nonadjacent-layout.patch

--- a/build/sections/07-resampling.tex
+++ b/build/sections/07-resampling.tex
@@ -330,7 +330,7 @@
 full padded box of $T$ entries, and each entry occupies $O(p)$ bits,
 then exposing and restoring all the lines costs
 \[
- O\bigl(Tp\,d^2(1+\ell^\tau)\bigr)
+ O\bigl(Tp\,d(1+\ell^\tau)\bigr)
 \]
 using the equal-width chunk-swap bound $O(Vu^\tau)$ and the fixed-digit
 movement primitive. Any fixed polynomial in $p$ of setup per line is
@@ -377,21 +377,49 @@
 tapes. Loop indices and the growing shape descriptor reside on tapes;
 they are not extra heads.
 
-The descriptor records which coordinate each physical address field
-holds. Thus moving fields changes their positions, but not the validity
-conditions above. To move an axis past its neighbor, interchange their
-$\ell$-bit or $(\ell-1)$-bit address fields. Equal widths use
-Lemma~\ref{lem:chunk-swap}. For unequal widths, write the longer field as
-a leading bit $e$ followed by $\ell-1$ bits. If it comes first, the order
-is $e\,x\,y$, with $x,y$ of equal width. Interchange $x,y$, then move $e$
-between them, obtaining $y\,e\,x$. If the longer field comes second,
-first move its leading bit left of the shorter field, then interchange
-the two equal-width pieces. The bit move uses
-Lemma~\ref{lem:elementary-streams}. Each adjacent interchange therefore
-costs $O(Tp(1+\ell^\tau))$. Exposing and restoring one axis needs
-$O(d)$ adjacent interchanges, hence all axes need $O(d^2)$. These are
-exact permutations of the padded box, so they change neither norms nor
-errors.
+The descriptor records which named coordinate each physical address field
+holds, together with its width and valid range. Before processing axis $i$,
+interchange its entire address field with the innermost field; after processing
+its lines, undo precisely that interchange. If it is already innermost, no
+interchange is needed. Thus each axis starts and ends in the original field
+order, and all axes together require at most $2(d-1)$ field interchanges.
+
+These interchanges need not be adjacent. Lemma~\ref{lem:chunk-swap} permits
+an arbitrary intervening field $[G]$, obtained by collapsing the intervening
+axes, and a suffix containing the complete coefficient word. For widths that
+differ by one, use the explicit nonadjacent construction following that lemma.
+Suppressing the prefix and suffix, its two cases are
+\[
+ (e,x,G,y)\longmapsto(e,y,G,x)\longmapsto(y,G,e,x),
+\]
+\[
+ (x,G,e,y)\longmapsto(e,x,G,y)\longmapsto(e,y,G,x),
+\]
+where $x,y$ have equal bit width and $e$ is one bit. Each uses one equal-width
+interchange and one single-bit move. Empty equal-width pieces require no swap.
+The intervening axes keep their internal order, and each complete record keeps
+its bit order. The cost of a full field interchange is
+$O(Tp(1+\ell^\tau))$, including workspace cleanup.
+
+At a completed interchange the descriptor moves the coordinate names, widths,
+and validity bounds with their fields. A processed line therefore still runs
+through the same named coordinate in increasing numerical order. Other named
+coordinates retain their values; their physical order is immaterial to the
+validity test. After the line algorithm, the inverse interchange restores the
+original field order. The padded box keeps its full physical size throughout.
+Invalid entries remain zero at the completed axis-map boundaries specified
+above; temporary bit moves and the internal arithmetic of a swap need not
+preserve an intermediate validity mask. The completed permutations and their
+inverses are exact, so the contraction and error arguments do not change.
+
+The list and collapsed descriptors are stored on tapes, not in new heads.
+Here $t_i\ge2$, so $d\le\log_2T$. With $V$ the physical bit volume,
+each collapsed length is at most $V$, and the list of axis names, ranges,
+and widths has length polynomial in $\log(2V)$. Ordinary integer arithmetic
+therefore prepares each call descriptor in $\operatorname{poly}(\log(2V))=O(V)$
+time. Each primitive reuses its fixed work tapes. Thus all exposure and
+restoration costs
+$O(Tp\,d(1+\ell^\tau))$.
 
 Since $t_i\geq r/2$, each tensor map invokes at most
 $\sum_iT/t_i\leq2dT/r$ lines. For any fixed setup degree $c$, work
--- a/build/sections/08-assembly.tex
+++ b/build/sections/08-assembly.tex
@@ -41,7 +41,7 @@
 \[
 \begin{gathered}
  \epsilon<\frac1{12},\quad \epsilon C_1<1,\quad
- \epsilon(2-\tau)<1-\tau,\quad
+ \epsilon(1-\tau)<1-\tau,\quad
  \frac34+\delta+\frac32\epsilon<1,\\
  \epsilon(1+c)<1,\qquad \epsilon+\delta<1.
 \end{gathered}
@@ -292,15 +292,28 @@
 offsets therefore cost $O(dTp^C/r)=o(Tp)$ to prepare, since
 $t_i\ge r/2$ and $r$ exceeds every fixed power of $p$.
 
-The axis widths are $\ell$ or $\ell-1$.  To exchange two adjacent axes
-of unequal width, move the longer axis's leading bit outside their
-equal-width pair, swap that pair by the proved chunk-swap procedure,
-and put the leading bit at its new position.  This adds a fixed number
-of linear passes to the equal-width swap cost.  Reversing all axes
-uses $O(d^2)$ such moves.  Therefore the map $\Phi$, its inverse, and
-their padding operations cost
-\[
- O\bigl(Tp\,d^2(1+\ell^\tau)\bigr).
+The axis widths are $\ell$ or $\ell-1$. Reverse the axis order by
+interchanging physical positions $j$ and $d+1-j$ for
+$j=1,\ldots,\lfloor d/2\rfloor$. Every pair is disjoint, so this uses
+$\lfloor d/2\rfloor$ full-field interchanges, not a sequence of adjacent moves.
+Each is covered by Lemma~\ref{lem:chunk-swap} and its subsequent one-bit-width
+extension, with all intervening fields collapsed into its spectator gap.
+Its cost is $O(Tp(1+\ell^\tau))$. Keep each axis name and width attached
+to that field in the descriptor. The final order is exactly
+$(a_1,\ldots,a_d)$, as required by the triangular updates above. Their target
+order and their requirement that controls precede targets are unchanged.
+The inverse uses the reversed interchange schedule after recovering the digits.
+
+The completed reversal and its inverse preserve complete records and their
+internal bit order. Padding is inserted before the forward reversal and
+removed only after the inverse reversal, so the insertion/deletion enumeration
+is unchanged. Since $d\le\log_2T$, the named-axis list and all collapsed
+lengths have polynomial length in $\log(2Tp)$. Each call descriptor can thus
+be prepared in $\operatorname{poly}(\log(2Tp))=O(Tp)$ time. The $d$ controlled
+rotations and the padding scans cost $O(dTp)$ as above. Therefore $\Phi$,
+its inverse, and their padding operations cost
+\[
+ O\bigl(Tp\,d(1+\ell^\tau)\bigr).
 \]
 This construction describes payload movement from address $a$ to its
 new address.  It is distinct from the operator convention
@@ -834,7 +847,7 @@
 Prefix-slot moves and individual butterfly rounds & $dK$ & $\epsilon(1+c)$\\
 Chunk exchanges for transform layout & $pK^{\tau-1}$ & $1-\epsilon c(1-\tau)$\\
 Simultaneous butterfly rounds & $\ell d^{\lambda'}$ & $1-\epsilon(1-\lambda')$\\
-CRT and axis layouts & $d^2(1+\ell^\tau)$ & $\tau+\epsilon(2-\tau)$\\
+CRT and axis layouts & $d(1+\ell^\tau)$ & $\tau+\epsilon(1-\tau)$\\
 Gaussian line maps & $dp^{1/2+\delta}\alpha$ & $3/4+\delta+3\epsilon/2$\\
 Chirps, twists and scalar products & $dp^\delta$ & $\epsilon+\delta$\\
 Packed polynomial products & $\log(rp)$ & $1-\epsilon$\\
@@ -853,7 +866,7 @@
  g_1&=1-\epsilon(1+c),\qquad
  g_2=\epsilon c(1-\tau),\qquad
  g_3=\epsilon(1-\lambda'),\\
- g_4&=1-\tau-\epsilon(2-\tau),\qquad
+ g_4&=(1-\tau)(1-\epsilon),\qquad
  g_5=1/4-\delta-3\epsilon/2,\\
  g_6&=1-\delta-\epsilon,\qquad g_7=\epsilon.
 \end{split}
@@ -866,7 +879,7 @@
 Thus $\min_i g_i=2^{-181}=2\kappa$.  This explicit slack will cover
 any remaining fixed power of $\log p$.
 
-Here $d^2$ is absorbed by $d^2\ell^\tau$, and
+Here $d$ is absorbed by $d\ell^\tau$, and
 $\log(rp)=O(p^{1-\epsilon}+\log p)=O(p^{1-\epsilon})$.
 The first three rows are the terms in
 \eqref{eq:synthetic-transform-cost}. The prefix-slot moves and
@@ -883,9 +896,12 @@
 deterministic repair, row padding and removal, all base-$m$ groups,
 and fixed-tape cleanup.
 
-Axis exposure and restoration require $O(d^2)$ adjacent axis moves.
-Each costs $O(V(1+\ell^\tau))$: peel a single excess bit when widths
-differ by one, exchange their equal-width parts, and restore the bit.
+Axis exposure and restoration require $O(d)$ nonadjacent full-field
+interchanges by Lemma~\ref{lem:tensor-resampling}; the CRT reversal also
+uses $O(d)$ such interchanges. Each costs $O(V(1+\ell^\tau))$, using the
+arbitrary spectator gap and the one-bit-width extension of
+Lemma~\ref{lem:chunk-swap}. Thus this row has exponent
+$\epsilon+\tau(1-\epsilon)$ rather than $2\epsilon+\tau(1-\epsilon)$.
 The $d$ CRT rotations copy valid and invalid intervals in $O(dV)$ time.
 Their offset setup was bounded above by $O(dTp^C/r)=o(Tp)$,
 using at most $T/t_i$ control prefixes for target $i$.