batched-assembly.tex

\subsection{An exact complex exponent after bulk recursion}
\label{sec:batched-complex-parameters}

Keep the paired complex producer of PR~\#7. Its constants are
\[
 h=28,\quad v_{\rm c}=\binom{28}{3}=3276,\quad m=21952,
 \quad R_{\rm c}=93838.
\]
There are $R_{\rm c}+h+1=93867$ auxiliary roles per invocation, including
the original central roles. Write
\[
 B_{\rm c}=v_{\rm c}^2(R_{\rm c}+h+1)=1007397164592,
\]
\[
 W_{\rm c}=2v_{\rm c}^2(v_{\rm c}+R_{\rm c}+h+1)=2085111546336,
\]
\[
 L_{\rm c}=3v_{\rm c}^2h(h+1)=26143580736,
 \qquad
 s_{\rm c}=W_{\rm c}m-2v_{\rm c}^3+2L_{\rm c}=45772350635112192.
\]
The two selected whole-residual classes have ranks $a_1=21896$ and
$a_2=21168$, each with $B_{\rm c}$ copies. The number of remaining
singleton children is
\[
 S_{\rm c}=s_{\rm c}-B_{\rm c}(a_1+a_2)=2389799139122304.
\]
Their time moment is
\[
 \Xi(\sigma)=\frac{S_{\rm c}}{W_{\rm c}m^\sigma}
    +\frac{B_{\rm c}}{W_{\rm c}}
      \left[\left(\frac{a_1}{m}\right)^\sigma
           +\left(\frac{a_2}{m}\right)^\sigma\right].
\]
At $\sigma=1$, this equals
$1-\eta_{\rm c}$ with $\eta_{\rm c}=5/12693352$.

Take
\[
 a_{\rm c}=\frac7{10^7},\qquad \sigma=1-a_{\rm c}.
\]
The same positive logarithm series used in the bit certificate gives
\[
 \log m<\frac{9997}{1000},\qquad
 \log\frac m{a_1}=\log\frac{392}{391}<\frac{2555}{10^6},
 \qquad
 \log\frac m{a_2}=\log\frac{28}{27}<\frac{36368}{10^6}.
\]
Let these upper bounds be $\ell_0,\ell_1,\ell_2$ and put
\[
 w_0=\frac{S_{\rm c}}{W_{\rm c}m},\qquad
 w_i=\frac{B_{\rm c}a_i}{W_{\rm c}m}\quad(i=1,2).
\]
The exact rational comparison is
\[
 \Xi(1-a_{\rm c})
 <\sum_{i=0}^2\frac{w_i}{1-a_{\rm c}\ell_i}<1.
\]
The gap in the last inequality is
\[
 \frac{24132873235669673777179671369131}
      {1525632551364197359955324563293421369131}
 >1.58\times10^{-8}.
\]
This certifies the stronger complex exponent with the unchanged
finite producer and the new whole-residual recursion.

\subsection{Final multiplication parameters and margins}
\label{sec:batched-final-assembly}

Use the controlled-basis projector-batched interchange saving
$a_{\rm b}=246/10^9$ and the preceding complex saving $a_{\rm c}=7/10^7$.
The complete rational parameter choice is
\[
\begin{aligned}
 \tau&=1-\frac{246}{10^9},&
 \sigma&=1-\frac7{10^7},&
 \epsilon&=\frac{7999999}{16000000},\\
 c&=1,& \beta&=\frac1{1000},& \delta&=\frac1{10^{10}},\\
 \zeta&=\frac1{10000},& C_1&=\frac{11999}{10000},\\
 \lambda&=1-\frac{246}{10^9}+\frac1{10^{16}},&
 \lambda'&=1-\frac{246}{10^9}+\frac2{10^{16}},\\
 \kappa&=\frac{6149999}{5\cdot10^{13}}.
\end{aligned}
\]
The value $c=1$ is allowed by the compact-control construction. Its
reservation exponent $\max\{1-c,0\}$ is zero, while
$1-\epsilon(1+c)=1/8000000>0$ still ensures $K=o(\ell)$ for $K=d$.
The larger fixed constants in the mixed-width row reservations do not
alter any power of $p$.

Since $\sigma<\tau$, the compact-control internal exponent is $\tau$.
Its leaf exponent is
\[
 \sigma+\beta(1-\sigma)=1-\frac{6993}{10^{10}}<\tau.
\]
The strict gaps of $10^{-16}$ between $\tau,\lambda,\lambda'$ absorb
the fixed powers of $\log p$ in the layer construction. The new
coefficient guard satisfies
\[
 1-\epsilon C_1=\frac{64008011999}{160000000000}>0.
\]
The fast Gaussian construction retains
$\alpha=\lfloor\sqrt{b/(8d)}\rfloor$ and $\gamma=2d\alpha^2\le b/4$.
Here $1-2\epsilon=1/8000000>0$, so its eventual hypotheses
$b\ge64d^2$, $\alpha^2\theta_i\ge1$ and the prime-interval growth
condition all continue to hold. The numerical maps, their target errors,
the final rounding step and the input/output format are unchanged.
For the retained chirp input generator, use
$F=2^{\lceil1.14\alpha^2\rceil}$ to bound both magnitude and shifted
absolute error; the corrected block argument still uses $P=34p$.

The seven margins below one in the assembly cost table are
\[
\begin{aligned}
 g_1&=1-\epsilon(1+c),&
 g_2&=\epsilon c(1-\tau),&
 g_3&=\epsilon(1-\lambda'),\\
 g_4&=(1-\epsilon)(1-\tau),&
 g_5&=1-\delta-2\epsilon,&
 g_6&=1-\delta-\epsilon,&
 g_7&=\epsilon.
\end{aligned}
\]
Their exact values are
\[
\begin{array}{c|c}
 \text{margin}&\text{value}\\\hline
 g_1&1/8000000\\
 g_2&983999877/8000000000000000\\
 g_3&9839998762000001/80000000000000000000000\\
 g_4&984000123/8000000000000000\\
 g_5&1249/10000000000\\
 g_6&312500039/625000000\\
 g_7&7999999/16000000
\end{array}
\]
All twenty-nine parameter constraints are strict. The minimum final
margin is $G=g_3$, and
\[
 G-\kappa=
 \frac{362000001}{80000000000000000000000}>0.
\]
This last fixed positive gap absorbs the remaining logarithmic factors.
The resulting conditional multiplication bound is
\[
 \boxed{
 T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
 \kappa=\frac{6149999}{5\cdot10^{13}}
       =1.2299998\times10^{-7}>2^{-23}.}
\]

The exact comparisons are reproduced by
\texttt{scripts/batched\_network.py} and recorded in
\texttt{certificates/batched-network.json}.