\rm eq}}(f)\rho_i
       =P\pi_{P_{\rm eq}}(f)r_i.
\]
Averaging over $f$, the remaining tagged masks, and the outer masks
returns the original row currency, because
\[
 \begin{aligned}
 \pi_{P_{\rm eq}}(f)\pi_{P_{\rm dis}}(m')&=\pi_{P_M}(m),\\
 \phi(v)&=\phi(V)\phi(P_M/m)\phi(P_I/a)\\
        &=P\phi(V)\pi_{P_M}(m)\pi_{P_I}(a),
 \end{aligned}
 \qquad m=fm',
 \quad v=VP/(ma).
\]
In particular,
\[
 r_iP\phi(V)\pi_{P_M}(m)\pi_{P_I}(a)=r_i\phi(v).
\]
The bounded number of arrangements and tiers only changes the constant.
\end{proof}

\begin{lemma}[Completion of ultra comparisons]\label{dir:ultra-count}
All ultra comparisons in Proposition~\ref{dir:birth-comparisons} have
total cost $O(W)+O_{k_0}(W^{1+c})$.
\end{lemma}

\begin{proof}
When a switch is available, division of the approximants of
Lemma~\ref{dir:ultra-switch} by $\sigma_0,\tau_0$ gives phase errors,
in units of $\psi_i,\psi_j$, bounded by
$\varepsilon e^{O(D')X_0}$; here
\eqref{eq:direct-size} absorbs the width ratio. These rationals qualify
in the birth tiers: the large broad tier in a medium broad comparison
is enabled by $\psi_i,\psi_j\ll\Delta$; a medium narrow comparison uses
the narrow tier; a high comparison uses its unique broad tier.
The choices $E\gg E'$, $Q_0\gg E$, and $E_g\gg E$ ensure the required
denominator, height, and error bounds. Their coupled determinant
identity gives exact modeled equality, covered by
Lemma~\ref{dir:exact-equalities}.

It remains to count nonexact pairs with no switch. Fix the group and the
two outer and tagged masks, treating broad and narrow comparisons
separately. There can be at most one coprime signature vector
$(\sigma_0,\tau_0)$ among these pairs. Indeed, two distinct coprime
vectors give an invertible pair of equations
\eqref{eq:direct-determinant}. Solving them with the integral
determinants as exact right sides gives rational replacements of the
two fixed phases, of denominators at most $2e^{2E'X_0}$, and errors
at most $e^{O(E')X_0}$ times the larger of the two tolerances.
At the pair with that larger tolerance, each error is at most the
corresponding numerator width, $\psi_i$ or $\psi_j$, times
$\varepsilon e^{O(E')X_0}$. Explicitly, put $M=e^{E'X_0}$.
The coefficient determinant is a nonzero integer of absolute value
at most $2M^2$, so the phase errors are at most $2M\Delta_{\max}$.
On the pair attaining $\Delta_{\max}$, both sides satisfy
\[
 \frac{\Delta_{\max}}{\psi_i}
      =C\varepsilon\sigma_0\frac{r_{\max}}{r_i}
      \le C\varepsilon M e^{CX_0},
\]
and the analogous inequality with $\tau_0,\psi_j$.
The same tier verification as above would make
that pair exact modeled, a contradiction.

For the one remaining signature, a fixed $V$ has at most one partner
$W'$. The folded determinant integer is unique, since the tolerance is
ultra. Retain only one low factor at each prime of $V$ not dividing $t$,
and average over its full common digit in the determinant cycle. The
resulting count is at most
\[
 \frac{V}{t}R(V)\prod_{p\mid t}R_p^{-1}
       =\frac{\phi(V)}{\phi(t)}
       \le\phi(V).
\]
Charging $\rho_i$ times this bound to the row on $V$ and restoring the
fold coefficients as in Lemma~\ref{dir:exact-equalities} gives $O(W)$.
This one-neighbour argument includes all widths at the fixed masks.
The ultra-switch exceptions supply the remaining
$O_{k_0}(W^{1+c})$ term.
\end{proof}

\subsection{The non-ultra row kernel}

All narrow comparisons have now been counted. For the remaining broad
comparisons, $f=1$. Use folded widths
$r_i'=Pr_i$, $r_j'=Pr_j$, width ratio
$\mathcal R=r_{\max}/r_{\min}$, and phases
$m_i\alpha_i,m_j\alpha_j$. The \emph{primary model} is the recognized
broad model, if present, and otherwise the recognized narrow model.
If neither is recognized, there is no primary model and all retained
low factors are ordinary scalars $R_p$. When a primary model exists,
retain its hard tests, with an ordinary scalar $R_p$ wherever it has no
active exclusion. This upper version costs at most a fixed multiplicative
constant even if two distinct medium models were recognized. In that
case the extra hard primes of the narrow model are above $e^{cB}$,
whereas low primes are at most $e^{JB}$, so their reciprocal sum is
bounded. Exact primary-modeled equalities have already been counted.

The homogeneous antecedent of this determinant-sieving strategy is the
Pollington--Vaughan overlap estimate~\cite{PV1990}, as restated in
\cite[Lemma~5.3]{KM2020}. The shifted, weighted row kernel with the
model-dependent restrictions used here is proved below; it is not
asserted to follow from that homogeneous estimate.

\begin{lemma}[Nonexact row kernel]\label{dir:row-kernel}
At fixed masks, the weighted folded count of the remaining broad
determinants is at most a constant times
\begin{equation}\label{eq:row-kernel}
 (r_i'\phi(V))(r_j'\phi(W'))
 \max(1,1/\eta_1)
 \prod_{\substack{p\mid st\\p>\eta_1^{c_1}}}R_p^{-C},
\end{equation}
where $c_1>0$ is a small absolute constant and mask probabilities are
applied outside the expression. This bound may retain the predicate
that there is an actual integral, surviving, nonexact cohit for the
specified masks and anchors, including its hole tests.
\end{lemma}

\begin{proof}
At a common prime not dividing $st$, the determinant cycle traverses a
full common digit. If two active forbidden residues coincide, the prime
divides the nonzero integer
\[
 q_iq_j\bigl\{s(Z_i+m_i\zeta_i)
                 -t(Z_j+m_j\zeta_j)\bigr\},
\]
where $q_i,q_j$ are the primary-model denominators. Each phase bias
contributes at most a constant times its relative height times
$\eta_1$, and $q_i,q_j$ are bounded by those heights. Thus this integer
has absolute value
$O(\eta_1\max(H_v,H_{\widehat v})^3)$, where $v,\widehat v$ are the
actual rows associated with the folded denominators $V,W'$.
Both hard restrictions require
$p>\max(H_v,H_{\widehat v})^{\epsilon_s}$. Above
$\eta_1^{c_1}$ only boundedly many prime divisors can satisfy both
thresholds. Indeed, writing $H=\max(H_v,H_{\widehat v})\ge3$ and
$M=\max(2,\eta_1^{c_1},H^{\epsilon_s})$,
\[
 \log(C\eta_1H^3)
 \le \log C+\log^+\eta_1+3\log H
 \ll_{C,c_1,\epsilon_s}\log M.
\]
The integer is nonzero precisely because this lifted determinant is not
an exact primary-modeled equality. If its upper size bound is below
one, no such integer occurs at all. Thus the coincidence loss is
uniform even when $\eta_1<1$ or $H$ is large. If either factor is
a scalar, there is no coincidence loss. At mismatch primes, pay the
displayed product.

When $\eta_1$ is large, average larger common digits first, keeping the
smaller ones fixed, then widen the determinant set and apply
Lemma~\ref{ar:sieve} through $\eta_1^{c_1}$. For bounded $\eta_1$ the
remaining small primes cost only a constant. Write $g=(V,W')$ and
$r_{\min}'=\min(r_i',r_j')$. The resulting width-weighted bound is
$g r_{\min}'\max(1,\eta_1)R(V)R(W')$ times the displayed mismatch
product. Since $\eta_1=[V,W']r_{\max}'$, its mass factor is
\[
 g r_{\min}'\max(1,\eta_1)R(V)R(W')
 =(r_i'\phi(V))(r_j'\phi(W'))\max(1,1/\eta_1).
\]
This proves \eqref{eq:row-kernel}. A nonnegative count vanishes when its actual-cohit
predicate fails, so the predicate may be retained throughout the
majorization.
\end{proof}

\paragraph{A preliminary linear charge.}
For small determinant scales use bins
$e^{-\lambda}\le\eta_1<e^{1-\lambda}$ with integer
$\lambda\ge\lambda_0$, and put $\eta=e^{-\lambda}$.
Before aggregating the broad comparisons in the medium range
$\ell_{\rm sh}B\le\lambda\le Q_0B+O(1)$, charge nonexact pairs for which
\begin{equation}\label{dir:small-signatures}
 s m_i,\ t m_j,\ \mathcal R\le e^{h'B},
\end{equation}
where the fixed exponent $h'$ is sufficiently small against $h_0$.
At fixed outer and tagged masks there is at most one such coprime
signature over the whole range. Otherwise the two-equation argument
of Lemma~\ref{dir:ultra-count} gives exact phase rationals of small
denominator, with relative errors at most $e^{O(h')B}$ in units of
the widths at the pair with the larger tolerance. At that pair
$\psi_i,\psi_j\ll e^{-\ell_{\rm sh}B}$, so the small broad tier, or
the larger tier, is enabled. The solved models qualify and force
exact modeled equality, a contradiction. The one-neighbour count
from Lemma~\ref{dir:ultra-count} therefore gives a linear charge
$O(W)$. We exclude this family from the remaining edge counts,
without changing the aggregate good-row lists.

For an exact width in a group, aggregate the low vertices $V$ with
normalized weights
\begin{equation}\label{dir:aggregate-vertices}
 r_i'\phi(V)\sum_{m,a}\pi_{P_M}(m)\pi_{P_I}(a)
 \mathbf1_{\{\text{the row }VP/(ma)\text{ is assigned here at width }r_i\}}.
\end{equation}
Only retained good masks enter this sum. Its total is the ordinary
block mass on the sublist. After division by $r_i'$, each weight is
at most $\phi(V)$: fixed $V,m,a$ prescribe one actual row, and the
unrestricted mask laws have total mass one. Edge weights below are
products of these normalized weights, with the mismatch factors and
necessary cohit predicates incorporated inside the mask and anchor
averages, excluding the pairs already charged in
\eqref{dir:small-signatures}. Their totals, multiplied by
$\max(1,1/\eta_1)$ on a scale,
bound Lemma~\ref{dir:row-kernel}.
For a fixed pair of normalized exact-width sublists, write $F,G$ for
their respective total weights.

For large $\eta_1$, edges with
$\sum_{p\mid st,\ p>\eta_1^{c_1}}1/p\le1$ cost only a constant times
the product of vertex weights. On a dyad $\eta_1\asymp H_1\ge2$,
give the other edges the extra factor
\[
 \exp\left(C'\log H_1
       \sum_{\substack{p\mid st\\p\gg H_1^{c_1}}}\frac1p\right),
\]
with its cutoff no higher than required by the preceding condition.
Together with the original mismatch product, this is an admissible
inflation $h$ for \eqref{eq:lcm}, uniformly in $H_1$:
$h_p-1=O((\log p)/p)$. Since the extra factor is at least a fixed
multiple of $H_1^{C'}$ on these edges, their uninflated contribution is
\[
 O\bigl(
 H_1^{-C'+2(1-u)}
 \mathcal R^{-(1-u)}(FG)^u
 \bigr).
\]
Here $u>1/2$ is fixed.
The scale sum converges. The width sum uses geometric decay and
Cauchy--Schwarz, and the group sum uses disjointness of row assignments.
Scales with $\eta_1$ bounded away from zero are similarly covered by
\eqref{eq:lcm}.

The only remaining task is therefore to obtain the saving
\eqref{eq:small-saving} at small $\eta_1$. That saving is summable even
after multiplication by $1/\eta_1$.

\subsection{A static discard near low rational grids}

We now define the additional birth rule promised in
Proposition~\ref{dir:birth-comparisons}. Its definition uses only the
aggregate good-row lists \eqref{dir:aggregate-vertices}, not an extracted
box and not any previous table deletion.

\begin{construction}[Low rational holes]\label{dir:hole-rule}
Fix a medium group and exact width, and let $F$ be its aggregate list
mass. For each integer $\lambda\ge\lambda_0$ put $\eta=e^{-\lambda}$.
For every dyad $[A,2A)$ with $A\ge\eta^{-16}$, test all positive integer
pivots $L_*$ satisfying
\[
 \eta^4\le D_0'=r_i'L_*A^2\le\eta^{-1}
\]
whose incidence class
\[
 \{V: V/(V,L_*)\in[A,2A)\}
\]
has list mass at least $\eta^{.1}D_0'$.

For each such pivot, test only rows in its incidence class for which
$n=V/(V,L_*)$ is squarefree. On these rows, discard a birth point $x_i$ if
\[
 \left\|L_*P x_i-\frac{b}{q}\right\|\le\frac1A
 \quad\hbox{for some }1\le q\le\eta^{-.1},\quad b\in\mathbb Z.
\]
\end{construction}

Only finitely many parameters are relevant in a finite block: a
nonempty incidence class bounds $A$ by the largest low denominator, then
$A\ge e^{16\lambda}$ bounds $\lambda$, and the interval for $D_0'$
bounds $L_*$. Although evaluating the rule uses the point's phase and
masks, the families of pivots are fixed by the original aggregate lists.
In particular, this definition makes no use of a tail-prime coordinate
of the numerator.

\begin{lemma}[Cost of low rational holes]\label{dir:holes}
For every fixed $\delta>0$, Construction~\ref{dir:hole-rule} loses at
most $\delta W$ in total birth first mass when $\lambda_0$ is sufficiently
large. The choice is uniform over all groups and widths. The rule
depends on numerator residues only through squarefree low moduli.
\end{lemma}

\begin{proof}
Fix a group, width, $\lambda$, and $A$. Group the pivots by dyads, so
that $D_0'$ has a common value $D$ up to fixed factors. For two pivots
$L',L''$ in such a dyad, put
\[
 g=(L',L''),\qquad k'=L'/g,\qquad l'=L''/g.
\]
Their incidence classes have intersection mass at most
\begin{equation}\label{dir:pivot-intersection}
 C D(k'l')^{-1/2}\tau(k')\tau(l').
\end{equation}
To see this, set
\[
 u=\frac{V}{(V,g)},\qquad k=(u,k'),\qquad l=(u,l'),\qquad u=nkl.
\]
Since $(k',l')=1$, the integers $k,l$ are coprime. The two incidence
conditions imply $nk,nl<2A$. For fixed $u$, all possible $V$ divide
$gu$, so their aggregate weight is at most
$r_i'\sum_{V\mid gu}\phi(V)=r_i'gu$. For fixed $k,l$, summing $nkl$
over $n\le2A/\max(k,l)$ costs $O(A^2)$. Summing over divisors
$k\mid k'$, $l\mid l'$ gives
$O(r_i'g A^2\tau(k')\tau(l'))$, which equals the upper bound
\eqref{dir:pivot-intersection} because $L',L''$ lie in the same dyad.

If $M_0$ pivots in this dyad qualify, incidence Cauchy--Schwarz and
Lemma~\ref{ar:tensor} give, for any sufficiently small fixed
$\epsilon>0$,
\[
 (M_0\eta^{.1}D)^2
       \ll F D\,M_0^{1+\epsilon}.
\]
The divisor factors in \eqref{dir:pivot-intersection} are absorbed by
an arbitrarily small loss in its exponent price. Thus
$M_0^{1-\epsilon}D\ll F\eta^{-.2}$. Since $F\le1$ and
$D\gg\eta^4$, putting $X=M_0D$ gives
\[
 X\ll
 F^{1/(1-\epsilon)}
 \eta^{-(.2+4\epsilon)/(1-\epsilon)}
 \ll F\eta^{-1}
\]
for, say, $\epsilon<.1$. Thus
\begin{equation}\label{dir:pivot-total}
 M_0D\ll F\eta^{-1}.
\end{equation}
Each individual incidence class also has mass $O(D)$.

On an actual tested row the values of $L_*P x_i$ form a translated
uniform grid of size
$n=V/(V,L_*)\asymp A$: indeed $P x_i=m_i(z_i+\alpha_i)/V$ and
$m_i$ is a unit modulo $V$. There are $O(\eta^{-.2})$ rational
residues of denominator at most $\eta^{-.1}$. A neighborhood of
radius $1/A$ around one residue captures $O(1/A)$ of this grid.
Conditioning on the hard birth laws inflates this proportion by at
most
\[
 R(n)^{-C}\ll A^{.1}.
\]
Here $n$ is squarefree; the tag coordinates are independent of its
low digits by CRT, and double exclusions occur only at large primes,
so the conditioning estimate follows prime by prime. Dropping the
soft tests can only enlarge this loss. The hole probability relative
to hard first mass is consequently
$O(\eta^{-.2}A^{-.9})$.

Combining with \eqref{dir:pivot-total}, and weakening exponents
slightly, gives the convenient bound
$O(F\eta^{-2}A^{-.8})$ per pivot dyad. For each $A$ there are
$O(1+\lambda)$ pivot dyads, by the permitted range of $D_0'$.
The sum over $A\ge\eta^{-16}$ and then over
$\lambda\ge\lambda_0$ is an arbitrarily small multiple of $F$.
Summing the disjoint group and width masses gives the result.
The query on a row uses only the squarefree modulus $n$; hence it
has the asserted numerator dependence.
Since the hard birth mean is at least $c_{\rm hard}$ times assigned
first mass, choosing $\delta$ smaller by this fixed factor makes the
loss an arbitrarily small fraction of that hard mean as well.
\end{proof}

\subsection{Excluding saturated small-scale boxes}

We use the small-scale bins and residual edge graph defined above.
First set notation common to the three cases below. Whenever the
saving \eqref{eq:small-saving} fails and we invoke
Proposition~\ref{ar:extraction}, it gives, for an arbitrarily small
fixed $\vartheta>0$, a saturated configuration
\begin{equation}\label{dir:saturated-form}
 \begin{aligned}
 V&=NU_0a_0/b_0,&\qquad
 W'&=NT_0c_0/d_1,\\
 s&=T_0c_0b_0,&
 t&=U_0a_0d_1,
 \end{aligned}
\end{equation}
with the coprimalities of that proposition and dyads
$a_0\asymp A$, $c_0\asymp C_0$. In particular,
$\mathcal R,U_0T_0\le\eta^{-\vartheta}$,
the four varying factors are pairwise coprime and squarefree, and
\[
 D_1=r_i'NU_0A^2,\qquad D_2=r_j'NT_0C_0^2,
 \qquad
 \eta^{2+\vartheta}\le D_i\le\eta^{-\vartheta}.
\]
The captured edge weight is at least
$\eta^\vartheta D_1D_2$. On its edges,
\[
 \|s m_i\alpha_i-t m_j\alpha_j\|\ll\eta.
\]

We use the unrestricted box sampling of
Corollary~\ref{ar:box-sampling}, together with the independent mask
laws, followed on each side by the conditional anchor law of its
sampled tagged mask. An event of probability $O(\eta^{10\vartheta})$ cannot account
for this saturation. The same is true of a first-list set of mass
$O(\eta^{10\vartheta}D_1)$. All switches below are made within these
unrestricted laws; the changed values need not themselves form edges.
We retain the necessary predicate of an actual cohit, passing all
filters and not an exact primary-modeled equality. Thus for relevant
masks and anchors there are actual integral numerators
$Z_i=m_i z_i$, $Z_j=m_j z_j$ with
\begin{equation}\label{dir:actual-small-determinant}
 |s(Z_i+m_i\alpha_i)-t(Z_j+m_j\alpha_j)|\ll\eta.
\end{equation}
The outer phases depend only on their respective tail masks, not on the
box's dyadic or defect parameters. These facts specify the sampling
interface needed in each of the three cases below. In the deeper medium
case, the small-signature family \eqref{dir:small-signatures} has already
received its linear charge and is absent from this graph.
The contradictions will come from tag exclusions on high type, density
or the static holes on shallow medium type, and broad-model recognition
on deeper medium type.

\begin{lemma}[High-type boxes]\label{dir:high-box}
No saturated configuration \eqref{dir:saturated-form} occurs on high
type among the remaining non-ultra comparisons.
\end{lemma}

\begin{proof}
Here $\lambda\le Q_0Y_d+O(1)$. Since $f=1$, the tagged masks are
disjoint; by the occupancy purge each has at least $c\kappa Y_d$
prime omissions. Switch the sampled anchor on each side, using the
prime alternative in Lemma~\ref{ar:rotation} with budget
$\epsilon'\lambda$, where
$C_1\epsilon'$ is small and $\vartheta\ll\epsilon'$. Stripping the
prime from its mask costs $O(1/p)$ under the deficit law. The anchor
selection probability is at most $O(1/(\kappa Y_d))$. Therefore the
failure probability on one side is at most
\[
 \frac{C e^{-\epsilon'\lambda}}{\kappa Y_d}
 \sum_{\substack{H:\ p\asymp H\\\text{in the dense bin}}}
       \frac1{\log H}
 \ll_\kappa e^{-\epsilon'\lambda}.
\]
The sum of reciprocal logarithms over these ordinary dyads is
$O(Y_d)$. This failure is negligible under the box sampling.

Successful switches give unique coupled rationals $A_i,A_j$ for
$s m_i\alpha_i,t m_j\alpha_j$, of denominators
$e^{O(\epsilon'\lambda)}$ and errors
$O(\eta e^{O(\epsilon'\lambda)})$. By the precision and
\eqref{dir:actual-small-determinant}, their exact difference is the
negative integral determinant. Each reduced numerator is divisible
by its own sampled anchor. As in Lemma~\ref{dir:tag-fold}, the
predictions
\[
 \mathcal L_i-\frac{S_i}{[V,W']}A_i,\qquad
 \mathcal L_j-\frac{S_j}{[V,W']}A_j
\]
have sizes and denominators at most $e^{O(Y_d)}$, by
\eqref{eq:direct-size}, and fit these anchor choices. Their exact
lifted relation contradicts survival at an anchor in the disjoint
masks. Saturation therefore cannot occur.
\end{proof}

\begin{lemma}[Shallow medium boxes]\label{dir:shallow-box}
No saturated configuration occurs on medium type with
$\lambda<\ell_{\rm sh}B$ after Construction~\ref{dir:hole-rule}.
\end{lemma}

\begin{proof}
The extracted size bounds imply, for small enough $\vartheta$,
\[
 \eta^2\le A/C_0\le\eta^{-2}.
\]
\emph{Bounded dyads: the density estimate.}
First suppose both $A,C_0\le\eta^{-20}$. For a fixed $a_0$ put
$L'=PNU_0a_0$. Every first-list row at the structured vertex divides
$L'$, and the lcm and width bounds give
\[
 \eta^{23}\le L'r_i\le C\eta.
\]
In detail,
\[
 L'r_i=\eta_1\frac{r_i/r_{\max}}{T_0c_0}
       \ge c\eta^{21+2\vartheta}\ge\eta^{23}
\]
for sufficiently small $\eta$, while $T_0c_0\ge1$ gives the upper
bound.
The largest original popular host of such a row has minus-log-scale
at least its minimized cost, which is at least $B$. Since
$23\lambda<B$, the integer $L'$ is at least that original host.
The density bound \eqref{eq:centre-density} consequently applies to
the assigned structured subfamily dividing $L'$. Its density is
$O_{k_0}((L'r_i)^{H_0})$ whenever it contributes.
Its entire first mass is therefore at most
$O_{k_0}(\eta^{H_0}L'r_i)$, including all its low defects and masks.
Summing over $a_0$ uses
$\sum_{a_0\asymp A}L'r_i\asymp r_i'NU_0A^2=D_1$.
The first-list mass is consequently
\[
 O_{k_0}(\eta^{H_0}D_1),
\]
which is negligible for saturation when $\vartheta$ is sufficiently
small and $\lambda_0$ sufficiently large.

\emph{Large dyads: a qualifying static hole.}
Otherwise the ratio bound implies $A,C_0\ge\eta^{-17}$.
Switch $a_0,c_0$ as whole factors, using the integer alternative in
Lemma~\ref{ar:rotation} with budget $\epsilon'\lambda$. Its failure
probability is negligible
under the box sampling. On the remaining configurations there are
rationals $\xi_i,\xi_j$ of denominators
$e^{O(\epsilon'\lambda)}$ satisfying
\[
 \left|T_0b_0m_i\alpha_i-\xi_i\right|
       \ll\frac{\eta e^{O(\epsilon'\lambda)}}{C_0},
 \qquad
 \left|U_0d_1m_j\alpha_j-\xi_j\right|
       \ll\frac{\eta e^{O(\epsilon'\lambda)}}{A}.
\]
Rational separation and \eqref{dir:actual-small-determinant} force
\begin{equation}\label{dir:common-rational}
 c_0(T_0b_0Z_i+\xi_i)
       =a_0(U_0d_1Z_j+\xi_j).
\end{equation}
Set $L_*=NU_0T_0$ and $y_i=P x_i$. The common rational
\[
 \frac{T_0b_0Z_i+\xi_i}{a_0}
       =\frac{U_0d_1Z_j+\xi_j}{c_0}
\]
has denominator at most $e^{O(\epsilon'\lambda)}$: its denominator
divides both $a_0$ times the denominator of $\xi_i$ and $c_0$ times
that of $\xi_j$, and $(a_0,c_0)=1$. Moreover,
$L_*y_i$ is within
$O(\eta e^{O(\epsilon'\lambda)}/(AC_0))$ of that rational.
The quotient
\[
 \frac{V}{(V,L_*)}=a_0
\]
is squarefree, even when $a_0$ shares primes with $N$; its additional
prime exponents are precisely the ones surviving this quotient.

This pivot qualifies in Construction~\ref{dir:hole-rule}.
Indeed $D_0'=T_0D_1$ lies between $\eta^4$ and $\eta^{-1}$ for small
$\vartheta$. If its incidence class had mass less than
$\eta^{.1}D_0'$, it would have mass at most
$\eta^{.1-\vartheta}D_1\le\eta^{10\vartheta}D_1$ when
$\vartheta<.1/11$. This is a restriction on aggregate vertex mass,
already integrated over the auxiliary masks and anchors, so
Corollary~\ref{ar:box-sampling} contradicts saturation. Also
$A\ge\eta^{-17}$ lies in the tested
range. Taking $\epsilon'$ sufficiently small makes the rational
denominator at most $\eta^{-.1}$ and the approximation error less
than $1/A$. Thus the actual point fails its hole test, contradicting
the retained cohit predicate.
\end{proof}

At this stage high boxes and shallow medium boxes have been excluded.
The remaining medium range has $\lambda$ comparable to $B$. This
allows small relative phase errors to activate a broad tier. The
preliminary charge has removed the case where all signature factors
and the width ratio are too small for a switch.

\begin{lemma}[Deeper medium boxes]\label{dir:deep-box}
For medium type with
$\ell_{\rm sh}B\le\lambda\le Q_0B+O(1)$, an additional nonexact
comparison count of size $O(W)$ may be charged directly. No
saturated configuration remains after that charge.
\end{lemma}

\begin{proof}
The $O(W)$ charge for \eqref{dir:small-signatures} was established
before defining the aggregate graph. It remains to exclude saturation
among its other edges.

\emph{The switch families.}
In a resulting saturated box, switch the following factors on side
$i$: the whole factor $c_0$ when $C_0\ge\eta^{-t_1}$; every divisor
of $b_0$ in dyads between $2\eta^{-t_1}$ and $\eta^{-J_1}$; and
every prime of $m_i$. On side $j$ use $a_0,d_1,m_j$.
Here $t_1>0$ is sufficiently small against $h'/Q_0$,
$J_1\gg J/\ell_{\rm sh}$ is fixed, then
$C_1\epsilon'$ is small against $t_1$, and finally $\vartheta$ is
sufficiently small.

For a divisor $h$ of the squarefree $b_0$, stripping under $\pi_N$
costs at most
\[
 \frac1{\phi(h)}
       \ll\frac{e^{\epsilon'\lambda/4}}h
       \qquad(h\le\eta^{-J_1}).
\]
At a prime $p\mid h$, the ratio of the single-omission law to the
zero-omission law is $1/p$ if $\nu_p(N)\ge2$, and $1/(p-1)$ if
$\nu_p(N)=1$. Their product is at most $1/\phi(h)$.
The ratio is applied only to valid original squarefree configurations;
dropping restrictions on the stripped parameters then enlarges the
upper count. No switched configuration is required to be good or
assigned.
The integer rotation estimate therefore makes the union of divisor
switch failures negligible, including the $O_{J_1}(\lambda)$ dyads.
Whole-factor switches have the same saving. A prime stripped from
$m_i$ costs $O(1/p)$; the prime rotation estimate, summed through
log-log heights $O(B)$, loses at most an additional factor $O(1+B)$.
Since $\lambda\ge\ell_{\rm sh}B$, these prime failures are also
negligible under the box sampling.

\emph{Qualification of the broad model.}
Outside all failures, any switch gives unique coupled approximants
to $s m_i\alpha_i,t m_j\alpha_j$, of denominators
$e^{O(\epsilon'\lambda)}$, errors
$O(\eta e^{O(\epsilon'\lambda)})$, and exact difference equal to
the negative determinant. After division by the whole signatures,
their denominators are at most $e^{O(t_1)\lambda}$. Here are all
the factors in this assertion. Every prime of $m_i$ divides the
first approximant numerator. Division by a switched $c_0$ leaves
a small denominator; if it is not switched, $c_0<2\eta^{-t_1}$.
Among prime factors of $b_0$ not dividing the numerator, the product
cannot reach the switch range: all are at most $e^{JB}$, so a
first-crossing product is still below $\eta^{-J_1}$ and would
contradict the small denominator after its switch. These factors
are coprime, and the additional factor $T_0$ is at most
$\eta^{-\vartheta}$. The second side is identical.

The errors after division, measured in $\psi_i,\psi_j$, are at
most $e^{O(\epsilon'+\vartheta)\lambda}$, using the extracted
width-ratio bound. Because $\lambda\le Q_0B+O(1)$ and the small
parameters were chosen in the stated order, the denominators and
relative heights fit the applicable broad tier. That tier is
enabled by $\psi_i,\psi_j\ll e^{-\ell_{\rm sh}B}$.
The exact coupled relation now makes the cohit an exact
primary-modeled equality, again a contradiction.

If no switch is available on either side, the same factor bounds
give $s m_i,t m_j\le e^{O(t_1)\lambda}$; the extracted width ratio
is at most $\eta^{-\vartheta}$. These fit the small-signature and
width bounds of the already charged family. This too is excluded,
and hence saturation is impossible.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{dir:birth-comparisons}]
Lemma~\ref{dir:holes} supplies the static birth discard, its loss,
and its squarefree low-coordinate dependence.
Lemma~\ref{dir:ultra-count} gives the asserted narrow estimate and
the same estimate for the ultra broad portion. For the remaining
broad portion, exact primary-modeled equalities have a linear
charge by Lemma~\ref{dir:exact-equalities}. The preliminary charge
for \eqref{dir:small-signatures} is also $O(W)$, and
Lemma~\ref{dir:row-kernel} reduces the residual counts to weighted
arithmetic edges.

The large and bounded-away-from-zero determinant scales were summed
above. At small scales, Lemmas~\ref{dir:high-box},
\ref{dir:shallow-box}, and \ref{dir:deep-box} exclude every
saturation alternative in this residual graph. Thus
\eqref{eq:small-saving} holds on all remaining edges. After the
factor $1/\eta_1$ in \eqref{eq:row-kernel}, its scale contribution
is
\[
 O\bigl(\eta^\chi\mathcal R^{-1/10}(FG)^u\bigr).
\]
The $\lambda$ sum converges; summing dyadic widths uses the
geometric factor $\mathcal R^{-1/10}$ and Cauchy--Schwarz, and
summing groups uses disjointness. Since $W\le1$ and $2u>1$, these
terms are bounded independently of the block. All steps used
unclipped birth weights, so none of the constants depends on
$N_0$. Finally, further thinning decreases each nonnegative
comparison count. This proves all assertions.
\end{proof}
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/direct.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/introduction.tex ====
\section{Introduction}\label{sec:introduction}

For a real number $t$, write $\|t\|$ for its distance to the nearest
integer, and let $\phi$ denote Euler's totient function. We study
approximation with a fixed additive shift and an arbitrary sequence of
tolerances. The exceptional null set in the result below is allowed to
depend on both of these data.

\begin{theorem}[Weak inhomogeneous Duffin--Schaeffer]
\label{thm:main}
Let $\gamma\in\mathbb R$ be fixed, and let
$\psi:\mathbb N\to[0,\infty)$ be finite-valued. If
\begin{equation}\label{eq:main-divergence}
  \sum_{q=1}^{\infty}\frac{\phi(q)}q\,\psi(q)=\infty,
\end{equation}
then, for Lebesgue-almost every $x\in\mathbb R$, the inequality
\[
  \|qx-\gamma\|<\psi(q)
\]
holds for infinitely many positive integers $q$.
\end{theorem}

Theorem~\ref{thm:main} resolves the weak inhomogeneous Duffin--Schaeffer
conjecture positively, including every prescribed irrational shift.
The integer nearest to $qx-\gamma$ is unrestricted: no coprimality
condition is imposed on the numerator. The conclusion is a divergence
statement, not a convergence converse or a quantitative asymptotic.

The mass transference principle of Beresnevich and Velani gives the
following Hausdorff-measure consequence.

\begin{corollary}[Hausdorff-measure divergence]\label{cor:hausdorff}
Fix $\gamma\in\mathbb R$ and a finite-valued
$\psi:\mathbb N\to[0,\infty)$. Let
$f:(0,\infty)\to(0,\infty)$ be continuous and nondecreasing, with
$f(r)\to0$ as $r\downarrow0$, and suppose that $f(r)/r$ is monotone.
Set $f(0)=0$ and
\[
  W_\gamma(\psi)=\{x\in\mathbb R:
    \|qx-\gamma\|<\psi(q)\text{ for infinitely many }q\}.
\]
If
\[
  \sum_{q=1}^{\infty}\phi(q)f\!\left(\frac{\psi(q)}q\right)=\infty,
\]
then, for every bounded interval $I$,
\[
  \mathcal H^f(I\cap W_\gamma(\psi))=\mathcal H^f(I),
\]
where $\mathcal H^f$ denotes Hausdorff $f$-measure.
\end{corollary}

The corollary is a sufficient divergence criterion for each fixed
shift, with unrestricted numerators. It requires no monotonicity of
$\psi$ and asserts neither a convergence converse nor a coprime-numerator
version. Its proof is given in Section~\ref{sec:consequences}.

\subsection{Background and prior work}

The starting point is the monotone theory of metric approximation.
Khintchine's theorem~\cite{Khintchine1924}, in its usual modern form,
gives a full-measure conclusion for $\|qx\|<\psi(q)$ when $\psi$ is
non-increasing and $\sum_q\psi(q)$ diverges. Szüsz extended the
monotone theory to each fixed inhomogeneous shift~\cite{Szusz1958};
see also the precise modern formulation in~\cite{Yu2021}.
Removing monotonicity changes the problem because many denominators
can describe strongly overlapping approximation intervals.
Duffin and Schaeffer~\cite{DS1941} exhibited this obstruction in the
homogeneous setting and proposed using reduced fractions with the
weighted sum $\sum_q\phi(q)\psi(q)/q$. Koukoulopoulos and
Maynard~\cite{KM2020} proved their conjecture. The weight $\phi(q)/q$
records the proportion of reduced residue classes; it also gives a
meaningful sufficient divergence criterion when all numerators are
allowed.

In the inhomogeneous problem, even unrestricted numerators do not
make unweighted divergence sufficient. Ramírez~\cite[Theorem~1]{Ramirez2017}
constructed nonmonotone counterexamples for every prescribed countable
collection of shifts. His constructions have finite totient-weighted
sum, as noted in that paper, and thus leave the weighted question open.
Chow, Hauke, Pollington and
Ramírez~\cite[Theorem~1.5]{CHPR2025} show that, for suitable shifts
including all rational shifts, even an approximation function that is
non-increasing on its support can have divergent unweighted sum and a
null approximation limsup. These results explain why a proof must
control arithmetic overlap on sparse supports, not merely sum the
lengths of the intervals.

The fixed-shift weighted question appears explicitly in
Yu~\cite[Question~1.2]{Yu2021} and under the name \emph{weak inhomogeneous
Duffin--Schaeffer conjecture} in Chow and
Technau~\cite[Conjecture~1.22]{ChowTechnau2024}. It is also
Conjecture~2 of Beresnevich, Hauke and Velani~\cite{BHV2024}.
The word ``weak'' concerns the conclusion: the integer $a$ in
$|qx-a-\gamma|<\psi(q)$ is arbitrary. The stronger formulation
requires $(a,q)=1$; this distinction is already explicit in
Ramírez~\cite{Ramirez2017}. Theorem~\ref{thm:main} addresses the weak
formulation for every prescribed real shift.

The coprime-numerator assertion is false in general: Hauke-Treuer,
Maynard and Pollington~\cite[Theorem~1]{HMP2026} construct
counterexamples for every nonzero rational shift and certain Liouville
shifts. Independent work of He and Liao~\cite[Theorems~1--2]{HL2026}
gives counterexamples for every nonzero rational shift and for a
residual set of real shifts.

Several earlier advances treat important parts of the nonmonotone
problem. Yu's Fourier approach~\cite{Yu2019} gives inhomogeneous
results under additional divergence assumptions, and his
Erdős--Vaaler type theorem~\cite[Theorem~1.3]{Yu2021} treats
$\psi(q)=O(1/(q(\log\log q)^2))$, with divergent unweighted sum,
for a class of irrational shifts containing all non-Liouville
irrationals. Chow and Technau~\cite[Theorem~1.23]{ChowTechnau2024}
prove the weak conjecture for certain structured nonmonotone
functions arising in multiplicative approximation on fibres.
Those theorems place restrictions on the shift, the approximation
function, or both; they do not supply the arbitrary-function,
arbitrary-fixed-shift assertion used here.

Beresnevich, Hauke and Velani~\cite[Theorem~14]{BHV2024} establish
the weak conjecture for every rational shift. More precisely, for
$\gamma=A/B$ in lowest terms with $B>0$, their solutions can be
required to satisfy $(A+aB,q)=1$. This is a restriction adapted to
the shift and is different from ordinary numerator coprimality
$(a,q)=1$. Their work also gives irrational-shift results under
additional block or approximation hypotheses, and constructs
classes of suitable shifts. The remaining task addressed by
Theorem~\ref{thm:main} is to retain a completely arbitrary fixed
irrational shift together with arbitrarily sparse, nonmonotone
approximation functions.

\subsection{Ancestry of the proof methods}

The arithmetic part adapts the common-pivot approach to prime
valuations developed in the homogeneous Duffin--Schaeffer work of
Koukoulopoulos and Maynard~\cite{KM2020}, simplified by Green and
Walker~\cite[Proposition~1.2 and Section~3]{GW2021}, and used in the
weighted argument of Hauke-Treuer, Vazquez and
Walker~\cite[Proposition~2.1]{HVW2026}. In that approach, failure of an
arithmetic estimate concentrates the exponents of two denominator
families around one common integer. The cited common-pivot
propositions concern hypothetical minimal counterexamples in their
respective arguments. We prove our weighted lcm estimate locally,
with its own totient capacities and multiplicative inflation
hypotheses, and then prove the small-scale extraction needed for
inhomogeneous overlaps. The one-coordinate concentration argument
is credited again at its point of use in Section~\ref{sec:preliminaries}.

Two further precedents help explain the organization. Chow and
Technau's shift-reduced numerator restrictions and localized full-measure
criteria~\cite[Definition~1.20 and Section~2.4]{ChowTechnau2024}
illustrate how numerator conditions can be adapted to inhomogeneous
arithmetic. Our conditions use rational models of the projected
phases and are specified independently below. Beresnevich, Hauke and
Velani obtain full measure by combining uniform distribution of selected
points $a/q$ with divergence or block-mass conditions and overlap
bounds~\cite[Theorems~11--12]{BHV2024}. Here the
transfer is proved directly for weighted finite arrays, using a
virtual comparison sum and a fixed finite-union approximation to an
avoided set. These are methodological comparisons; all the
construction and transfer estimates used in this paper are proved
here.

\subsection{Proof strategy and reusable estimates}

Put $r(q)=\psi(q)/q$. A row is the periodic array of points
$(z+\gamma)/q$, $z\in\mathbb Z$, with spatial half-width $r(q)$.
The proof works on arbitrarily late finite collections of rows whose
totient-weighted mass lies in a fixed small interval. On each collection
we construct a nonnegative weighted sum of interval indicators supported
inside the original approximation intervals. The construction changes
weights through prescribed prime-step updates and additional deletions.
For comparison, we omit only these additional deletions, keeping the
same realized prime-step update factors. This comparison sum is
equidistributed on fixed spatial
intervals; the expected discarded mass is small, and the retained sum
has a uniform second-moment bound. These bounds contradict the existence
of a positive-measure set avoiding all sufficiently late rows: approximate
that set by one fixed finite union of intervals, then use the second
moment to control the approximation error. Proposition~\ref{tab:conditional-reduction}
gives this transfer after the construction is defined.

There are two different obstructions to the second-moment bound. First,
nearness of two shifted points constrains an integer combination of
their numerators to an interval too short for direct averaging. The
inverse estimate of Proposition~\ref{ar:extraction} extracts denominators
concentrated near a common integer in prime-exponent coordinates.
Varying permitted factors then either bounds the overlap weight by a
rotation estimate or forces precise rational approximations to the grid
offsets. Section~\ref{sec:direct} treats both alternatives. Exact modeled
coincidences have total weight bounded linearly by the block's
totient-weighted mass, using explicit restrictions on numerator gcds.

Second, large common primes can create correlated alignments repeatedly
as different collections of rows begin to interact. Each row is assigned
an integer multiple of its denominator, called its centre. We expose the
prime factors of centres in an increasing sequence of scales. Before all factors have
been exposed, the row is represented by a finer projected grid containing
its eventual true points; its labels retain the original row and the
unexposed divisor data. At each prime step, short spatial cells group
projected points of the same width. Some entries are retained only when
their numerator is divisible by that prime. We pair portions of their
weights with portions of other entries in the cell; when a divisibility
test succeeds, the linked portion of the other weight is suppressed.
The precise rule in Section~\ref{sec:tables} preserves the first-moment
normalization while cancelling the most tightly correlated transfers.

The remaining correlations are paid by a squared-mass accounting
argument. Labels acquire equivalence classes through actual close
alignments, and later partitions preserve those classes even when
intermediate labels disappear. Deterministic interval grids record the
old connecting segments, allowing new short-cell partitions to respect the
history. At the first interaction of two previously separate collections,
a backward comparison transfers covariance terms to linked points that
passed their divisibility tests. The total incoming matching weight is
bounded by the weight available at each such point, as proved in
Lemma~\ref{join:capacity}. The averages use common residue translations;
they do not assume independent test outcomes or randomize the shift.

The reusable estimates are stated separately: small-scale extraction
with auxiliary weights in Proposition~\ref{ar:extraction} and
Corollary~\ref{ar:box-sampling}, the linear bound for exact rational-model
coincidences in Lemma~\ref{dir:exact-equalities}, and the variance budget
for history-preserving projected arrays in Lemma~\ref{tab:variance-budget}.

\subsection{Organization}

Section~\ref{sec:preliminaries} gives the finite-block reduction and
the elementary tensor and residue estimates used to select centres.
Sections~\ref{sec:centres}
and~\ref{sec:prescriptions} assign centres and prescribe the initial
weights. Section~\ref{sec:tables} gives the projected-array construction
and reduces the theorem to three precisely stated comparison and
partition assertions. Section~\ref{sec:arithmetic} develops the
sieve, rotation, and small-scale concentration estimates for close pairs.
Section~\ref{sec:direct} applies them to the comparisons
for adjacent simultaneous insertions. Section~\ref{sec:sparse} controls alignments
during a prime step, and Section~\ref{sec:partitions} constructs the
required partitions. Section~\ref{sec:joins} proves the remaining
first-interaction estimates and completes the finite induction.
Section~\ref{sec:consequences} proves the Hausdorff-measure corollary.
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/introduction.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/joins.tex ====
\section{First joins of previously constructed tables}\label{sec:joins}

The direct comparisons in Section~\ref{sec:direct} cover adjacent
simultaneous-birth pairs. We now bound all remaining first-sharing pairs,
including nonadjacent pairs born at the same height. The principal difficulty is that the old
weights depend on earlier matchings. We retain their actual prefixes in a
backward calculation. A covariance term is then charged to the two
positions that tested the corresponding deletions, using the capacities
of the earlier matching matrices.

\begin{proposition}[Remaining first-join comparisons]
\label{join:remaining-first-joins}
Consider a finite block and the retained centres, masks, and table
operations of Sections~\ref{sec:centres}--\ref{sec:partitions}, with the
stated parameter hierarchy. At each first sharing of a component, omit
the simultaneous-birth comparisons already treated in
Section~\ref{sec:direct}. For the remaining comparisons, the following
estimates hold uniformly over the permitted prior table history.
\begin{enumerate}
\item The sum of broad pair counts in $\lambda\otimes\lambda$, at
distance at most $C r_{\max}$ and with width weight $r_{\min}$, is $O(1)$.
Here $C$ is any fixed positive constant.
\item The same-width, same-remaining-mask counts in common new cells,
weighted by the single factor $r_i\pi_I(a)$ and the product of the two
raw weights, have total
\[
 O(W)+O_{k_0}(W^{1+c})
\]
for some fixed $c>0$.
\end{enumerate}
The constants are independent of the slot cap $N_0$ and of the finite
block. Thus, together with the direct comparisons, these estimates
establish Proposition~\ref{tab:first-join-input}.
\end{proposition}

We prove the proposition in stages. All point counts below are per unit
length, with one first-point lift chosen and the second point ranging
over the relevant near lifts. We use the centre notation
\[
 G=(L,M),\qquad U=L/G,\qquad T=M/G,\qquad
 \Theta_{ij}=G r_{\min},\qquad K=[L,M]r_{\max}.
\]
When a bound is summed over masks, its outer-mask coefficients are kept
separate: they are independent probability factors for a broad count,
and one common probability factor for a clipping count.

\subsection{The two kinds of first join}

\begin{lemma}[Classification of the remaining pairs]
\label{join:classification}
Suppose centres $i,j$ first share a component at height $Y$, and their
comparison is not a simultaneous-birth comparison treated in
Section~\ref{sec:direct}. Either
\begin{equation}\label{join:distant}
 \log(UT)>\beta Y/2,
\end{equation}
or, after interchanging the centres, $i$ is old, $j$ is a medium-type
newborn, and
\begin{equation}\label{join:mixed}
 Y=A_0 B_j.
\end{equation}
In this case $B_i/B_j$ is a sufficiently small fixed fraction,
$\log(UT)=O(\beta A_0 B_j)$, and the sum of the
reciprocals of all prime divisors of both centres below bin $Y$ is at most
$s_{\rm low}B_j$, with the arbitrarily small constant specified in the
parameter hierarchy. The same alternatives apply after replacing an old
endpoint by a centre in one of its earlier components.
\end{lemma}

\begin{proof}
If both centres are newborn but their pair was not treated by the
direct comparisons, then $\log(UT)>\beta Y$: those comparisons cover
the adjacent newborn pairs. Such a pair satisfies
\eqref{join:distant}, even if a path of other newborn centres puts
the two centres in one component.

If both centres are old, they belonged to different components at the
preceding height $Y/2$. The previous adjacency threshold therefore gives
\eqref{join:distant}. This also defines the distant case when one or both
centres are newborn.

Suppose \eqref{join:distant} fails. The two centres are then directly
adjacent at activation. A high-type newborn has the same last dense bin
as every neighbour at that height, since an exponent discrepancy at a
prime in that bin already exceeds the edge log-cost. Such neighbours
start together, so they give a simultaneous-birth comparison. The
remaining newborn is consequently medium type. Initially its start is
at most $2D_0 B_j$ up to the fixed onset convention, and hence
$\log(UT)=O(\beta D_0B_j)$. The centre exponents agree at every prime
whose bin has height at least $\ell B_j$.

Centres in the same band would have the same start. A centre in a larger
retained band cannot start earlier. Thus the old centre belongs to a
smaller band, separated from $B_j$ by the prescribed large factor. If a
common dense bin existed at height at least $\ell B_j$, its height would
exceed $D_0 B_i$, contradicting the retained medium type of $i$. There is
therefore no such dense bin. In particular the newborn starts at
$A_0B_j$, rather than at twice a later dense height. The bins below
$\ell B_j$ contribute $O(\ell B_j)$ to the reciprocal-prime sum; all
later bins below $Y$ are sparse and contribute $O(\kappa A_0 B_j)$.
The choices of $\ell$ and $\kappa A_0$ give the assertion.

An earlier component of an old centre is contained in its old child at
the current join. Replacing that centre inside the earlier component
keeps it old and keeps the two endpoints in different old children, or
keeps it opposite the same newborn. The preceding argument applies to
the replaced pair as well.
\end{proof}

Call the alternatives \emph{distant} and \emph{mixed}, respectively.
For a comparison at height $Y$, fix its processed masks $d,e$ and write
\[
 v'=L/d,\qquad w'=M/e.
\]
All prime factors of $d,e$ are below the present bin. Write $a,b$ for
the outer masks, with either their independent laws or their single
common law. Here $I$ is the component's current common remaining part:
the broad coefficient is $\pi_I(a)\pi_I(b)$, whereas clipping requires
$a=b$ and uses just $\pi_I(a)$. Set
\[
 f_0=(Td,Ue),\qquad
 H=B_i+B_j+\log(2UT),\qquad P_0=\rho H,
\]
where the fixed constant $\rho>0$ is sufficiently small.

At relative distance at most $S r_{\max}$, the determinant integer lies
in an interval of length $O(SK/f_0)$. Its cycle has width-weighted size
\begin{equation}\label{join:bare-cycle}
 r_{\min}(v',w')=\Theta_{ij}\frac{f_0}{de}.
\end{equation}
The bare main term is consequently $O(S k_L k_M/(de))$. In a broad
comparison $S$ is a fixed constant. In the clipping comparison the
preliminary partition tests permit
\begin{equation}\label{join:narrow-distance}
 S\ll \exp\bigl(-E_g\max(B_i,B_j)\bigr),\qquad r_i=r_j.
\end{equation}
We drop actual common-cell membership after using this distance bound.
This removes the history-dependent cell predicate from all the
subsequent counts.

For a projected denominator $v'$, put
\[
 R_{\rm done}(v')=
 \prod_{\substack{l\mid v'\\l\ \mathrm{processed}}}R_l.
\]
The identity
\begin{equation}\label{join:processed-currency}
 \frac{R_{\rm done}(v')}{d}=\pi_{\rm done}(d)
\end{equation}
uses the deficit law on the prime powers already processed at this
join. We will return the main terms to this currency, preserving the
outer coefficients separately.

\subsection{A backward expansion of the actual weights}

For an old side, its \emph{adaptive range} consists of the primes from
its start up to, but not including, the present bin. The adaptive range
of a newborn is empty. Above $P_0$, the reciprocal sum of mismatch
primes dividing $UTde$ is bounded uniformly. For $UT$ this follows from
its log-size; for $d,e$ it follows from the good-mask reciprocal purge,
since $d_0\ll\rho$. In particular, for every fixed $C$,
\begin{equation}\label{join:mismatch-restoration}
 \prod_{\substack{p>P_0\\p\mid UTde}}R_p^{-C}\ll 1.
\end{equation}
The constant can depend on the fixed cutoffs, but not on the centres or
masks in the block.

A processed prime $p>P_0$ is called a \emph{bad omission} for the fixed
pair if a side adaptive at $p$ omits it and either the two centre
exponents at $p$ differ or the other side omits $p$ as well.

\begin{lemma}[Backward decomposition]
\label{join:backward-decomposition}
Read in decreasing order every prime divisor of either centre above
$P_0$ whose size lies in the union of the two adaptive ranges. For the
fixed current grids, masks, and distance
predicate, the weighted pair count is bounded by the sum of the
following nonnegative terms:
\begin{enumerate}
\item a continuing product term, stopped at the low fields or at the
first bad omission;
\item one separately stopped joint-loss term at each common full factor
where both deleted parts are nonconstant and at least one is an
adaptive linked deletion.
\end{enumerate}
On the continuing term, already read primes supply, up to a uniform
constant, the product of the corresponding ordinary $R_p$ factors on
the two rows. Each adaptive joint-loss term has the actual raw link
masses as coefficients. No condition inherited from a larger-prime
alignment remains on the continuing term.
\end{lemma}

\begin{proof}
Keep the pair domain on its current grid throughout this expansion.
The only position restriction is the distance predicate; row and mask
restrictions remain allowed. On a side adaptive at a prime $p$, denote
the actual ancestor weights just before and just after its update by
$w_x^-$ and $w_x^+$. Between successive primes read backwards, discard
any intervening clipping, preliminary tests, or further multipliers
from the upper bound. Monotonicity then bounds the current weight by
$w_x^+$, and after processing the update the continuing term retains
$w_x^-$. If the current row omitted $p$, its locations already lie on
the successful ancestor subgrid; no unrestricted success probability
is inserted at this point.

When this procedure leaves a side's adaptive range, its actual birth
weight is an upper bound for the remaining prefix. Replace that weight
by the product upper version of Proposition~\ref{tab:virtual-marginals}.
In each fixed static anchor choice, keep one active birth exclusion at
each prime, chosen for that vertex independently of its position or
partner; if no exclusion is active, keep the actual scalar. Drop the
other filters. Later-processed masks use primes disjoint from the birth
digits and hence multiply the current numerator by units at all these
static primes. The retained exclusions are therefore fixed residues
in the current numerator. Average the static anchor choices only after
the corresponding upper calculation, independently for the two
vertices.

If both adaptive ranges are empty, make these static replacements
immediately. There are no bad-omission or joint-loss terms in
that case. The continuing term goes directly to the remaining-field
count, which retains all high static factors above its low cutoff.
This includes nonadjacent newborn pairs in the distant case.

Stop the continuing term at its first bad omission. At any other
mismatch, whether a centre-exponent discrepancy or an actual omission,
drop the field at $p$. Equation~\eqref{join:mismatch-restoration} pays
for restoring its ordinary factor throughout this branch.

It remains to describe a common full factor $p^\nu$. Simultaneous
translation by $1/p^\nu$ preserves the two current grids and their
distance. On adaptive sides it preserves the actual pre-$p$ prefixes
and transports the links equivariantly. On a static side the smaller
retained factors are invariant. Write the adaptive multiplier as
$1-\mathcal D_p$, where $\mathcal D_p$ is the linked-success deletion
fraction plus $1/p$ times the unpaired fraction. Its orbit mean is
$1/p$. A static field is either a fixed forbidden colour or the
constant $R_p$.

More explicitly, at an adaptive full-factor point $x$, the centered
deleted raw mass is
\begin{equation}\label{join:centered-link-loss}
 \sum_{\xi}\mathfrak m_{x,\xi}
 \left(\mathbf 1_{\{\xi\ \mathrm{succeeds}\}}-\frac1p\right),
\end{equation}
where $\mathfrak m_{x,\xi}$ is the actual raw mass linked from $x$ to
the plus entry $\xi$. These masses are transported unchanged on the
joint orbit. Indeed, writing
$M_x=\sum_\xi\mathfrak m_{x,\xi}\le w_x^-$ and
$I_\xi=\mathbf1_{\{\xi\ \mathrm{succeeds}\}}$, the deleted raw mass is
\[
 \frac{w_x^--M_x}{p}+\sum_\xi\mathfrak m_{x,\xi}I_\xi.
\]
Consequently
\[
 w_x^+=R_pw_x^-
       -\sum_\xi\mathfrak m_{x,\xi}(I_\xi-1/p).
\]
For two adaptive full-factor sides, orbit averaging gives the exact
identity
\begin{equation}\label{join:raw-covariance}
 \mathbb E(w_x^+w_y^+)
 =R_p^2w_x^-w_y^-
 +\sum_{\xi,\eta}\mathfrak m_{x,\xi}\mathfrak m_{y,\eta}
       \left(\mathbb E(I_\xi I_\eta)-\frac1{p^2}\right).
\end{equation}
The centered part of a static forbidden field has the
same form with one indicator; a scalar field has no centered part.
Expanding the product of the two updates, the orbit average is at most
$R_p^2$ times the two pre-prefixes plus the average joint term of their
nonconstant success-losses. Indeed the covariance expansion of
\eqref{join:centered-link-loss} contains the joint-success kernel minus
the nonnegative product-of-means kernel.

Continue only the product-main term. Stop each joint-loss term at
this prime, retaining the actual link masses and, for a static
indicator, its smaller static factors. At least one side is adaptive
at a prime in the union being read, so a nonzero separately stopped
term has an adaptive linked side. Summing over locations turns the
joint orbit averages into the corresponding counts of actual
successes. Such a stopped term is not subsequently expanded into
further adaptive covariance terms.

This construction never assumes independent match plans. It retains
the actual prefix whenever a link decision is needed, and each
nonconstant coefficient is precisely $\mathfrak m_{x,\xi}$, not a
fractional coefficient multiplied by an enlarged prefix. All the
ordinary factors from larger read primes are now constants. Their
mismatch restoration is uniformly bounded by
\eqref{join:mismatch-restoration}, proving the stated form of every
branch.
\end{proof}

The expansion has three kinds of terminal term: the continuing product
reaches the low fields, it stops at its first bad omission, or a separate
joint-loss term stops where it splits off. The first two keep the
original centre pair and require a periodic count of the remaining
fields. The third requires an additional calculation before it can be
charged to the positions that tested the corresponding deletions.

\subsection{Counting terms without adaptive links}

\begin{lemma}[Remaining-field count]\label{join:endpoint-count}
The retained fields through $P_0$ on the two projected grids have
periods dividing those denominators, of sizes at most
$\exp(O(P_0))$, and their separate product means are the corresponding
ordinary factors. Remaining high static fields, if present, increase
the main-term constant by a bounded factor and the roundoff by at most
$\exp(C\Sigma)$, where
\[
 \Sigma=\sum_{\substack{p\mid LM\\ \log\log p<Y}}\frac1p.
\]
The period loss can be taken at most $\exp(\epsilon H)$ by choosing
$\rho$ sufficiently small after $\epsilon>0$ is fixed.
\end{lemma}

\begin{proof}
For adaptive low fields, the modulus and separate product means follow
from Corollary~\ref{tab:intermediate-fields} and
Equation~\eqref{eq:core-loss}. If only static factors remain, their prime
residue periods give the same bound directly.
Apply Lemma~\ref{ar:cycle} to the product of the two retained low-field
weights. Their separate means give the independent residue mean in that
lemma, and the product of their periods is at most $\exp(O(P_0))$.

Suppose some high static factors remain. Then their low companion
fields are static and have coprime moduli. Drop high mismatch factors
at the uniformly bounded restoration cost. At a high common full
factor, the two forbidden residues either coincide on the determinant
cycle or do not. Their cycle density, relative to the two ordinary
factors, is at most
\begin{equation}\label{join:static-coincidence}
 \prod_p\left(1+\frac Cp
       \mathbf 1_{\{n\equiv n_p\pmod p\}}\right),
\end{equation}
where $n$ is the determinant integer and $n_p$ is the possible
coincidence residue. With only one static factor at $p$ there is no
extra factor. The digit CRT separates these high factors from the low
fields.

Expand \eqref{join:static-coincidence}. A term imposes one congruence
on $n$ modulo a squarefree product $Q'$. Since $Q'$ is coprime to the
low periods, the progression form of Lemma~\ref{ar:cycle} gives the interval length
divided by $Q'$, plus $O(\exp(O(P_0)))$ roundoff, both multiplied by
the ordinary mean of the low fields. The main extra product has local
increments $O(1/p^2)$ and is bounded. The roundoff extra product is at
most $\exp(C\Sigma)$.

There is no unhandled combination of high static fields and adaptive
low fields. If either side is adaptive at or below $P_0$ on the
continuing term, the union of adaptive ranges covers every higher
prime below the join; those factors have already been read. Replace
the remaining actual prefix by its product upper version through
$P_0$ on the current grid. If a bad omission has stopped the branch
and lower adaptive fields are dropped, only static residue fields are
needed. Finally $P_0=\rho H$ and a sufficiently small $\rho$ give the
claimed period loss.
\end{proof}

Consider first the continuing term when it reaches the low fields.
Lemma~\ref{join:endpoint-count} restores
$R_{\rm done}$ on both sides in the main term. By
\eqref{join:processed-currency}, summing the masks with their assignment
restrictions gives $O(m_Lm_M)$. In the clipping count the small factor
$S$ in \eqref{join:narrow-distance} absorbs the possible inverse
assignment density arising from the common outer mask.
To see this explicitly, for side $i$ let
\[
 A_i(a)=\sum_{\substack{d:\,da\text{ assigned and good}}}
               \pi_{{\rm done},i}(d).
\]
Then $0\le A_i\le1$ and $\mathbb E_a A_i\le\delta_{L_i}$.
Independent outer laws give at most $\delta_L\delta_M$. For one
outer law,
\[
 \mathbb E_a(A_i(a)A_j(a))\le\min(\delta_L,\delta_M),\qquad
 \frac{S\mathbb E_a(A_iA_j)}{\delta_L\delta_M}
 \le\frac{S}{\max(\delta_L,\delta_M)}\ll1,
\]
by $\delta_{L_i}\ge e^{-.64B_i}$ and the choice of $E_g$.

Now suppose the term stops at a bad omission $p$, in bin $Y_p$. Drop
adaptive cuts at $p$ and below, while keeping the upper ordinary
factors already obtained. Keep the static upper fields through this
point and apply Lemma~\ref{join:endpoint-count}. For this bound use a
broad distance even in the clipping count. Since the adaptive range
starts in sparse bins, restoring the dropped adaptive factors costs
at most $\exp(C\kappa Y_p)$.

Designate one adaptive omitted side, say $L$. Drop its assignment
restriction. Summing its processed mask gives a factor $1/p$ while
the other side's assignment remains in force. This assertion is
uniform in the fixed outer mask: the unrestricted processed law has
$\Pr(p\mid d)=1/p$. Thus for the common outer law the remaining sum
is at most $\mathbb E_a A_j(a)/p\le\delta_M/p$, and the same bound
holds with independent outer laws. If $p\mid UT$, then
$p>P_0$ and
\[
 \sum_{\substack{p\mid UT\\p>P_0}}\frac{\log p}{p}
 \le \frac{\log(UT)}{P_0}\ll_\rho1.
\]
Since $\log\log p\ge A_0B_i$ on the designated adaptive side and
$\exp(C\kappa Y_p)\le(\log p)^{C\kappa}$, this implies
\begin{equation}\label{join:bad-prime-saving}
 \sum_{\substack{p\mid UT,\ p>P_0\\
                  \log\log p\ge A_0B_i}}
 \frac{\exp(C\kappa Y_p)}p
 \ll \exp(-cA_0B_i).
\end{equation}
If instead $p$ divides the partner mask $e$, the same bound holds at
each fixed good partner mask. Indeed $p>P_0>d_0B_j$, and its good-mask
bound controls $\sum_{p\mid e}(\log p)^{.3}/p$ in this range. Choose
$C\kappa<.3$ to obtain the exponential saving from the same lower
cutoff. This saving pays for $\delta_L^{-1}$ in
\eqref{eq:centre-lower}, also for a coupled outer law. The resulting
main terms again sum to $O(m_Lm_M)$: before this last restoration
their bound is
$k_Lk_M\delta_M e^{-cA_0B_i}
 =m_Lm_M\delta_L^{-1}e^{-cA_0B_i}$.

For either kind of terminal term just described, the roundoff per fixed mask choice,
with its outer coefficients still outside, is at most
\begin{equation}\label{eq:no-link-roundoff}
 C\Theta_{ij}\frac{f_0}{de}
 \exp(\epsilon H+C\Sigma)
 \bigl(1+\omega(d)+\omega(e)\bigr).
\end{equation}
Here density gains have been dropped. The omega factor allows every
possible bad stopping prime; there is only one continuing branch.

\subsection{Charging joint-loss terms to successful plus entries}

At a separately stopped term at $p>P_0$, move each adaptive linked
full-factor portion fractionally along its link to the plus entry
whose success removed it. This transfers an upper-bound counting
predicate; it does not move any retained table location. Denote the
image centres by $L',M'$, leaving an unmapped static centre unchanged.
On a mapped first side, let $d'$ be the below-$p$ processed deficit of
the image plus row, whose pre-success denominator is $L'/d'$; use $e'$
analogously on a mapped second side. These image deficits need not
equal the original minus deficits $d_{<p},e_{<p}$.
Here $d_{<p},e_{<p}$ are the parts of the original masks $d,e$
supported below $p$.

Each image has the same width and belongs to the same component as its
origin at the $p$-step. Each origin and its image have the same prime powers from $p$
upward, and the matching preserves the entire mask after $p$. Cell
span allows us to enlarge the distance predicate to $O(r_{\max})$ on
the image pair. On a static partner, retain its current-grid forbidden
residue at $p$ and its smaller static factors instead.

At this stage it is only the distance predicate that has moved. The
original full-factor locations still satisfy the divisibility
conditions imposed by their later projections. We first average those
original-grid restrictions, then sum the origins using matching
capacities, and finally count the image grids and restore their
first-mass factors. The following exact averaging supplies the first
of these steps.

\begin{lemma}[Averaging later omissions]\label{join:later-omissions}
For each mapped side, write
\[
 d=d_{<p}b_i^+,
\]
where $b_i^+$ is the processed portion of its actual mask above $p$
and below the join bin. In the linked comparison, expansion to the
full minus grid at the $p$-step, followed by averaging its later
projection congruences, costs exactly $1/b_i^+$. These factors multiply
over the mapped sides.
\end{lemma}

\begin{proof}
The full ancestor on this side has denominator $L/d_{<p}$, phase
$b_i^+a\gamma$, and numerator $b_i^+z$, where $z$ is the current
numerator. The post-success plus image has the same phase. Expand back to
this minus grid while retaining the condition that its numerator is
zero modulo $b_i^+$. This is exactly the original current projection,
not a condition on the linked image.

Let $l\mid b_i^+$. All these later omissions are adaptive. Since the
branch did not stop at a larger bad omission, the two original centres
have the same exponent $\nu_l$ and the partner has no omission at $l$.
Translate both sides and all participating links by multiples of
$1/l^{\nu_l}$. Both expanded grids are closed under these
translations, including the current grid of an unmapped static
partner. The translations preserve the pre-$p$ prefix data and links,
success at $p$, the retained factors through $p$ on a static side,
and the broadened distance between the image locations. They also
preserve the congruences at other later-omission primes.

More precisely, put $V_0=L/d_{<p}$, so $\nu_l(V_0)=\nu_l$.
The designated numerator increment is $V_0/l^{\nu_l}$, a unit
modulo $l$ and a multiple of every other later-omission prime and
of $p$. On the pre-success plus grid the denominator is $L'/d'$,
which also has its full $p$-power; its numerator increment under
the same translation is therefore a multiple of $p$, preserving
the success residue. The partner is full at $l$, so it has the same
grid-closure property. Thus all the other congruences and $p$-success remain
unchanged, whereas the designated numerator runs uniformly modulo $l$. Its
divisibility condition therefore averages to $1/l$. Good adaptive
masks have deficit at most one at these primes, and omissions above
$p$ on two mapped sides are disjoint. Successive primewise averaging
gives the claimed product. Equivalently, the translations permute
tuples consisting of minus points and their links, with any static
forbidden entry. All upper ordinary factors are already constants.
No survival condition from a later prime or partition remains in
these tuples, so the image need not survive any later projection.
\end{proof}

\begin{lemma}[Capacity and uniqueness of the charge]
\label{join:capacity}
After the averaging of Lemma~\ref{join:later-omissions}, summing all
originating full-factor labels charges at most the weight of each
successful image plus entry. With two mapped endpoints the bound is
the product of these capacities. The charged image pair first shares
a component at the same join as the original pair, and there is no
extra sum over origin masks or comparison epochs. This holds for
both the product-law broad count and the one-law clipping count.
\end{lemma}

\begin{proof}
Fix an image plus label at its $p$-step, its width, and its post-$p$
mask. Its incoming link masses sum to at most its own raw weight,
and hence to at most one. The fixed post-$p$ mask, split at the
current join height, determines both
$b_i^+$ and the current outer mask $a$. Every originating centre in
this $p$-component has the same prime powers from $p$ upward.
Consequently its carried upper ordinary factors, the factor
$1/b_i^+$, and its current outer-law coefficient depend only on these
preserved data, not on the individual preimage.

Below-$p$ origin masks are already labels in this one link matrix;
they are included in its column sum. Moreover each involved minus
ancestor already has an original assigned-row label, including its
entire future mask. It therefore has only one possible current row
and mask labelling, with its later numerator fixed by the projection
condition before that condition was averaged. Thus no additional processed-mask sum occurs
after the capacity bound. If both sides are mapped, widening the
domain to a predicate on the image pair alone permits the two column
bounds to be multiplied. If a side is static, keep its same upper
static choice throughout this summation and then average its anchors.
Other origin restrictions may be discarded.

In particular, for two fixed successful image labels $\xi,\eta$
with the same outer mask $a$, let $c_\xi,c_\eta$ denote their
preserved ordinary and later-omission factors. Nonnegativity and
the column capacities give
\begin{equation}\label{join:one-law-capacity}
 \begin{split}
 &\sum_{x,y}\pi_I(a)c_\xi c_\eta
           \mathfrak m_{x,\xi}\mathfrak m_{y,\eta}\\
 &\qquad\le
 \pi_I(a)c_\xi c_\eta
       \left(\sum_x\mathfrak m_{x,\xi}\right)
       \left(\sum_y\mathfrak m_{y,\eta}\right)
 \le \pi_I(a)c_\xi c_\eta w_\xi w_\eta.
 \end{split}
\end{equation}
The original pairs can be a restricted subset of the displayed
Cartesian product. There is one outer coefficient on both sides
of this inequality, not its square. Independent outer laws give
the same calculation with $\pi_I(a)\pi_I(b)$ instead.

The earlier $p$-component of each mapped centre is contained in the
appropriate old child of the current join. Hence an image pair could
not have shared a component earlier: that would already have joined
the two old children. In a mixed comparison the newborn is unchanged
and has no earlier table. Thus each image pair is charged at its own
first-sharing epoch. The choice of mapped sides has only finitely
many possibilities. Finally the outer coefficient is preserved,
whether it is $\pi_I(a)\pi_I(b)$ or the single $\pi_I(a)$.
\end{proof}

Bound the mapped plus raw weights by one, retaining the smaller static
factors on any unmapped side. The joint-loss term is then bounded by a
count on two grids with denominators
\[
 \frac{L'}{pd'},\qquad \frac{M'}{pe'}.
\]
On a mapped side $d'$ is the below-$p$ processed mask of the new plus
row. On a static side put $e'=e$; its forbidden residue at $p$ turns
the current grid into one of denominator $M/(pe)$ with a modified
phase. This transformation multiplies the smaller static residues by
a $p$-unit and translates them, so their counts and means remain
unchanged. There is at most one static side in this branch.

Primed centre parameters now refer to the image pair. The
path log-costs to the image centres are
$\ll\exp(h_*Y_p/10)$ by \eqref{eq:component-path}, whereas a mapped
centre has band at most $Y_p/A_0$. The original inequality $p>\rho H$
therefore implies
\[
 p>\rho'H'
\]
for some fixed $\rho'>0$. Use a smaller smoothing cutoff against
$H'$ in Lemma~\ref{join:endpoint-count}. Any high mismatches of the
image pair restore at bounded relative cost by its good masks, after
taking $d_0$ small enough. Retained static fields need no adaptive
averaging here.

The image determinant cycle has width-weighted size
\[
 \Theta'\frac{(T'd',U'e')}{pd'e'},
\]
and its determinant interval has length
\[
 O\left(\frac{[L',M']r_{\max}}
                  {p(T'd',U'e')}\right).
\]
Its main term is therefore
\begin{equation}\label{join:linked-main}
 \frac{k_{L'}k_{M'}}{p^2d'e'},
\end{equation}
multiplied by the carried upper ordinary factors, the factors
$1/b_i^+$ on the mapped sides, and the ordinary factors supplied by
the remaining static tests.

For a mapped side the first-mass quantity to restore is
\[
 k_{L'}\pi_{\rm done}(d'pb_i^+).
\]
Its geometric factor $k_{L'}/(pd'b_i^+)$ and its ordinary factors
above $p$ are exactly present in \eqref{join:linked-main}. Explicitly,
if $D'$ is the processed prime-power part of $L'$ at the join, then
\[
 \pi_{D'}(d'pb_i^+)
 =\frac{R(D'/(d'pb_i^+))}{pd'b_i^+}.
\]
Only its ordinary factors through $p$ remain to be restored, at cost
at most $\exp(CY_p)$. Drop this side's assignment restriction and
sum the deficit law: the required deficit one at $p$ has unrestricted
probability at most $1/p$. Thus the geometric $1/p$ becomes the
forced-omission probability; it does not disappear in this sum. On a
static partner keep the actual assignment restriction on $e'$; its
forbidden $p$-residue supplies an additional $O(1/p)$. The same
calculation works with a coupled outer mask, because mapped sides
have dropped their assignment restrictions while the static side
retains its own. At a fixed common outer mask $a$, the mapped
processed sum is at most $1/p$ and the static side contributes at
most $A_j(a)/p$ after its field means are restored. Averaging $a$
gives at most $\delta_{M'}/p^2$. Two mapped sides instead give
at most $1/p^2$ before restoring their assignment densities.

For each mapped centre, $Y_p\ge A_0B'$. Summing the resulting
$1/p^2$ over these primes more than pays both $\exp(CY_p)$ and the
inverse assignment densities from \eqref{eq:centre-lower}. Thus all
such main terms are charged by $O(m_{L'}m_{M'})$ to the image pair.
This conclusion uses the capacities and unique comparison epoch of
Lemma~\ref{join:capacity} before summing centre pairs.

The roundoff, per image masks and stopping prime, is at most
\begin{equation}\label{eq:linked-roundoff}
 C\Theta'\frac{(T'd',U'e')}{pd'e'}
 \prod_{\mathrm{mapped}\ i}(b_i^+)^{-1}
 \exp(\epsilon H'+C\Sigma').
\end{equation}
We have dropped ordinary-factor gains, but kept the outer coefficients.
Unlike the main-term restoration, this bound has no
$\exp(CY_p)$ loss.

\subsection{Summing roundoff and completing the comparisons}

All main terms are now bounded by products of first masses. It remains
to sum the two roundoff bounds \eqref{eq:no-link-roundoff} and
\eqref{eq:linked-roundoff}. The charged pair is the original pair in
the first bound and the image pair in the second. Write its parameters
without primes in either formula. At common centre exponents the
local unrestricted mask sum for the gcd ratio is $1+O(1/p)$; at an
exponent discrepancy it is bounded by a fixed divisor power times
$1+O(1/p)$, as in Section~\ref{sec:direct}. The same calculation
permits the factor $2^{\omega(d)+\omega(e)}$, which bounds the
omega factor in \eqref{eq:no-link-roundoff}. Additional upper masks
sum reciprocally, and in \eqref{eq:linked-roundoff} the stopping
primes are summed with their factor $1/p$. These sums give
\begin{equation}\label{eq:summed-roundoff}
 C\Theta_{ij}\tau(UT)^C
       \exp(\epsilon H+C\Sigma).
\end{equation}
The outer laws, including the one-law version, are probability laws
throughout. The coefficients of $\Sigma$ in this estimate are
absolute.

\paragraph{Distant pairs.}
Here $\Sigma\ll Y$ and \eqref{join:distant} holds. Take $\beta$
sufficiently large and then $\epsilon$ sufficiently small. Substituting
\eqref{eq:centre-kernel} and \eqref{eq:centre-lower} into
\eqref{eq:summed-roundoff}, the factor $\exp(CY)$ is paid by a small
part of the available negative power of $UT$. With mass exponent
$u=.75$, a strict margin remains for the tensor estimate, as does a
fixed residual exponential decay in $Y$. The factor $\tau(UT)^C$
costs an arbitrarily small further power. The residual decay pays the
shell multiplicities and the height sum, while a positive power of
the width ratio remains for summing exact dyadic widths. These
roundoffs consequently total $O_{k_0}(W^{1+c})$ for some $c>0$.

\paragraph{Mixed pairs.}
Let $j$ be the newborn larger-band centre. By
Lemma~\ref{join:classification}, $\log(UT)=O(\beta A_0B_j)$ and
$\Sigma$ is arbitrarily small relative to $B_j$. If
\[
 \log(r_i/r_j)>.65 B_j^*,
\]
the width-gain consequence following \eqref{eq:exterior-gain} applies
to \eqref{eq:summed-roundoff}. With sufficiently small harmonic and
period exponents, it leaves an exponential saving in $B_j$ and a
positive residual width power, again summable with a mass power
beyond one.

Otherwise the exterior alternative in \eqref{eq:exterior-gain}
holds for every assigned row of $j$. Keep its processed mask $c_j$
fixed and set $W'=M/c_j$. Its good-mask condition gives
$\tau(c_j)\le e^{s'B_j}$, since the entire processed segment has the
small reciprocal-prime sum. Divide \eqref{eq:no-link-roundoff} by
this row's first-mass quantity
$r_jW'R_{\rm done}(W')$, leaving its outer law in place. Summing
only the old side's processed mask gives
\begin{equation}\label{join:mixed-relative-error}
 C\exp(\epsilon H+C\Sigma)\frac{r_{\min}}{r_j}
 \left(\frac{W'}{(W',L)}\right)^{-1}
 \tau(U)^C\tau(c_j)^C.
\end{equation}
For completeness, before this sum the geometric ratio is
$\frac{r_{\min}}{r_j}(W',L/d)/W'$. Its sum is bounded exactly as in
the conditional target-grid calculation of
\eqref{eq:target-grid}; the extra omega factor is absorbed by fixed
divisor moments. The inverse ordinary density costs at most
$\exp(C\Sigma)$.
Writing $E_{\rm ext}=W'/(W',L)$, the primewise sum just used is
\[
 \sum_d\frac{(W',L/d)}{W'}
 \ll E_{\rm ext}^{-1}\tau(U)^C\tau(c_j)^C\exp(C\Sigma).
\]
At a prime, the free deficit allowance before the denominator of
this ratio grows is at most $\nu_p(U)+\nu_p(c_j)$; beyond that
allowance the terms decrease geometrically. This proves the bound
also with the fixed divisor moments for the omega factor.

In a mixed joint-loss term the newborn side is necessarily static;
its centre and assigned row are unchanged. Its
fictitious restricted denominator is $W'/p$, and the other counted
denominator also contains the factor $1/p$. Relative to the original
newborn first mass this gives the same bound
\eqref{join:mixed-relative-error}, with an additional $1/p$ whose
sum is absorbed in $\exp(C\Sigma)$. The exterior ratio is unchanged
by dividing this common full factor from both denominators: at the
stopping prime $p$ the two image centres have the same full exponent
and $p$ divides neither $d'$ nor $c_j$, so
\[
 \left(\frac{L'}{pd'},\frac{W'}p\right)
   =\frac1p(L'/d',W'),\qquad
 \frac{W'/p}{(W'/p,L'/p)}=\frac{W'}{(W',L')}.
\]
The classification of Lemma~\ref{join:classification} still applies
to the image pair, so its old centre has the required smaller band.
Dropping origin restrictions did not change the newborn assigned
row or its divisibility by $W'$.

An assigned true denominator $v$ of $j$ divides $W'$. Therefore
\[
 \frac{W'}{(W',L)}\ge \frac{v}{(v,L)},
\]
and \eqref{eq:exterior-gain} makes the product of the width factor
and the inverse exterior ratio in
\eqref{join:mixed-relative-error} exponentially small in $B_j$.
Choose first $s'$ small enough for $\tau(c_j)^C$, then the small
harmonic allowance, and finally the period and divisor-function
powers sufficiently tiny relative to $\beta A_0$. There remains an
exponential saving more than sufficient to sum all smaller-band
centres $i$ by their centre count. The resulting contribution is
$O(W)$, and can be made a small fixed multiple of $W$ where needed.

All terminal main terms were bounded by the product of the original or
charged image masses, so their total is $O(W^2)$. The distant and
width-separated roundoffs are $O_{k_0}(W^{1+c})$, and the oriented
mixed roundoffs are $O(W)$. These conclusions apply separately to
the broad and narrow counts with their respective mask laws. Since
$W\le1$, they prove Proposition~\ref{join:remaining-first-joins}.
In particular none of the constants used for clipping depends on
$N_0$.

\subsection{Closure of the finite construction}

We finish by explaining the order in which the proved estimates are
used. This also records the parameter margins needed to make the
finite construction uniform over arbitrarily late blocks.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The elementary initial reduction deals with widths for which
$vr(v)$ does not tend to zero. In the remaining case fix a putative
positive-measure set avoided by a whole tail, dyadically reduce the
positive widths, and use the finite late blocks of mass between
$w$ and $2w$ described in Section~\ref{sec:preliminaries}.

The row and centre selections and the birth prescriptions are fixed
from the block before the adaptive history starts. The additional
birth holes in Section~\ref{sec:direct} depend only on those assigned
lists. They therefore do not require any estimate for an already
running table. Direct comparisons establish the adjacent simultaneous-birth
part of Proposition~\ref{tab:first-join-input}.

Proceed through the finitely many component starts, component joins,
and sparse-prime steps in chronological order. At a component change,
the histories from earlier steps and their deterministic danger
trains are already defined. Section~\ref{sec:partitions} constructs
the new padded partitions from these histories, proving
Proposition~\ref{tab:partition-input} with negligible first-mass
loss. The preliminary tests give the cell spans used in the present
section. Proposition~\ref{join:remaining-first-joins} then supplies
the remaining broad and narrow comparisons for the actual old
prefixes, before the new clipping operation. The narrow comparison
bound permits that clipping. At each following sparse-prime update,
Section~\ref{sec:sparse} applies to the existing component and
partitions; it supplies Proposition~\ref{tab:sparse-input} and the
new danger trains for later component changes. Its nonexceptional
tight pairs coarsen the accounting classes as prescribed. Thus no
stage needs an estimate depending on a future partition or a future
adaptive matching. Induction completes the finite construction.

We recall the quantitative margins. In sparse bins the harmonic
losses that compete with a height saving are at most
$\exp(C(\sigma+\kappa)Y)$, with absolute $C$, whereas the off-gap
window supplies decay with exponent $(1-\sigma_g)Y$. The choices of
$\sigma,\kappa$ leave a strict margin. The switching budgets in
the sparse and direct comparisons are taken sufficiently large for
their crude unneutralized costs; sparse errors may carry
$\tau(UT)^C$ separately. At foreign-child partition tests and
distant first joins, the coefficients of $C\Sigma$ or $CY$ are
absolute, so the initial large choice of $\beta$ pays them. At
mixed comparisons the divisor loss needs only absolute smallness
of $Cs'$. After these choices the period and exterior powers can be
taken as small as required relative to $\beta A_0$.

Take $k_0$ small enough to absorb all bounded-height effects and to
make the exponentially decaying selection and history losses small
relative to the selected mass and the fixed avoided-set measure.
The colour, type, and hard-mean constants have already been fixed.
The later choice of the large fixed threshold $\lambda_0$ makes
the birth-hole losses small. Its holes use squarefree low moduli,
so this choice does not enlarge the prime powers required for core
equivariance. Finally choose $w$ sufficiently small for the terms
$O_{k_0}(W^{1+c})$ and choose the fixed cap $N_0$ sufficiently large
for the narrow comparison losses. All these choices are independent
of the location of the finite block in the tail.

We have now proved all three inputs to
Proposition~\ref{tab:conditional-reduction}. The energy bound of
Proposition~\ref{tab:energy-bound}, the virtual marginals of
Proposition~\ref{tab:virtual-marginals}, and their equidistribution
give the contradiction to the positive avoided set exactly as in
that reduction. Consequently, for the fixed shift and width
sequence, almost every point belongs to infinitely many of the
prescribed interval trains. This proves the theorem.
\end{proof}
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/joins.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/partitions.tex ====
\section{Construction of the operational partitions}
\label{sec:partitions}

We now prove Proposition~\ref{tab:partition-input}.  At a component
creation or change, the new cells must be short and padded, must contain
the inherited accounting classes, and must not put surviving points from
two different cells of one old child into a common new cell.  The danger
trains of Construction~\ref{sp:danger-trains} allow us to satisfy these
requirements simultaneously.  We first bound the cost of keeping target
points away from too much inherited danger history, and then choose new
boundaries outside the inherited class hulls.

Throughout this section, every comparison is between centres of the same
exact width $r$.  We work chronologically, conditional on the entire
construction before a component creation or change at height $Y_P$.
All old partitions, weights, classes and histories are therefore fixed.
We assume the partition assertions already hold for the old children.
We retain the stronger induction that the actual arc length of an
occupied old cell, at any surviving centre-$j$ point, is at most
$2r\exp(-2E_gB_j)<1/4$.  The two directional boundary tests below
give this bound at each creation; later projections and deletions
do not enlarge a cell.  At the first creation there are no old cells.
The sparse estimates in Proposition~\ref{sp:sparse-estimates} apply to this
prior history, and Construction~\ref{sp:danger-trains} supplies its
deterministic trains.  No choice made below changes an earlier partition
or an earlier coarsening edge.

\subsection{Weighted target-grid counts}

Let $M=L_j$ be a target centre.  At the current projection write
\[
 W'=M/c,
 \qquad
 R_{\mathrm{done}}(W')
   =\prod_{\substack{p\mid W'\\p\ \mathrm{processed}}}R_p,
\]
where $c$ is the processed part of one of its retained good row masks.
Fix the remaining mask and hence its phase.  Its current grid can be written as
$x_z=(z+\xi)/W'$, with $z$ running through $W'$ consecutive integers.
Denote the actual raw weight there by $w_z$, setting $w_z=0$ at a dead
entry.  The remaining-mask probability is not included in $w_z$.

A source train generated by centre $L=L_i$ in an earlier sparse bin $Y'$
has denominator $V'=L/d$.  Here $d$ includes the single omission $q$
at its generating step, and $q$ is the largest prime divisor of $d$.
For one of its prescribed offsets $y_0$, its centres are
$(n+y_0)/V'$, $n\in\mathbb Z$.  As usual, put
\[
 G=(L,M),\qquad U=L/G,\qquad T=M/G,
 \qquad m=\frac{W'}{(W',V')}.
\]
In particular $m\mid Td$.  Counts involving the two grids use periodic
lifts, so a source centre is counted every time it lies in the indicated
interval, even if several such centres are near the same target point.

\begin{lemma}[A target grid against a source train]
\label{part:target-grid}
For every sufficiently small fixed $\epsilon>0$, every fixed constant
$C_0$, and $0<R\le C_0$, one has
\begin{equation}
\begin{split}
 \frac{1}{W'}\sum_{z\bmod W'}w_z
 \sum_{n\in\mathbb Z}
 \mathbf 1_{\{|x_z-(n+y_0)/V'|\le C_0rR\}}
 \ll_{\epsilon,C_0}
 R_{\mathrm{done}}(W')e^{CY'}
 \left(V'rR+\frac{(2UT)^\epsilon}{m}\right).
\end{split}
\label{eq:target-grid}
\end{equation}
The constant $C$ in the harmonic factor is absolute.  The estimate is
uniform in the preceding history, fixed masks and phases.  It holds also
at a new insertion, where only static target fields are present.
\end{lemma}

\begin{proof}
Use the single-row product upper fields from
Corollary~\ref{tab:intermediate-fields}.  First fix the static anchor choices, select
one active birth exclusion at each applicable prime, and work inside
their average.  Soft tests, holes and intervening discards can be
dropped.  The resulting product upper version dominates the actual
weight after the static average is taken.  Each retained field at a
processed prime divisor of $W'$ has its prescribed one-row mean $R_p$.

Set $P_0=\rho\log(2UT)$, where $\rho>0$ will be sufficiently small in
terms of $\epsilon$.  The proximity multiplicity on the left of
\eqref{eq:target-grid} is a query modulo $m$ in $z$.  Drop the fields
at processed primes $p\mid m$ with $p>P_0$.  Relative to their ordinary
means, this costs at most
\[
 \prod_{\substack{p\mid m\\p>P_0}}R_p^{-1}
 \le \exp\bigl(CY'+O_\rho(1)\bigr).
\]
Indeed $m\mid Td$, primes of $d$ lie below the end of bin $Y'$, and
their reciprocal sum is $O(Y')$.  For primes of $T$ above
$\rho\log(2UT)$, the bounds on their size and on
$\sum_{p\mid T}\log p$ give a bounded reciprocal sum, with constant
depending only on $\rho$.

Keep every multiplier as the function defined by the actual prior
history; removing a field from an upper product does not recompute
any matching plan.  Average the retained large fields in decreasing
prime order.  At an adaptive full factor $p^\nu$, the current grid
still contains that full power.  Since $p\nmid m$, $p^\nu\mid V'$
also, so translation by $1/p^\nu$ preserves both the grid and the
proximity query.  Every retained earlier field is invariant on this
orbit, and the adaptive field supplies exactly its mean $R_p$.
An adaptive omission supplies only the scalar $R_p$ when its residual
power is positive, and supplies no field when that power is zero.
At a retained static prime of residual exponent $\mu>0$, translation
by $1/p^\mu$ fixes the query and all other retained static residues,
while its numerator step is a unit modulo $p$; hence its single
exclusion again averages to $R_p$.  All static primes of this target
precede its adaptive range, so no unaveraged adaptive field remains
when this static averaging is performed.  No independence of the
matching plans is used.

Deleting any of the high fields has left the low fields themselves
unchanged.  If $p_*\le P_0$ is the last adaptive update at such a
prime, let $Q_*$ be its historical post-update core.  The low fields
are individually invariant under translation by $1/Q_*$.  Intersect
this group with the current grid: the intersection consists of
translations by multiples of $1/(W',Q_*)$.  Thus their joint numerator
period divides
\[
 D=\frac{W'}{(W',Q_*)}\le\frac{M}{Q_*}
   \le\exp(O(p_*))\le\exp(O(P_0)).
\]
Here $Q_*\mid M$ and $W'\mid M$ give the first inequality prime by
prime, and \eqref{eq:core-loss} gives the second.  If no low adaptive
update exists, only static residue fields and scalars remain; one can
take $D=\prod_{p\mid W',\ p\le P_0}p$, which divides $W'$ and has
the same bound.  Hence the period estimate remains valid after
projection and after arbitrary deletion of the specified high fields.
Choose $\rho$ small in terms of $\epsilon$ to obtain
$D\ll_\epsilon(2UT)^\epsilon$.  Bounded $P_0$, including $U=T=1$,
is harmless; with no nonconstant low fields take $D=1$.

Write $g=(W',V')$.  In units of the source grid, the target coordinate
has the form
\[
 V'x_z-y_0
   =\frac{V'/g}{m}z+\text{a fixed shift},
 \qquad (V'/g,m)=1.
\]
Conditional on a residue of $z$ modulo $D$, its residues modulo $m$
run uniformly through a coset whose mesh, after this unit
multiplication, is $(D,m)/m$.  Thus the average number of source
centres in the query is
\[
 O\left(V'rR+\frac{(D,m)}{m}\right),
\]
uniformly in that residue modulo $D$.  This is the ordinary uniform
grid count, including multiplicities.  Multiply by the low product
field and average over its residues.  Its product mean, together with
the already averaged large fields, supplies the ordinary factors in
$R_{\mathrm{done}}(W')$, apart from the bounded restoration cost
already displayed.  Finally average the static anchor choices.  This
proves \eqref{eq:target-grid}.
\end{proof}

The next mask calculation separates an unrestricted source sum from
the target assignment.  This distinction will be important when the
target density is too small to discard its assignment restriction.
For a processed mask $c$, let $c_{\le Y'}$ be its part supported on
primes up through the end of bin $Y'$.

\begin{lemma}[Source masks and target deficit currency]
\label{part:mask-sums}
For the source masks of bin $Y'$,
\[
 \sum_d\frac1d\le e^{CY'}.
\]
At every fixed target mask $c$, put
\[
 E_{\mathrm{ext}}=
 \frac{W'}{(W',L)}=\frac{T}{(T,c)}.
\]
Then
\begin{equation}
 \sum_d\frac1m
 \ll E_{\mathrm{ext}}^{-1}
       \tau(U)\tau(c_{\le Y'})e^{CY'}.
\label{part:conditional-mask-sum}
\end{equation}
Under the unrestricted deficit law on the processed target mask,
\begin{equation}
 \mathbb E\bigl[(T,c)\tau(c_{\le Y'})\bigr]
 \le \tau(T)^C e^{CY'}.
\label{part:target-mask-moment}
\end{equation}
All the harmonic exponents here have absolute coefficients.
\end{lemma}

\begin{proof}
We may extend each nonnegative source sum to all divisors of $L$
supported below the end of bin $Y'$.  The first inequality follows by
multiplying the geometric sums and using the reciprocal-prime bound.
The generating prime introduces no additional sum: it is the largest
prime divisor of $d$, so is determined by that mask.

For the second inequality, fix a prime and write
$n=\nu_p(L)$ and $w=\nu_p(W')$.  If the exponent of $p$ in $d$ is
$j$, then its exponent in $m$ is
\[
 (w-n)_++\bigl(j-(n-w)_+\bigr)_+.
\]
The first term is the exponent in $E_{\mathrm{ext}}$.  Writing
$a=(n-w)_+$, the local sum after removal of that term is at most
\[
 \sum_{j=0}^n p^{-(j-a)_+}
 \le \frac{a+1}{1-1/p}.
\]
Moreover
\[
 a\le \nu_p(U)+\nu_p(c),
 \qquad
 a+1\le (\nu_p(U)+1)(\nu_p(c)+1).
\]
Only primes in the indicated source segment contribute the second
factor involving $c$.  Multiplication proves
\eqref{part:conditional-mask-sum}.  The formula for
$E_{\mathrm{ext}}$ is also immediate prime by prime from $W'=M/c$.

To prove \eqref{part:target-mask-moment}, use independence of the
deficit coordinates.  At $p^t\parallel T$, with deficit exponent
$J$, its geometric tail gives
\[
 \mathbb E\bigl[p^{\min(t,J)}(J+1)\bigr]
 \ll (t+1)^2.
\]
For example, the terms with $J<t$ sum to at most a constant times
$\sum_{j<t}(j+1)$, and the geometric tail with $J\ge t$ costs
$O(t+1)$.  If the divisor factor is absent, this estimate only
improves.  At primes not dividing $T$, the needed divisor moment is
$1+O(1/p)$.  Their product over the source segment is $e^{CY'}$,
while the other factors are bounded by a fixed power of $\tau(T)$.
This proves the assertion.
\end{proof}

Let $b$ denote a target outer mask and $I$ its remaining part.  Its
baseline first-mass coefficient satisfies
\begin{equation}
 rW'R_{\mathrm{done}}(W')\pi_I(b)
       =k_M\pi_M(cb).
\label{part:target-currency}
\end{equation}
Summing this over the original assigned masks gives $m_M=k_M\delta_M$.
Consequently the main term in \eqref{eq:target-grid} may always retain
the target assignment.  In contrast, when we sum its error without
that restriction, \eqref{part:target-mask-moment} introduces the
relative factor $1/(T\delta_M)$, up to arbitrarily small powers of
$2UT$.  At a fixed assigned target mask we can instead use
\eqref{part:conditional-mask-sum}, with its factor
$E_{\mathrm{ext}}^{-1}\tau(c_{\le Y'})$, and incur no inverse
assignment density.

\subsection{History tests and their first-mass cost}

For an already generated source centre-bin history $(i,Y')$, put
\[
 \ell'=\exp\bigl(-.3\exp(\sigma_gY')\bigr),
 \qquad Z'=\exp\bigl(\exp(h_aY')\bigr).
\]
Use the complete deterministic supply from
Construction~\ref{sp:danger-trains}: for every permissible good
processed mask $d$ generated in this bin, take its list of at most
$Z'$ offsets, and put an interval of radius $r\ell'$ around each
point of the corresponding grid of denominator $L_i/d$.  Denote the
resulting periodic family of intervals by $\mathcal H_{i,Y'}$.
Completeness here means all permissible masks and prescribed offsets
of this already generated history, not future or unprocessed bins.
We do not restrict the supply to trains supporting edges that
happened to survive; later deletion of an entry never deletes a
part of this history.

For a target point $x$ and a specified relative scale $0<t\ll1$,
call a tested bin \emph{threatening at scale $t$} if
\[
 |\log\ell'|\le 100|\log t|.
\]
Apply the following test separately for each tested source
centre-bin pair $(i,Y')$:
\begin{enumerate}
 \item In a threatening bin, discard $x$ if it lies within
 $(\ell'+2t)r$ of any centre of a train in
 $\mathcal H_{i,Y'}$.
 \item In a nonthreatening bin, discard $x$ if the sum of the lengths
 of the intervals of $\mathcal H_{i,Y'}$ meeting
 $[x-2r,x+2r]$ exceeds $e^{-Y'}tr$.
\end{enumerate}
The second sum may count overlaps with multiplicity, which only
strengthens the test.  Both tests use periodic lifts.

At the present component change there are two kinds of tests.  Every
target centre $j$ uses its intrinsic scale
\[
 t_j=\exp(-2E_gB_j)
\]
against all its already generated histories in the component.  In
addition, a point belonging to an old child $D$ uses the child-gap
scale
\[
 t_D=\exp(-1.5E_2Y_D),
\]
where $Y_D$ is that child's creation or last change height, against
histories of \emph{different} old children.  In such a foreign-child
comparison, with $Y_{\mathrm{prev}}=Y_P/2$, the prior component
separation gives
\begin{equation}
 \log(UT)>\beta Y_{\mathrm{prev}},
 \qquad
 Y_{\mathrm{prev}}\ge\max(Y',Y_D),
 \qquad B_i,B_j\le Y_{\mathrm{prev}}/A_0.
\label{part:foreign-separation}
\end{equation}

\begin{lemma}[Cost of a history test]
\label{part:history-test-cost}
For either kind of test, the first mass discarded from the tested
target array on $M=L_j$ by one source centre-bin pair $(i,Y')$ is at
most $e^{-cY'}m_M$.  Here $c>0$ can be taken sufficiently large for
the centre-count summations below, after the fixed parameter choices
and then a sufficiently small $k_0$.  The bound is uniform in the
prior history and does not depend on the slot cap $N_0$.
\end{lemma}

\begin{proof}
\emph{Threatening main term.}\quad
In a threatening bin, apply \eqref{eq:target-grid} with
$R=\ell'+2t$, sum the at most $Z'$ offsets for each $d$, and then
sum $d$.  Keeping the target assignment in the main term and using
$V'=L/d$ and Lemma~\ref{part:mask-sums} gives the relative main cost
\begin{equation}
 \ll e^{CY'}Z'k_L(\ell'+2t).
\label{part:threat-main}
\end{equation}
The threatening inequality implies
$t\le\exp(-.003\exp(\sigma_gY'))$.  Also
$B_i\le Y'/A_0$ and $\log k_L\le .2B_i+O(1)$.
Since $h_a<\sigma_g$, expression
\eqref{part:threat-main} is smaller than any required fixed
exponential in $-Y'$ once the heights are large.

\emph{Nonthreatening tests.}\quad
For a nonthreatening bin, an interval of radius $r\ell'$ meeting
$[x-2r,x+2r]$ has its centre within $3r$ of $x$.
Apply \eqref{eq:target-grid} at this constant relative radius, and
use Markov's inequality for the length threshold $e^{-Y'}tr$.
The length per counted interval is $2r\ell'$.  The factor $e^{Y'}$
from the threshold is absorbed in the harmonic exponential.  In the
main term retain the target assignment; only in the error extend
the target-mask sum to the unrestricted law.  By
\eqref{part:target-mask-moment} and arbitrarily small divisor powers,
the relative cost is at most
\begin{equation}
 C_\epsilon\frac{\ell'}{t}e^{CY'}Z'
 \left(k_L+\frac{(2UT)^\epsilon}{T\delta_M}\right).
\label{part:nonthreat-cost}
\end{equation}
All small powers of $2UT$ here may be enlarged and renamed
$\epsilon$.

We check that \eqref{part:nonthreat-cost} is negligible for both
scales.  Equal widths imply
\begin{equation}
 \log U-\log T=\log k_L-\log k_M,
 \qquad |\log U-\log T|\ll B_i+B_j.
\label{part:equal-width-ratios}
\end{equation}
Thus, for $\epsilon<1/2$, the ratio
$(2UT)^\epsilon/T$ is bounded by an exponential in a small fixed
multiple of $B_i+B_j$.  By \eqref{eq:centre-lower}, the density loss is at most
$\delta_M^{-1}\le \exp(.32B_j^*)\le\exp(.64B_j)$.
Write $A=|\log\ell'|$ and $a=|\log t|$.  In the present case
$A>100a$, so $\ell'/t=\exp(-(A-a))$ has a fixed large margin over
$\exp(-a)$.  At the intrinsic scale, $a=2E_gB_j$; at the
child-gap scale, $a=1.5E_2Y_D$ and $B_j\le Y_D/A_0$.
In both cases this margin pays the displayed density and ratio
losses.  It also pays $e^{CY'}Z'$ and $k_L$, because
$\log Z'=\exp(h_aY')=o(\exp(\sigma_gY'))$.

\emph{Threatening error term.}\quad
It remains to estimate the threatening error, where the factor
$\ell'/t$ is unavailable.  Its unrestricted relative bound is
\begin{equation}
 C_\epsilon e^{CY'}Z'
       \frac{(2UT)^\epsilon}{T\delta_M}.
\label{part:threat-error}
\end{equation}
For a foreign-child test, the threatening inequality gives
$\exp(\sigma_gY')=O(Y_D)$ and hence
$\log Z'=o(Y_D)$.  Equations
\eqref{part:foreign-separation} and
\eqref{part:equal-width-ratios} show that the denominator $T$ pays
\eqref{part:threat-error}: its logarithmic saving is
$(1/2-\epsilon)\log(UT)$ up to $O(B_i+B_j)$, whereas the
remaining height costs are $CY'+o(Y_D)+O(B_j)$.
Taking $\beta$ sufficiently large absolutely absorbs these costs,
uniformly since $Y',Y_D\le Y_{\mathrm{prev}}$.

For an intrinsic test, the threatening inequality instead gives
\[
 Y'=O(1+\log B_j)=o(B_j),\qquad
 \log Z'=o(B_j),\qquad B_i/B_j=o(1).
\]
If $\log(UT)\ge C_0B_j$ for a sufficiently large fixed $C_0$,
the same use of \eqref{part:equal-width-ratios} makes
\eqref{part:threat-error} exponentially small in $B_j$.
Suppose therefore that $\log(UT)<C_0B_j$.  Retain the target
assignment and apply \eqref{part:conditional-mask-sum} separately
at each of its masks.  For every true assigned row $v$ at $M$,
equal widths rule out the width alternative in
\eqref{eq:exterior-gain}, so
\[
 \frac{v}{(v,L)}>\exp(.015B_j^*).
\]
Since $v\mid W'$, it follows that
\[
 E_{\mathrm{ext}}=\frac{W'}{(W',L)}
       \ge \frac{v}{(v,L)}\ge \exp(.01B_j).
\]
The reciprocal-prime sum in the initial segment through bin $Y'$
is $O(Y')=o(B_j)$.  The good-mask initial-segment condition
therefore gives $\tau(c_{\le Y'})\le e^{s'B_j}$.
Relative to this assigned mask's own baseline
\eqref{part:target-currency}, its error is consequently at most
\[
 C_\epsilon e^{CY'}Z'(2UT)^\epsilon
 E_{\mathrm{ext}}^{-1}\tau(c_{\le Y'})
 \le \exp\bigl(-.01B_j+s'B_j+\epsilon C_0B_j+o(B_j)\bigr).
\]
Here the factor $\tau(U)$ has been absorbed by enlarging the
arbitrarily small power.  Choose that power sufficiently small;
$s'$ was already chosen small absolutely.  This gives the required
saving without a factor $\delta_M^{-1}$.  Summing the retained
target assignments completes all cases.
\end{proof}

We next sum these estimates without charging an unchanged history
repeatedly.  At a fixed target centre, its intrinsic scale never
changes.  For each source centre-bin pair, perform the intrinsic
comparison at the first partition change after that history has
been generated and has joined the target's component.  A later
position is a descendant at exactly the same location, and the
tested family is the same full deterministic supply, so it cannot
newly fail that intrinsic test.  For foreign-child comparisons,
the charge is indexed by the fixed triple $(j,i,Y')$, not by a
child's changing name.  Components merge but never split.  If this
source history exists before the first join of $i$ and $j$, that
join is its only possible foreign comparison with the old target
child.  Afterward $i$ and $j$ belong to the same old child at every
later change.  If the history is generated after their join, it is
never foreign to $j$.  Thus changing child-gap scales introduce no
repeated charge.  No monotonicity of the foreign test as $t_D$
changes is being used.

At height $Y'$, the number of possible source centres is at most
$\exp(4Y'/A_0+O(1))$.  Summing
Lemma~\ref{part:history-test-cost} over sources and dyadic heights,
then over target first masses $m_M$, gives $o(W)$ as $k_0\to0$.
The same count, without target weights, bounds the total physical
length per unit period of all danger intervals by
\begin{equation}
 \sum_{i,Y'} e^{CY'}Z'k_{L_i}\ell'<.01,
\label{part:global-danger-length}
\end{equation}
after decreasing $k_0$ if necessary.  Indeed each source mask
contributes at most $2Z'(L_i/d)r\ell'$, and
$\sum_d1/d\le e^{CY'}$.  The double-exponential decay of $\ell'$
pays both this harmonic factor and the centre count.

All history tests are equivariant under the current core.  Each
source supply was invariant under its post-generation core, and
the current common core divides that earlier core.  The predicates
use only distances, lengths and the fixed centre scales.  The
preliminary survivors after these tests therefore retain the
required core symmetry.

\subsection{Inherited hulls and new boundaries}

We have bounded the mass needed to keep short segments around
surviving points mostly outside the inherited histories.  We now
use those free portions to place boundaries.  The construction is
performed separately for each exact width $r$ in the component.

For each nonempty inherited accounting class, order its preliminary
surviving positions within its occupied old cell, whose actual arc
length is less than $1/4$ by the stronger induction.  Define its
\emph{short hull} to be the closed arc between the first and last
of these positions.  Omit empty classes and use a singleton hull
when all positions coincide.

The preliminary positions, inherited hulls and preliminary gaps
between occupied old cells are determined before any jitter is
sampled; they are not recomputed after the subsequent pointwise
purges.  The following observation ensures
that avoiding these hulls is affordable.

\begin{lemma}[Location of inherited class hulls]
\label{part:inherited-hulls}
Every positive-length inherited class hull lies in the union of
the danger intervals belonging to its old child.  It lies in its
old cell, and the hull operation respects current-core translations.
\end{lemma}

\begin{proof}
An inherited equivalence can be witnessed by a historical chain
of actual coarsening edges.  The endpoints never change location
under subsequent projections.  Construction~\ref{sp:danger-trains}
covers each connecting segment by that child's stored history.
If intermediate labels have disappeared, retain their segments
in the witnessing chain: the full deterministic supply has not
been reduced.  The historical chains may backtrack, but their
images lie in one connected component $K$ of the old child's
finite closed danger union.  By
\eqref{part:global-danger-length}, $K$ is an arc of length less
than $.01$; a chain with nonzero winding would cover the circle
and is impossible.

The current class also lies in one occupied old cell $C$.  By the
stronger induction specified at the start of the section, its
actual arc length is at most $2r\exp(-2E_gB_j)<1/4$ at any
surviving centre-$j$ point.  Choose a cut outside $C\cup K$.
Both become real intervals, and the interval between the extreme
surviving class positions lies in both.  Its length is less than
$.01$, so it is the uniquely determined short circular hull.
Old padding keeps it inside the old cell, including with the
half-open endpoint convention.  All classes and histories are
equivariant, and this hull operation commutes with the current
core translations.  Singleton hulls require no historical segment
and satisfy the equivariance assertion directly.
\end{proof}

Write $Q$ for the core at the start of the present bin and put
$T_{\mathrm{core}}=1/Q$.  Subdivide one period into $N$ equal
microintervals, where
\[
 N=\left\lceil\frac{T_{\mathrm{core}}}{r\exp(-E_2Y_P)}\right\rceil,
 \qquad a_P=T_{\mathrm{core}}/N\asymp r\exp(-E_2Y_P).
\]
The stated comparability holds because $Q\mid M$ at every target
centre and hence
\[
 Qr\exp(-E_2Y_P)
 \le k_M\exp(-E_2Y_P)\ll1.
\]
Choose one independent uniform candidate boundary point, called a
\emph{raw line}, in each microinterval, and replicate the
configuration with period $1/Q$.  Skip a line if it
belongs to any closed inherited hull of the preliminary survivors.
The hulls used for this decision are now fixed; they are not
shrunk and the lines are not redrawn after later discards.
Figure~\ref{fig:inherited-hull} illustrates the distinction between
surviving labels and the hull protected by this rule.

\input{figures/inherited-hull}

The unskipped lines define the new cyclic partition.  If none
remain, use the single circle cell.  Use half-open cells with a
fixed orientation when assigning endpoints.  First discard every
entry within
\[
 \delta_P=r\exp(-1.5E_2Y_P)
\]
of a raw line, including a skipped one.  Conditional on the prior
history, each fixed entry has probability
$O(\delta_P/a_P)=O(\exp(-E_2Y_P/2))$ of this discard.

For each preliminary target point $x$ of centre $j$, require an
unskipped line in each directional segment of length $t_jr$
starting at $x$, and discard the entry on failure.  At least half
of either segment is available for such a line.  To see this,
Lemma~\ref{part:inherited-hulls} puts all the skipped hulls in the
danger histories.  The threatening tests exclude their intervals
from these segments: a train centre more than
$(\ell'+2t_j)r$ from $x$ cannot have its radius-$\ell'r$ interval
meet them.  For nonthreatening histories, the total covered length
is at most
\[
 t_jr\sum_{i,Y'}e^{-Y'}<\tfrac12t_jr.
\]
The source-centre count and the large minimum height justify the
last inequality.  Singleton hulls have zero length.

We record a quantitative non-repetition condition for both the
intrinsic segments and the child-gap segments used below.  Write
$\log k_M\le .2B_j+C_k$, where $C_k$ is an absolute constant.
Since $Qr\le k_M$, the choices of scales give
\[
 \begin{aligned}
  \frac{a_P}{T_{\mathrm{core}}}&\le
       \exp\bigl(C_k-(E_2-.2/A_0)Y_P\bigr),\\
  \frac{t_jr}{T_{\mathrm{core}}}&\le
       \exp\bigl(C_k-(2E_g-.2)B_j\bigr),\\
  \frac{t_Dr}{T_{\mathrm{core}}}&\le
       \exp\bigl(C_k-(1.5E_2-.2/A_0)Y_D\bigr)
       \quad\text{when }B_j\le Y_D/A_0.
 \end{aligned}
\]
Take the minimum bands and heights sufficiently large that each
right-hand side is at most $1/4$.  For either segment length
$s=t_jr$ or $s=t_Dr$, this ensures
$s+a_P\le T_{\mathrm{core}}/2<T_{\mathrm{core}}$.
Distinct periodic copies of one microinterval are separated by
a gap $T_{\mathrm{core}}-a_P>s$.  Thus the tested segment meets at most one
copy of each independently sampled microinterval, even when it
crosses the chosen fundamental-period boundary.  For any
measurable available subset $A$ of the segment, the probability
that no raw line falls in $A$ is
\[
 \prod_J\left(1-\frac{|A\cap J|}{a_P}\right)
 \le \exp\left(-\frac{|A|}{a_P}\right),
\]
where the product is over those microintervals.  Hence the
two-direction failure probability is at most
\begin{equation}
 2\exp\bigl(-c t_j\exp(E_2Y_P)\bigr).
\label{part:intrinsic-jitter}
\end{equation}
This estimate permits any fragmentation of the available set.
Since $B_j\le Y_P/A_0$ and $E_2\gg E_g$, it is uniformly
negligible.  Every occupied new cell now has actual arc length at
most $2t_jr<1/4$ as measured from each surviving point of centre $j$.
This proves the stronger occupied-cell induction used for the hulls.
In particular it has the required intrinsic span
$\ll r\exp(-E_gB_j)$.

At an insertion containing a high-type newborn, the entire
component is newborn.  Indeed the last dense bin agrees across
each component edge at that height: the edge ratio is too small
to change any exponent in that bin.  Every active centre in the
component consequently has the same last dense bin and starts
now.  There are thus no inherited hulls to skip.  With one raw
line in each microinterval, every cell has length at most
$2a_P\ll r\exp(-E_gY_P)$, which is the stronger span needed
for high-type direct births.

\subsection{Separation of old cells and completion of the induction}

It remains to ensure that one new cell cannot contain surviving
points from two different old cells of the same child.  Fix an
old child $D$ and its nonempty old cells, where nonemptiness is
measured using the preliminary survivors.  If there is only one
such cell there is nothing to prove.  Otherwise list these cells
cyclically.  Between consecutive ones, the arc from the last
preliminary surviving point of the first to the first of the
next has length at least $2t_Dr$.  Indeed the old padding places
both points at distance at least $t_Dr$ from the old boundaries
between them.

The child's own current hulls do not enter this gap, by
Lemma~\ref{part:inherited-hulls} and the choice of the surviving
extrema.  Consider the segment of length $t_Dr$ running from
either endpoint into the gap.  The foreign-child history tests
at that endpoint leave at least half its length outside all
other children's hulls, by exactly the coverage argument used
for the intrinsic segments.  The non-repetition condition proved
above applies here, since
\[
 Qt_Dr\le t_Dk_M\ll1,
 \qquad B_j\le Y_D/A_0
\]
for its endpoint centre $j$; in particular $t_Dr+a_P<T_{\mathrm{core}}$.

For each nonempty old cell test its two adjacent gaps in these
directions.  If either test finds no unskipped line, discard all
entries of this child in that old cell.  Always use the fixed
preliminary positions and gaps, even if other subsequent tests
have already removed some points.  The same independent-jitter
calculation gives a failure probability per old cell of at most
\begin{equation}
 2\exp\bigl(-c t_D\exp(E_2Y_P)\bigr).
\label{part:child-jitter}
\end{equation}
Since $Y_P\ge2Y_D$,
\[
 t_D\exp(E_2Y_P)
   =\exp\bigl(E_2Y_P-1.5E_2Y_D\bigr)
   \ge\exp(E_2Y_P/4).
\]
Thus this failure probability is uniformly negligible as well.

Two old cells of this child that still contain survivors now
have an unskipped line between their preliminary live positions
in each direction around the circle.  This remains true if
intermediate old cells have lost all their entries: the separating
lines were selected using the original gaps and are never
removed or redrawn.  Consequently surviving points from this
child in a common new cell come from one old cell.

Let $\mathcal F$ denote all data after the preliminary history tests
and before the current jitter draws.  The pointwise bounds for
raw-line padding and \eqref{part:intrinsic-jitter} multiply the
corresponding fixed preliminary weights.  For an old cell $C$, let
$M_C^0$ be its preliminary first mass, including the outer-mask
factors, and let $E_C$ be its gap-test failure event defined using
the fixed preliminary gaps and hulls.  Any earlier pointwise
discards only reduce the mass left in $C$, so its subsequent
cell-purge loss is at most $\mathbf1_{E_C}M_C^0$ for every
realization.  Consequently
\[
 \mathbb E[\text{cell-purge loss in }C\mid\mathcal F]
 \le \mathbb P(E_C\mid\mathcal F)M_C^0.
\]
We never condition on the outcomes of the other tests.  Their
correlation with $E_C$ is therefore irrelevant, and overlapping
loss estimates only overcount the actual discard.  The preliminary
old-cell label sets are disjoint, and their total first mass is at
most $W$.  Thus \eqref{part:child-jitter} can be summed with these
mass weights.  The total entering first mass at each height is
also at most $W$.  Summation over the finite dyadic chronology gives
$o(W)$, uniformly in its length.  Together with the history-test
loss this is the required negligible partition discard.

No unskipped boundary enters an inherited closed hull, so
restricting a class to survivors does not split it across new
cells.  Every surviving point has distance at least $\delta_P$
from all new boundaries.  At a later sparse step of this
unchanged component, of height $Y'\ge Y_P$, the relaxed
coarsening scale satisfies
\[
 r\exp\bigl(-\tfrac12\exp(\sigma_gY')\bigr)
   \ll r\exp(-1.5E_2Y_P)=\delta_P.
\]
Thus the new good coarsening edges cannot cross a boundary.
The inequalities hold from the chosen minimum height onward,
because $\exp(\sigma_gY_P)\gg E_2Y_P$.

Finally, all parts of the construction respect the core
symmetry.  Old objects remain equivariant for the smaller common
core, the hull and coverage operations commute with its
translations, and the realized raw-line configuration is copied
with exactly that period.  The directional and old-cell tests
are transported under the same translations.  This establishes
the equivariance of the realized partitions and discards, not
merely invariance of their probability law.

The new partitions contain inherited classes, separate surviving
old cells within each child, and have the required span, padding
and realized core equivariance.  We may now apply the prescribed
cap clipping inside these cells.  The old-child separation supplies the
clipping inequality from Section~\ref{sec:tables}; the clipping
loss is accounted for separately there.  Subsequent sparse
updates retain the cells, their padding and their inherited
classes as described above.  An unchanged component needs no
partition replacement and no fresh clipping.  The construction
therefore continues through every component creation and change,
and proves Proposition~\ref{tab:partition-input}.
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/partitions.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/preliminaries.tex ====
\section{Finite blocks and elementary conventions}
\label{sec:preliminaries}

We first reduce Theorem~\ref{thm:main} to uniformly controlled tests on
finite collections of rows. Throughout the proof the shift $\gamma$ is
fixed. We work on $\mathbb T=\mathbb R/\mathbb Z$ with normalized
Lebesgue measure, using periodic lifts to $\mathbb R$ when counting pairs.
In a lifted pair count the first point is counted per unit length and the
second ranges over all its lifts. After the small-width reduction below,
the widths are sufficiently small that the circle and lifted near-pair
counts agree.

\subsection{Reduction to small fixed-mass blocks}

For $v\ge1$ set $r(v)=\psi(v)/v$, and let
\[
 E_v=\bigcup_{z\in\mathbb Z}
 \bigl((z+\gamma)/v-r(v),(z+\gamma)/v+r(v)\bigr)
 \pmod1.
\]
Thus $x\in E_v$ is exactly the approximation inequality in
Theorem~\ref{thm:main}. We discard rows of width zero.

\begin{lemma}\label{pre:large-widths}
If $v r(v)$ does not tend to zero, then $\limsup_v E_v$ has full measure.
\end{lemma}
\begin{proof}
There are a constant $0<c<1/2$ and an unbounded sequence of $v$ for which
$r(v)\ge c/v$. Let $F_v$ be the union of the row intervals of half-width $c/v$.
Its indicator is bounded by one, has mean $2c$, and for every fixed
interval $J\subset\mathbb T$ satisfies
\[
 \int_J\mathbf1_{F_v}=2c|J|+O(1/v).
\]
The error comes only from the two incomplete periods of length $1/v$.
The same convergence holds for a fixed finite union of intervals.
For a measurable set $A$, approximate $A$ in measure by such a union;
the bound $0\le\mathbf1_{F_v}\le1$ gives
$\int_A\mathbf1_{F_v}\to2c|A|$ along this sequence.
Consequently no positive-measure set avoids a whole tail of the $F_v$,
and therefore none avoids a whole tail of the $E_v$. Since the complement
of a limsup is the countable union of these tail-avoiding sets, the
conclusion follows.
\end{proof}

Henceforth suppose $v r(v)\to0$. Replace every positive width by a
dyadic number between half its value and its value, retaining the notation
$r(v)$. This preserves divergence of $\sum_v\phi(v)r(v)$ and can only
shrink the approximation sets. If Theorem~\ref{thm:main} fails, there is a measurable
set $A\subset\mathbb T$ of measure $\mu>0$ that avoids every $E_v$ in
some tail. Indeed the complement of the limsup is the union over $n$ of
$\bigcap_{v\ge n}E_v^c$.

\begin{lemma}\label{pre:blocks}
For every fixed $0<w\le1/2$ there are arbitrarily late, pairwise disjoint
finite sets $\mathcal B$ of positive-width rows such that
\begin{equation}\label{eq:block-mass}
 w\le W:=\sum_{v\in\mathcal B}\phi(v)r(v)\le2w.
\end{equation}
Their least denominator tends to infinity, and all their spatial widths
tend uniformly to zero.
\end{lemma}
\begin{proof}
The individual terms satisfy $\phi(v)r(v)\le v r(v)\to0$.
In a sufficiently late tail each is at most $w$. Starting after the
previous block, stop at the first partial sum reaching $w$; divergence
ensures that this occurs, and the last summand bounds the overshoot by
$w$. The final assertion follows from $r(v)\to0$.
\end{proof}

All subsequent constructions are on one such finite block. Constants may
depend on $\mu$ and on the fixed hierarchy parameters, but not on the
block, its denominators, or its widths. We choose the hierarchy first and
then take $w$ sufficiently small. The last approximation of $A$ by a union
of intervals will occur only after those constants and the fixed bound on
cell loads introduced in Section~\ref{sec:tables} have been chosen.

\subsection{Arithmetic notation and divisor laws}

For positive integers $d,e$, write $(d,e)$ and $[d,e]$ for their gcd and
lcm. Let $\nu_p(n)$ be the exponent of a prime $p$ in $n$, let $\tau(n)$
be the number of positive divisors, and let $\omega(n)$ be the number of
distinct prime divisors. Put
\[
 R(n)=\frac{\phi(n)}n,\qquad R_p=1-\frac1p.
\]
All logarithms are natural unless a dyadic index is explicitly specified.
The symbols $O$, $\ll$, and $\asymp$ have constants uniform in the varying
block data; dependence on fixed parameters will be indicated when needed.

\begin{definition}\label{pre:deficit-law}
A mask on an integer $L\ge1$ is a divisor $d\mid L$. Its deficit law is
\begin{equation}\label{eq:deficit-law}
 \pi_L(d)=\frac{\phi(L/d)}L.
\end{equation}
An exponent in $d$ records the corresponding deficit from $L$ in $L/d$.
\end{definition}

The identity $\sum_{d\mid L}\phi(d)=L$ makes this a probability law.
Euler factorization makes its prime coordinates independent. More
explicitly, if $p^\nu\parallel L$, then
\[
 \Pr(\nu_p(d)=j)=
 \begin{cases}p^{-j}R_p,&0\le j<\nu,\\p^{-\nu},&j=\nu,
 \end{cases}
 \qquad
 \Pr(\nu_p(d)\ge j)=p^{-j}\quad(0\le j\le\nu).
\]
These probabilities weight row denominators. They are not probabilities
that a numerator lies in a residue class. Numerator counts at a fixed mask
will always be justified separately by a complete period or a specified
translation orbit. Other auxiliary probability laws used later are also
explicit finite weights; the shift itself is never randomized.

\begin{lemma}\label{pre:elementary-prime-bounds}
Uniformly for $z\ge2$, $b\ge a\ge2$, and integers $n\ge1$,
\begin{align}
 \sum_{p\le z}\log p&\ll z,
 &\sum_{p\le z}\frac{\log p}{p}&\ll\log(2z),\label{eq:prime-log-sums}\\
 \sum_{a\le p\le b}\frac1p
 &\ll \frac1{\log a}+\log\frac{\log b}{\log a},
 &\prod_{p\le z}R_p^{-1}&\ge\sum_{1\le m\le z}\frac1m.
 \label{eq:prime-reciprocals}
\end{align}
For every fixed $C,\epsilon>0$,
\begin{equation}\label{eq:divisor-small-power}
 \tau(n)^C+R(n)^{-C}\ll_{C,\epsilon}n^\epsilon.
\end{equation}
\end{lemma}
\begin{proof}
The primes in $(N,2N]$ divide $\binom{2N}{N}\le4^N$.
Summing over dyadic intervals gives the first bound in
\eqref{eq:prime-log-sums}; partial summation gives the second. The same
binomial estimate gives $O(t/\log t)$ primes in a dyadic interval at
scale $t$. Decomposing $[a,b]$ into multiplicative intervals and summing
their reciprocal costs gives the first bound in
\eqref{eq:prime-reciprocals}, including the possible endpoint interval.
The Euler product on its right expands into reciprocals of all positive
integers with prime factors at most $z$, including every integer at most
$z$.

For \eqref{eq:divisor-small-power}, at all sufficiently large primes the
factor $(j+1)^C$ is bounded by $p^{\epsilon j/2}$ for every $j\ge1$,
and $R_p^{-C}\le p^{\epsilon/2}$. Each of the finitely many remaining
primes has a bounded supremum after division by its positive exponential
factor $p^{\epsilon j}$, or contributes a fixed constant in the
$R(n)$ bound. Multiplication proves the assertion.
\end{proof}

We will use Cauchy--Schwarz, H\"older, the Chinese remainder theorem,
Markov's inequality, and finite orbit averaging. Size parameters and
vertex totals in independent arithmetic estimates are local to those
estimates; the centre and table notation is introduced only when needed.

\subsection{A tensor kernel for integer families}

The centre construction will use a kernel bound for pairs of integer
families. We derive it from a concentration observation on a single prime
coordinate, retaining the concentration conclusions for the weighted lcm
estimate in Section~\ref{sec:arithmetic}.
The diagonal-concentration argument is an adaptation of Green and
Walker~\cite[Lemma~2.1]{GW2021} and Hauke-Treuer, Vazquez and
Walker~\cite[Lemma~3.2]{HVW2026}. We include the proof, together with
the kernel-total and off-pivot estimates needed below.
Exponent indices in this subsection belong to $\mathbb Z_{\ge0}$.

\begin{lemma}[One-coordinate concentration]\label{ar:pivot}
Fix $1/2<u<1$ and $A\ge1$. For probability vectors $\alpha,\beta$ and
$0<q<1/2$, put
\[
 T_q(\alpha,\beta)=\sum_{i,j}
 q^{|i-j|}A^{\mathbf1_{i\ne j}}(\alpha_i\beta_j)^u.
\]
Then
\[
 T_q(\alpha,\beta)\le 1+O_{u,A}(q^{1/(1-u)}).
\]
Moreover, if a probability array $(m_{ij})$ satisfies
\[
 m_{ij}\le q^{|i-j|}A^{\mathbf1_{i\ne j}}
 (\alpha_i\beta_j)^u,
\]
there is, for sufficiently small $q$, an index $k$ such that
\[
 d:=(1-\alpha_k)+(1-\beta_k)\ll_{u,A}q^{1/(1-u)}.
\]
For this index the mass on $i,j\ne k$ is $O_{u,A}(d^{2u})$, and the mass
on pairs with exactly one index equal to $k$ and the other at distance at
least $a\ge1$ is $O_{u,A}(q^a d^u)$.
\end{lemma}

\begin{proof}
The off-diagonal convolution kernel has $\ell^1$ norm $O_A(q)$, whereas
$\|\alpha^u\|_2,\|\beta^u\|_2\le1$. Hence its contribution is $O_A(q)$.
The diagonal contribution is at most
\[
 \left(\sum_i\alpha_i^{2u}\sum_j\beta_j^{2u}\right)^{1/2}.
\]
If the total is at least one, the latter expression is $1-O_A(q)$.
Since
\[
 \sum_i\alpha_i^{2u}\le(\max_i\alpha_i)^{2u-1},
\]
both vectors have atoms of mass $1-o(1)$. These atoms have the same index:
the actual diagonal sum is at least $1-O_A(q)$, whereas different
dominant indices would bound it by
$O(d_\alpha^u+d_\beta^u+(d_\alpha d_\beta)^u)=o(1)$, with
$d_\alpha,d_\beta$ their off-atom masses.
Call the common index $k$ and define $d$ as in the statement.

The central term is at most $1-c_ud$ when $d$ is small. The single-off
terms are $O_{u,A}(qd^u)$ by geometric summation and Holder's inequality.
More generally, without any support-size assumption,
\[
 \sum_{|j-k|\ge a}q^{|j-k|}\beta_j^u
 \le\left(\sum_{j\ne k}\beta_j\right)^u
 \left(\sum_{|j-k|\ge a}q^{|j-k|/(1-u)}\right)^{1-u}
 \ll_u q^a d^u.
\]
The double-off terms are $O_{u,A}(d^{2u})$ by the convolution bound applied
to the two off-parts. Consequently
\[
 T_q(\alpha,\beta)
 \le 1-c_ud+O_{u,A}(d^{2u}+qd^u).
\]
Absorb $d^{2u}$ using $2u>1$. Maximizing the remaining expression in $d$
proves the asserted excess bound. If a probability array is dominated by
the kernel, its total is one, so the same inequality forces
$d\ll q^{1/(1-u)}$. Restricting the single-off geometric sum to distances
at least $a$ gives its last asserted bound. When the kernel total is less
than one, the first conclusion is immediate.
For $q$ in a compact subinterval $[q_0,1/2)$, the convolution norm is
bounded by $1+2Aq/(1-q)\le1+2A$. Increasing the constant therefore
extends the first bound to all $0<q<1/2$.
\end{proof}

\begin{lemma}[Tensor kernel]\label{ar:tensor}
For $1/2<u<1$, $s>1-u$, and nonnegative sequences $(x_L),(y_M)$ on the
positive integers,
\[
 \sum_{L,M\ge1}
 \frac{(x_Ly_M)^u}{\bigl(L/(L,M)\cdot M/(L,M)\bigr)^s}
 \ll_{u,s}\left(\sum_Lx_L\sum_My_M\right)^u.
\]
\end{lemma}

\begin{proof}
It suffices first to consider finite supports and positive totals, and
normalize both totals to one. At each sufficiently large prime $p$,
Lemma~\ref{ar:pivot} with
$q=p^{-s}$ bounds the optimal constant in the corresponding
one-coordinate estimate by
$1+O_{u,s}(p^{-s/(1-u)})$. Each of the finitely many small primes also has
a bounded optimal constant, directly from the $\ell^1$ convolution bound.

To tensorize, fix the two exponents at the first prime and apply the
remaining-coordinate inequality to the corresponding slices. Its right
side is the product of their slice totals to the power $u$. The
one-coordinate bound then sums those totals. Repeating this argument gives
the product of these one-prime constants. This product converges because
$s/(1-u)>1$. Zero totals are trivial, and monotone convergence removes
the finite-support restriction.
\end{proof}

\subsection{Residue averages for rational grids}

For positive integers $v,w$, consider the two lifted grids
$(z+\xi)/v$ and $(z'+\xi')/w$, with fixed real phases and positive
half-widths $r_1,r_2$. Put
\[
 g=(v,w),\qquad s=w/g,\qquad t=v/g.
\]
The integer determinant $n=sz-tz'$ records their relative positions.
In particular, distance at most $C\max(r_1,r_2)$ restricts $n$ to an
interval of length $O_C(\eta)$, where
$\eta=[v,w]\max(r_1,r_2)$.

\begin{lemma}[Residue averages on determinant cycles]\label{ar:cycle}
Fix divisors $D_1\mid v$, $D_2\mid w$ and a nonnegative function of
$z\bmod D_1,z'\bmod D_2$. At every integer determinant there are $g$
solutions per unit first-point translate; let $A(n)$ be their average
value. Then $A$ has a period dividing $D_1D_2$, and its mean $\overline A$
is the independent uniform residue mean of the function. For any real
interval $J$,
\[
 \sum_{n\in J\cap\mathbb Z}A(n)
 =|J|\overline A+O(D_1D_2\overline A).
\]
If one also requires $n\equiv n_0\pmod Q$ and $(Q,D_1D_2)=1$, the main
term is $|J|\overline A/Q$ with the same error bound.
\end{lemma}

\begin{proof}
Since $(s,t)=1$, all solutions at a fixed determinant differ by $(t,s)$.
Taking the first numerator over one period of length $v$ gives a cycle of
length $g$. A Bezout solution for an increment of $D_1D_2$ in the
determinant preserves both tested residues, proving periodicity. Averaging
also over $n\bmod[v,w]$ enumerates all $vw$ pairs of numerator residues
modulo $v,w$, proving the mean assertion.

A nonnegative periodic sequence of period $P$ and mean $\overline A$ has
total mass $P\overline A$ in one period. Splitting an interval into whole
periods and two end pieces therefore gives an error $O(P\overline A)$.
Take $P=D_1D_2$. Finally, stepping by a number coprime to $P$ permutes its
residues, so the same proof applies on the stated progression.
\end{proof}
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/preliminaries.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/prescriptions.tex ====
\section{Masks and initial numerator prescriptions}
\label{sec:prescriptions}

The centre assignments of Section~\ref{sec:centres} determine which
rows are available. We now remove a negligible set of their deficit
masks and define the weights at their first projections. The hard
weights have a uniform lower mean comparable to the totient density;
the additional soft conditions have arbitrarily small relative
first-mass cost. All prescriptions here are fixed before the
adaptive tables are constructed.

\subsection{Good deficit masks}

For a centre $L=L_i$ of band $B=B_i$, the assigned masks are
$d=L/v\in\mathcal F_i$. Probabilities below refer to the unrestricted
deficit law $\pi_L$, not to a uniform distribution on assigned rows.
Recall that its prime coordinates are independent and
$\Pr(\nu_p(d)\ge j)=p^{-j}$ for $j\le\nu_p(L)$.

\begin{definition}[Good masks]
\label{pre:good-masks}
An assigned mask $d$ is \emph{good} if it satisfies all applicable
conditions below.
\begin{enumerate}
\item Its excess prime-power part obeys
\[
 \prod_p p^{(\nu_p(d)-1)_+}\le e^{C_{\rm mask}B}.
\]
\item On every initial segment of the prime divisors of $L$ whose
reciprocal sum is at most $s_{\rm low}B$, the part of $d$ supported
on that segment has divisor count at most $e^{s'B}$.
\item Its large-prime harmonic cost satisfies
\[
 \sum_{\substack{p\mid d\\p>d_0B}}\frac{(\log p)^{.3}}p\le1.
\]
\item On high type, the last dense bin contains at least
$c\kappa Y_d$ prime divisors of $d$, where $c>0$ is a sufficiently
small fixed absolute constant.
\end{enumerate}
\end{definition}

An \emph{omitted} prime means a prime dividing the deficit mask:
one copy is absent from the row denominator, not necessarily the
entire prime power of the centre. In all tagged or tail ranges below,
good masks have deficit at most one at each prime. Indeed these
primes exceed $e^{JB}$, or are still larger in terms of their starting
height, and $J>C_{\rm mask}$.

\begin{lemma}[Negligible mask deletion]
\label{pre:mask-purge}
With the order of choices in Section~\ref{cen:hierarchy}, deleting
all assigned masks that are not good loses $o(W)$ as $k_0\to0$.
Uniformly for each retained centre, its unrestricted failure
probability is $o(\delta_L)$.
\end{lemma}

\begin{proof}
For the first condition take a fixed small positive moment of the
excess prime-power part. The geometric deficit tails give a bounded
Euler product: the local contribution beyond one starts at exponent
two and is $1+O(p^{-2+\epsilon})$. With this fixed moment exponent,
choose $\epsilon C_{\rm mask}>2$. Markov's inequality then gives
failure probability $\ll e^{-\epsilon C_{\rm mask}B}$.

For the second condition it is enough to test the largest allowed
initial segment, or the entire support if its reciprocal sum is
small enough. Every fixed moment of the divisor count on this
segment is at most
\[
 \exp\left(C_{\rm moment}\sum_{p\text{ in the segment}}\frac1p\right).
\]
Choose a fixed moment $M$ with $Ms'>3$, then choose $s_{\rm low}$
so small that $Ms'-C_Ms_{\rm low}>2$. Markov's inequality makes
the failure probability at most $e^{-2B}$. This moment choice
occurs between the choices of $s'$ and $s_{\rm low}$ and does not
depend on any later tag or model parameter.

For the third condition the logarithm of the moment generating
function with coefficient $2B$ is at most
\[
 \sum_{p>d_0B}
 \frac{\exp(2B(\log p)^{.3}/p)-1}{p}=o(B).
\]
To see the last bound, split at $B^2$. Below that point one can
bound even the integer sum by
$O(\log B\exp(C_{d_0}(\log B)^{.3}))=o(B)$. Above it the
exponent is small and the integer sum is
$O(B\sum_{n>B^2}(\log n)^{.3}/n^2)=o(B)$.
Thus the failure probability is $e^{-2B+o(B)}$. Summing over all
integers has made the estimate uniform over every centre prime
support: no factor depending on $\log\log L$ occurs. The fixed
$d_0$ affects only how large $B$ must be for the $o(B)$ estimate,
which is permitted because $k_0$ is chosen last.

For the last condition, independence and the reciprocal sum in
the dense bin give
\[
 \mathbb E\exp\left(-\sum_{\substack{p\mid L\\p\in\mathcal P(Y_d)}}
 \mathbf1_{\{p\mid d\}}\right)
 =\prod_{\substack{p\mid L\\p\in\mathcal P(Y_d)}}
 \left(1-\frac{1-e^{-1}}p\right)
 \le e^{-c_1\kappa Y_d}.
\]
For sufficiently small $c$, failure has probability
$e^{-c_2\kappa Y_d}$. Since $Y_d>D_0B$ on high type, choose
$D_0$ after $\kappa$ so that $c_2\kappa D_0>2$.

Take $C_{\rm mask}$ large in the first estimate. All four
probabilities are then $o(e^{-.64B})$, uniformly as the minimum
band tends to infinity. Since $B_i^*<2B_i$,
\eqref{eq:centre-lower} gives $\delta_L\ge e^{-.64B}$.
Multiplying each failure probability by $k_L$, and using
$m_L=k_L\delta_L$, proves the claimed relative and total losses.
\end{proof}

From now on assigned masks are good. Nonnegative upper bounds may
still extend sums to the unrestricted mask laws.

\subsection{The first projection and its currency}

At centre $L_i$, let $P_I$ be the product of the full prime powers
of $L_i$ with $\log\log p\ge Y_0(i)$. Define the tagged factor
$P_M$ as follows. On high type it contains precisely the full prime
powers in the last dense bin. On medium type it contains all full
prime powers at $p>e^{JB}$ below the tail $P_I$. For medium type
partition the tagged primes into the ordinary bins
$\ell B\le Y'<Y_0(i)$ and the extra bin
\[
 e^{JB}<p<\exp(\exp(\ell B)).
\]
Empty bins require no action. Put $P=P_MP_I$ and write $L_i=S_iP$;
these factors have disjoint prime supports.

The first projection restores only factors omitted in the unprocessed
tail; the low and tagged deficits remain in its denominator.
For an assigned row, decompose its denominator and its deficit into
\[
 v=V(P_M/m)(P_I/a),\qquad
 V\mid S_i,\quad m\mid P_M,\quad a\mid P_I.
\]
Its \emph{birth projection} is the grid
\begin{equation}
 \begin{gathered}
 x=\frac{z+a\gamma}{VP/m},\quad z\in\mathbb Z,
 \\
 r_i\pi_{P_I}(a)(VP/m)R(VP_M/m)=r_i\phi(v).
 \end{gathered}
 \label{pre:birth-grid}
\end{equation}
The true row points are exactly the subarray $a\mid z$. The identity
in \eqref{pre:birth-grid} follows from multiplicativity across the
three disjoint prime supports. It is the first-mass normalization:
the entire projection receives the single coefficient
$\pi_{P_I}(a)$, while the baseline numerator density is
$R(VP_M/m)$. The coefficient is not an additional numerator test.

\subsection{Tagged hard factors}

Write
\[
 \alpha=a\gamma,\qquad d^0=(S_i/V)m,\qquad
 \mathcal L_i=\lfloor d^0\alpha\rfloor.
\]
These quantities depend on the current row and its masks, not just
on the centre whose index appears in $\mathcal L_i$.
For a rational whose denominator is a unit modulo a prime, its
congruence class is interpreted by inversion of that denominator.

\begin{construction}[Tagged numerator weights]
\label{pre:tag-prescription}
In each tag bin in which $m$ has at least one prime divisor, sample
an anchor uniformly from those divisors. In an ordinary bin of
height $Y'$, try to find a reduced rational $F_i$ satisfying
\[
 F_i\equiv\mathcal L_i\pmod {p_{\rm anchor}},\qquad
 |F_i|,\ \operatorname{den}(F_i)
 \le\exp(\exp(h_HY')).
\]
In the extra medium bin, replace this bound by $e^{EB}$.
The key $F_i$ is local to this bin and anchor choice; different bins
may use different keys despite the suppressed bin index.
If such a rational is found, then at every full tag factor in that
bin, namely $p\mid P_M$ with $p\nmid m$, forbid
\[
 d^0z+\mathcal L_i\equiv F_i\pmod p.
\]
If reconstruction fails, or if the bin has no anchor because its
tag mask is empty, use the scalar weight $R_p$ at those full
factors. At every omitted tag factor with positive residual
exponent, use the scalar $R_p$ as well. Average jointly over the
independent anchor choices in the different bins.
\end{construction}

There is at most one reconstructed rational. Write the height bound
in its bin as $Q$. If $F_1=n_1/q_1$ and $F_2=n_2/q_2$ were two
candidates, then $q_i\le Q$ and $|n_i|\le Qq_i$, so
\[
 |n_1q_2-n_2q_1|\le 2Qq_1q_2\le2Q^3.
\]
The anchor prime exceeds $2Q^3$: this follows from $h_H<1$ in
ordinary bins and $J>4E$ in the extra bin, once the bands are large.
It also exceeds $Q$, so the candidate denominators are units.
The congruence forces the displayed integer to vanish, proving
uniqueness. The same size comparison applies to every prime where
the candidate is used. At a full tag factor $d^0$ is also a unit,
so a successful reconstruction forbids exactly one numerator
residue. Anchor averaging makes the final prescription
deterministic.

\subsection{Low rational models}

The numerator restrictions in this subsection use a rational model of
the fixed projected phase. Shift-adapted primitivity also occurs in
Chow and Technau~\cite[Definition~1.20]{ChowTechnau2024}; here the
models, thresholds and gcd bounds are defined for the present rows
and their projections.

For the projected numerator width and relative rational height put
\[
 \psi=(VP/m)r_i,\qquad
 H_v(\zeta)=2+q+\frac{|\alpha-\zeta|}{\psi},
 \quad \zeta=n_0/q\text{ reduced},\quad q\ge1.
\]
Here the scalar $\psi$ is the projected numerator width, not the
original tolerance function in Theorem~\ref{thm:main}.
The subscript $v$ includes the row's fixed width and birth
projection. In particular $\psi\le k_L/m$.

\begin{definition}[Recognized models]
\label{pre:recognized-models}
The two medium broad tiers are alternatives; the narrow tier is an
additional test. High type uses only its single broad tier.
On medium type, use the large broad tier if
$\psi\le e^{-4EB}$, recognizing a rational when
$H_v(\zeta)\le e^{EB}$. Otherwise use the small broad tier only if
$\psi\le e^{-4h_0B}$, with threshold $H_v(\zeta)\le e^{h_0B}$.
Also test the narrow tier
\[
 q\le e^{EB},\qquad |\alpha-\zeta|\le\psi e^{-6EB}.
\]
On high type use just one broad tier,
\[
 H_v(\zeta)\le\exp(\exp(h_HY_d)).
\]
\end{definition}

\begin{lemma}[Uniqueness and separation of models]
\label{pre:model-uniqueness}
Each tier recognizes at most one rational. The same statement
holds across rows with a fixed $\alpha$ and the same tier
parameters: a fixed $B$ on medium type, or a fixed $Y_d$ on high
type, even if the latter rows have different bands. At fixed
$\alpha,B$, a large broad model and a narrow model recognized
on different medium rows must also coincide. If a medium row has
distinct broad and narrow models,
the broad tier is the small one and the narrow model has
denominator at least $e^{cB}$ for a fixed $c>0$.
\end{lemma}

\begin{proof}
Distinct reduced rationals of denominators at most $Q$ differ by
at least $Q^{-2}$. In a medium broad tier with exponent
$h\in\{E,h_0\}$, the error from $\alpha$ is at most $e^{-3hB}$
and the denominator is at most $e^{hB}$, proving uniqueness.
In the narrow tier, the centre scale bound gives
\[
 \psi\le k_L\le e^{.2B+O(1)},
\]
so its error is at most $e^{-(6E-.2)B+O(1)}$, again smaller than
half the rational separation. This estimate is uniform across
rows at that fixed band.

For high type, the good-mask occupancy condition supplies a tag
prime in $m$, and hence
\[
 \psi\le\exp(.2B+O(1)-\exp(Y_d)).
\]
For all such rows at fixed $Y_d$, put
$Q=\exp(\exp(h_HY_d))$. Their model errors are uniformly at most
\[
 \exp\bigl(.2Y_d/D_0+O(1)-\exp(Y_d)+\exp(h_HY_d)\bigr)
 <\tfrac12 Q^{-2},
\]
for sufficiently large $Y_d$. This proves the asserted cross-row
uniqueness because $h_H<1$.

The large medium broad model and narrow model cannot be distinct,
even on different rows with the same $\alpha,B$: their two error
bounds sum to less than $e^{-2EB}$, and both denominators are at
most $e^{EB}$. If the small broad model has denominator $q_b$
and a distinct narrow model has denominator $q_n$, then
\[
 \frac1{q_bq_n}
 \le |\zeta_b-\zeta_n|
 \le e^{-3h_0B}+e^{-(6E-.2)B+O(1)}.
\]
Using $q_b\le e^{h_0B}$ gives $q_n\gg e^{2h_0B}$, which is
the last assertion.
\end{proof}

For every recognized model $n_0/q$, require at each $p\mid V$
\begin{equation}
 p\nmid qz+n_0
 \quad\text{if}\quad p>H_v(\zeta)^{\epsilon_s}
 \text{ and }p\nmid q.
 \label{pre:low-hard-tests}
\end{equation}
If no exclusion is active at $p$, multiply by $R_p$ instead. When
two models impose exclusions, impose both. The resulting low
factors, together with Construction~\ref{pre:tag-prescription},
are called the \emph{hard birth weights}.

\begin{lemma}[Hard first-mass bounds]
\label{pre:hard-mean}
There is $c_{\rm hard}>0$, independent of $k_0$ once it is
sufficiently small, such that the mean hard birth weight on a row's
projected numerator grid lies in
\[
 [c_{\rm hard}R(VP_M/m),\ R(VP_M/m)].
\]
Consequently its first mass, including the single coefficient
$\pi_{P_I}(a)$ and the spatial width, lies between
$c_{\rm hard}\phi(v)r(v)$ and $\phi(v)r(v)$.
\end{lemma}

\begin{proof}
Fix the anchor choices. At a prime with one active exclusion the
uniform residue mean is exactly $R_p$; the same is true of its
scalar replacement. At an omitted tag factor the scalar is
present exactly when that prime remains in the projected
non-tail denominator. The prime coordinates of the numerator
are uniform and independent by CRT.

Only two distinct medium models can reduce the mean below this
one-factor product. By Lemma~\ref{pre:model-uniqueness}, their
additional narrow-model hard primes exceed $e^{c\epsilon_sB}$.
On medium type every prime of $V$ is at most $e^{JB}$. Thus the
reciprocal sum of these additional primes is bounded by a
constant independent of $B$ and $k_0$. They are also all large
when $k_0$ is small; in particular two distinct active exclusions
cannot occur at $p=2$. The local mean with two exclusions is at
least $1-2/p$, whose ratio to $R_p$ is $1-1/(p-1)$.
Multiplying these ratios gives a positive uniform lower constant.
The upper mean is always at most the one-factor product.
These bounds persist after averaging the anchor choices.
The last assertion is \eqref{pre:birth-grid}.
\end{proof}

\subsection{Soft tests and the remaining initial deletion}

In addition to the hard weights, require for every recognized model
\begin{equation}
 (qz+n_0,V)\le H_v(\zeta)^{h_s}.
 \label{pre:soft-tests}
\end{equation}

\begin{lemma}[Relative cost of the soft tests]
\label{pre:soft-loss}
Under the probability measure obtained by normalizing a row's
hard birth weights, the probability that at least one of the tests
\eqref{pre:soft-tests} fails is $O(\epsilon_s/h_s)$, with an absolute
implied constant. In particular their relative first-mass loss can
be made arbitrarily small compared with $\mu$ by the stated
choice of $\epsilon_s$.
\end{lemma}

\begin{proof}
Fix a recognized model $n_0/q$ and the static anchor choices.
For $p\mid q$, reducedness gives $p\nmid n_0$, so that prime does
not divide $qz+n_0$. For $p\nmid q$ above the threshold
$H_v^{\epsilon_s}$, the model's own hard exclusion already makes
its valuation zero. At a remaining prime, at most the other
model can impose a nonconstant exclusion. Conditioning on it
inflates each geometric valuation tail by at most an absolute
factor. The scalar factors and the tag factors do not change
this conclusion. Therefore
\[
 \mathbb E_{\mathrm{hard}}\log(qz+n_0,V)
 \ll\sum_{p\le H_v^{\epsilon_s}}\frac{\log p}{p-1}
\ll\epsilon_s\log H_v.
\]
For the last inequality, if $H_v^{\epsilon_s}<2$ the prime sum
is empty; otherwise $\log(2H_v^{\epsilon_s})\ll
\epsilon_s\log H_v$. At $p=2$ conditioning on the one possible
other exclusion costs at most a factor two, so the same absolute
tail bound applies. If $n_0=0$, reducedness gives $q=1$; the
valuation tails are still the same geometric tails, truncated at
the exponents in $V$, including the numerator residue zero.
Markov's inequality gives failure probability
$O(\epsilon_s/h_s)$ for this model. There are at most two
recognized models, so the union bound proves the assertion.
For each fixed anchor choice this bounds failed hard mass by
$O(\epsilon_s/h_s)$ times its hard mass. Integrate this inequality
over anchor choices before normalizing. The normalized average
is a mixture of the conditioned laws, and has the same relative
bound, without an additional factor $c_{\rm hard}^{-1}$.
\end{proof}

One additional medium-type birth deletion is needed: it removes
small neighborhoods of rational points after specified integer
dilations. Its explicit definition and cost proof are given in
Section~\ref{sec:direct}. It uses only the aggregate lists of good
rows and squarefree moduli of the low numerator. In particular it
is specified before, and independently of, all running-table
deletions or estimates.

All birth requirements are multiplied on each entry, with weight
zero when a required exclusion or test fails; the anchor average
is joint over that entry's prescriptions. No static test uses a
tail-prime coordinate of the projected numerator $z$. Thus the
birth arrays retain all unprocessed-prime translation symmetries,
which are the input to the table construction in
Section~\ref{sec:tables}.
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/prescriptions.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/sparse.tex ====
\section{Alignments in sparse prime steps}\label{sec:sparse}

We now estimate the alignment terms that arise during a sparse-prime
update in an existing component.  The estimates use only the previously
constructed tables, their core equivariance, and the static birth
prescriptions.  In particular, they do not require a choice of the next
operational partition.  Besides paying for alignments outside a very small
distance scale, we construct deterministic interval trains covering the
segments by which accounting classes will be joined.  These trains are
the input to the partition construction in Section~\ref{sec:partitions}.

Fix a component at height $Y$ and a common prime $q$ in its current bin.
For two centres $L=L_i$ and $M=L_j$, retain the notation
\[
 G=(L,M),\qquad U=L/G,\qquad T=M/G,\qquad
 \Theta_{ij}=G r_{\min},\qquad K=[L,M]r_{\max}.
\]
At a successful plus endpoint the denominator has the form
\[
 V'=\frac{L}{dq},\qquad\text{or}\qquad W'=\frac{M}{eq},
\]
where $d,e$ are processed deficits below $q$.  Write $a,b$ for the
remaining masks above $q$, and put
\begin{equation}\label{sp:signatures}
 f_0=(Td,Ue),\qquad s_0=Td/f_0,\qquad t_0=Ue/f_0.
\end{equation}
Thus the successful positions are $(z+a\gamma)/V'$ and
$(z'+b\gamma)/W'$.  Throughout this section their mask coefficients are
the \emph{product} of the two after-$q$ probabilities, including when the
two masks happen to be equal.  They are not the single mask probability
used in clipping.

The endpoint train test below has the following concrete form.  For each
successful denominator $V'$, a deterministic list contains at most
$Z_Y=\exp(\exp(h_aY))$ offsets $y_0\pmod1$.  An endpoint of width $r$
passes if it lies within $r\exp(-.4\exp(\sigma_gY))$ of some point
$(z+y_0)/V'$ with $z\in\mathbb Z$ and a listed offset.  The lists depend
only on the centre, prime and processed mask, not on the actual table
history.  Construction~\ref{sp:danger-trains} specifies them using
rational residues and quantile marks.

\begin{proposition}[Sparse-step estimates]\label{sp:sparse-estimates}
Consider any finite table history up to a sparse-prime update that satisfies
the centre and good-mask prescriptions and the preceding chronological
core-equivariance rules.  Suppose its operational cells have the span and
padding required in Section~\ref{sec:tables}.  Uniformly in that history
and in the slot cap, the following costs, for a fixed centre pair and bin,
are at most
\[
 C\Theta_{ij}e^{-cY}\tau(UT)^C,
\]
where $c>0$ and $C$ are fixed independently of the block:
\begin{enumerate}
 \item the width-weighted product-mask counts of actual successful plus
 pairs at distances between
 $r_{\max}\exp(-\exp(\sigma_gY))$ and $C_0r_{\max}$;
 \item the corresponding counts of before-$q$ plus pairs at distance at
 most $C_0r_{\max}$, multiplied by $1/q^2$, with after-$q$ mask coefficients;
 \item for equal widths $r$, the actual successful plus pairs at distance
 at most $r\exp(-\tfrac12\exp(\sigma_gY))$ that fail the deterministic
 train test of Construction~\ref{sp:danger-trains} at either endpoint.
\end{enumerate}
Here each count is summed over the primes of the bin and uses the factor
$r_{\min}$, or $r$ in the third case.  The first and third counts may use
the actual raw arrival weights immediately after the update; the second
uses the before-update raw plus weights.  The constant $C_0$ can be any
fixed positive constant.

Join accounting classes by all actual surviving equal-width successful
plus pairs at the relaxed scale in the third case that pass both train
tests.  Each connecting segment is covered by the deterministic train
history, which is invariant under the post-generation core translations.
Summing the displayed errors over centre pairs, bins, and components is
bounded.  Consequently this construction proves
Proposition~\ref{tab:sparse-input}.
\end{proposition}

We prove the proposition in stages.  All upper estimates below may drop
assignment restrictions and adaptive filters when indicated.  The actual
arrival weights are bounded by the static factors retained in those upper
estimates, since a surviving label follows same-location ancestors with
nonincreasing raw weight.

\subsection{Bare counts and switching}

A pair at distance at most $C_0r_{\max}$ satisfies
\begin{equation}\label{sp:determinant}
 \left|s_0(z+a\gamma)-t_0(z'+b\gamma)\right|
 \le \frac{C_0K}{qf_0}.
\end{equation}
By \eqref{eq:component-path}, the right side is less than $1/2$ after
increasing the fixed lower cutoff.  Thus there is at most one possible
integer $s_0z-t_0z'$ for fixed masks.  The gcd of the two successful
denominators is $Gf_0/(qde)$, so their width-weighted bare count is at most
\begin{equation}\label{sp:bare-count}
 \Theta_{ij}\frac{f_0}{qde}.
\end{equation}
The same primewise gcd-ratio calculation used in
Lemma~\ref{dir:ultra-switch} bounds the unrestricted sum of $f_0/(de)$
by $\exp(CY)\tau(UT)^C$.  Outer masks still integrate as independent
probabilities.

For the second assertion of Proposition~\ref{sp:sparse-estimates}, use the
before-$q$ denominators instead.  The number of determinant integers is
$O(1+K/f_0)$, and the width-weighted pre-count is therefore at most
\[
 C\Theta_{ij}\frac{K+f_0}{de}.
\]
Multiply by $q^{-2}$ and sum over $q\ge\exp(\exp Y)$.  The resulting
double-exponential saving absorbs both the crude mask cost and the bound
on $K$ from \eqref{eq:component-path}.  This proves the required
$\Theta_{ij}e^{-cY}\tau(UT)^C$ estimate.

For the remaining assertions, use the switching argument with budget
$u=D''Y$, where $D''$ is sufficiently large absolutely.  Test divisors
$h\mid(d,s_0)$ in the ordinary dyads between
$\exp(C_1u)$ and $\exp(3C_1u)$, with the cutoff rounded as in
Lemma~\ref{dir:ultra-switch}, and test every such prime in dyads above the
lower cutoff.  Make the symmetric tests on $(e,t_0)$.

On a fiber with $d/h,e,a,b,q$ fixed, stripping $h$ leaves $f_0$ unchanged.
Indeed, at a prime $l$, write $A_l=\nu_l(Td)$, $B_l=\nu_l(Ue)$,
and $j_l=\nu_l(h)$.  The divisibility $h\mid(d,s_0)$ gives
$j_l\le A_l-\min(A_l,B_l)$, so
\[
 \min(A_l-j_l,B_l)=\min(A_l,B_l).
\]
It replaces $s_0$ by $s_0/h$ and extracts a factor
$1/h$ from \eqref{sp:bare-count}.  The tolerance in
\eqref{sp:determinant} stays fixed, while $h$ occurs linearly in the
phase.  The integer rotation estimate therefore applies on this fiber.
The stripped row need not remain assigned or good, because those
restrictions have been discarded in this nonnegative upper bound.
Summing its exceptional alternatives over the tested dyads, stripped
masks, and current primes gives
\begin{equation}\label{eq:sparse-switch}
 \ll \Theta_{ij}e^{-cD''Y}\tau(UT)^C.
\end{equation}
There are only $\exp(O(Y))$ prime-size dyads below the end of the bin;
the divisor-test dyads are fewer.  The crude mask and reciprocal-prime
sums have the same $\exp(O(Y))$ allowance.  These costs are absorbed by
the choice of $D''$, leaving the divisor factor displayed in
\eqref{eq:sparse-switch}.

The argument is also valid if the tolerance in
\eqref{sp:determinant} is multiplied by one of the smaller gap factors
used below.  Outside the union of these failures, any available switch
gives coupled, unique rational approximants $A_i,A_j$ to
$s_0a\gamma,t_0b\gamma$.  Their denominators are at most $e^{O(u)}$,
their errors are at most $e^{O(u)}$ times the tolerance, and
\[
 A_i-A_j=-(s_0z-t_0z').
\]
Division by a switched divisor retains a denominator bounded by
$e^{O(u)}$.  In particular a sufficiently large switched prime divides
the relevant numerator.  The divisor-assembly argument in
Lemma~\ref{dir:ultra-switch}, together with the good-mask prime-power
bound, then gives the corresponding small denominators after division
by the full signatures.  If no switch is available, the signatures
themselves are at most $\exp(\exp(h_*Y))$.  Indeed
$s_0/(d,s_0)\mid T$ and $t_0/(e,t_0)\mid U$, the unmatched prime-power
excess is bounded by the purge, and a larger uncancelled squarefree
product would contain a divisor in the tested range.  The remaining
centre-ratio cost is covered by \eqref{eq:component-path}.

\subsection{Neutralizing earlier dense bins}

Call the intermediate-distance range in the first assertion of
Proposition~\ref{sp:sparse-estimates} the \emph{off-gap} range.
The crude factor $\exp(CY)$ is too large for its sum.  We now
remove the contribution of any earlier dense bins near the current
height by using the birth prescriptions.

The centre exponents agree in every bin of height $Y'\ge\sigma Y$,
by \eqref{eq:component-path}.  If no such bin is dense, the reciprocal
sum through the current bin is already $O((\sigma+\kappa)Y)$: bins
below $\sigma Y$ have the absolute harmonic bound, and all remaining
bins are sparse.  Otherwise the last dense height $Y_d\ge\sigma Y$
is common to both centres, since bins after the start are sparse.
The starts are also common.  On medium type,
\[
 \frac{\sigma Y}{D_0}\le B_i,B_j\le\frac{Y}{A_0};
\]
the retained band separation forces $B_i=B_j$.  On high type each start
is $2Y_d$.  On medium type all bins with
$\sigma Y\le Y'<Y_0$ are ordinary tag bins.

Write $d=d^0u_i$ and $e=e^0u_j$, where $d^0,e^0$ are below this common
start.  At birth, the ancestors of the current successful points had
phase and numerator
\[
 \alpha_i^0=u_iqa\gamma,\qquad z_i^0=u_iqz,
 \qquad
 \alpha_j^0=u_jqb\gamma,\qquad z_j^0=u_jqz'.
\]
Consequently the birth integer on the first side is
$\mathcal L_i=\lfloor dqa\gamma\rfloor$, and similarly on the second.
Switch every tag mismatch prime in the indicated bins of height at
least $\sigma Y$; on high type use its tag bin $Y_d$.  Such a prime is
eligible for \eqref{eq:sparse-switch}: centre exponents agree, its good
deficit has exponent at most one, and it divides the relevant signature
when it occurs on just one side.

If a mismatch occurs, form the predicted tag lifts
\begin{equation}\label{sp:predicted-tags}
 F_i^{\mathrm{pred}}=\mathcal L_i-\frac{qf_0}{T}A_i,
 \qquad
 F_j^{\mathrm{pred}}=\mathcal L_j-\frac{qf_0}{U}A_j.
\end{equation}
Their log sizes and log denominators are at most $\exp(h_*Y)$.
For the size bound, the large phase cancels:
$(qf_0/T)s_0a\gamma=dqa\gamma$, so the remaining size is controlled
by the approximation error times $qf_0/T$, plus the floor error.
The denominator bound follows from the same approximant bounds and
\eqref{eq:component-path}.  These bounds fit the recognized tag heights
because $h_*\ll\sigma h_H$.

At every included anchor on the first side,
$F_i^{\mathrm{pred}}\equiv\mathcal L_i$.  A common omission supplies
the prime through $f_0$; a mismatched omission supplies it through the
approximant numerator.  The denominators are units there.  Thus every
included anchor reconstructs this prediction.  The coupled exact
identity gives
\[
 T(d^0z_i^0+\mathcal L_i-F_i^{\mathrm{pred}})
 =U(e^0z_j^0+\mathcal L_j-F_j^{\mathrm{pred}}).
\]
Reduction at a mismatch prime contradicts the full side's tag exclusion
if both tag masks in that bin are nonempty.  Hence each such bin has
identical masks or one empty mask.  On high type the occupancy purge
forces the tag masks to be the same nontrivial mask $m$.

\paragraph{Earlier low bins on high type.}
On high type it remains to handle bins
$\sigma Y\le Y'<Y_d$.  Use the birth signatures
\[
 f_B=(Td^0,Ue^0),\qquad s_B=Td^0/f_B,
 \qquad t_B=Ue^0/f_B.
\]
Their birth determinant has tolerance
$C_0K/f_B\le C_0K/m$, which is sufficiently small by the size of a
prime in the common nontrivial tag mask and
\eqref{eq:component-path}.  Make the same divisor and prime switches
on $(d^0,s_B)$ and $(e^0,t_B)$, holding the birth phases fixed.
Stripping such a factor preserves both $f_B$ and $f_0$, since its primes
are below the common start: at each of them $u_i,u_j$ have zero
valuation, so the preceding primewise minimum calculation applies
to both gcds.  Fixing $q,u_i,u_j,a,b,e^0,d^0/h$ fixes both birth
phases during the rotation test.  The exact rescaling is
\[
 s_Bu_iq=\frac{qf_0}{f_B}s_0,
 \qquad t_Bu_jq=\frac{qf_0}{f_B}t_0.
\]
Thus the birth determinant is the current determinant multiplied by
$qf_0/f_B$, and its tolerance is $C_0K/f_B$, unchanged on the
stripping fiber.  The charge is still the current success count
\[
 \Theta_{ij}\frac{f_0}{q d^0u_i e^0u_j}:
\]
stripping extracts $1/h$ without removing its factor $1/q$.
The rotation error is uniform in the fixed birth phases, even though
those phases vary when $q$ is subsequently summed.  Hence
\eqref{eq:sparse-switch}, including its $1/q$ summation, still applies.

When a switch occurs, divide the resulting coupled approximants by
$s_B,t_B$.  Their denominators are at most
$\exp(\exp(h_*Y))$: use
$s_B/(s_B,d^0)\mid T$, the analogous divisibility on the other side,
the prime-power purge, and the divisor-assembly consequences of the
switches.  If $\psi_i^0=(L/d^0)r_i$ is the first birth numerator
width, then
\[
 \frac{C_0K/f_B}{s_B}
 =\frac{C_0Lr_{\max}}{d^0}
 =C_0\psi_i^0\frac{r_{\max}}{r_i}.
\]
Thus its approximation error divided by $\psi_i^0$ is at most
$e^{O(D''Y)}r_{\max}/r_i\le\exp(\exp(h_*Y))$, and similarly
on the other side.  The models concern the transported birth phases
$\alpha_i^0=u_iqa\gamma$, not the current phases $a\gamma$;
no division of a birth-model denominator by $u_iq$ is required.
Since $Y_d\ge\sigma Y$ and $h_*\ll\sigma h_H$, the high birth
models therefore qualify
and yield exact equality for the birth determinant.  At a mismatch
prime in these bins one signature is divisible by the prime and the
model denominators are units.  Exact equality then violates the hard
exclusion on the full side.  This prime lies above the hard threshold
by $h_*\ll\sigma h_H$ and $h_H\ll\sigma$.
If there is no switch, the signatures themselves are at most
$\exp(\exp(h_*Y))$, so they cannot contain a mismatch prime of height
at least $\sigma Y$.  Thus those masks are identical as well.

\paragraph{Local factors after reconstruction.}
We can now sum the neutralized coordinates using only static upper
weights, dropping other processed filters.  On an identical bin, the
gcd-ratio coordinate is $1/p$ at a common omission and $1$ at a full
factor.  If a common residual power remains, retain one side's scalar
or hard factor.  Its mean is $R_p$ on the residual common digit, since
$s_0,t_0,u_iq$ are units at $p$.  The summed local cost is at most
\[
 R_p+\frac1p R_p^{[\text{common exponent at least }2]}\le1.
\]
For a low static prime on high type, retain any one active hard
exclusion, or the actual scalar if no exclusion is active; discard the
other low tests for this upper count.

In a one-empty tag bin the empty side uses the ordinary scalar factors
$R_p$ prescribed at birth.
Neither-omitted coordinates cost at most $R_p^2$ on averaging the
common digit.  The two possible one-sided omissions have bare cost
$1/p$ each.  Even allowing both choices bounds the total by
$1+O(p^{-2})$.  Fix masks and anchor choices before using digit CRT,
as in Lemma~\ref{dir:tag-fold}.  There are only a bounded number of
bins between $\sigma Y$ and $Y$, so summing their identical/empty
arrangements costs a constant.  All unneutralized bins together now
cost only
\begin{equation}\label{sp:reduced-harmonic-cost}
 \exp\bigl(C(\sigma+\kappa)Y\bigr)\tau(UT)^C,
\end{equation}
with absolute coefficients in the harmonic exponent.

The earlier dense bins have therefore been removed from the harmonic
cost.  We next sum the remaining factor $1/q$ over the distance window
for off-gap pairs, then estimate failures of the endpoint train tests
for the tightly aligned equal-width pairs.

\subsection{The off-gap prime window}

We retain the factor $1/q$ in \eqref{sp:bare-count}.  To make the
remaining variables independent of the position of $q$ inside its bin,
extend $d,e$ through the end of that bin.  Write $a=a_Ya_>$ and
$b=b_Yb_>$ for their within-bin and above-bin parts.  Let $I_{>q}$
be the common prime-power part above $q$, and let $I_>$ be its part
above the bin.  The actual outer laws
satisfy the pointwise bound
\[
 \pi_{I_{>q}}(a)\pi_{I_{>q}}(b)
 \le \frac{\pi_{I_>}(a_>)\pi_{I_>}(b_>)}{a_Yb_Y},
\]
because $\pi_{p^\nu}(p^j)=R(p^{\nu-j})p^{-j}\le p^{-j}$.
Pad $a_Y,b_Y$ to arbitrary divisors of the whole bin product.  At
each common bin prime $p^\nu$, the total cost of the four extended
coordinates is bounded by
\[
 \left(\sum_{j,k=0}^{\nu}p^{-\max(j,k)}\right)
 \left(\sum_{j=0}^{\nu}p^{-j}\right)^2
 \le 1+\frac Cp.
\]
Thus padding costs at most $\exp(C\kappa Y)$.  Actual allowed
choices still require that $q$ be absent from all four masks and
that their before/after supports have the proper order; dropping
these restrictions later only enlarges the prime window.

We also need a bound for each fixed padded tuple that is independent
of $q$, not merely a bound after summing masks at a fixed $q$.
At a neutralized earlier prime, write $j,k\in\{0,1\}$ for its two
deficits.  After static digit averaging, the identical-bin mode is
bounded by the local majorant
\[
 Q_p(0,0)=R_p,\qquad
 Q_p(1,1)=p^{-1}R_p^{[\nu\ge2]},\qquad
 Q_p(1,0)=Q_p(0,1)=0.
\]
In the one-empty mode use instead
\[
 Q_p(0,0)=R_p^2,\qquad
 Q_p(1,0)=Q_p(0,1)=p^{-1},\qquad Q_p(1,1)=0.
\]
The latter permits either one-sided omission separately at each
prime and hence enlarges a bin with one globally empty mask.
The coordinate sums are at most $1$ and $1+O(p^{-2})$, respectively.
These majorants hold for every fixed anchor choice: their derivation
uses only one test's mean, or an ordinary scalar $R_p$ together with the other
test's mean, and is independent of the forbidden offsets.  Thus
the birth offsets may vary with $q$ without changing $Q_p$.
Adaptive weights have already been dropped.  Sum over the bounded
number of bin-mode patterns, not over a separate union of modes at
each prime.  Together with the unneutralized gcd coordinates and
the padded outer-law bound, this gives a $q$-independent majorant
for every fixed tuple, whose sum is bounded by
\eqref{sp:reduced-harmonic-cost} with the padding allowance.

For a fixed choice of these integers, the normalized distance of an
actual cohit is
\[
 \frac{\text{distance}}{r_{\max}}
 =\|s_0a\gamma-t_0b\gamma\|\frac{qf_0}{K}.
\]
The nearest determinant is unique.  If the norm vanishes the pair is
not an off-gap pair; otherwise the required distance range restricts
$q$ to a multiplicative window whose logarithmic width is
$O(1+\exp(\sigma_gY))$.  Elementary prime counting in ordinary dyads
therefore gives
\[
 \sum_{q\text{ in this window and bin}}\frac1q
 \ll \exp\bigl(-(1-\sigma_g)Y\bigr),
\]
since $\log q\ge\exp Y$.  The neutralized digit factors above are
independent of this $q$ summation: they belong to earlier bins.  We may
first enlarge the actual allowed prime support to this whole window
and then sum those local bounds.  Combining with
\eqref{sp:reduced-harmonic-cost}, and using the choices of $\sigma$ and
$\kappa$, proves the first estimate of
Proposition~\ref{sp:sparse-estimates}.

\subsection{Train coverage of tightly aligned pairs}

Only the relaxed tight equal-width scale remains.  We construct its
exceptional set by means of lists fixed before the table history is
used.  The purpose of the construction is twofold: its failures have
small product-mask cost, while every accepted connecting segment has
a deterministic spatial cover that can be protected at future joins.

\begin{construction}[Deterministic danger trains]\label{sp:danger-trains}
For a centre $i$, current prime $q$ in bin $Y$, and processed deficit
$d$ such that $dq$ is the processed portion of some good mask, put
$V'=L_i/(dq)$ and use a list of residues $y_0\pmod1$ of size at most
\[
 Z_Y=\exp(\exp(h_aY)).
\]
Include in the list every rational residue of denominator at most
$\exp(\exp(h_*Y))$.  Also include $0$ and generalized inverse
cumulative quantiles for the distribution of $a\gamma\pmod1$ under
the unrestricted deficit law on the prime powers of $L_i$ above $q$,
using
\[
 N=\left\lceil\exp(\exp(2h_*Y))\right\rceil
\]
equal probability increments.  Repeated quantiles are included only
once.  Each open arc disjoint from the marks then has probability
$O(1/N)$.

The endpoint test requires $x$ to lie within
\[
 r_i\exp\bigl(-.4\exp(\sigma_gY)\bigr)
\]
of a point $(z+y_0)/V'$, for an integer $z$ and a listed residue.
The stored train history uses the same grids with the larger radius
\[
 r_i\exp\bigl(-.3\exp(\sigma_gY)\bigr).
\]
The supply consists of all these grids for the indicated good processed
masks, whether or not any actual arrival uses them.
\end{construction}

The list-size bound follows because the rational residues number at most
$O(\exp(2\exp(h_*Y)))$ and the quantiles number $O(N+1)$, while
$3h_*<h_a$.  The outer law used for the quantiles depends only on the
centre and the current prime, not on matching decisions, partitions, or
which rows survive.  Each train is invariant under the post-$q$ core:
that core divides $V'$, by the construction underlying
\eqref{eq:core-loss}.  Thus this invariance persists at every later
component change.

Consider an actual successful pair of common width $r$ with distance at
most $r\exp(-\tfrac12\exp(\sigma_gY))$.  Apply the switching estimate
\eqref{eq:sparse-switch} at this smaller tolerance.  If a switch is
available outside its paid failures, the recovered rational models for
$a\gamma,b\gamma$ have denominators at most
$\exp(\exp(h_*Y))$.  Their errors are respectively at most
\[
 V'r\exp(-.4\exp(\sigma_gY)),\qquad
 W'r\exp(-.4\exp(\sigma_gY)).
\]
Indeed division by the signature converts the determinant tolerance
to the corresponding phase radius, and all switch losses are absorbed
by the gap between the exponents $-.5$ and $-.4$.  These rationals are
on the lists, so both endpoint tests hold.

If no switch is available, $s_0,t_0\le\exp(\exp(h_*Y))$.
Fix $b$.  The determinant condition restricts $a\gamma\pmod1$ to
at most $s_0$ arcs, each of radius
\[
 O\bigl(V'r\exp(-\tfrac12\exp(\sigma_gY))\bigr).
\]
To see the factor $V'$, divide the determinant tolerance by $s_0$;
the lcm of the successful denominators equals $s_0V'$.
Every such arc has diameter smaller than the phase padding in the
endpoint test.  This also handles marked endpoints: a closed
admissible arc meeting a mark places all its phases within that
diameter of the mark.  Arcs crossing $0$ may be split at $0$, which
is itself marked.  For the possibly atomic phase law, every component
of the complement of the quantile marks has mass at most $1/N$;
otherwise its cumulative distribution would cross some $k/N$
inside that component, placing a generalized inverse mark there.
In particular every atom of mass greater than $1/N$ is marked.
Thus an admissible arc failing the endpoint test is disjoint from
the marks and has unrestricted mass $O(1/N)$.  Failure on the first
side therefore has probability
$O(s_0/N)$, and failure on the second side has probability $O(t_0/N)$.
We may drop assignment restrictions for this estimate, keeping the
two outer laws independent.  These superexponentially small
probabilities absorb the crude mask costs and the current-prime sum.
Together with \eqref{eq:sparse-switch}, they prove the third estimate
of Proposition~\ref{sp:sparse-estimates}.

Declare precisely these pairs failing at least one endpoint test to
be the exceptions to coarsening.  Join every actual surviving
nonexceptional successful pair at the relaxed scale.  Its connecting
segment lies in the stored trains: the distance to an endpoint train
centre is at most the test radius plus the pair distance, which is
smaller than the stored radius for large $Y$.  This remains a valid
historical cover if an intermediate label later disappears.

Finally, every error obtained above is bounded by
$C\Theta_{ij}e^{-cY}\tau(UT)^C$.  Since
$Y\ge A_0\max(B_i,B_j)$, the centre-kernel summation using
\eqref{eq:centre-kernel} and \eqref{eq:centre-lower} applies, retaining
exponential decay for shells and heights.  This proves all assertions
of Proposition~\ref{sp:sparse-estimates}.  We have both the required
within-component energy bounds and a history-independent family of
train intervals covering every class-joining segment; the next section
uses exactly that family to preserve classes when components change.
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/sparse.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/tables.tex ====
\section{Projected tables and the second-moment reduction}
\label{sec:tables}

We now turn the selected rows and their birth prescriptions into weighted
interval tests.  A row initially appears on a finer grid than its true
numerator grid.  As the remaining primes are read, we restrict that finer
grid and adjust its weights.  The adjustment preserves first mass and
cancels many close-pair fluctuations.  The uncancelled fluctuations are
controlled by squared masses of classes of labels.

The arithmetic estimates needed for this procedure are stated precisely
below as Propositions~\ref{tab:first-join-input}--\ref{tab:partition-input}.
Their proofs occupy Sections~\ref{sec:direct}--\ref{sec:joins}.  The present
section defines the procedure and proves the second-moment reduction
conditional on those inputs.  In particular, none of their counting
conclusions is presumed to follow merely from the table notation.

\subsection{Chronology, labels, and the translation core}

Activate a centre $L_i$ at its start height $Y_0(i)$.  At the beginning of
each dyadic height $Y$, join active centres $L_i,L_j$ by an edge when
$\log UT\le\beta Y$, where, as before,
\[
 G=(L_i,L_j),\qquad U=L_i/G,\qquad T=L_j/G.
\]
The resulting graph components grow and coalesce as $Y$ increases.  All
prime powers in a component at primes with $\log\log p\ge Y$ are equal:
an exponent difference at such a prime would already exceed the edge
threshold.  Equality propagates along paths.

Every active band satisfies $B_i\le Y/A_0$.  The centre count from
Section~\ref{sec:centres} therefore bounds the number of vertices by
$\exp(4Y/A_0+O(1))$.  Sum edge log-ratios along a path and use
$|\log k_{L_i}|=O(B_i)$ to obtain, for any two vertices of one component,
\begin{equation}
 \log\bigl(2+UT+K+r_{\max}/r_{\min}\bigr)
 \le \exp(h_*Y/10).
 \label{eq:component-path}
\end{equation}
Here $K=[L_i,L_j]r_{\max}$, and the parameter hierarchy makes $A_0$
sufficiently large for this bound.  Read the common tail primes of the
bin in increasing order.  The block is finite, so finitely many height
steps, including empty ones if necessary, process every prime and start
and eventually join every centre.

At an intermediate time let $I$ be the common remaining prime-power
part of a component.  A row assigned to $L_i$ has a processed deficit
$d$ and a remaining deficit $a\mid I$, with
\[
 v=\frac{L_i}{da}.
\]
Its current projected grid is
\begin{equation}
 x=\frac{z+a\gamma}{L_i/d},\qquad z\in\mathbb Z.
 \label{tab:projected-grid}
\end{equation}
We distinguish labels even when their locations coincide.  A label
records the centre, the assigned row, and the numerator coordinate.
Each label has a raw weight $w_x\in[0,1]$.  Its weighted point mass is
\[
 \lambda_x=\pi_I(a)w_x.
\]
We also write $\lambda$ for the atomic point measure with these
label masses; coincident labels retain their multiplicities.
All point sums are per unit length.  The first mass of an array is
$\sum_x r_x\lambda_x$, with $r_x=r_i$ on a row of centre $i$.
We call the collection of these labelled arrays and raw weights for
one component its \emph{projected table}.

The role of the mask law is expressed by the exact identity
\begin{equation}
 r_i\pi_I(a)\frac{L_i}{d}
 R\left(\frac{L_i/I}{d}\right)
 =r_i\phi(v)=k_{L_i}\pi_{L_i}(da).
 \label{tab:first-currency}
\end{equation}
Thus the single coefficient $\pi_I(a)$ in first mass is distinct from
a numerator-density factor.  We sum only over the assigned good masks;
we do not interpret their mask coefficients as automatic numerator
exclusions.

At birth the raw weights are the static prescriptions of
Section~\ref{sec:prescriptions}, including the additional static test
constructed in Section~\ref{sec:direct}.  Subsequent locations descend
from labels of the same row at the same physical point.  For example,
if a further mask portion $b$ is processed, the old denominator, phase,
and numerator are all $b$ times their new values.  Raw weights never
increase.  Dead labels may be omitted or assigned weight zero.  We use
the circle for partitions and the periodic lifts for pair counts;
these give the same nearby pairs because all physical widths are small.

The order at a height is as follows: insert newborn projections,
perform the partition tests when a component is created or changes,
clip the resulting cell loads if needed, and then perform its
sparse-prime updates.  We call the birth factors \emph{static}; the
prime updates from its start onward are \emph{adaptive}.

For the translations used below we need a common symmetry group.
Immediately before a prime $p$ in bin $Y$, put
\[
 X=\exp(2h_HY).
\]
Start with the gcd of the centres in the component.  At each already
passed prime $l$, including all primes in earlier bins, reduce its
exponent by
\[
 4+\left\lfloor\frac{CX}{\log l}\right\rfloor,
\]
clipping at zero, where $C$ is a sufficiently large fixed constant.
The resulting integer is the \emph{pre-$p$ core}.  After the update,
make this reduction at $p$ as well; the result is the
\emph{post-$p$ core}, denoted $\mathrm{core}_p$.  At bin start only
earlier primes have been reduced.  Cores decrease by divisibility
through this chronology, including at joins.

The structural requirement on every operation is invariance, or
equivariance of its labelled objects, under translation by the
reciprocal of the current core.  The birth prescriptions have this
property.  Indeed, a processed deficit uses at most
$1+\lfloor C_{\rm mask}B_i/\log l\rfloor$ powers at $l$, a hard
exclusion costs one residue digit, and a soft condition need only know
the numerator modulo
\[
 l^{\min\{\nu_l(V),\,1+\lfloor h_s\log H_v/\log l\rfloor\}}.
\]
Here $\log H_v\ll X$ for every model in use.  The extra static tests
use squarefree low numerator moduli.  These observations show both
that the core divides the projected denominator and that reciprocal
core translation preserves the birth data.  The construction of
partitions and the matching rule below preserve this symmetry.

The loss of the core is bounded by
\begin{equation}
 \log\frac{L_i}{\mathrm{core}_p}
 \ll \exp(h_*Y/10)+p+\exp(O(X))\ll p.
 \label{eq:core-loss}
\end{equation}
For the gcd loss, sum the edge log-ratios over a spanning tree.  At a
fixed prime the maximal exponent loss needed to reach the common gcd
is at most the sum of the absolute exponent changes on that tree.
The remaining loss follows from the prime-log bound and the reductions
just specified.  For instance, primes not exceeding $\exp(O(X))$
give the last term; above that threshold the sum of the terms
$4\log l+CX$ up to $p$ is $O(p)$.  Since $\log p\ge\exp(Y)$ and
$2h_H<1$, the last inequality follows.  The path term is bounded using
\eqref{eq:component-path} and the choice of $A_0$.

In particular, if $p^\nu\parallel L_i$ is a currently unprocessed
tail prime power, then $p^\nu$ divides the pre-$p$ core.  All previous
decisions therefore preserve translation by $1/p^\nu$.  This includes
the transport of earlier projection data along the same-location
ancestor of a current label: at every earlier time that prime power
belonged to the core of its then-component.

\subsection{Cells, clipping, and a prime matching rule}

The construction uses, separately for each exact width in a component,
a partition of the circle into cells.  A partition is replaced only
at creation or change of that component.  The precise existence and
loss estimate for these partitions is
Proposition~\ref{tab:partition-input}.  We first state the properties
needed to define the update.
At a change, the previous components contributing existing tables are
called the \emph{old children}; newly inserted centres are not old
children.

After the partition tests, any surviving points in the same cell,
across all masks, are separated by
\begin{equation}
 O\bigl(r_i e^{-E_gB_i}\bigr)
 \label{tab:cell-span}
\end{equation}
when measured at an entry of centre $i$ there.  For a high-type
component having direct newborn comparisons at height $Y$, the
stronger bound $O(r_i e^{-E_gY})$ holds.  Cells are padded at surviving
points: in a later sparse step at height $Y'$ of the same unchanged
component, same-width pairs at distance at most
\[
 r_i\exp\left(-\tfrac12\exp(\sigma_gY')\right)
\]
lie in the same cell.  At a component change, survivors from one old
child that lie in one new cell come from one of that child's old
cells.

For each current remaining mask separately, cap the total raw load
in a cell at $N_0$ by scaling all its weights if necessary.  If $x_0$
is its pre-clip load and $x_D$ is the contribution of an old child $D$,
then the inherited cap gives
\begin{equation}
 \sum_Dx_D^2\le N_0x_0.
 \label{tab:clip-child}
\end{equation}
When $x_0>N_0$, it follows that
\[
 x_0-N_0
 \le\frac{x_0^2-\sum_Dx_D^2}{N_0}.
\]
Consequently the first mass lost in clipping is at most $N_0^{-1}$
times the same-width, same-mask pair count in the new cells, excluding
pairs from the same old child and including newborn diagonals.  Its
currency is
\begin{equation}
 r_i\pi_I(a)w_xw_y,
 \label{tab:one-law}
\end{equation}
with \emph{one} mask-law factor.  This is the narrow count in
Proposition~\ref{tab:first-join-input}.

There is also an equivalence relation on surviving labels, separately
for each exact width but not separately for each mask.  Its
equivalence classes, called \emph{accounting classes}, lie inside
cells.  At a component change retain each old-child class restricted
to its surviving labels; newborn labels start as singletons.  At
each sparse-prime step coarsen these classes only after updating
weights, using the tightly aligned successful plus pairs selected in
Proposition~\ref{tab:sparse-input}.  At every thinning or projection
we restrict a class to its surviving descendants without splitting
its equivalence relation further.  In particular, deleting an old
connecting label does not split a class.  The new partitions must
respect all these inherited equivalences.

Fix a tail prime power $p^\nu\parallel L_i$.  Good masks have no
deficit of exponent at least two at this prime.  For each mask after
$p$, and in each cell of a fixed width, form two lists before the
update.  Write $I=p^\nu I'$ and fix an after-prime mask $a\mid I'$.
The \emph{plus list} has before-prime mask $pa$, with the single
omission at $p$; the \emph{minus list} has before-prime mask $a$,
with the full factor.  First multiply plus weights by $R_p$ if $\nu\ge2$.
A plus label succeeds exactly when its before-prime numerator is
divisible by $p$, so that it lies on its next projection.

Fractionally match the residual-scaled plus weights to the minus
weights in that cell.  Concretely, use a finite nonnegative matrix
whose row and column sums are bounded by the endpoint weights and
whose total mass is the smaller of the two list loads.  Product
coupling provides such a matrix equivariantly.  Retain a matched
plus portion precisely on success, and retain its matched minus
portion precisely on failure of that plus label.  Retain unpaired
plus mass on success, and multiply unpaired minus mass by $R_p$.
This rule never moves a retained location and is post-core
equivariant.

\begin{lemma}[One-prime mass and cap preservation]
\label{tab:prime-update}
The matching rule preserves the cap on raw cell load per after-prime
mask.  On averaging over pre-core $p$-translations, each transported
label's weighted mass after the update has mean equal to its
before-prime weighted mass.  In particular, the total first mass per
unit length is preserved by every sparse update.
\end{lemma}

\begin{proof}
Let $P$ be the residual-scaled plus load and $M$ the minus load.
Matched portions contribute their full matched mass between the two
lists.  Only the larger list has unpaired mass, which can only
decrease.  The post-update load is therefore at most $\max(P,M)$,
and hence at most $N_0$.

For these fixed before-prime and after-prime masks, the probabilities
satisfy
\begin{equation}
 \pi_I(a)=R_p\pi_{I'}(a),\qquad
 \pi_I(pa)=\frac{R_p^{[\nu\ge2]}}p\pi_{I'}(a).
 \label{tab:mask-transition}
\end{equation}
Translation by $1/p^\nu$ changes the plus numerator by a unit
modulo $p$.  Its success indicator $S$ thus has orbit mean $1/p$.
The pre-link weights and the matching matrix transport equivariantly
on this orbit.  A matched portion of residual-scaled size $m$ has,
in units of $\pi_{I'}(a)$, before-prime masses $m/p$ and $mR_p$ and
after-prime masses $mS$ and $m(1-S)$.  Unpaired portions give the
same marginal identities.  Summing over the finite translation
orbits proves first-mass preservation.  The orbit is a simultaneous
translation of all lists; the assertion does not require independent
success colors.
\end{proof}

\subsection{The three quantitative inputs}

We specify the counts with their different pair-weight conventions
explicit.  For labels $x,y$ of widths $r_x,r_y$, write
$r_{xy}^- =\min(r_x,r_y)$ and $r_{xy}^+=\max(r_x,r_y)$.
The broad pair currency is
\begin{equation}
 r_{xy}^-\lambda_x\lambda_y
 =r_{xy}^-\pi_I(a_x)\pi_I(a_y)w_xw_y.
 \label{tab:product-law}
\end{equation}
The product of probabilities is used even when $a_x=a_y$.  A centre
pair has a \emph{first-sharing time} when its two centres first lie
in one component; a self-pair has its birth as first-sharing time.
At a join, counts involving new labels are taken before the fresh
cap clipping and after its preliminary partition tests.  For the
broad majorants those tests can be dropped.  For the narrow
majorants they first supply the cell-span restriction, after
which the point weights can be bounded by their incoming values.

The distinction between these currencies is essential. Broad energy
compares two weighted arrays and therefore multiplies their two mask
coefficients. Clipping controls raw load at one fixed mask, so it
uses that mask coefficient once. Table~\ref{tab:currency-map} records
these conventions for reference throughout the remaining proof.

\begin{table}[htbp]
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}p{.25\linewidth}p{.40\linewidth}p{.27\linewidth}@{}}
\toprule
Quantity & Weight per entry or ordered pair & Restriction \\
\midrule
First mass & $r_x\pi_I(a_x)w_x$ & One label \\
Broad pair count & $r_{xy}^-\pi_I(a_x)\pi_I(a_y)w_xw_y$
  & Both mask factors, even if $a_x=a_y$ \\
Clipping pair count & $r\pi_I(a)w_xw_y$
  & $r_x=r_y=r$, $a_x=a_y=a$ \\
\bottomrule
\end{tabular}
\caption{Mass conventions. Here $I$ is the current common remaining
prime-power part, $a_x$ is the remaining deficit of label $x$, and
$w_x$ is its raw weight. The clipping estimate also uses the new-cell
and old-child restrictions stated below.}
\label{tab:currency-map}
\end{table}

\begin{proposition}[First-sharing comparison input]
\label{tab:first-join-input}
For the selected finite blocks, there is a static additional birth
test, depending only on the row data and squarefree low numerator
moduli and not on the running tables, with arbitrarily small relative
first-mass loss against the hard birth means, such that the following
statements hold.

For any fixed required absolute $C$, the sum, over first-sharing
times and centre pairs, of the broad currencies
\eqref{tab:product-law} for point pairs at distance at most
$Cr_{xy}^+$ is $O(1)$.

For exact same-width, same-remaining-mask point pairs in one new
cell, sum the currencies \eqref{tab:one-law}, excluding pairs from
the same old child and including all newborn self and diagonal
contributions.  This sum is
\begin{equation}
 \ll W+O_{k_0}(W^{1+c})
 \label{tab:narrow-input}
\end{equation}
for a fixed $c>0$.  Both bounds are uniform in the finite block and
in prior table histories obeying the stated operations and core
symmetries.  The constants in \eqref{tab:narrow-input} may depend on
the preceding fixed cutoffs but not on $N_0$.
\end{proposition}

The direct adjacent simultaneous-birth part, including the static additional
test, is proved in Proposition~\ref{dir:birth-comparisons}.  The
remaining first-sharing times are treated in
Section~\ref{sec:joins}.  For the narrow estimates these proofs use
the span bounds \eqref{tab:cell-span}, rather than any unspecified
independence of cell membership.

\begin{proposition}[Sparse-alignment input]
\label{tab:sparse-input}
For each sparse-prime step in bin $Y$, there is an equivariant
selection of exceptional same-width successful plus pairs at
distance at most
\[
 r_i\exp\left(-\tfrac12\exp(\sigma_gY)\right).
\]
Coarsen accounting classes by every nonexceptional pair in this
range whose two arriving masses are retained.  Uniformly under the
allowed prior histories, the following three sums over components,
steps, applicable centre pairs, masks, and locations are bounded.
\begin{enumerate}
\item Actual simultaneous plus successes at distances between
$r_{xy}^+\exp(-\exp(\sigma_gY))$ and $Cr_{xy}^+$, weighted by
$r_{xy}^-$ times the product of their raw arrival weights and their
two after-prime mask probabilities.
\item Before-prime plus pairs at distance at most $Cr_{xy}^+$,
weighted by $r_{xy}^-/p^2$ times the product of their before-prime
raw weights and their two after-prime mask probabilities.
\item The selected exceptional actual successful pairs in the
relaxed same-width range above, with the same product-law currency
as in the first item.
\end{enumerate}
The constants are uniform in the block, in $N_0$, and in the
allowed preceding histories.  In the first and third items one may
use raw arrival weights immediately after the update and before
coarsening.  The successful linked and unpaired portions of a plus
entry have total mass at most that raw arrival weight.  The first
two items impose no equality of widths, and $C$ may be any required
fixed absolute constant.
\end{proposition}

Section~\ref{sec:sparse} gives an explicit selection by deterministic
trains of short intervals and proves these bounds.  Thus the
exception selection is part of the construction, not a choice made
after seeing the total energy.

\begin{proposition}[Chronological partition input]
\label{tab:partition-input}
The cells and accounting classes described above can be constructed
chronologically for every finite block.  The construction preserves
current-core equivariance, the cell spans and padding stated in
\eqref{tab:cell-span} and the paragraph following it, all inherited
accounting classes without splitting, and the property that one
new cell meets at most one surviving old cell of each old child.
After clipping, the raw cap per cell and remaining mask is $N_0$.
It supports the coarsening rule of
Proposition~\ref{tab:sparse-input} at each subsequent sparse step.

Aside from the cap clipping separately bounded by
\eqref{tab:narrow-input}, its total first-mass loss can be made
arbitrarily small relative to the selected block mass by the fixed
parameter cutoffs.  An expected loss bound over the specified
finite partition jitters suffices; the structural assertions and
core equivariance hold for every realized construction.  The
comparison bounds apply to the actual running tables or to the
nonnegative majorants indicated in their proofs.
\end{proposition}

This proposition is proved in Section~\ref{sec:partitions}.  Its
construction only uses previously generated interval trains and
inherited classes.  Conversely, the sparse estimates apply to the
history before the current coarsening.  The two inputs are therefore
used by finite chronological induction: the partition at a change
is constructed from the past, and the next sparse estimate is
applied before its new classes are joined.  No future partition is
used to establish a past estimate.

More explicitly, the order
is inherited tables and newborns, preliminary history tests,
realized padded cells and their further tests, cap clipping,
prime matching updates, and then class coarsening.

The history
tests use deterministic supplies of past interval trains and
single-row upper fields, not the cap's value.  The conditional
jitter bounds at each height charge a proportion of entering
first mass, which at that height is at most $W$; their
height-dependent failure probabilities are summable.  The
first-sharing and sparse
comparison bounds use weights at most one and their indicated
prefix or upper-field structure.  Thus their constants and the
non-clipping loss budgets are fixed before $N_0$ is chosen.

Figure~\ref{fig:chronology} separates the operations from the estimates
that justify them. Its return to previously stored history is the reason
that the partition and sparse-step arguments can be used in one finite
induction.

\input{figures/chronology}

\subsection{Accounting energy and broad pair energy}

We now show what these inputs accomplish.  Write
\[
 \mathcal B
 =\sum_{\text{current components}}
   \sum_{\substack{x,y\text{ in that component}\\
               \operatorname{dist}(x,y)\le10r_{xy}^+}}
       r_{xy}^-\lambda_x\lambda_y
\]
for the broad energy.  Discards decrease it.  Births and first joins
inject the pairs counted in
Proposition~\ref{tab:first-join-input}.  We must bound the remaining
increments from sparse updates.

For this purpose define the accounting energy
\begin{equation}
 \mathcal A=\sum_A r_A\lambda(A)^2,
 \qquad \lambda(A)=\sum_{x\in A}\lambda_x,
 \label{tab:accounting-energy}
\end{equation}
where $A$ ranges over current accounting classes, all of whose
labels have the same width $r_A$.  Since a class is inside one cell,
the cap for each mask and $\sum_{a\mid I}\pi_I(a)=1$ give
$\lambda(A)\le N_0$.  This applies to inherited old classes even
before a fresh clipping.

\begin{lemma}[Variance budget]
\label{tab:variance-budget}
Let $J_p$ be the increment of \eqref{tab:accounting-energy} on old
classes during a sparse-prime update, before coarsening, and let
$K_p'\ge0$ be the increment from the subsequent coarsening.  Then
\[
 J_p\ge0,\qquad
 \sum_p(J_p+K_p')\ll N_0W,
\]
where the sum includes all component steps.
\end{lemma}

\begin{proof}
Transport old labels and classes by the pre-core translations
$1/p^\nu$, assigning zero post-weight to vanished labels.
For a residual-scaled matched portion let $\mu$ denote its raw
mass times its after-mask probability.  Its contribution to the
increment of a class $A$ is
\[
 \mu\bigl(\mathbf1_{\{\text{plus}\in A\}}
          -\mathbf1_{\{\text{minus}\in A\}}\bigr)(S-1/p).
\]
For an unpaired plus portion omit the minus indicator; an unpaired
minus portion has zero increment.  This formula incorporates the
factor $R_p^{[\nu\ge2]}$ already absorbed into the residual-scaled
mass, and applies separately with each portion's own mask probability.
Thus it also applies to classes mixing masks and to links crossing
classes.

Let $\mathcal G$ be the finite group generated by $1/p^\nu$.
Write $Y_A(g)$ for the post-mass on the transported class $gA$,
and $m_A$ for its old mass, which is constant on its class orbit.
The formula above gives $\mathbb E_{g\in\mathcal G}Y_A(g)=m_A$.
Since each $g$ permutes the complete set of old classes per unit
length,
\[
 J_p=\sum_A r_A\bigl(\mathbb E_gY_A(g)^2-m_A^2\bigr)
     =\sum_A r_A\operatorname{Var}_g Y_A(g)\ge0.
\]
Possible stabilizers cause no change: averaging over the whole group
repeats every element of a class orbit equally.  This is an identity
for the actual summed energy increment, not an extra randomized
construction.  Coarsening adds the cross products between distinct
old classes that are joined, giving $K_p'\ge0$.

The total first mass inserted at births is at most $W$, by the
hard mean upper bound and \eqref{tab:first-currency}.
Lemma~\ref{tab:prime-update} preserves first mass at sparse steps,
so the total first mass lost in all discards is at most $W$.
Discarding class mass $d_A$ drops its square by at most
$2N_0d_A$.  Restriction to survivors creates no further drop through
splitting, because the equivalence relation is retained.  New
singleton classes contribute nonnegative injections, and the final
accounting energy is at most $N_0W$.  Telescoping the energy gives
the asserted budget.
\end{proof}

\begin{proposition}[Broad-energy bound]
\label{tab:energy-bound}
Assuming Propositions~\ref{tab:first-join-input}--\ref{tab:partition-input},
the final broad energy is bounded uniformly down the finite-block
tail.  The bound may depend on the fixed cap $N_0$ and preceding
hierarchy parameters, but not on the block.
\end{proposition}

\begin{proof}
\emph{Orbit expansion.}\quad
Index the centered fluctuations at a prime by residual-scaled plus
portions.  A portion of raw size $m$ and after-mask probability
$\pi$ has centered coefficient
\[
 \pi m(S-1/p).
\]
For an unpaired portion it acts only at its plus endpoint.  For a
matched portion it acts there and with the opposite sign at its
minus endpoint of the same width.  Indeed, in units of $\pi$, the
matched pre-masses are $m/p,m(1-1/p)$ and the post-masses are
$mS,m(1-S)$.

The broad distance kernel is invariant under simultaneous
translation.  On expanding the bilinear energy and averaging over
the pre-core orbit, the linear centered terms vanish.  The
quadratic terms are the joint-success kernel on plus portions minus
$1/p^2$ times the unrestricted kernel on the same pre-link
portions, with the signed endpoint kernels just described.  Both
terms carry the \emph{product} of their after-mask probabilities.
After summing locations, orbit-averaged joint successes can
equivalently be counted as actual simultaneous successes.

Whenever an endpoint pair contributes to a broad signed kernel,
the two corresponding plus positions are within $O(r_{xy}^+)$,
by the cell-span bound.  The total link fractions emanating from
one plus label are at most its mass.  Therefore all the absolute
$1/p^2$ terms and all joint-success terms outside the tightest
range are bounded by the first two parts of
Proposition~\ref{tab:sparse-input}.

\emph{Tight cancellation.}\quad
At distance at most
$r_{xy}^+\exp(-\exp(\sigma_gY))$ between the plus positions,
all endpoint distances, including those involving matched minus
positions, qualify for the broad kernel.  More explicitly, choose
$k_0$ small enough that the constant in \eqref{tab:cell-span}
times $e^{-E_gB_i}$ is at most $1/10$ for every retained centre.
Each linked displacement is then at most one tenth of its own
width.  All four endpoint distances are at most
$(1+1/10+1/10)r_{xy}^+<10r_{xy}^+$.  Matching preserves each
endpoint's width, so their four kernel values, including the
factor $r_{xy}^-$, are identical.  The signed contribution
of a transfer consequently cancels.  Only products of genuinely
unpaired plus arrivals remain in this range.

\emph{One-width self cost.}\quad
First fix one exact width $r$.  Consider the self cost consisting
of products of aligned unpaired plus arrivals within the relaxed
range $r\exp(-\tfrac12\exp(\sigma_gY))$, multiplied by $r$.
Pairs in different old classes are paid either by exceptions in
Proposition~\ref{tab:sparse-input} or by the coarsening increment
$K_p'$.

For the within-class pairs, let $P_A$ be the successful arriving
plus mass in an old class $A$ that is either unpaired or matched
to a minus portion outside $A$.  Let $N_A$ be the mass removed
from minus portions in $A$ by successful plus portions outside
$A$.  These are weighted masses, including their after-mask
probabilities.  Internal transfers cancel, and unpaired minus
weights have no centered fluctuation.  The old class fluctuation
is therefore the centered version of $P_A-N_A$.  Orbit
expectation gives
\begin{align}
 r\sum_A\mathbb E P_A^2
 &\le J_p+
 r\sum_A\left\{\bigl(\mathbb E(P_A-N_A)\bigr)^2
                    +2\mathbb E(P_AN_A)\right\}.
 \label{tab:self-cost}
\end{align}
To check the inequality, expand
$P_A^2=(P_A-N_A)^2+2P_AN_A-N_A^2$ and discard the last,
nonpositive term.  The variance part is included in $J_p$.

\emph{Capacity bounds.}\quad
Here are explicit capacity bounds for both charges.  Index plus
sources by $\xi$, with old class $B_\xi$, success indicator
$S_\xi$, and mass $c_\xi$ that it would carry upon success
(its residual-scaled raw weight times its after-mask probability).
Let $q_{\xi A}$ be the weighted mass of its links to minus
portions in $A\ne B_\xi$, and put $q_{\xi B_\xi}=0$.  Then
\[
 \sum_Aq_{\xi A}\le c_\xi,\qquad
 P_A\le\sum_{B_\xi=A}c_\xi S_\xi,
 \qquad N_A=\sum_\xi q_{\xi A}S_\xi.
\]
Every link uses the same after-mask at its two ends; the
inequalities therefore remain true when $A$ contains different
masks.  With
$a_{\xi A}=c_\xi\mathbf1_{\{B_\xi=A\}}+q_{\xi A}$,
we have $\sum_Aa_{\xi A}\le2c_\xi$.  Consequently,
\begin{align*}
 \sum_A\bigl|\mathbb E(P_A-N_A)\bigr|^2
 &\le p^{-2}\sum_{\xi,\eta}\sum_Aa_{\xi A}a_{\eta A}\\
 &\le 4p^{-2}\sum_{\substack{\xi,\eta\\
                  \operatorname{dist}(\xi,\eta)=O(r)}}c_\xi c_\eta,\\
 \sum_AP_AN_A
 &\le\sum_{B_\xi\ne B_\eta}
       c_\xi q_{\eta B_\xi}S_\xi S_\eta\\
 &\le\sum_{\substack{B_\xi\ne B_\eta\\
                  \operatorname{dist}(\xi,\eta)=O(r)}}
       c_\xi c_\eta S_\xi S_\eta.
\end{align*}
The distance restrictions are justified before enlarging the
sums: a nonzero term in a common class $A$ has both source
locations in its cell, either directly or through a link.
The first bound is paid by the before-plus counts times
$1/p^2$.  In the second, each ordered source pair occurs only
once, regardless of how many destination classes receive parts
of one source.  The far or exceptional pairs are paid by
Proposition~\ref{tab:sparse-input}.  Each remaining pair is in
the relaxed range and joins two old classes, so its charge is
at most the corresponding product of retained post-class masses
included in $K_p'$.  Summing \eqref{tab:self-cost} over widths,
components, and primes and applying
Lemma~\ref{tab:variance-budget} bounds the total relaxed self cost.

\emph{Unequal widths.}\quad
Finally take unequal widths.  Bin the actual post-projection
locations in bins of length $\delta=r_{\max}e^{-E_Y}$, where
$E_Y=\exp(\sigma_gY)$.  By \eqref{eq:component-path},
\[
 \log(r_{\max}/r_{\min})+\log3
 \le \exp(h_*Y/10)+\log3<\tfrac12E_Y
\]
for all permitted heights, so $3\delta$ is below the smaller
width's relaxed self scale.  Let $X_{i,b}$ be the unpaired
arriving mass of width $r_i$ in bin $b$, and let $\mathcal S_i$
be its full relaxed self cost at this step.  A tight pair is
in the same or adjacent bins, whence
\[
 r_{\min}\sum_{|b-c|\le1}X_{i,b}X_{j,c}
 \le3r_{\min}
       \left(\sum_bX_{i,b}^2\sum_cX_{j,c}^2\right)^{1/2}
 \le3\sqrt{\frac{r_{\min}}{r_{\max}}}
       \sqrt{\mathcal S_i\mathcal S_j}.
\]
First apply Cauchy--Schwarz over all steps and components for
each fixed pair of widths.  Then sum the resulting kernel
$2^{-|i-j|/2}$ over dyadic width indices; its summable convolution
bounds the total by a constant times the total self cost.  No
accounting class has to mix widths.  This bounds every sparse
increment of the broad energy; its birth and join injections are
bounded by Proposition~\ref{tab:first-join-input}.
\end{proof}

\subsection{Virtual weights and first-mass losses}

The energy argument supplies a uniform second moment.  To obtain
a positive integral on a fixed set, we also need to understand the
spatial distribution of first mass.  Adaptive weights need not be
fixed residue exclusions.  Their exact orbit marginals provide a
replacement for that simpler description.

\begin{proposition}[Virtual product marginals]
\label{tab:virtual-marginals}
For each retained good row, there is a nonnegative virtual weight
on its true numerator grid with the following properties.
\begin{enumerate}
\item It dominates the actual final weight, and its first mass
lies between $c_{\rm hard}\phi(v)r(v)$ and $\phi(v)r(v)$.
\item Its total excess over the actual final first mass is exactly
accounted for by the first-moment discard losses in the projected
tables, when the virtual weight omits the soft and additional
static tests, cap clipping, and partition discards.
\item At fixed static anchor choices, every subset of the
actual hard and adaptive fields has mean equal to the product
of its local means.  An actual hard field has mean $1-b_p/p$
when it forbids $b_p\le2$ distinct residues, or mean $R_p$
when it is a prescribed scalar; adaptive full fields have mean
$R_p$.  For upper counts one may retain just one hard birth
exclusion at each applicable prime, or the scalar $R_p$ when
there is no exclusion.  Any subset of the resulting hard and
adaptive fields has the product of its ordinary local means.
\item Fields through a prime $p$, restricted to a later projection
of denominator $v'$, have a joint period dividing $v'$ of size
at most $\exp(O(p))$.
\end{enumerate}
Domination in the first item is after the static anchor average;
it is not required separately for each auxiliary anchor choice.
\end{proposition}

\begin{proof}
We first construct the fields and prove domination and the first-mass
bounds.  We then verify subset means, identify discard losses by
telescoping, and control the periods.

\emph{Field construction and domination.}\quad
Fix a row and a location on its true grid.  At each full adaptive
prime $p$, take the actual fractional multiplier of its minus
ancestor at the $p$-update as the normalized field at that
location.  If the entry or plan is unavailable there, extend the
field by $R_p$.  Unavailability is pre-$p$-equivariant.
For clarity, if its positive raw prefix weight is $w_x$, its
link masses are $m_{x\xi}$, and
$u_x=w_x-\sum_\xi m_{x\xi}$ is unpaired, its multiplier is
\[
 M_x=1-\sum_\xi\frac{m_{x\xi}}{w_x}S_\xi
          -\frac{u_x}{p w_x}.
\]
The coefficients and the positive-prefix condition transport
invariantly on the pre-$p$ orbit.  Since each $S_\xi$ has mean
$1/p$, the field has orbit mean $R_p$; the extension has the
same mean on unavailable or zero-prefix orbits.  These are
translations by $1/p^\nu$, where $p^\nu\parallel L_i$.
All earlier fields are invariant along this orbit.

Because this row is full at $p$, its later grid is preserved by
the same translations.  In particular, restricting to later
omission subarrays does not destroy the orbit: the increment of
the earlier numerator is divisible by their prime factors,
which are coprime to $p$.  At an adaptive omission, the later
grid itself implements the finer projection; the only remaining
field at this prime is the scalar $R_p$ if its residual power is
positive.

Multiply these adaptive fields by the full hard birth weight,
omitting soft tests, the additional static discard, clipping,
and partition discards.  To handle hard anchor averaging, first
fix all static anchor choices but use the \emph{same} adaptive
multiplier functions computed from the actual averaged tables.
Every such fixed earlier hard field is invariant along each
future prime orbit.  Independence of the anchor choices from
the adaptive plans is therefore unnecessary.  Average the
resulting virtual products over the anchor choices at the end.

For the actual averaged hard weight, the actual final weight is
its product with these multipliers and the intervening discard
fractions.  All discard fractions lie in $[0,1]$, so the virtual
weight dominates it.  Reverse orbit averaging removes the
adaptive fields one by one.  CRT then gives the hard birth
means, bounded between $c_{\rm hard}$ and $1$ times the ordinary
totient density.  The true-grid omission proportions supply
exactly the missing mask factors, as in
\eqref{tab:first-currency}.  This proves the first item.

\emph{Subset means.}\quad
The same reverse averaging works for any selected subset of
the actual fields: remove the selected adaptive fields in
reverse order, retaining all selected earlier hard fields,
then use CRT on the hard fields.  Their residue colours may
depend on the fixed anchors, but their means are $1-b_p/p$
with $b_p\le2$, or $R_p$ for a scalar.  All these means are
positive: two active distinct exclusions occur only at large
primes by the model-separation estimate.  No lower bound on
their full product is needed for this exact subset identity.
For an upper version choose, within each fixed static
anchor prescription, one of the exclusions at a prime with an
active exclusion, independently of the partner in a pair
count.  Its mean is $R_p$; if none is active keep the prescribed
scalar $R_p$.  Dropping other exclusions and filters gives a
pointwise upper bound.  Any retained subset of these fields and
of the adaptive fields again has product mean, proving the
third item.

\emph{Discard telescoping.}\quad
For the second item, pull each true-grid label back through
its same-location ancestors and order its finite operations.
Let $f_j$ denote the nondiscard multiplier at operation $j$,
and $d_j\in[0,1]$ its retained discard fraction, inserting
identity factors when only one operation occurs.  Take the
anchor-averaged hard birth weight as the initial factor.
The exact finite identity is
\[
 \prod_{j=1}^n f_j-\prod_{j=1}^n(f_jd_j)
 =\sum_{j=1}^n
    \left(\prod_{k<j}f_kd_k\right)f_j(1-d_j)
    \left(\prod_{k>j}f_k\right).
\]
Thus every term has the actual prefix and the virtual suffix,
even when adaptive plans depend on the actual preceding
tables.  Future full multipliers eliminate in
reverse because the discard decision and its prefix are
invariant under the corresponding unprocessed-prime
translations.  Future omitted coordinates restrict to their
true numerator subarrays and contribute their residual
scalars.  More explicitly, for a remaining mask $a\mid I$,
averaging the future full fields on that subarray leaves the
factor $R(I/a)$.  The current prefix-weighted discard array
meets the subarray in exactly the fraction $1/a$, by its
still-full unprocessed core.  The factor recovered at the
discard time is thus
\begin{equation}
 \frac{R(I/a)}a=\pi_I(a).
 \label{tab:discard-currency}
\end{equation}
This is precisely its present first-mass currency.  Summing
these telescoping terms proves the assertion.  Soft-test losses
may accordingly be compared to hard birth means; future fields
restore those same means.

\emph{Period control.}\quad
Finally an adaptive field through $p$ is invariant under the
post-$p$ core translations.  Intersect this translation group
with a later grid of denominator $v'$.  Its period in the later
numerator coordinate divides $v'$ and is at most
\[
 \frac{v'}{(v',\mathrm{core}_p)}
 \le\frac{L_i}{\mathrm{core}_p}\le\exp(O(p)),
\]
by \eqref{eq:core-loss}.  Earlier field cores are divisible
by the post-$p$ core, so this also gives a common period for
all fields through $p$.  The constant is fixed by the hierarchy
and independent of the block.  Static hard fields use only
residue colours modulo their respective primes, so their joint
modulus divides $\prod_{l\le p}l\le\exp(O(p))$.  The potentially
large rational model numerator and denominator specify the
colour, not its modulus; if the denominator is divisible by
the prime, that exclusion is inactive.  Soft and hole tests
are omitted from the virtual product.  Thus their larger
allowed moduli and the hole cutoff $\lambda_0$ do not affect
this period bound.  This proves
the fourth item.
\end{proof}

We will also use the preceding argument before the last projection.
The following version makes explicit the interface needed to count an
existing table against a fixed spatial query in
Sections~\ref{sec:partitions} and~\ref{sec:joins}.

\begin{corollary}[Upper fields on an intermediate projection]
\label{tab:intermediate-fields}
Fix an admissible realized table history and a row whose current
projected grid is $(z+a\gamma)/(L_i/d)$, where $I$ is the remaining
prime-power part and $a\mid I$. On this grid the actual raw weight
is bounded, after averaging static anchors, by a product of fields
at the processed primes. Each nonconstant field lies in $[0,1]$.
At a full adaptive prime use the multiplier computed from the actual
matching, extended by $R_p$ where its prefix is unavailable; at a
birth prime retain one active hard exclusion, or its prescribed scalar
if none is active. An adaptive omission contributes only its residual
scalar when a positive power remains.

For every fixed choice of the static anchors, every subset of these
fields has mean equal to the product of its ordinary local means.
In particular the full upper product has mean
\[
 R_{\mathrm{done}}(L_i/d)
 :=R\bigl((L_i/I)/d\bigr).
\]
The joint numerator period of any retained fields through a prime $p$
divides the current projected denominator and is at most $\exp(O(p))$,
with the same fixed-parameter uniformity as in
Proposition~\ref{tab:virtual-marginals}.
\end{corollary}

\begin{proof}
Stop the field construction in
Proposition~\ref{tab:virtual-marginals} at the present projection.
Omit all discard fractions and keep the adaptive multiplier functions
from the realized tables. Choosing one hard exclusion can only enlarge
the fixed-anchor birth weight; average these enlarged products after
multiplying by the same adaptive fields. This gives the claimed
pointwise upper bound on the actual raw weight.

Reverse orbit averaging applies to every selected adaptive field
already processed. Later omissions, now only those before the present
time, use other primes and preserve its full-prime translation orbit.
The remaining static fields have their ordinary means by CRT. A
processed adaptive omission leaves precisely the residual scalar prescribed by
the prime update. Thus every subset has its product mean, and the
product over all primes is $R((L_i/I)/d)$. Finally intersect the
historical post-$p$ core translations with the present grid, exactly
as in the period-control argument above. Removing some fields does
not alter the remaining field functions or enlarge their joint period.
\end{proof}

\begin{lemma}[Equidistribution of virtual first mass]
\label{tab:equidistribution}
On every fixed nondegenerate spatial interval, the virtual
weighted mass is the interval's length times the total virtual
row mass, with relative error tending to zero uniformly as the
denominators tend to infinity.  The same statement holds for
their weighted trains
of interval indicators of half-width $r(v)/2$.
\end{lemma}

\begin{proof}
On the true grid of denominator $v$, first retain only fields
at primes $p\le c_0\log v$, with $c_0>0$ sufficiently small
and fixed.  Proposition~\ref{tab:virtual-marginals} gives a
joint period dividing $v$ and at most $v^{1/2}$.  For a
nonnegative periodic array, complete periods inside a fixed
interval contribute exactly its period mean; the two possible
incomplete periods give relative error $O(v^{-1/2})$.  This
constant may depend on the fixed interval's positive length;
the bound does not require a lower bound on the mean of one period.

Deleting the remaining fields changes the total mean by at
most a relative constant times
\[
 \sum_{\substack{p\mid v\\p>c_0\log v}}\frac1p=o(1).
\]
Indeed there are at most
$\log v/\log(c_0\log v)$ such prime divisors, so the sum is
$O(1/\log\log v)$.  To see the relative comparison explicitly,
fix anchors and denote the full and truncated means by $M$ and
$M_0$.  The actual-field subset identity of
Proposition~\ref{tab:virtual-marginals} gives
\[
 \frac{M_0-M}{M}
 =\prod_{\substack{p\mid v\\p>c_0\log v}}\mu_p^{-1}-1
 \le \exp\left(C\sum_{\substack{p\mid v\\p>c_0\log v}}\frac1p\right)-1,
\]
where $\mu_p\ge1-2/p$ is the relevant local mean.  All low
means cancel from this ratio, even if their product is very
small.  This uniform fixed-anchor estimate survives anchor
averaging.  Deleting factors in $[0,1]$ increases the array, so
this is an $L^1$ comparison and proves equidistribution for the
full virtual array.  Work first with fixed anchor choices and
then average; all estimates are uniform in those choices.

Since $r(v)\to0$, integrals of the weighted interval trains are
squeezed between the corresponding point counts in slightly
enlarged and slightly shortened fixed intervals.  This gives
the final assertion.
\end{proof}

\subsection{Completion of the conditional reduction}

\begin{proposition}[Finite-block reduction]
\label{tab:conditional-reduction}
Assume the selections and birth prescriptions of
Sections~\ref{sec:centres} and~\ref{sec:prescriptions}, and
Propositions~\ref{tab:first-join-input}--\ref{tab:partition-input}.
Then no measurable set of positive measure can avoid the
original approximation intervals in a whole tail.  Consequently
these inputs imply Theorem~\ref{thm:main}.
\end{proposition}

\begin{proof}
Let the proposed avoided set $E$ have measure $\mu>0$.  The row
selection and full-mask purges retain a fixed fraction $c_*>0$
of the block mass.  Indeed, after the raw-row case has been
excluded, the nonraw rows have a fixed positive share; the
negligible purges and selection of one of finitely many types
and colours preserve a fixed share.  The constant
$c_{\rm hard}>0$ is fixed by the hard prescriptions before
the later loss budgets are chosen.  Let $F^{\rm vir}$ be the sum of the virtual
weighted indicators with half-width $r(v)/2$, and let $F$ be
the corresponding realized sum on the true grids.  An indicator
has integral $r(v)$, so its mean integral agrees with the first
mass convention.  Proposition~\ref{tab:virtual-marginals} gives
\[
 M_-\le\int F^{\rm vir}\le M_+,\qquad
 M_-=c_{\rm hard}c_*w,\quad M_+=2w.
\]
These bounds are uniform in the block and every realized
partition history; we will not need to assume that its virtual
mass is deterministic.

The soft loss must be compared with the actual virtual first mass,
rather than with its uniform lower bound. Put
\[
 V=\mathbb E\int F^{\rm vir},\qquad M_-\le V\le M_+.
\]
By Lemma~\ref{pre:soft-loss} and the discard telescoping identity,
its contribution to the expected final loss is at most $\eta_s V$,
where $\eta_s=C\epsilon_s/h_s\le\mu/20$. This is the choice of
$\epsilon_s$ already made in the hierarchy; it does not require
$\epsilon_s$ to be selected after the resulting $c_{\rm hard}$.

With $c_{\rm hard}$ now fixed, make the additional static and partition
losses small using Propositions~\ref{tab:first-join-input}
and~\ref{tab:partition-input}. The clipping loss is at most
\[
 \frac1{N_0}\{O(W)+O_{k_0}(W^{1+c})\},
\]
by \eqref{tab:clip-child} and \eqref{tab:narrow-input}.
Choose the fixed cap only after those comparison constants.
Since $W\le2w$, the later loss budgets and the cap can be chosen so
that the combined non-soft loss is at most $\mu M_-/20$. Thus
\[
 \mathbb E\|F^{\rm vir}-F\|_1
 \le \eta_s V+\mu M_-/20.
\]
If no jitters are used, omit the expectation. The telescoping identity
\eqref{tab:discard-currency} ensures that these are exactly the
relevant final losses, including the soft loss measured against the
hard birth means.

Proposition~\ref{tab:energy-bound} gives a uniform bound on
$\|F\|_2$.  Indeed, the intersection length of two half-width
intervals is at most $r_{xy}^-$ and vanishes unless their
centres are at distance at most $r_{xy}^+$.  It is therefore
bounded by the final broad energy.  If jitters are used, the
same bound holds for $\mathbb E F$ by Jensen's inequality.
Fix a uniform constant $C_2$ with $\|\mathbb E F\|_2\le C_2$.

All constants, including $N_0,C_2,M_-$, are now fixed.  Choose
a fixed finite union of intervals $U$ with
\[
 |U\mathbin{\triangle}E|
 \le \min\left\{\frac\mu2,\,
           \left(\frac{\mu M_-}{4C_2}\right)^2\right\}.
\]
In particular $|U|\ge\mu/2$.  Uniform relative equidistribution
from Lemma~\ref{tab:equidistribution}, applied only after this
choice, gives on every sufficiently late block
\[
 \mathbb E\int_U F^{\rm vir}
 \ge (|U|-\mu/10)V\ge .4\mu V.
\]
Subtracting the loss bound gives
\[
 \int_U\mathbb E F
 \ge (.4\mu-\eta_s)V-\mu M_-/20
 \ge .3\mu M_-.
\]  However, every realized
summand is supported inside an original approximation interval,
so $\int_E\mathbb E F=0$.  Cauchy--Schwarz then gives
\[
 \int_U\mathbb E F
 \le C_2|U\mathbin{\triangle}E|^{1/2}\le .25\mu M_-,
\]
a contradiction.  This completes the
conditional reduction.
\end{proof}

To finish the argument, it remains to establish
Propositions~\ref{tab:first-join-input}, \ref{tab:sparse-input},
and~\ref{tab:partition-input}; Section~\ref{sec:joins} assembles
these inputs in the required chronological order.
==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//sections/tables.tex ====

==== BEGIN The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//references/main.bib ====
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==== END The-weak-inhomogeneous-Duffin-Schaeffer-conjecture//references/main.bib ====