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\title[A \(23/33+\eps\) Goldbach exceptional-set bound]
{A computer-assisted \(23/33+\eps\) bound for the exceptional set\\
in the binary Goldbach problem}

\author{Lorenzo Schiavone}
\date{18 July 2026}
\subjclass[2020]{Primary 11P32; Secondary 11M26, 11Y40}
\keywords{binary Goldbach problem, exceptional set, Dirichlet
\(L\)-functions, zero-density estimates, computer-assisted proof}

\begin{document}

\begin{abstract}
Let \(E(X)\) denote the number of even integers not exceeding \(X\)
which are not a sum of two primes.  Building on the zero-packet
framework of Zhao and the exceptional-set reduction of Pintz, we prove
\[
   E(X)\ll_\eps X^{23/33+\eps},
\]
and hence \(E(X)\ll X^{69697/100000}\).
The new ingredient is not a stronger zero-density theorem.  It is an
exact use of information already present in the fixed-class and
unrestricted zero-density inequalities.  First, an exhaustive
early-or-late split retains the positive density charge supplied by
each character class's own first zero.  Second, character classes are
kept in their common first-zero order while the \(R\)- and \(T\)-parts
of Zhao's packet are recombined.  A cap-and-mass majorization lemma
then gives the exact quadratic maximum over the resulting enlarged
polytope.

At the exponential coefficient \(A=33/10\), exact rational interval
arithmetic proves, in every discretized non-near-Siegel branch,
\[
 \Qpacket_A<\frac{198479}{200000}=0.992395;
\]
the directed limiting value is at most \(0.992394547694\), with slack
at least \(0.007605452306\).  The near-Siegel branch has a separate
positive gap depending on the fixed lower zero defect.  A
coefficient-stable form of the Pintz--Zhao bridge gives
\[
 E(X)\ll_\eps X^{23/33+\eps}.
\]
The exact inequality
\[
 \frac{69697}{100000}-\frac{23}{33}
 =\frac1{3300000}>0
\]
then gives the stated exponent.  All transcendental comparisons in
the finite certificate use outward rational enclosures; floating
point is used only to select trial parameters which are subsequently
verified exactly.  Source identifiers and reproduction commands are
included.
\end{abstract}

\maketitle

\section{Introduction}

Write
\[
 E(X)
 :=
 \#\{\,n\leq X:n\equiv0\pmod 2,\ 
               n\neq p+p'\text{ for all primes }p,p'\,\}.
\]
Pintz proved \(E(X)<X^{0.72}\) for sufficiently large \(X\)
\cite{Pintz2018}.  Zhao subsequently obtained
\[
 E(X)=O(X^{7/10})
 \tag{1.1}\label{eq:zhao-seven-tenths}
\]
and, by the same zero-density architecture, the Linnik-type bound
\(P(q)=O(q^5)\) \cite{Zhao2026}.  The implicit constants in these
exceptional-set results are ineffective.

The purpose of this paper is to refine Zhao's Goldbach packet while
leaving its analytic zero-density inputs unchanged.  The refinement
has three parts.

\begin{enumerate}[label=\textup{(\roman*)}]
\item Inside a fixed relative-conductor class, the first zero of that
class contributes a positive \(D\)-term to Zhao's fixed-class
inequality.  An exhaustive partition of its possible location yields
a universal class cap without assigning an invalid upper position
from a cumulative global count.
\item The unrestricted inequality orders the possible character
classes by their first zeros.  This gives rankwise \(R\)-caps, while
the unrestricted estimates give total \(R\)- and \(T\)-mass budgets.
\item The two cap sequences refer to the same ordered classes.  They
must therefore remain coupled.  An exact aligned majorization lemma
maximizes \(\sum_i(R_i+T_i)^2\) and prevents the loss created by
separately maximizing the square and cross terms.
\end{enumerate}

The coefficient in Zhao's packet is \(10/3\).  We work at
\[
 A=\frac{33}{10}.
\tag{1.2}\label{eq:A-value}
\]
This is a strengthening: lowering \(A\) increases every exponential
zero weight.

\begin{theorem}[Main theorem]\label{thm:main}
Let \(E(X)\) be the binary Goldbach exceptional set defined above.
For every \(\eps>0\),
\[
 \boxed{E(X)\ll_\eps X^{23/33+\eps}.}
\tag{1.3}\label{eq:main-result}
\]
The implicit constant is ineffective.
\end{theorem}

\begin{corollary}\label{cor:decimal}
One has
\[
 E(X)\ll X^{69697/100000}=X^{0.69697}.
\tag{1.4}\label{eq:decimal-result}
\]
\end{corollary}

The exact improvement in the base exponent over
\cref{eq:zhao-seven-tenths} is
\[
 \frac7{10}-\frac{23}{33}=\frac1{330}.
\tag{1.5}\label{eq:exact-improvement}
\]

The proof is computer-assisted only in a finite collection of explicit
inequalities.  The analytic inputs are quoted from Zhao
\cite[Lemma~3.1, Lemma~3.3, (3.14)--(3.43)]{Zhao2026} and Pintz
\cite[\S2, Theorems~I--K]{Pintz2018}.  The elementary majorization
argument is proved in full in \cref{sec:aligned}; the exact
analytic-to-certificate correspondence is stated in
\cref{sec:certificate}.

\begin{remark}[What the theorem does not prove]\label{rem:not-goldbach}
\Cref{thm:main} still permits a power-sized exceptional set.  It does
not prove the binary Goldbach conjecture and it gives no effective
threshold beyond which every even integer is Goldbach.
\end{remark}

\section{The zero packet and the exponent bridge}
\label{sec:packet-bridge}

\subsection{Packet notation}

Let \(Q\) be a modulus and let
\(\{\Kclass_\nu\}_{\nu\in I}\) be any finite family of nonempty,
pairwise-disjoint sets of Dirichlet characters modulo \(Q\).  We call
their union the relevant characters and call the \(\Kclass_\nu\)
packet classes.  For fixed truncation height \(H\), let
\(\Zset_\nu\) be the multiset of zeros
\(\rho=\beta+i\gamma\), counted with multiplicity, of the primitive
\(L\)-functions inducing the characters in \(\Kclass_\nu\) which
satisfy
\[
 1-\frac{H}{\log Q}\leq\beta\leq1,
 \qquad |\gamma|\leq H.
\]
Set
\[
 \lambda_\rho=(1-\beta)\log Q,
 \qquad
 S_{\nu,A}=\sum_{\rho\in\Zset_\nu}\e^{-A\lambda_\rho},
\]
and label defects within a nonempty class and the classes themselves by
\[
 \lambda_{\nu,1}\leq\lambda_{\nu,2}\leq\cdots,
 \qquad
 \lambda_{1,1}\leq\lambda_{2,1}\leq\cdots.
\]
Thus \(\lambda_{1,1}\) is the distinguished smallest defect and
\(\mu_\nu:=\lambda_{\nu,1}\) is the first defect of class \(\nu\).
Define the quadratic packet
\[
 \Qpacket_A=\sum_\nu S_{\nu,A}^{\,2}.
\tag{2.1}\label{eq:packet}
\]

\begin{definition}[Coefficient-\(A\) packet property]\label{def:ZA}
For fixed \(A>0\), write \(\mathsf Z(A)\) for the following assertion.
For \(c>0\), put
\[
 H_c:=\max\left\{5.68,\ 1.09\log\frac1c\right\}.
\]
For every \(c>0\) there exists
\[
 \kappa=\kappa(A,c)>0
\]
such that, for every \(H\geq H_c\) and \(C\geq1\), there exists
\[
 Q_0=Q_0(A,c,H,C)
\]
with the following property.  For every modulus \(Q\geq Q_0\) and
every finite family of nonempty, pairwise-disjoint character sets
\(\{\Kclass_\nu\}_{\nu\in I}\) modulo \(Q\), if the associated zero
multisets satisfy
\[
 \lambda_\rho\geq c
 \quad(\rho\in\Zset_\nu),
 \qquad
 \max_{\chi,\chi'\in\Kclass_\nu}
 \cond(\chi\overline{\chi'})\leq C,
\tag{2.2}\label{eq:packet-hypotheses}
\]
for every \(\nu\in I\), then
\[
 \Qpacket_A\leq1-\kappa.
\tag{2.3}\label{eq:packet-gap}
\]
\end{definition}

\begin{proposition}[Separation of defect and conductor parameters]
\label{prop:two-parameter}
Fix \(A>0\) and \(c>0\).  Suppose that the limiting
coefficient-\(A\) packet calculation obtained from Zhao's finite
argument closes, at zero source error, with a gap
\(\kappa_0=\kappa_0(A,c)>0\), uniformly for every \(H\geq H_c\).
Here uniformity means that all contributions above the finite detector
levels are controlled by \(H\)-independent tail majorants and that the
finite operations admit a common perturbation tolerance
\(\delta=\delta(A,c)>0\).  Then the analytic estimates underlying that
calculation may be made uniform for every \(H\geq H_c\) and every fixed
relative-conductor bound \(C\) in the precise order
\[
 (A,c)\longmapsto(\kappa_0,\delta),\qquad
 \forall H\geq H_c:\ H\longmapsto\epsilon_*(A,c,H),\qquad
 \forall C\geq1:\ (H,C)\longmapsto Q_0.
\tag{2.4}\label{eq:parameter-order}
\]
In particular, one may take the final packet gap
\(\kappa=\kappa_0/2\), independently of \(H\) and \(C\); those
parameters affect only \(Q_0(A,c,H,C)\).
\end{proposition}

\begin{proof}
We distinguish throughout the \emph{target perturbation}
\(\delta>0\) tolerated by the finite packet calculation from the
\emph{source conductor exponent} \(\epsilon_*>0\) in Pintz's density
theorems.  They are not the same parameter.

The relative-conductor hypothesis \(M\leq C\) is used to place one
class in Zhao's \(M=O(1)\) alternative.  In that alternative,
Lemma~3.1 uses
\[
 k=2(\phi+3x+y+z),\qquad \phi=\frac13,
\tag{2.5}\label{eq:fixed-k}
\]
and the fixed-class form of Lemma~3.3 is
\[
 (\Delta^2-\eps_z)N+2\Delta D\leq1.
\tag{2.6}\label{eq:fixed-density}
\]
After this alternative has been selected, neither
\cref{eq:fixed-k} and \cref{eq:fixed-density}, nor Zhao's displayed constant
\(\mathcal C(x,y,z,\Lambda,\lambda_0)\) contains the numerical value
of \(M\).  The same is true of Zhao's fixed-class integer
consequences, Abel summation, the \(R\)-decomposition, and the two
fixed-class \(D\)-alternatives.  Thus \(C\) is absent from every
limiting finite inequality.

By hypothesis, the exact certificate consists of finitely many rational
inequalities, finitely many continuous operations whose denominators are
strictly separated from zero, and tail inequalities which are uniform in
\(H\).  Its common target tolerance
\(\delta=\delta(A,c)>0\) is such that perturbing all source inequalities
by at most \(\delta\) changes the final packet ceiling by less than
\(\kappa_0/2\).  This choice precedes \(H\) and \(C\).

Now fix \(H\geq H_c\).  At the source of Zhao's fixed-class estimates,
Pintz's Theorems~D and~I are applied under
\[
 \cond(\chi\overline{\chi'})\leq Q^{\epsilon_*}.
\tag{2.7}\label{eq:source-conductor}
\]
Pintz's Theorem~D, specifically his (4.3), and Theorem~I, specifically
his (4.34), enter Zhao's Lemmas~3.1 and~3.3 with errors of the form
\[
 O_H(\epsilon_*)+o_{H,\epsilon_*}(1)
 \qquad(Q\longrightarrow\infty);
\tag{2.8}\label{eq:source-error}
\]
in particular, the multiplier \(1+C_D(H)\epsilon_*\) in
Theorem~D is height-dependent.  Choose
\(\epsilon_*=\epsilon_*(A,c,H)>0\) sufficiently small that every
\(O_H(\epsilon_*)\)-term, after its finitely many uses in the packet
calculation, consumes less than \(\delta/2\).

Only now fix \(C\).  If
\[
 Q\geq C^{1/\epsilon_*},
\tag{2.9}\label{eq:C-threshold}
\]
then \(M\leq C\) implies \cref{eq:source-conductor}.  Enlarge \(Q_0\)
until all the \(o_{H,\epsilon_*}(1)\)-terms in
\cref{eq:source-error}, together with the fixed-height zero-labeling
and Deuring--Heilbronn errors, consume less than \(\delta/2\).
The resulting finite packet ceiling is at most
\(1-\kappa_0/2\).

For fixed \(A\), changing the exponential coefficient affects only
fixed numerical weights and detector parameters.  None of the limiting
conductor reductions introduces the numerical value of \(C\).
This proves the asserted order
\cref{eq:parameter-order} and the quantifiers in \cref{def:ZA}.
\end{proof}

\begin{remark}
Zhao states the lower-defect and relative-conductor restrictions using
one parameter.  \Cref{prop:two-parameter} records the separation of
their roles needed below.  It is essential here that \(C\) is fixed
as \(Q\to\infty\); the statement would be false with no further
information if \(C=C(Q)\) were allowed to grow.
\end{remark}

\subsection{The Pintz--Zhao bridge}

\begin{lemma}[Conjugation and common-modulus domination]
\label{lem:common-modulus}
Let \(0<\theta<1/A\), \(Y=\lfloor X^\theta\rfloor\), \(q\leq Y\), and
\(C_0\geq1\).  Let \(\mathcal R\) be a submultiset of the natural
ordered Cartesian product of two finite zero multisets: each ordered
pair may occur only with the product of its zero multiplicities.  Write
its elements as \((\rho_i,\rho_j)\), with
\(\rho_\ell=\beta_\ell+i\gamma_\ell\) belonging to a primitive
character \(\chi_\ell^*\) of conductor \(r_\ell\mid q\), and suppose
\[
 \cond\bigl((\chi_i^*\chi_j^*)^*\bigr)<C_0
 \qquad ((\rho_i,\rho_j)\in\mathcal R).
\tag{2.10}\label{eq:original-product-condition}
\]
Put \(Q=q\lfloor Y/q\rfloor\).  After conjugating the second
zero variable, let \(\mathscr X\) be the set of distinct primitive
characters occurring in either coordinate, induce them to modulus \(Q\),
and
partition them into the connected components generated by
\[
 \cond\bigl((\chi^*\overline{\chi'^*})^*\bigr)<C_0.
\tag{2.11}\label{eq:relative-edge}
\]
Fix any \(U\geq1\) for which the selected and conjugated zeros lie in
the Zhao rectangle
\[
 1-\frac{U}{\log Q}\leq\beta\leq1,\qquad |\gamma|\leq U.
\]
If \(\Zset_\nu(U)\) denotes the full zero multiset of the characters in
component \(\nu\) in that rectangle, and
\(\lambda_\rho=(1-\beta)\log Q\), then
\[
 \sum_{(\rho_i,\rho_j)\in\mathcal R}
 X^{-(1-\beta_i)-(1-\beta_j)}
 \leq
 \sum_\nu
 \left(\sum_{\rho\in\Zset_\nu(U)}
       \e^{-A\lambda_\rho}\right)^2.
\tag{2.12}\label{eq:common-modulus-domination}
\]
Every component has the fixed within-class bound
\[
 \cond\bigl((\chi^*\overline{\chi'^*})^*\bigr)
 \leq C_{\mathrm{rel}}:=C_0^{K-1},
\tag{2.13}\label{eq:class-conductor}
\]
where \(K=\lvert\mathscr X\rvert\) (and the assertion is void when
\(\mathscr X=\varnothing\)).
\end{lemma}

\begin{proof}
If \(L(\rho_j,\chi_j^*)=0\), then
\(L(\overline{\rho_j},\overline{\chi_j^*})=0\).
The involution
\[
 (\rho_j,\chi_j^*)\longmapsto
 (\overline{\rho_j},\overline{\chi_j^*})
\tag{2.14}\label{eq:conjugation-involution}
\]
preserves \(1-\beta_j\), height, conductor, multiplicity, and the
condition \(r_j\mid q\).  It converts
\cref{eq:original-product-condition} bijectively into
\cref{eq:relative-edge}.  This is the reindexing between Pintz's
(2.13) and (2.36)--(2.37).

Since \(q\leq Y\), one has
\[
 q\mid Q,\qquad Y/2<Q\leq Y.
\tag{2.15}\label{eq:Q-range}
\]
If \(\chi^*\) has conductor dividing \(Q\), its induction \(\chi_Q\)
satisfies
\[
 L(s,\chi_Q)=L(s,\chi^*)
 \prod_{\substack{p\mid Q\\p\nmid\cond\chi^*}}
 \left(1-\chi^*(p)p^{-s}\right).
\tag{2.16}\label{eq:induced-L}
\]
The primitive \(L\)-function inducing \(\chi_Q\) is
\(L(s,\chi^*)\), so the primitive packet zero multiset is unchanged by
definition.  The additional Euler-factor zeros of the imprimitive
function lie on \(\Re s=0\) and do not enter that multiset.  Moreover,
\[
 \cond\bigl((\chi_Q\overline{\chi'_Q})^*\bigr)
 =
 \cond\bigl((\chi^*\overline{\chi'^*})^*\bigr).
\tag{2.17}\label{eq:primitive-conductor-preserved}
\]

Conductor submultiplicativity along a simple path of at most \(K-1\)
edges gives \cref{eq:class-conductor}.  For
\(\delta_\ell=1-\beta_\ell\) and \(B=1/\theta>A\),
\cref{eq:Q-range} gives
\[
 X^{-(\delta_i+\delta_j)}
 \leq Q^{-B(\delta_i+\delta_j)}
 =\e^{-B(\lambda_i+\lambda_j)}
 \leq\e^{-A(\lambda_i+\lambda_j)}.
\tag{2.18}\label{eq:weight-conversion}
\]
Every retained ordered occurrence is therefore bounded by the
corresponding ordered occurrence in its component square.  The
Cartesian-product multiplicity hypothesis ensures that no occurrence
is used more often than it appears in that square.  Enlarging the
selected zeros to the full multisets \(\Zset_\nu(U)\) only adds
nonnegative terms, which proves
\cref{eq:common-modulus-domination}.
\end{proof}

\begin{lemma}[Explicit-formula positivity ledger]
\label{lem:positivity-ledger}
Fix Pintz's explicit-formula parameter \(\tau>0\), and let
\(H_0,T_0\) be fixed truncation parameters.  Let \(K\) be an upper
bound for the number of selected zeros and let \(\eta>0\) be the
small-singular-series cutoff in Pintz's (2.12).  Set
\[
 \mathfrak s_0
 :=2\prod_{p\geq3}\left(1-\frac1{(p-1)^2}\right)>0.
\tag{2.19}\label{eq:singular-lower}
\]
For an even \(m\in[X/2,X]\), let \(\mathcal C_X(m)\) be the retained
zero--zero sum in Pintz's (2.21), with its original \(X\)-weights and
with the pole--pole pair removed.  Assume that no retained pole--zero
or zero--pole pair occurs.  Then, for all sufficiently large \(X\),
\[
\begin{split}
 \frac{R_1(m)}{\mathfrak S(m)m}
 \geq{}&
 1-(1+\omega_X)\mathcal C_X(m)
 -\frac{2\eta(K+1)^2}{\mathfrak s_0}\\
 &-\frac{2C_{\mathrm{EF}}(\tau)}{\mathfrak s_0}
 \left(
 \e^{-c_{\mathrm{EF}}H_0}
 +\frac1{\sqrt{T_0}}+X^{-\tau}
 \right),
\end{split}
\tag{2.20}\label{eq:positivity-ledger}
\]
where
\(\omega_X=O_{H_0,T_0}(1/\log X)\).
Consequently, if
\(\mathcal C_X(m)\leq1-\kappa\) for a fixed
\(0<\kappa\leq1/2\), then the parameters may be chosen in the order
\[
 \kappa\longmapsto H_0,T_0
 \longmapsto K
 \longmapsto\eta
 \longmapsto X_0
\tag{2.21}\label{eq:ledger-order}
\]
so that
\[
 R_1(m)\geq\frac{\kappa}{2}\mathfrak S(m)m>0
 \qquad(X\geq X_0).
\tag{2.22}\label{eq:positive-major-arc}
\]
\end{lemma}

\begin{proof}
For even \(m\), the Euler product in Pintz's (2.9) gives
\(\mathfrak S(m)\geq\mathfrak s_0\).  The pole--pole term in
Pintz's Theorem~A is exactly \(\mathfrak S(m)m\).
For every selected non-pole--pole singularity pair, Pintz's
beta-integral estimate (2.20) gives, uniformly over the fixed
rectangle,
\[
 \left|
 \frac{\Gamma(\rho_i)\Gamma(\rho_j)}
      {\Gamma(\rho_i+\rho_j)}
 \right|
 m^{-(\delta_i+\delta_j)}
 \leq
 (1+\omega_X)X^{-(\delta_i+\delta_j)}.
\tag{2.23}\label{eq:normalization-error}
\]
Indeed, if \(s=\delta_i+\delta_j\leq2H_0/\log X\), then
\[
 m^{-s}\leq2^sX^{-s}
 =\left(1+O_{H_0}\!\left(\frac1{\log X}\right)\right)X^{-s},
\]
which is the precise use of \(m\geq X/2\).

By hypothesis the retained non-pole terms are exactly the zero--zero
terms counted by \(\mathcal C_X(m)\).  There are at most
\((K+1)^2\) non-pole--pole pairs.  For every pair
failing one of Pintz's three retention conditions, his (2.12) gives
\(\lvert\mathfrak S(\chi_i,\chi_j,m)\rvert\leq\eta\); after increasing
\(X_0\), the beta factor in \cref{eq:normalization-error} is at most
two.  Their normalized total is therefore at most the second error
term in \cref{eq:positivity-ledger}.  Dividing the remainder in
Pintz's Theorem~A by
\(\mathfrak S(m)m\geq\mathfrak s_0X/2\) gives the final line of
\cref{eq:positivity-ledger}.

Choose \(H_0,T_0\) so that the first two explicit-formula terms
contribute at most \(\kappa/16\) each.  The log-free density bound
then fixes \(K=O_\tau(\e^{2H_0})\).  Choose \(\eta\) so that its term
is at most \(\kappa/16\).  Finally enlarge \(X_0\) until the
\(X^{-\tau}\)-term is at most \(\kappa/16\) and
\(\omega_X\leq\kappa/4\).  Since
\[
 (1+\kappa/4)(1-\kappa)\leq1-\frac{3\kappa}{4},
\]
\cref{eq:positivity-ledger} yields
\cref{eq:positive-major-arc}.
\end{proof}

\begin{theorem}[Coefficient-stable bridge]\label{thm:bridge}
Assume \(\mathsf Z(A)\).  If
\[
 0<\theta<\min\left\{\frac25,\frac1A\right\},
\tag{2.24}\label{eq:theta-range}
\]
then
\[
 E(X)\ll_{A,\theta}X^{1-\theta}.
\tag{2.25}\label{eq:theta-bound}
\]
Consequently, for every \(\eps>0\),
\[
 E(X)\ll_{\eps,A}
 X^{1-\min\{2/5,\,1/A\}+\eps}.
\tag{2.26}\label{eq:A-exponent}
\]
In particular, if \(A\geq5/2\), then
\[
 E(X)\ll_{\eps,A}X^{1-1/A+\eps}.
\]
\end{theorem}

\begin{proof}
It is enough to count exceptions in \(X/2<m\leq X\).
Fix, independently of \(X\),
\[
 0<\tau<
 \min\left\{\tau_0,\frac{2/5-\theta}{2},\frac{2}{45}\right\},
\tag{2.27}\label{eq:tau-choice}
\]
where \(\tau_0\) is the small absolute upper bound in Pintz's
Theorem~A.  Set Pintz's preliminary short-interval parameter
\(\varepsilon_0=2\tau\), and apply the explicit formula with
\[
 \vartheta_{\mathrm P}=\theta+2\tau,
 \qquad \eps_{\mathrm P}=\tau.
\tag{2.28}\label{eq:Pintz-substitution}
\]
Its hypotheses hold because
\[
 \tau<\vartheta_{\mathrm P}<\frac25<\frac49-\tau,
\]
and it supplies
\[
 X^{\theta+\tau}\leq P\leq X^{\theta+2\tau}.
\tag{2.29}\label{eq:P-range}
\]
The fact that this one \(P\) works simultaneously for every
\(m\leq X\), rather than being selected separately for each target,
is the uniformity assertion in Pintz's published
Theorem~1 \cite{Pintz2023}.
Pintz's minor-arc estimate (2.6) therefore leaves
\[
 O(XP^{-1}\log^{10}X)
 =o(X^{1-\theta})
\tag{2.30}\label{eq:minor-exceptions}
\]
targets.

Invoke Pintz's Theorem~B with \(\eps'=\tau\).  If its exceptional
real zero exists, then
\[
 E(X)\ll X^{3/5+\tau}=O(X^{1-\theta}),
\]
because \(\theta+2\tau<2/5\).  We may henceforth work in the
complementary alternative.  The classical primitive zero-free region
\cite[Ch.~14]{Davenport}, together with the absence of that single
possible exceptional real zero, supplies a fixed \(h=h(\tau)>0\)
such that every relevant bounded-height zero satisfies
\[
 1-\beta\geq\frac h{\log X}.
\tag{2.31}\label{eq:lower-beta}
\]
The constant \(h\) is chosen from the absolute zero-free-region
constant and Pintz's Theorem~B before any truncation height below is
chosen.  For every later fixed height \(U\),
\[
 \log(r(U+2))\leq(2/5)\log X+O_U(1).
\]
Thus taking \(U=H_Z\) below affects only \(X_0\), not \(h\); there is
no circular dependence.

Set \(c=h\theta/2\), and let
\(\kappa=\kappa(A,c)>0\) be furnished by \(\mathsf Z(A)\), decreased
if necessary so that \(\kappa\leq1/2\).  Choose \(H_0,T_0,K,\eta\)
in the order \cref{eq:ledger-order}, decreasing \(\eta\) further so
that \(\eta\leq1\).  Write
\[
 C_0=\eta^{-3},
 \qquad C_{\mathrm{rel}}=C_0^{K-1}.
\tag{2.32}\label{eq:relative-constants}
\]
Here Pintz's selected labelled-zero multiset is conjugation-stable, and
the number of distinct primitive characters is no larger than the
number of labelled zeros.  Thus the log-free density estimate supplies
one \(K\geq1\) which bounds both quantities (even if the selected set is
empty).  These are fixed numbers.
If one writes
\(\eta=\varepsilon_{\mathrm{cut}}/K^2\), then
\(\eta^{-3}=K^6/\varepsilon_{\mathrm{cut}}^3\); this is the
reciprocal forced by
Pintz's (2.13) and (2.24), rather than the opposite parenthetical
quantity printed in his (2.34).

Pintz partitions the targets into at most \(2^K\) subsets according
to which selected primitive conductors divide \(C_1(\eta)m\);
see his (2.26).  For one subset let \(q\) be their least common
multiple, with the convention \(\operatorname{lcm}(\varnothing)=1\),
and put \(Y=\lfloor X^\theta\rfloor\).  For the empty subset the
retained zero sum is empty.  If \(q>Y\), Pintz's
divisibility count (2.29) gives \(O_{\eta,K}(X^{1-\theta})\) targets
in that subset.  We may therefore suppose \(q\leq Y\) and form the
modulus \(Q\) in \cref{lem:common-modulus}.

The retained critical sum contains no pole--zero pair once \(X\) is
large.  Indeed, such a pair would force the zero character to have
primitive conductor \(<C_0\).  There are only finitely many such
primitive \(L\)-functions, and none vanishes at \(1\); their
bounded-height zeros eventually lie outside
\(\beta\geq1-H_0/\log X\).  We may therefore apply
\cref{lem:common-modulus} to the retained zero--zero pairs.  These
pairs form a submultiset of the natural ordered zero product, with
the multiplicities required there.

Pintz's horizontal truncation and \cref{eq:Q-range} give
\[
 (1-\beta)\log Q\leq\theta H_0,
 \qquad |\gamma|\leq T_0.
\]
Thus the selected zeros lie in a fixed Zhao rectangle with
\[
 H_Z:=\max\left\{
 \theta H_0,\ T_0,\ 5.68,\ 1.09\log\frac1c
 \right\}.
\tag{2.33}\label{eq:packet-height}
\]
Moreover, \(\log Q\geq(\theta/2)\log X\) for all sufficiently large
\(X\), and \cref{eq:lower-beta} gives
\[
 \lambda_\rho=(1-\beta)\log Q\geq c.
\tag{2.34}\label{eq:uniform-defect}
\]
The same bound holds for every additional zero of the selected
primitive characters in this full Zhao rectangle.  In particular,
\(H_Z\geq H_c\), so the moving cutoff required in the near-Siegel
branch is contained in the rectangle before \(\mathsf Z(A)\) is
invoked.

All parameters in
\[
 Q_0(A,c,H_Z,C_{\mathrm{rel}})
\]
are now fixed, and \(Q>Y/2\to\infty\).  Hence, for sufficiently large
\(X\), \(\mathsf Z(A)\) and
\cref{eq:common-modulus-domination} give
\[
 \mathcal C_X(m)
 \leq
 \sum_\nu
 \left(\sum_{\rho\in\Zset_\nu(H_Z)}
       \e^{-A\lambda_\rho}\right)^2
 \leq1-\kappa.
\tag{2.35}\label{eq:critical-packet-bound}
\]
\Cref{lem:positivity-ledger} now yields \(R_1(m)>0\) outside the
minor-arc exceptions and the large-\(q\) subsets.  Summing their
bounds over the at most \(2^K\) subsets proves the desired count:
indeed, on every remaining target,
\[
 R_1(m)\geq\frac{\kappa}{2}\mathfrak S(m)m
 \geq\frac{\kappa\mathfrak s_0}{4}X,
\]
which dominates Pintz's minor-arc bound
\(\lvert R_2(m)\rvert\leq X/\sqrt{\log X}\).  Hence
\(R(m)=R_1(m)+R_2(m)>0\) for sufficiently large \(X\).
This proves
\cref{eq:theta-bound} on \([X/2,X]\), and dyadic summation proves it
up to \(X\).

Finally put
\(\alpha=\min\{2/5,1/A\}\).  Given \(\eps>0\), choose once and for
all a fixed \(\theta<\alpha\) with
\(\alpha-\theta<\eps\).  Then
\[
 X^{1-\theta}\leq X^{1-\alpha+\eps},
\]
which proves \cref{eq:A-exponent}.  No parameter is allowed to vary
with \(X\) in this endpoint passage.
\end{proof}

\begin{remark}[Logical scope of the bridge]\label{rem:bridge-scope}
\Cref{thm:bridge} is conditional only on the packet property
\(\mathsf Z(A)\).  It does not itself prove the conductor-uniform
packet statement.  The quantifier order in \cref{def:ZA} is used
precisely after \(H_Z\) and \(C_{\mathrm{rel}}\) have been fixed and
before the final modulus threshold is imposed.
\end{remark}

\section{Coefficient stability of Zhao's analytic inputs}
\label{sec:coefficient-stability}

\(\eps_z>0\) denotes Zhao's auxiliary analytic error.  For every
fixed positive choice of \(\eps_z\), the source estimates below hold
once the modulus exceeds a threshold depending on the fixed detector
parameters; equivalently, the discarded source errors tend to zero as
the modulus tends to infinity.  The certificate first works at
\(\eps_z=0\) with strict rational margins and then chooses one common
positive value in \cref{eq:epsilon-separations}.

Fix a cutoff \(\Lambda\), and decompose one class as
\[
\begin{split}
 T_i&=\sum_{\rho\in\Zset_i}
       \e^{-A\max(\lambda_\rho,\Lambda)},\\
 R_i&=\sum_{\rho\in\Zset_i}
 \left(\e^{-A\lambda_\rho}
       -\e^{-A\max(\lambda_\rho,\Lambda)}\right).
\end{split}
\tag{3.1}\label{eq:RT}
\]
Then \(S_{i,A}=R_i+T_i\) and
\[
 \Qpacket_A=\sum_i(R_i+T_i)^2.
\tag{3.2}\label{eq:RT-square}
\]
When the cutoff is variable, we write explicitly
\[
 T_i(L)=\sum_{\rho\in\Zset_i}
          \e^{-A\max(\lambda_\rho,L)}.
\]

\begin{lemma}[\(T\)-stability]\label{lem:T-stability}
Suppose Zhao's Lemma~3.1 gives
\[
 \sum_j \e^{-k\max(\lambda_j,\Lambda)}
 \leq(1+\eps_z)\mathcal C(x,y,z,\Lambda,\lambda_0)
\tag{3.3}\label{eq:zhao-T-input}
\]
with \(k\leq A\).  Then
\[
 T_A(\Lambda)
 \leq
 (1+\eps_z)\e^{-(A-k)\Lambda}
 \mathcal C(x,y,z,\Lambda,\lambda_0).
\tag{3.4}\label{eq:T-general-A}
\]
\end{lemma}

\begin{proof}
For every \(\lambda\),
\[
 \e^{-A\max(\lambda,\Lambda)}
 =
 \e^{-(A-k)\max(\lambda,\Lambda)}
 \e^{-k\max(\lambda,\Lambda)}
 \leq
 \e^{-(A-k)\Lambda}
 \e^{-k\max(\lambda,\Lambda)}.
\]
Sum and apply \cref{eq:zhao-T-input}.
\end{proof}

Zhao's fixed-class tail gives \(100\e^{-2.22\Lambda}\) at
\(A_0=10/3\) for \(\Lambda\geq5.2\).  Reapplying Zhao's Lemma~3.1
with the same detector parameters and using
\cref{lem:T-stability}, at \(A\leq A_0\) we obtain the adjusted tail
\[
 x=\frac{29}{1000},\quad y=\frac{83}{1000},\quad
 z=\frac{52}{1000},\quad
 k=2\left(\frac13+3x+y+z\right)=\frac{833}{750},
\]
for which \(A_0-k=1667/750>2.22\).  Hence the adjusted tail is
\[
 100\exp\left(
 -\left[2.22-\left(\frac{10}{3}-A\right)\right]\Lambda
 \right).
\tag{3.5}\label{eq:adjusted-tail}
\]

\begin{lemma}[\(R\)-stability]\label{lem:R-stability}
Zhao's \(R\)-comparison remains valid at coefficient \(A\) provided
the detector parameter satisfies
\[
 x>
 \max\left\{\frac2A,\frac{4\lambda_0}{5}\right\}.
\tag{3.6}\label{eq:x-condition}
\]
\end{lemma}

\begin{proof}
The kernel support is \(0\leq u\leq2/x\).  For
\(d=\Lambda-\lambda_j\geq0\), Zhao compares
\(\e^{ud}-1\) with \(\e^{Ad}-1\).  For \(0\leq u\leq A\),
\[
 d\longmapsto\frac{\e^{ud}-1}{\e^{Ad}-1}
\]
is nonincreasing.  Thus \(2/x\leq A\) supplies the ratio
monotonicity, while Zhao's density inequality independently requires
\(x\geq4\lambda_0/5\).  Every certified detector is chosen strictly
above both thresholds, which is also the strict support hypothesis in
Pintz's Theorem~K.
\end{proof}

No interpolation in \(A\) is used.  Every detector selected
numerically is substituted back into
\cref{eq:T-general-A} and \cref{eq:x-condition}, and Zhao's \(R\)-formulas using
outward rational arithmetic.

\section{The positional fixed-class cap}
\label{sec:positional}

Zhao uses the compactly supported kernel
\[
 g(u)=\frac{(2-u)^3(4+6u+u^2)}{30}
 \quad(0\leq u\leq2),
\tag{4.1}\label{eq:g}
\]
and \(g(u)=0\) otherwise.  Let
\[
 G(s)=\int_0^2g(u)\e^{-su}\,du.
\]
For fixed \(x,\lambda_0\), define
\[
 \psi(v)=
 \frac{G((v-\lambda_0)/x)}{G(-\lambda_0/x)}.
\tag{4.2}\label{eq:psi}
\]
Put \(f_x(u)=xg(xu)\), \(F_x(s)=G(s/x)\), and
\[
 \xi=
 \frac{\phi f_x(0)}{2F_x(-\lambda_0)}
 =\frac{8\phi x}{15G(-\lambda_0/x)},
 \qquad \phi=\frac13.
\]
For a fixed character class \(i\) and detector level \(u\), write
\[
\begin{split}
 N_i(u)&=\#\{\,\rho\in\Zset_i:\lambda_\rho\leq u\,\},\\
 D_i(u)&=\sum_{\substack{\rho\in\Zset_i\\\lambda_\rho\leq u}}
       \bigl(\psi(\lambda_\rho)-\psi(u)\bigr),\\
 \Delta(u)&=\psi(u)-\xi.
\end{split}
\tag{4.3}\label{eq:positional-definitions}
\]
Since \(g\geq0\), \(G'(s)<0\), and hence \(\psi\) is decreasing.

At a detector level \(u\), Zhao's fixed-class inequality is
\[
 (\Delta(u)^2-\eps_z)N_i(u)+2\Delta(u)D_i(u)\leq1.
\tag{4.4}\label{eq:fixed-class}
\]

\begin{lemma}[Directed positional count]\label{lem:positional-count}
Take \(\lambda_0=a\) in \cref{eq:psi} and
\cref{eq:positional-definitions}.
Assume every zero in one fixed class has defect at least \(a\), and
that the class contains zeros at positions at most
\(v_1,\ldots,v_m\leq u\).  Then
\[
 D_i(u)\geq
 \sum_{j=1}^m\bigl(\psi(v_j)-\psi(u)\bigr).
\tag{4.5}\label{eq:D-lower}
\]
Let \(B\) be a nonnegative integer.  If rational lower enclosures
\(0<\Delta_-\leq\Delta(u)\) and
\(0\leq D_-\leq D_i(u)\) satisfy
\[
 (B+1)\Delta_-^2+2\Delta_-D_- -1>0,
\tag{4.6}\label{eq:integer-separation}
\]
then, after taking \(\eps_z>0\) sufficiently small,
\[
 N_i(u)\leq B.
\tag{4.7}\label{eq:integer-bound}
\]
\end{lemma}

\begin{proof}
Every certified zero contributes at least
\(\psi(v_j)-\psi(u)\) because \(\psi\) is decreasing.  If
\(N_i(u)\geq B+1\), the left side of
\cref{eq:fixed-class} exceeds one by
\cref{eq:integer-separation} for all sufficiently small
\(\eps_z>0\), a contradiction.
\end{proof}

\begin{definition}[Certified Abel cap]\label{def:abel-cap}
Fix an interval \([a,b]\subseteq[0.92,\Lambda]\).  Choose a finite
rational mesh
\[
 \Lambda=w_0<w_1<\cdots<w_L=5.2.
\]
At every \(w_\ell\), set the detector base parameter
\(\lambda_0=a\), use the same-class first-zero position \(b\), and
choose a rational detector \(x_\ell\) in
\cref{lem:positional-count} to certify
\(N_i(w_\ell)\leq B_\ell\).  Replace these raw bounds by the
nondecreasing suffix-minimum envelope
\[
 \widetilde B_\ell=\min_{j\geq\ell}B_j.
\]
This remains valid because
\(N_i(w_\ell)\leq N_i(w_j)\leq B_j\) for \(j\geq\ell\).
Relabel \(\widetilde B_\ell\) as \(B_\ell\).  The resulting directed
cap is
\[
\begin{split}
 c_A(\Lambda;a,b)
 :=\;&B_0\e^{-A\Lambda}
 +\sum_{\ell=1}^L(B_\ell-B_{\ell-1})
       \e^{-Aw_{\ell-1}}\\
 &+100\exp\left(
 -\left[2.22-\left(\frac{10}{3}-A\right)\right]5.2
 \right).
\end{split}
\tag{4.8}\label{eq:abel-cap-definition}
\]
Every exponential in the certificate is replaced by an outward
rational enclosure.  The definition of
\(c_A^\infty(\Lambda)\) is identical, except that the same-class
first-zero charge is omitted and every zero has lower defect
\(\Lambda\).
\end{definition}

\begin{proposition}[Universal early-or-late cap]
\label{prop:universal-cap}
Let \(\mu_i\) be the first zero in a high class, and suppose
\(\mu_i\geq0.92\).  Set
\[
 \Lambda=\frac{733}{500}=1.466
\tag{4.9}\label{eq:Lambda}
\]
and use the partition
\[
 0.92,\ 1,\ 1.05,\ 1.10,\ 1.15,\ 1.20,\ 1.25,\
 1.30,\ 1.40,\ 1.466.
\tag{4.10}\label{eq:high-partition}
\]
Let \(\mathcal P\) denote the nine consecutive closed intervals
determined by this list.  For the caps of \cref{def:abel-cap}, every
high class satisfies
\[
 T_i\leq C_A(\Lambda)
 :=
 \max\left\{
 c_A^\infty(\Lambda),
 \max_{[a,b]\in\mathcal P}
 c_A(\Lambda;a,b)
 \right\}.
\tag{4.11}\label{eq:universal-cap}
\]
\end{proposition}

\begin{proof}
If \(\mu_i\in[a,b]\), every zero in the class is at least \(a\), and
the class itself supplies a zero no later than \(b\).  Thus
\cref{lem:positional-count} applies with the positive charge
\(\psi(b)-\psi(u)\).  Abel summation of the resulting certified
integer envelope gives \(c_A(\Lambda;a,b)\).  If
\(\mu_i>\Lambda\), the ordinary cap at lower defect \(\Lambda\)
applies.  The cases in \cref{eq:high-partition}, together with this
late case, exhaust every possible \(\mu_i\).
\end{proof}

\begin{remark}[Direction of information]\label{rem:no-upper-from-global}
The cumulative unrestricted zero count gives rankwise lower bounds
for first zeros.  It does not give an upper position for an individual
class.  In \cref{prop:universal-cap}, the actual first zero selects its
own interval; no upper position is inferred from a global count.
\end{remark}

In \cref{eq:abel-cap-definition}, every unresolved jump is
deliberately placed at the adverse left endpoint.  The adaptive
tolerance measures only the excess of this safe placement; it is not
an unaccounted numerical error.

\section{Aligned cap-and-mass majorization}
\label{sec:aligned}

The next elementary lemma is responsible for the decisive saving in
the secondary branches.

\begin{theorem}[Aligned cap-and-mass lemma]\label{thm:aligned}
Let
\[
 a_1\geq a_2\geq\cdots\geq0,
 \qquad
 b_1\geq b_2\geq\cdots\geq0,
\]
on a finite or countable index set.  Suppose
\[
 0\leq r_i\leq a_i,\quad \sum_i r_i\leq U,
 \qquad
 0\leq t_i\leq b_i,\quad \sum_i t_i\leq V.
\tag{5.1}\label{eq:cap-mass}
\]
Define the left-greedy fills
\[
\begin{split}
 g_i^R&=\min\left\{a_i,
       \left(U-\sum_{j<i}a_j\right)_+\right\},\\
 g_i^T&=\min\left\{b_i,
       \left(V-\sum_{j<i}b_j\right)_+\right\}.
\end{split}
\tag{5.2}\label{eq:greedy}
\]
Then
\[
 \sum_i(r_i+t_i)^2
 \leq
 \sum_i(g_i^R+g_i^T)^2.
\tag{5.3}\label{eq:aligned-square}
\]
The right side is attained in the enlarged polytope
\cref{eq:cap-mass}.
\end{theorem}

\begin{proof}
We first prove the bilinear statement
\[
 \sum_i r_it_i\leq\sum_i g_i^Rg_i^T.
\tag{5.4}\label{eq:aligned-bilinear}
\]
Decreasing rearrangement preserves feasibility.  Indeed, if the
\(k\)-th largest coordinate of \(r\) exceeded \(a_k\), then at least
\(k\) original coordinates would exceed \(a_k\), while only the first
\(k-1\) cap positions can do so.  The same holds for \(t\).
The rearrangement inequality gives
\[
 \sum_i r_it_i\leq\sum_i r_i^\downarrow t_i^\downarrow.
\]
For every \(k\),
\[
 \sum_{i\leq k}r_i^\downarrow
 \leq\min\left\{U,\sum_{i\leq k}a_i\right\}
 =\sum_{i\leq k}g_i^R.
\tag{5.5}\label{eq:prefix-majorization}
\]
Abel summation against the decreasing nonnegative sequence
\(t^\downarrow\) gives
\[
 \sum_i r_i^\downarrow t_i^\downarrow
 \leq\sum_i g_i^Rt_i^\downarrow.
\]
Apply the corresponding \(b,V\) prefix majorization against the
decreasing sequence \(g^R\) to obtain
\cref{eq:aligned-bilinear}.

Apply \cref{eq:aligned-bilinear} to two copies of \(r\), two copies
of \(t\), and to \(r,t\).  Adding the resulting bounds proves
\cref{eq:aligned-square}.  The greedy pair itself is feasible, so
the bound is attained.  The countable case follows by truncation,
using the finite mass budgets.
\end{proof}

If an explicit finite prefix is followed by a repeated tail cap, the
tail is included before taking the right-to-left suffix maximum.  This
produces the least nonincreasing coordinatewise majorant and avoids
lowering a raw tail cap.  Each residual mass is then filled by complete
tail caps and at most one remainder.

\section{Ordering the character classes}
\label{sec:ordering}

For a chosen unrestricted detector, write
\[
 N(u)=\#\{\,\rho:\lambda_\rho\leq u\,\},
 \qquad
 D(u)=\sum_{\lambda_\rho\leq u}
       \bigl(\psi(\lambda_\rho)-\psi(u)\bigr),
\]
where the sums run over all packet classes.  In the active branch,
the at most two zeros certified at or below \(0.92\) are called the
designated low zeros, and the classes containing them are the low
classes.  Every remaining nonempty class is a high class and has
first defect at least \(0.92\).  Order these high classes by their
first defects.

In the active branch, let
\[
 0.92=u_0<u_1<\cdots<u_J=1.24,
 \qquad u_j-u_{j-1}=\frac1{200}.
\tag{6.1}\label{eq:global-mesh}
\]
The unrestricted form of Zhao's Lemma~3.3 is
\[
 (\Delta^2-\xi-\eps_z)N(u)+2\Delta D(u)\leq1-\xi.
\tag{6.2}\label{eq:unrestricted-count}
\]
For an active box \(a\leq\lambda_{1,1}\leq b\), use \(a\) for every
adverse lower-defect or exponential estimate, and use \(b\) only for
the positive positional term.

If \cref{eq:unrestricted-count} certifies \(N(u_j)\leq B_j\),
subtract the explicitly known low zeros.  The number of additional
high classes beginning by \(u_j\) is then bounded by a nondecreasing
integer envelope \(H_j\).  Ranks
\[
 H_{j-1}<i\leq H_j
\]
may be assigned first-zero lower endpoint \(u_{j-1}\).  Zhao's
fixed-class \(R\)-formula supplies a raw cap at that endpoint.  A
right-to-left suffix maximum, formed together with the repeated
\(u_J\)-tail, gives a decreasing coordinatewise majorant.

The unrestricted estimates give total budgets
\[
 \sum_iT_i\leq V_T,
 \qquad
 \sum_iR_i\leq V_R.
\tag{6.3}\label{eq:mass-budgets}
\]
Because Zhao introduces the unrestricted \(R\)-formula under
\(N\geq3\), \(V_R\) is replaced by the maximum of that unrestricted
bound and
\[
 2\left(\e^{-0.92A}-\e^{-A\Lambda}\right)
\tag{6.4}\label{eq:two-zero-guard}
\]
to cover \(N\leq2\).  The ordered caps and
\cref{eq:mass-budgets} are now exactly the hypotheses of
\cref{thm:aligned}.

\subsection{Allocation-compatible low classes}

In the active range, Zhao's alternatives allow at most two low zeros.
There are exactly three allocations:
\[
\begin{array}{c|c|c}
\text{allocation}&\text{global known positions}
 &\text{same-class positions}\\ \hline
\text{two classes, two zeros}&(b,0.92)&(b)\ \text{and}\ (0.92)\\
\text{one class, one zero}&(b)&(b)\\
\text{one class, two zeros}&(b,0.92)&(b,0.92).
\end{array}
\tag{6.5}\label{eq:allocations}
\]
The two-class case never assigns \((b,0.92)\) to a single class.
Applying \cref{lem:positional-count} separately to each identified
low class gives its \(T\)-cap; the compatible fixed \(R\)-cap then
gives the exact low-class square.  The high-class square is bounded
by \cref{thm:aligned}.

\begin{proposition}[Exhaustion of Zhao's alternatives]
\label{prop:branch-exhaustion}
Let \(H\geq H_c\).  The following rows cover every
\(c\leq\lambda_{1,1}\leq H\).  The middle column is the applicable
consequence of Zhao's Lemma~2.4; the last column records where the
corresponding packet is certified.
\begin{center}
\small
\begin{tabular}{p{0.17\textwidth}p{0.51\textwidth}p{0.20\textwidth}}
\toprule
range of \(\lambda_{1,1}\)&zero alternative&certificate branch\\
\midrule
\([c,0.01]\)&
\(N(\max\{5.68,1.09\log(1/\lambda_{1,1})\})\leq1\)&
near-Siegel\\
\([0.01,0.10]\)&\(N(3.08)\leq1\)&scalar\\
\([0.10,0.30]\)&\(N(1.58)\leq1\)&scalar\\
\([0.30,0.40]\)&\(N(1.29)\leq1\)&scalar\\
\([0.40,0.60]\)&\(N(0.92)\leq2\)&active \(63\) rows\\
\([0.60,0.62]\)&
\(N(0.85)\leq1\) or \(N(0.91)\leq2\)&secondary\\
\([0.62,0.64]\)&\(N(0.85)\leq2\)&secondary\\
\([0.64,0.68]\)&\(N(0.74)\leq2\)&secondary\\
\([0.68,0.702]\)&\(N(0.702)\leq2\)&secondary\\
\([0.702,H]\)&\(\lambda_{i,1}\geq0.857\) for \(i\geq5\)&
aligned suffix\\
\bottomrule
\end{tabular}
\end{center}
\end{proposition}

\begin{proof}
The zero alternatives are precisely Zhao's Lemma~2.4, including its
three unconditional final assertions.  If \(c>H\), the packet is empty
and there is nothing to prove.  Otherwise, intersect the displayed
closed intervals with \([c,H]\) and discard empty intersections; the
resulting rows cover \([c,H]\), with harmless overlaps at their
endpoints.  In the active interval,
\cref{eq:allocations} exhausts the possible placements of the at most
two low zeros.  In every secondary interval the verifier checks each
admissible allocation \((m,k)\), where \(m\) is the number of known
low zeros and \(k\) the number of classes which contain them.  The
remaining cases are the four scalar rows and the near-Siegel
majorant \cref{eq:near-siegel-majorant}.
\end{proof}

\section{The finite packet theorem}
\label{sec:certificate}

\begin{theorem}[Computer-assisted packet theorem at \(A=33/10\)]
\label{thm:certified-packet}
The coefficient-\(33/10\) packet property
\(\mathsf Z(33/10)\) holds.
More precisely, every discretized non-near-Siegel branch satisfies
\[
 \Qpacket_{33/10}
 <\frac{198479}{200000}=0.992395,
\tag{7.1}\label{eq:finite-packet}
\]
while the near-Siegel branch has a positive gap depending only on the
fixed lower zero defect \(c\).
\end{theorem}

The proof uses Zhao's inequalities in the forms transcribed in
\cref{app:formulas}, the branch exhaustion in
\cref{prop:branch-exhaustion}, and the exact certificate described
below.

\subsection{Exact arithmetic model}

Every terminating decimal used as an input denotes the exact
corresponding rational number.  All final comparisons are made in
\(\mathbb Q\).  Displayed packet bounds are rounded upward, while
displayed positive separations and slacks are rounded downward.
For \(x\geq0\),
the verifier encloses
\[
 \e^x=\sum_{k=0}^{m}\frac{x^k}{k!}+R_m(x)
\]
using the rational bound
\[
 0\leq R_m(x)
 \leq
 3^{\lceil x\rceil}\frac{x^{m+1}}{(m+1)!}.
\tag{7.2}\label{eq:exp-remainder}
\]
Negative arguments are handled by reciprocating a positive enclosure.
For the kernel transform, write
\[
 G(s)=\sum_{k\geq0}\frac{(-s)^km_k}{k!},
\]
where the moments are the exact rationals
\[
 m_k=
 \frac1{30}\left(
 \frac{32\,2^{k+1}}{k+1}
 -\frac{40\,2^{k+3}}{k+3}
 +\frac{20\,2^{k+4}}{k+4}
 -\frac{2^{k+6}}{k+6}
 \right).
\tag{7.3}\label{eq:moments}
\]
The omitted series is bounded by
\[
 \frac89\sum_{j>m}\frac{(2|s|)^j}{j!}.
\tag{7.4}\label{eq:G-remainder}
\]
Floating point performs only coarse scans and one-dimensional searches
for detector parameters.  The selected decimal rational is accepted
only after exact substitution into all support, positivity, and
next-integer separation inequalities.

\subsection{Active branches}

The active interval is covered by
\[
 [0.40,0.405],\ [0.405,0.41],\ [0.41,0.42],\ldots,[0.59,0.60].
\tag{7.5}\label{eq:active-cover}
\]
Each of the \(21\) boxes is checked in all three allocations from
\cref{eq:allocations}, for \(63\) directed rows.

The common high-class quantities are
\[
\begin{split}
 C_A(\Lambda)&\leq0.066553314396,\\
 V_T&\leq6.650837008243,\\
 V_R&\leq0.455946922433.
\end{split}
\tag{7.6}\label{eq:active-common}
\]
The exhaustive cap split in \cref{prop:universal-cap} is recorded in
\cref{tab:universal}.
\begin{table}[ht]
\centering
\small
\begin{tabular}{cccc}
\toprule
\([a,b]\)&\(c_A(\Lambda;a,b)\)&nodes&minimum separation\\
\midrule
\([0.92,1]\)&\(0.066553314396\)&203&\(\geq0.000019922993\)\\
\([1,1.05]\)&\(0.065749401858\)&195&\(\geq0.000019546894\)\\
\([1.05,1.10]\)&\(0.065434575306\)&191&\(\geq0.000018216157\)\\
\([1.10,1.15]\)&\(0.065122210666\)&185&\(\geq0.000055196136\)\\
\([1.15,1.20]\)&\(0.064818914259\)&182&\(\geq0.000004433480\)\\
\([1.20,1.25]\)&\(0.064511514566\)&177&\(\geq0.000013802128\)\\
\([1.25,1.30]\)&\(0.064214254658\)&174&\(\geq0.000007323372\)\\
\([1.30,1.40]\)&\(0.064292723507\)&170&\(\geq0.000058651699\)\\
\([1.40,1.466]\)&\(0.063449399440\)&162&\(\geq0.000038035052\)\\
\(\mu_i>1.466\)&\(0.065103393638\)&173&\(\geq0.000000293236\)\\
\bottomrule
\end{tabular}
\caption{Universal high-class \(T\)-cap case split.  Every displayed
decimal is rounded outward.}
\label{tab:universal}
\end{table}
The active maxima, separated by allocation, are recorded in
\cref{tab:active}.
\begin{table}[ht]
\centering
\begin{tabular}{lcc}
\toprule
allocation&packet upper bound&slack\\
\midrule
two classes/two zeros&\(0.982040541993\)&\(\geq0.017959458007\)\\
one class/one zero&\(0.774022703391\)&\(\geq0.225977296609\)\\
one class/two zeros&\(0.961300315347\)&\(\geq0.038699684653\)\\
\bottomrule
\end{tabular}
\caption{Maximal active packets, separated by low-zero allocation.}
\label{tab:active}
\end{table}
All three maxima occur in the first half-box.

\subsection{Secondary branches}

If \(m\) known low zeros occupy \(k\) classes, the number of classes
is at most
\[
 N_{\mathrm{zeros}}-(m-k),
\tag{7.7}\label{eq:class-count}
\]
because \(m-k\) excess zeros cannot begin new classes.  The positional
count therefore constructs a common ordered \(T\)-cap list.  In the
same order, the \(R\)-caps have the form
\[
 (\underbrace{a_R,\ldots,a_R}_{k},r_2,r_2,\ldots).
\tag{7.8}\label{eq:secondary-R-caps}
\]
Here \(N_{\mathrm{zeros}}\) is the certified unrestricted count
\(N(u)\) at the branch's separation level \(u\), \(r_2\) is the
fixed-class \(R\)-cap when every defect is at least \(u\), and \(a_R\)
is the compatible cap for a class containing designated low zeros.

For a secondary lower endpoint \(\ell\), let
\(r_{11}^{\mathrm{fixed}}\) be the larger of
\cref{eq:R-fixed-small} and \cref{eq:R-fixed-large} with prescribed lower
endpoints \((\ell,\ell)\), and let
\(r_{12}^{\mathrm{fixed}}\) be the analogous cap with endpoints
\((\ell,u)\).  Let \(U_{11}\) and \(U_{12}\) denote the maximum of,
respectively, the corresponding unrestricted bound from
\cref{eq:R-unrestricted} and the direct two-zero guard.  The endpoint
\(u\) in the \(12\)-bound
represents the worst possible next zero when only one low zero is
known; an absent next zero only decreases the true sum.
For allocations \((m,k)=(1,1),(2,1),(2,2)\), respectively, the
compatible fixed cap and unrestricted budget are
\[
 (r_{12}^{\mathrm{fixed}},U_{12}),\quad
 (r_{11}^{\mathrm{fixed}},U_{11}),\quad
 (r_{12}^{\mathrm{fixed}},U_{11}).
\tag{7.9}\label{eq:secondary-map}
\]
Applying \cref{thm:aligned} gives \cref{tab:secondary}.
\begin{table}[ht]
\centering
\small
\begin{tabular}{lccc}
\toprule
branch&\((1,1)\)&\((2,1)\)&\((2,2)\)\\
\midrule
\([0.60,0.62]\), \(N(0.91)\leq2\)
&\(0.941335795526\)&\(0.979388257544\)&\(0.981239614797\)\\
\([0.60,0.62]\), \(N(0.85)\leq1\)
&\(0.967962977997\)&inadmissible&inadmissible\\
\([0.62,0.64]\), \(N(0.85)\leq2\)
&\(0.939895062527\)&\(0.967038727055\)&\(0.972356429912\)\\
\([0.64,0.66]\)
&\(0.976868013140\)&\(0.987983379260\)&\(0.992394547694\)\\
\([0.66,0.68]\)
&\(0.950821675538\)&\(0.957609181221\)&\(0.962392175459\)\\
\([0.68,0.702]\)
&\(0.955998283891\)&\(0.956640496475\)&\(0.958101358231\)\\
\bottomrule
\end{tabular}
\caption{Certified upper bounds for the full secondary packet.}
\label{tab:secondary}
\end{table}
The limiting row is \([0.64,0.66]\), \((m,k)=(2,2)\), with
\[
 \Qpacket_{33/10}\leq0.992394547694,
 \qquad
 1-\Qpacket_{33/10}\geq0.007605452306.
\tag{7.10}\label{eq:limiting-slack}
\]

\begin{remark}[Necessity of alignment]
If the \(T^2\)-energy and the \(R^2+2RT\) terms are maximized
separately, the \([0.64,0.66]\) rows \((2,1)\) and \((2,2)\) have
negative slacks \(-0.005953317963\) and \(-0.010066017325\).
\Cref{thm:aligned} changes the latter full packet to
\(0.992394547694\).  Thus simply replacing \(10/3\) by \(33/10\) in
Zhao's scalar table does not prove the result.
\end{remark}

\subsection{The final scalar and near-Siegel branches}

For \(\lambda_{1,1}\geq0.702\), Zhao's class separation allows at most
four classes before the later level \(0.857\).  Use aligned cap lists
\[
 t(v)=\text{the certified fixed-class \(T\)-cap under }\mu_i\geq v,
 \qquad
 r(v)=\text{the corresponding fixed-class \(R\)-cap},
\]
after replacing each raw list, together with its repeated tail, by
the coordinatewise suffix-maximum majorant.  Thus
\[
\begin{split}
 \mathbf t&=(t(0.702),t(0.702),t(0.702),t(0.702),t(0.857),\ldots),\\
 \mathbf r&=(r(0.702),r(0.702),r(0.702),r(0.702),r(0.857),\ldots).
\end{split}
\tag{7.11}\label{eq:tail-lists}
\]
The old separated scalar is \(1.002462657342\); the aligned packet is
\[
 \Qpacket_{33/10}\leq0.981677779591.
\tag{7.12}\label{eq:aligned-tail-value}
\]
The remaining distinguished-zero branches are listed in
\cref{tab:scalar}.
\begin{table}[ht]
\centering
\begin{tabular}{lcc}
\toprule
branch&packet upper bound&slack\\
\midrule
\([0.702,H]\) (aligned)&\(0.981677779591\)&\(\geq0.018322220409\)\\
\([0.01,0.10]\)&\(0.946741143646\)&\(\geq0.053258856354\)\\
\([0.10,0.30]\)&\(0.884075525138\)&\(\geq0.115924474862\)\\
\([0.30,0.40]\)&\(0.974798896903\)&\(\geq0.025201103097\)\\
\bottomrule
\end{tabular}
\caption{Aligned suffix and distinguished-zero scalar branches.}
\label{tab:scalar}
\end{table}

For the near-Siegel branch, set
\[
 b_A=2.22-\left(\frac{10}{3}-A\right).
\]
Write
\[
 L=\Lambda(\lambda)
 =\max\left\{5.68,\ 1.09\log\frac1\lambda\right\},
 \qquad
 \mathcal T_A^{\mathrm{all}}(L)=\sum_i T_i(L).
\]
If \(c\leq\lambda=\lambda_{1,1}\leq0.01\), Zhao's
first-zero alternative gives
\[
 N(L)\leq1.
\]
Indeed,
\[
 L\leq
 \max\left\{5.68,\ 1.09\log\frac1c\right\}
 =H_c\leq H,
\]
so this cutoff lies inside the packet rectangle.
Consequently, in the decomposition \cref{eq:RT},
\[
 R_1=\e^{-A\lambda}-\e^{-AL},
 \qquad R_i=0\quad(i\geq2).
\tag{7.13}\label{eq:near-siegel-R}
\]
It follows that
\[
\begin{split}
 \Qpacket_A
 &=R_1^2+2R_1T_1(L)+\sum_iT_i(L)^2\\
 &\leq
 \e^{-2A\lambda}
 +\left(\max_iT_i(L)\right)
  \left(2+\sum_iT_i(L)\right).
\end{split}
\tag{7.14}\label{eq:near-siegel-decomposition}
\]

For the fixed-class detector
\[
 x=\frac{29}{1000},\qquad
 y=\frac{83}{1000},\qquad
 z=\frac{52}{1000},\qquad
 k=\frac{833}{750},
\]
the exact rational enclosure used by the verifier gives
\[
 \mathcal C(x,y,z,5.2,0)
 <99.729.
\tag{7.15}\label{eq:near-siegel-C}
\]
For \(\lambda_0=0\), the source kernel simplifies to
\[
 \mathcal B(t,L,0)
 =\frac{\phi}{2}\frac{1-\e^{-2Lt}}L
  +\left(\frac{1-\e^{-Lt}}L\right)^2.
\]
Both terms are nonincreasing in \(L>0\), since
\[
 \frac{1-\e^{-aL}}L=\int_0^a\e^{-Lu}\,du
\]
is nonincreasing.  Thus \(\mathcal C(x,y,z,L,0)\) is nonincreasing in
its cutoff.  Moreover,
\[
 (1+10^{-3})\,99.729=99.828729<100,
 \qquad
 A-k-b_A=\frac1{375}>0.
\]
Hence, for
\(0<\eps_z\leq10^{-3}\), Zhao's fixed-class estimate and the
coefficient shift of \cref{lem:T-stability} give
\[
 \max_iT_i(L)\leq100\e^{-b_AL}.
\tag{7.16}\label{eq:near-siegel-maxT}
\]
The unrestricted exact enclosure, with the same source-error
allowance, gives
\[
 \mathcal T_A^{\mathrm{all}}(L)
 \leq\mathcal T_A^{\mathrm{all}}(5.68)
 <0.016308621213<\frac1{50}.
\tag{7.17}\label{eq:near-siegel-totalT}
\]
Substitution in \cref{eq:near-siegel-decomposition} proves
\[
 \Qpacket_A
 \leq
 \e^{-2A\lambda}
+202\e^{-b_A\Lambda(\lambda)}.
\tag{7.18}\label{eq:near-siegel-majorant}
\]

The exact verifier establishes
\[
\begin{gathered}
 b_A=\frac{164}{75},\qquad
 1.09b_A=\frac{4469}{1875}>2,\\
 d_0:=2A-\frac{2A^2+202}{200}
 =\frac{54811}{10000}>0,\\
 \e^{-2A/200}+202\e^{-5.68b_A}
 \leq0.968353823265<1.
\end{gathered}
\tag{7.19}\label{eq:near-siegel-checks}
\]
For \(0<\lambda\leq1/200\),
\(L\geq1.09\log(1/\lambda)\), so
\[
 \e^{-2A\lambda}
 \leq1-2A\lambda+2A^2\lambda^2,
 \qquad
 \lambda^{1.09b_A}\leq\lambda^2,
\]
give
\[
 \Qpacket_A\leq1-d_0\lambda.
\tag{7.20}\label{eq:near-siegel-small}
\]
On \(1/200\leq\lambda\leq1/100\), no monotonicity assertion is
needed: termwise,
\[
 \e^{-2A\lambda}\leq\e^{-2A/200},
 \qquad
 \e^{-b_AL}\leq\e^{-5.68b_A},
\]
and the last line of \cref{eq:near-siegel-checks} applies.  Thus the
explicit fixed-\(c\) gap
\[
 \kappa_{\mathrm{NS}}(33/10,c)
 :=
 \min\left\{
 \frac{54811}{10000}\min\left(c,\frac1{200}\right),
 \ \frac{6329235347}{200000000000}
 \right\}
 >0
\tag{7.21}\label{eq:near-siegel-gap}
\]
is valid.  No uniformity as \(c\downarrow0\) is claimed; the
exceptional-real-zero alternative is removed in
\cref{thm:bridge} before this estimate is used.

\subsection{Absorbing the auxiliary error}

The certificate evaluates the limiting \(\eps_z=0\) formulas only
after proving a finite collection of strict separations:
\[
\begin{gathered}
 A-k>0,\qquad
 \Delta^2-\xi>0,\\
 (B+1)(\Delta^2-\xi)+2\Delta D-(1-\xi)>0,\\
 (B+1)\Delta^2+2\Delta D-1>0,\\
 1-\Qpacket_A^{\mathrm{upper}}>0.
\end{gathered}
\tag{7.22}\label{eq:epsilon-separations}
\]
Every bound is continuous in \(\eps_z\), with denominators separated
from zero.  Since the list is finite, one may choose a common
\(\eps_z^*>0\) for which every non-near-Siegel row remains valid.
Choose once and for all
\[
 0<\eps_z\leq\min\left\{\eps_z^*,10^{-3}\right\},
\]
and only then take the maximum of the finitely many modulus
thresholds.  This same positive source-error parameter is used in the
near-Siegel bounds \cref{eq:near-siegel-maxT} and
\cref{eq:near-siegel-totalT}.
Since
\[
 \frac1{250}=0.004<0.007605452306,
\]
the non-near-Siegel branches retain gap \(1/250\).  Together with the
near-Siegel gap \(\kappa_{\mathrm{NS}}(A,c)>0\), this gives
\[
\Qpacket_{33/10}
 \leq1-\min\left\{\frac1{250},
                \kappa_{\mathrm{NS}}(33/10,c)\right\}.
\tag{7.23}\label{eq:final-packet-gap}
\]
The finite detector levels and the tail estimate
\cref{eq:T-input-appendix} are independent of the outer rectangle
height \(H\).  Hence \cref{eq:final-packet-gap} is a common zero-source
gap for every \(H\geq H_c\), and the strict separations in
\cref{eq:epsilon-separations} provide the common perturbation tolerance
required by \cref{prop:two-parameter}.  Applying that proposition for
an arbitrary fixed \(C\geq1\) moves all conductor dependence into
\(Q_0(33/10,c,H,C)\).  This is precisely the quantifier order in
\cref{def:ZA}, and proves \cref{thm:certified-packet}.

\section{Proof of the main theorem}

\begin{proof}[Proof of \cref{thm:main}]
By \cref{thm:certified-packet}, the packet property
\(\mathsf Z(33/10)\) holds.  Apply \cref{thm:bridge}:
\[
 E(X)\ll_\eps X^{1-10/33+\eps}
 =X^{23/33+\eps}.
\tag{8.1}\label{eq:base-exponent}
\]
Let
\[
 \gamma=\frac{69697}{100000}.
\]
The exact margins are
\[
 \gamma-\frac{23}{33}=\frac1{3300000}>0,
\qquad
 \frac1{1-\gamma}-\frac{33}{10}
 =\frac1{303030}>0.
\tag{8.2}\label{eq:bridge-margins}
\]
This is \cref{thm:main}.  Choose the \(\eps\) in
\cref{eq:base-exponent} smaller than \(1/3300000\).  Then
\(E(X)\ll X^\gamma\), proving \cref{cor:decimal}.
\end{proof}

\section{Reproducibility and source correspondence}
\label{sec:reproducibility}

\subsection{Verifier architecture}

The release separates parameter search, exact verification, witness
export, and witness checking.  \Cref{tab:code-map} records the
proof-to-code correspondence.
\begin{table}[ht]
\centering
\small
\begin{tabular}{p{0.35\textwidth}p{0.56\textwidth}}
\toprule
Analytic object&Implementation\\
\midrule
release integrity and orchestration&
\texttt{verify\_release.py}\\
active \(63\)-row packet&
\texttt{zhao\_069697\_full\_active\_audit.py}\\
independent active/scalar optimization check&
\texttt{export\_active\_scalar\_witness.py},
\texttt{check\_active\_scalar\_witness.py}\\
secondary aligned construction&
\texttt{zhao\_secondary\_aligned\_trial.py}\\
independent secondary optimization check&
\texttt{export\_secondary\_aligned\_witness.py},
\texttt{check\_secondary\_aligned\_witness.py}\\
final scalar rows&
\texttt{zhao\_secondary\_scalar\_audit.py}\\
near-Siegel branch&
\texttt{zhao\_near\_siegel\_exact.py}\\
early-or-late cap&
\texttt{zhao\_high\_class\_positional\_trial.py}\\
aligned majorization and tail fill&
\texttt{zhao\_aligned\_rt\_staircase.py}\\
same-class positional caps&
\texttt{zhao\_positional\_fixed\_density.py}\\
outward rational kernel arithmetic&
\texttt{zhao\_packet\_certificate.py}\\
\bottomrule
\end{tabular}
\caption{Analytic-to-code correspondence.}
\label{tab:code-map}
\end{table}

\subsection{Commands}

From the project root, the complete manifest-driven release check is
\begin{verbatim}
python3 -u \
  preprints/goldbach-exception-069697/verify_release.py
\end{verbatim}
A fast integrity-only pass is
\begin{verbatim}
python3 -u \
  preprints/goldbach-exception-069697/verify_release.py --hash-only
\end{verbatim}
The three standard-library witness checks can also be run directly:
\begin{verbatim}
python3 -u experiments/check_active_scalar_witness.py \
  preprints/goldbach-exception-069697/active-scalar-witness.json

python3 -u experiments/check_secondary_aligned_witness.py \
  preprints/goldbach-exception-069697/secondary-aligned-witness.json

python3 -u experiments/zhao_near_siegel_exact.py \
  --witness preprints/goldbach-exception-069697/\
near-siegel-witness.json
\end{verbatim}
The exact primary command-line arguments are stored in
\texttt{certificate-manifest.json}, rather than duplicated in the
paper.  The entry points refuse to run when
\texttt{sys.flags.optimize} is nonzero: the verifier must not be
invoked with \texttt{python -O}, because Python optimization removes
assertions used as proof guards.

\subsection{Frozen release identifiers}

The machine-readable file \texttt{certificate-manifest.json} records
the exact constants, the complete source dependency closure, every
theorem-critical command and argument, expected row counts, success
markers, and SHA-256 digests of the exact standard output.  The
companion \texttt{MANIFEST.sha256} identifies the manuscript and all
release artifacts.  Keeping these identifiers in machine-readable
files avoids an unaudited discrepancy between a printed hash table and
the executable release.

The cold runs were performed on macOS arm64 with Python~3.9.6 and
NumPy~2.0.2.  NumPy is used for detector search, but every accepted
detector is substituted into outward rational inequalities.  The
active/scalar, secondary, and near-Siegel witness checkers use only the
Python standard library.

\subsection{Verification boundary}

The finite witnesses contain the exact ordered class coordinates, raw
\(R/T\) caps, suffix majorants, mass budgets, greedy fills, objectives,
and slacks for all \(16\) rows in \cref{tab:secondary}, all \(63\)
active rows, and all four scalar rows.  Their standard-library checkers
import none of the detector, staircase, interval-kernel, or
optimization modules.  They first verify that the witness has exactly
the theorem row cover, then recompute the class-count recurrences,
majorants, fills, and objectives from the exported boundary data.  The
active/scalar checker also executes an adversarial upward-tail test;
both finite-row checkers reject a missing row.  The near-Siegel checker
independently recomputes the rational enclosures and the gap in
\cref{eq:near-siegel-gap}.

The finite-row checkers deliberately treat the analytic derivation of
the raw caps, integer count inequalities, and mass budgets from Zhao's
estimates as their trusted boundary.  That derivation is specified in
\cref{sec:coefficient-stability}, \cref{sec:positional},
\cref{sec:ordering}, and \cref{app:formulas}
and checked by the primary rational verifier; it is not made
independent merely by orchestration through
\texttt{verify\_release.py}.  Likewise, the computer certificate does
not replace the source-level argument in
\cref{prop:two-parameter} and \cref{thm:bridge}.  These boundaries are
recorded so
that an independent specialist audit can target the remaining analytic
correspondence rather than repeat the elementary polytope calculation.

\section{Concluding remarks}

The gain from \(7/10\) to \(0.69697\) is not obtained from a new
density theorem.  It comes from preserving dependence which is lost
when character classes or the \(R/T\) decomposition are decoupled.
This distinction matters: the old separated estimates actually cross
one in two branches at \(A=33/10\), whereas the aligned common-order
polytope remains below one with visible room.

The next substantial improvement is unlikely to come from a still
finer scalar mesh alone.  It would require either a stronger
within-class two-zero density statement, a new restriction on
cross-class concentration, or a different use of the common class
geometry.  Eliminating the exceptional set altogether remains a
qualitatively different problem.

\appendix

\section{Analytic formulas used by the certificate}
\label{app:formulas}

This appendix records the limiting forms of the Zhao inequalities
which are evaluated by the certificate.  It fixes the direction of
every endpoint substitution and makes the dependence on \(A\)
explicit.

Fix \(x>0\), \(0\leq\lambda_0<\Lambda\), and set
\[
 F_x(s)=G(s/x),\qquad
 \psi(v)=\frac{F_x(v-\lambda_0)}{F_x(-\lambda_0)},
 \qquad
 \xi=\frac{\phi xg(0)}{2F_x(-\lambda_0)},\quad
 \phi=\frac13.
\]
Assume
\[
 \Delta=\psi(\Lambda)-\xi>0.
\]
Let
\[
 w_A(v)=\e^{-Av}-\e^{-A\Lambda},
 \qquad 0\leq v\leq\Lambda,
\]
and suppose \(m\in\{1,2\}\) zeros have prescribed lower endpoints
\(\lambda_1,\ldots,\lambda_m\leq\lambda^*<\Lambda\), while every
later zero satisfies \(\lambda_j\geq\lambda^*\).

The unrestricted \(R\)-bound used in the certificate is
\[
\begin{split}
 R_A(\Lambda)\leq&
 \sum_{j=1}^m w_A(\lambda_j)\\
 &+
 \frac{w_A(\lambda^*)}
      {\psi(\lambda^*)-\psi(\Lambda)}
 \left\{
 \frac{1-\xi}{2\Delta}
 -\sum_{j=1}^{m}
   \bigl(\psi(\lambda_j)-\psi(\Lambda)\bigr)
 \right\}.
\end{split}
\tag{A.1}\label{eq:R-unrestricted}
\]
Inside one fixed class, choose an integer \(N^*\geq2\).  In the
alternative \(N\leq N^*\),
\[
 R_A(\Lambda)
 \leq
 \sum_{j=1}^m w_A(\lambda_j)
 +(N^*-m)w_A(\lambda^*).
\tag{A.2}\label{eq:R-fixed-small}
\]
In the alternative \(N\geq N^*+1\),
\[
\begin{split}
 R_A(\Lambda)\leq&
 \sum_{j=1}^m w_A(\lambda_j)\\
 &+
 \frac{w_A(\lambda^*)}
      {\psi(\lambda^*)-\psi(\Lambda)}
 \left\{
 \frac{1-(N^*+1)\Delta^2}{2\Delta}
 -\sum_{j=1}^{m}
   \bigl(\psi(\lambda_j)-\psi(\Lambda)\bigr)
 \right\}.
\end{split}
\tag{A.3}\label{eq:R-fixed-large}
\]
At positive \(\eps_z\), the numerator in the final fraction is
\[
 1-(N^*+1)(\Delta^2-\eps_z).
\tag{A.4}\label{eq:R-epsilon}
\]
The fixed-class cap is the larger of
\cref{eq:R-fixed-small} and \cref{eq:R-fixed-large}; the unrestricted mass uses
\cref{eq:R-unrestricted} and the direct small-\(N\) guard.

Write
\[
 N=N(\Lambda)=\#\{j:\lambda_j\leq\Lambda\},
 \qquad
 D=D(\Lambda)=
 \sum_{\lambda_j\leq\Lambda}
 \bigl(\psi(\lambda_j)-\psi(\Lambda)\bigr).
\]
The zero-count inequalities are, respectively,
\[
\begin{split}
(\Delta^2-\xi-\eps_z)N+2\Delta D&\leq1-\xi,
 &&\text{unrestricted},\\
(\Delta^2-\eps_z)N+2\Delta D&\leq1,
 &&\text{one fixed class}.
\end{split}
\tag{A.5}\label{eq:two-density-input}
\]
The positive \(D\)-term is retained only when the corresponding zeros
are known to lie in that same class.  Global known positions are used
only in the unrestricted line.

For completeness, define Zhao's auxiliary function (with continuous
extension at removable singularities) by
\[
\begin{split}
 \mathcal B(t,\lambda,\lambda_0)
 :=\;&\frac{\phi}{2}
 \frac{1-\e^{-2(\lambda+\lambda_0)t}}{\lambda+\lambda_0}\\
 &+\frac{1-\e^{-\lambda t}}{\lambda(\lambda+\lambda_0)}
 +\frac{\e^{-2(\lambda+\lambda_0)t}-\e^{-\lambda t}}
 {(\lambda+\lambda_0)(\lambda+2\lambda_0)},
\end{split}
\]
and
\[
 \mathcal C(x,y,z,\Lambda,\lambda_0)
 =
 \frac1{xz}\left(\frac12+\frac{\phi+x}{y}\right)
 \sqrt{
 \mathcal B(\phi+x+y,\Lambda,\lambda_0)
 \mathcal B(z,\Lambda,\lambda_0)}.
\]
Finally, Zhao's Lemma~3.1 supplies
\[
 \sum_j\e^{-k\max(\lambda_j,\Lambda)}
 \leq(1+\eps_z)\mathcal C(x,y,z,\Lambda,\lambda_0),
\tag{A.6}\label{eq:T-input-appendix}
\]
with
\[
 k=
 \begin{cases}
 2(\phi+3x+y+z),&\text{fixed class},\\
 2(2\phi+3x+y+z),&\text{unrestricted union}.
 \end{cases}
\tag{A.7}\label{eq:k-two-cases}
\]
The smaller coefficient is never used for an unrestricted mass.
\Cref{lem:T-stability} converts \cref{eq:T-input-appendix} to
coefficient \(A\).

\section{Tail filling and branch inventory}
\label{app:tail}

Suppose the explicit cap prefix is followed by constant tail caps
\(\alpha,\beta>0\), and the residual masses after the prefix are
\[
 R=q\alpha+\rho,\qquad
 T=p\beta+\sigma,
\qquad
 0\leq\rho<\alpha,\quad0\leq\sigma<\beta.
\]
The exact tail contribution to the aligned cross term is
\[
 C_{\mathrm{tail}}=
 \begin{cases}
 q\alpha\beta+\rho\sigma,&q=p,\\
 q\alpha\beta+\rho\beta,&q<p,\\
 p\alpha\beta+\alpha\sigma,&p<q.
 \end{cases}
\tag{B.1}\label{eq:tail-cross}
\]
This is obtained by writing out the two left-greedy fills.  The same
construction simultaneously gives the two square terms.

For raw listed caps \(c_1,\ldots,c_K\) and repeated tail \(d\), the
decreasing coordinatewise majorant is formed as
\[
 (\widetilde c_1,\ldots,\widetilde c_K,\widetilde d)
 =
 \operatorname{suffixmax}(c_1,\ldots,c_K,d).
\tag{B.2}\label{eq:suffixmax}
\]
The tail must be inserted before this operation; replacing it by the
minimum of the last listed cap and \(d\) can lower a valid raw cap.

The complete branch inventory checked by the manifest-driven release is:
\[
\begin{array}{lr}
\text{active directed rows}&63,\\
\text{secondary aligned rows}&16,\\
\text{scalar rows}&4,\\
\text{near-Siegel analytic branch}&1.
\end{array}
\tag{B.3}\label{eq:branch-inventory}
\]
The active driver asserts the \(21\)-box cover and all three low-zero
allocations.  The secondary theorem command must explicitly request
the full scope, as recorded in \cref{sec:reproducibility}.

\begin{thebibliography}{99}

\bibitem{Davenport}
H. Davenport,
\emph{Multiplicative Number Theory},
3rd ed., revised by H. L. Montgomery,
Graduate Texts in Mathematics 74, Springer, 2000.

\bibitem{Pintz2018}
J. Pintz,
\emph{A new explicit formula in the additive theory of primes with
applications II. The exceptional set in Goldbach's problem},
arXiv:1804.09084v2 (2018).
\url{https://arxiv.org/abs/1804.09084v2}.

\bibitem{Pintz2023}
J. Pintz,
\emph{A new explicit formula in the additive theory of primes with
applications I. The explicit formula for the Goldbach problem and the
generalized twin prime problem},
Acta Arith. \textbf{210} (2023), no.~1, 53--94.
\url{https://doi.org/10.4064/aa220728-31-3}.

\bibitem{Zhao2026}
G. Zhao,
\emph{The exceptional set of Goldbach problem and Linnik's constant},
arXiv:2511.05631v2 (2026).
\url{https://arxiv.org/abs/2511.05631v2}.

\end{thebibliography}

\end{document}
