same-network-129.patch

--- a/build/main.tex
+++ b/build/main.tex
@@ -27,7 +27,7 @@
 \begin{abstract}
 We give a deterministic algorithm that multiplies two $n$-bit integers
 in $O(n(\lg n)^{1-\kappa})$ worst-case time, with
-$\kappa=2^{-182}$, on one fixed finite-alphabet Turing machine with a
+$\kappa=2^{-129}$, on one fixed finite-alphabet Turing machine with a
 fixed finite number of one-dimensional tapes. The algorithm is exact
 for every input length and disproves the $n\log n$ optimality conjecture
 of Sch\"onhage and Strassen in this model.
--- a/build/sections/00-introduction.tex
+++ b/build/sections/00-introduction.tex
@@ -25,7 +25,7 @@
 inputs. Its worst-case running time satisfies
 \[
  T_A(n)=O\!\left(n(\lg n)^{1-\kappa}\right),
- \qquad \kappa=2^{-182}.
+ \qquad \kappa=2^{-129}.
 \]
 \end{theorem}
 
--- a/build/sections/08-assembly.tex
+++ b/build/sections/08-assembly.tex
@@ -19,10 +19,10 @@
 Take the following fixed rational numbers:
 \begin{equation}\label{eq:fixed-parameters}
 \begin{gathered}
- \tau=\sigma=1-2^{-50},\qquad \beta=\frac12,\qquad \delta=\frac1{16},\\
- \lambda=1-2^{-52},\qquad c=2^{-56},\qquad
- \lambda'=1-2^{-54},\\
- C_1=20,\qquad \epsilon=2^{-75},\qquad \kappa=2^{-182}.
+ \tau=\sigma=1-2^{-42},\qquad \beta=\frac12,\qquad \delta=\frac1{16},\\
+ \lambda=1-3\cdot2^{-86},\qquad c=2^{-43},\qquad
+ \lambda'=1-2^{-85},\\
+ C_1=20,\qquad \epsilon=2^{-43},\qquad \kappa=2^{-129}.
 \end{gathered}
 \end{equation}
 These constants are deliberately conservative; no optimization is claimed.
@@ -30,11 +30,11 @@
 The layer conditions hold with room to spare:
 \[
  \max\{\tau,\sigma\}<\lambda<1,\qquad
- \tau(1+c/\beta)<1-31\cdot2^{-55}<\lambda,
+ \tau(1+c/\beta)=1-2^{-84}<1-3\cdot2^{-86}=\lambda,
 \]
 and
 \[
- \sigma+\beta(1-\sigma)=1-2^{-51}<\lambda<\lambda'<1.
+ \sigma+\beta(1-\sigma)=1-2^{-43}<\lambda<\lambda'<1.
 \]
 The strict final inequality absorbs the logarithmic number of groups in
 a base-$m$ partition.  The dimension exponent also satisfies
@@ -184,8 +184,8 @@
 All remaining scalar setup, including comparisons for fixed rational
 powers, modular inverses,
 prime products and shape descriptors, has cost polynomial in $p$.
-For example, $d=\lfloor b^{2^{-75}}\rfloor$ is found by binary search
-using exact comparisons $d^{2^{75}}\le b$; the exponent is fixed.
+For example, $d=\lfloor b^{2^{-43}}\rfloor$ is found by binary search
+using exact comparisons $d^{2^{43}}\le b$; the exponent is fixed.
 The same procedure computes $K$ and all other fixed rational powers.
 All primes and length parameters are therefore generated by the machine;
 they are not advice.
@@ -860,10 +860,10 @@
 \end{equation}
 Direct substitution gives
 \[
- g_1>\frac12,\quad g_2=2^{-181},\quad g_3=2^{-129},\quad
- g_4>2^{-51},\quad g_5>\frac18,\quad g_6>\frac12,\quad g_7=2^{-75}.
-\]
-Thus $\min_i g_i=2^{-181}=2\kappa$.  This explicit slack will cover
+ g_1>\frac12,\quad g_2=2^{-128},\quad g_3=2^{-128},\quad
+ g_4>2^{-44},\quad g_5>\frac18,\quad g_6>\frac12,\quad g_7=2^{-43}.
+\]
+Thus $\min_i g_i=2^{-128}=2\kappa$.  This explicit slack will cover
 any remaining fixed power of $\log p$.
 
 Here $d^2$ is absorbed by $d^2\ell^\tau$, and
--- a/build/sections/03-motifs.tex
+++ b/build/sections/03-motifs.tex
@@ -679,15 +679,16 @@
 \]
 For both recurrences we may use the fixed rational exponent
 \begin{equation}
- \tau=\sigma=1-2^{-50}.
+ \tau=\sigma=1-2^{-42}.
  \label{eq:explicit-motif-exponents}
 \end{equation}
-Both exact deficits displayed above are greater than $20\cdot2^{-50}$.
-Since $m<2^{20}$ and $\log 2<1$, we have $\log m<20$, whence
-\[
- m^{1-2^{-50}}
- =m\exp(-2^{-50}\log m)
- >m(1-20\cdot2^{-50})>m(1-\eta)
+Both exact deficits displayed above are greater than $14\cdot2^{-42}$.
+The rational comparison $\sum_{j=0}^{18}14^j/j!>m$
+gives $m<e^{14}$ and hence $\log m<14$, whence
+\[
+ m^{1-2^{-42}}
+ =m\exp(-2^{-42}\log m)
+ >m(1-14\cdot2^{-42})>m(1-\eta)
 \]
 for either $\eta=\eta_{\rm b}$ or $\eta=\eta_{\rm c}$.
 Consequently
--- a/build/sections/04-swap.tex
+++ b/build/sections/04-swap.tex
@@ -270,7 +270,7 @@
 \end{proof}
 
 
-We use the rational value $\tau=1-2^{-50}$ established in
+We use the rational value $\tau=1-2^{-42}$ established in
 \eqref{eq:explicit-motif-exponents}. Its verification uses fixed integer
 comparisons and the displayed exponential inequality; it needs no
 real-arithmetic oracle.
--- a/build/sections/05-layers.tex
+++ b/build/sections/05-layers.tex
@@ -827,7 +827,7 @@
 \label{prop:simultaneous-layer}
 Fix
 \[
- \tau=\sigma=1-2^{-50},\qquad \beta=\frac12,\qquad C_1=20.
+ \tau=\sigma=1-2^{-42},\qquad \beta=\frac12,\qquad C_1=20.
 \]
 Let $c,\epsilon,\lambda,\lambda'$ be fixed positive rational numbers
 satisfying