diff --git a/coordinate_sweep_contraction_v2.txt b/coordinate_sweep_contraction_v2.txt
index 8eda38b..3224a34 100644
--- a/coordinate_sweep_contraction_v2.txt
+++ b/coordinate_sweep_contraction_v2.txt
@@ -45,47 +45,47 @@
 
 
 
-\allowdisplaybreaks[1]
+allowdisplaybreaks[1]
 
-# Row--column symmetry and contraction of coordinate sweeps\\
-       \large a clear and complete account
+# Row--column symmetry and contraction of coordinate sweeps
+       large a clear and complete account
 
 
 
-\maketitle
+maketitle
 
 Abstract: 
 The Thorp shuffle on $N=2^d$ labeled cards mixes the full permutation to
-uniform in $Θ(\log N)$ physical shuffles. The upper bound uses an
+uniform in $Θ(log N)$ physical shuffles. The upper bound uses an
 absolute number of coordinate sweeps (each sweep = $d$ physical
 shuffles). This document gives the complete proof, following the
 weighted-contraction route. Everything the proof uses from representation
 theory is collected in Part~part:toolkit as a precise, quotable list.
 
 
-\tableofcontents
+tableofcontents
 
 %=========================================================================
-\part{What is the problem}
+part{What is the problem}
 %=========================================================================
 
 # The shuffle, in plain words
 
 Take a deck of $N=2^d$ cards. Label the $d$ positions of a card by bits
-$x=(x_1,...,x_d)\in\mathbb F_2^d$. One physical Thorp shuffle
+$x=(x_1,...,x_d)inmathbb F_2^d$.bbF_2^d$. One physical Thorp shuffle
 does the following.
 
 
-\item Deal the deck into the $N/2$ unordered pairs of positions that differ
+item Deal the deck into the $N/2$ unordered pairs of positions that differ
       only in the first bit, and in each pair independently swap the two
       cards with probability $1/2$. (This is a "riffle" along the first
       coordinate: it is the classical Thorp move.)
-\item Rotate the $d$ coordinates cyclically, so that the next shuffle
+item Rotate the $d$ coordinates cyclically, so that the next shuffle
       riffles along what was the second coordinate.
 
 
-In coordinates a physical shuffle sends $(x_1,u)\mapsto(u,x_1+\xi_u)$,
-where $u=(x_2,...,x_d)$ and the bits $\xi_u$ are independent and fair.
+In coordinates a physical shuffle sends $(x_1,u)mapsto(u,x_1+xi_u)$,
+where $u=(x_2,...,x_d)$ and the bits $xi_u$ are independent and fair.
 After $d$ physical shuffles the
 rotation has come back to the identity, so $d$ physical shuffles are the
 same as the same matching applied independently along each of the
@@ -96,51 +96,51 @@ A single card is already uniform after one sweep (it has been swapped
 according to $d$ independent fair coins). The subtlety is that the
 joint law of all the cards is not uniform: the cards have used the
 same switches, so they are correlated. The theorem says that an
-absolute constant number of sweeps (i.e.\ $O(d)$ physical shuffles)
+absolute constant number of sweeps (i.e. $O(d)$ physical shuffles)
 destroys every trace of this correlation, measured in total variation
 distance from the uniform permutation.
 
 # Statement of the theorem
 
-Let $S_N$ be the symmetric group. A permutation $g\in S_N$ acts on
+Let $S_N$ be the symmetric group. A permutation $gin S_N$ acts on
 positions, and the product $gh$ means "apply $h$ first, then $g$".
 Let $q_d$ be the law of one physical shuffle; it is a probability measure
 on $S_N$. Then $K_d=q_d^{*d}$, the law of one coordinate sweep, is the
-$d$-fold convolution. We write $K_d(\lambda)$ for the expectation of the
-representing matrix $\rho_\lambda(g)$ of the irreducible unitary
-representation labelled by the partition $\lambda⊢ N$:
+$d$-fold convolution. We write $K_d(lambda)$ for the expectation of the
+representing matrix $rho_lambda(g)$ of the irreducible unitary
+representation labelled by the partition $lambda⊢ N$:
 
-K_d(\lambda)=\sum_{g\in S_N} K_d(g)\,\rho_\lambda(g).
+K_d(lambda)=sum_{gin S_N} K_d(g),rho_lambda(g).
 
-Write $D_\lambda=\dim\rho_\lambda$. Because one matching layer averages
+Write $D_lambda=dimrho_lambda$. Because one matching layer averages
 the (self-adjoint unitary) action of a product of disjoint transpositions,
 its Fourier matrix is an orthogonal projection; therefore every
-$K_d(\lambda)$ is a contraction.
+$K_d(lambda)$ is a contraction.
 
 [Main; weighted sweep moments]
-There exist positive weights $W_\lambda$ (one per partition
-$\lambda⊢ 2^d$) and an absolute constant $p_*<∞$ such that, for
-every $d\ge1$ and every $\lambda⊢ 2^d$,
+There exist positive weights $W_lambda$ (one per partition
+$lambda⊢ 2^d$) and an absolute constant $p_*<∞$ such that, for
+every $dge1$ and every $lambda⊢ 2^d$,
 
-D_\lambda^{3/4}\ \le\ W_\lambda\ \le\ D_\lambda,
-\qquad
-W_\lambda\,\|K_d(\lambda)\|_{p_d}^{p_d}\le 1
-\quadfor some 2\le p_d\le p_*.
+D_lambda^{3/4} le W_lambda le D_lambda,
+qquad
+W_lambda,|K_d(lambda)|_{p_d}^{p_d}le 1
+quadfor some 2le p_dle p_*.
 
-Consequently $\|K_d(\lambda)\|_{\op}\le D_\lambda^{-3/(4p_*)}$, and there
+Consequently $|K_d(lambda)|_{op}le D_lambda^{-3/(4p_*)}$, and there
 is an absolute $q=4p_*/3<∞$ with the unweighted bound
-$D_\lambda\|K_d(\lambda)\|_q^q\le1$.
+$D_lambda|K_d(lambda)|_q^qle1$.
 
 
 [Optimal order of mixing]
-Let $t_{\rm mix}(d)$ be the least number of physical shuffles after which
+Let $t_{rm mix}(d)$ be the least number of physical shuffles after which
 the distance from uniform is at most $1/4$, uniformly over the starting
 deck, and put $N=2^d$. There is an absolute constant $C$ with
-\[
- \Big\lceil(2)/(N)\log_2(3N!)/(4)\Big\rceil
- \ \le\ t_{\rm mix}(d)\ \le\ C\,d .
-\]
-In particular $t_{\rm mix}(d)=Θ(\log N)$.
+[
+ Biglceil(2)/(N)log_2(3N!)/(4)Bigrceil
+  le t_{rm mix}(d) le C,d .
+]
+In particular $t_{rm mix}(d)=Θ(log N)$.
 
 
 The upper bound is the hard direction and occupies Parts
@@ -149,7 +149,7 @@ Part~part:lower.
 
 # The plan, in plain words
 
-Split the $d$ coordinate directions into two blocks of $\approx d/2$ each.
+Split the $d$ coordinate directions into two blocks of $approx d/2$ each.
 The positions become a rectangular board of $m$ rows by $n$ columns
 ($mn=N$). The first block of directions shuffles within rows; the
 second shuffles within columns. Each block is itself a collection of
@@ -159,19 +159,19 @@ row, and one inside each column).
 Induction on $d$. Suppose we already understand the two smaller
 sweeps on the row group and the column group. The full sweep is the
 product of the row part and the column part. In any irreducible
-representation $V_\lambda$, the operator norm of a product is not simply
+representation $V_lambda$, the operator norm of a product is not simply
 the product of the norms; the overlap between the row decomposition and
 the column decomposition gets in the way. Part~part:overlap is
 exactly the estimate of this overlap: how much of a vector that lives in
 a single row-type can survive projection onto a single column-type inside
-$V_\lambda$.
+$V_lambda$.
 
-The key engineering idea is the weight $W_\lambda$. It is a number
-between $D_\lambda^{3/4}$ and $D_\lambda$, built so that the ratio
-$W_\lambda/(W_\mu W_\nu)$ exactly pays for the overlap cost of going from
-a row-type $\mu$ and a column-type $\nu$ up to $\lambda$. This is what
+The key engineering idea is the weight $W_lambda$. It is a number
+between $D_lambda^{3/4}$ and $D_lambda$, built so that the ratio
+$W_lambda/(W_mu W_nu)$ exactly pays for the overlap cost of going from
+a row-type $mu$ and a column-type $nu$ up to $lambda$. This is what
 makes the induction close with a bounded Schatten exponent
-$p_d\le p_*$, independent of $d$.
+$p_dle p_*$, independent of $d$.
 
 Three independent ways to prove the overlap estimate are known. This
 document gives one of them completely: the weighted contraction
@@ -183,27 +183,27 @@ Remarks~rem:frames--rem:coherent but do not develop them,
 in order to keep this account short enough to actually read.
 
 %=========================================================================
-\part{Notation and the standard tools we use}
+part{Notation and the standard tools we use}
 %=========================================================================
 
 # Conventions
 
-All vector spaces are over $\mathbb C$ and all representations are unitary.
-For a partition $\lambda$ of $N$ (a Young diagram with $N$ boxes) let
-$V_\lambda$ be the irreducible $S_N$-representation and
-$D_\lambda=\dim V_\lambda$. On a product of groups, $\mu,\nu,...$
-denote tuples of partitions and $V_\mu,D_\mu$ are the tensor products and
+All vector spaces are over $mathbb C$bbC$ and all representations are unitary.
+For a partition $lambda$ of $N$ (a Young diagram with $N$ boxes) let
+$V_lambda$ be the irreducible $S_N$-representation and
+$D_lambda=dim V_lambda$. On a product of groups, $mu,nu,...$
+denote tuples of partitions and $V_mu,D_mu$ are the tensor products and
 products of dimensions. Traces on a single matrix space are
 unnormalised; Schatten norms are
-$\|A\|_p^p=\Tr(|A|^p)$ for $2\le p<∞$ and $\|A\|_∞$ is the
+$|A|_p^p=Tr(|A|^p)$ for $2le p<∞$ and $|A|_∞$ is the
 operator norm. All logarithms in estimates are natural; we write
-$f\lesssim g$ or $f=O(g)$ when $f\le Cg$ for an absolute constant $C$,
+$flesssim g$ or $f=O(g)$ when $fle Cg$ for an absolute constant $C$,
 and constants are relabelled freely.
 
 We use the symmetric group $S_L$ of degree $L$ and allow $L=0$ (the
 trivial group) and the empty partition (dimension $1$). Box coordinates
 are $(i,j)$ with $i$ the row and $j$ the column; the width of
-$\lambda$ is $\lambda_1$ and $k:=N-\lambda_1$ counts the boxes below the
+$lambda$ is $lambda_1$ and $k:=N-lambda_1$ counts the boxes below the
 first row.
 
 # The toolkit
@@ -212,152 +212,152 @@ These are the classical facts used in the proof. Each is precise and ready
 to quote.
 
 
-\item[(T1)] Regard irreducibles on subgroups. Restricting $V_\lambda$
-  to a Young subgroup $S_{L_1}\times...\times S_{L_r}$ decomposes into
-  tensor products $V_{\lambda^{(1)}} ⊗ ... ⊗  V_{\lambda^{(r)}}$,
-  with the $\lambda^{(i)}$ running over all tuples with
-  $|\lambda^{(1)}|+...+|\lambda^{(r)}|=N$ and a nonnegative integer
+item[(T1)] Regard irreducibles on subgroups. Restricting $V_lambda$
+  to a Young subgroup $S_{L_1}times...times S_{L_r}$ decomposes into
+  tensor products $V_{lambda^{(1)}} ⊗ ... ⊗  V_{lambda^{(r)}}$,
+  with the $lambda^{(i)}$ running over all tuples with
+  $|lambda^{(1)}|+...+|lambda^{(r)}|=N$ and a nonnegative integer
   multiplicity (the Littlewood--Richardson coefficient). In particular
-  restriction to $S_{N-1$ is "delete one box"}: $V_\lambda$
-  restricted to $S_{N-1}$ is the direct sum of $V_{\lambda^-}$ over all
+  restriction to $S_{N-1$ is "delete one box"}: $V_lambda$
+  restricted to $S_{N-1}$ is the direct sum of $V_{lambda^-}$ over all
   ways to remove a corner box. (Branching; see Sagan 2001, Thm 2.8.3,
   or James 1978.)
 
-\item[(T2)] Two subgroups at once. If $V_\sigma$ is an irreducible of
-  $S_{m}$ and $V_\tau$ of $S_{n}$, then
-  $\Ind_{S_m\times S_n}^{S_{m+n}}(V_\sigma ⊗  V_\tau)$ contains
-  $V_\chi$ with multiplicity given by the Littlewood--Richardson
-  coefficient for the skew shape $\chi/\sigma$ filled by the content of
-  $\tau$. (Littlewood--Richardson; James 1978, Thm 16.4.)
+item[(T2)] Two subgroups at once. If $V_sigma$ is an irreducible of
+  $S_{m}$ and $V_tau$ of $S_{n}$, then
+  $Ind_{S_mtimes S_n}^{S_{m+n}}(V_sigma ⊗  V_tau)$ contains
+  $V_chi$ with multiplicity given by the Littlewood--Richardson
+  coefficient for the skew shape $chi/sigma$ filled by the content of
+  $tau$. (Littlewood--Richardson; James 1978, Thm 16.4.)
 
-\item[(T3)] Pieri's rule. Induction of a single row (or single
+item[(T3)] Pieri's rule. Induction of a single row (or single
   column) adds boxes to the first (or first column) subject to the diagram
   staying a partition. (Stanley EC2, Thm 7.15.7.)
 
-\item[(T4)] Hook-length formula, with the correction we keep. For
-  $\lambda⊢ L$,
-  $D_\lambda=L!/\prod_{boxes}hook(b)$. If $\lambda$ is the
-  diagram $\alpha$ decorated by a single extra long top row of length
-  $L-|\alpha|$, the ratio of the top-row factorial to the hooks it creates
-  is at least $\binom{L}{|\alpha|}\exp(-|\alpha|/(L-|\alpha|+1))$, obtained
-  by comparing the top-row part with $\binom{L}{|\alpha|}$ and bounding the
+item[(T4)] Hook-length formula, with the correction we keep. For
+  $lambda⊢ L$,
+  $D_lambda=L!/prod_{boxes}hook(b)$. If $lambda$ is the
+  diagram $alpha$ decorated by a single extra long top row of length
+  $L-|alpha|$, the ratio of the top-row factorial to the hooks it creates
+  is at least $binom{L}{|alpha|}exp(-|alpha|/(L-|alpha|+1))$, obtained
+  by comparing the top-row part with $binom{L}{|alpha|}$ and bounding the
   hook perturbation in the first row. We use this repeatedly. (Sagan
   2001, Thm 3.10.2.)
 
-\item[(T5)] Ordinary Schur--Weyl duality. On the tensor power
-  $(\mathbb C^h)^{ ⊗  L}$ the groups $U(h)$ and $S_L$ act and commute,
+item[(T5)] Ordinary Schur--Weyl duality. On the tensor power
+  $(mathbb C^h)^bbC^h)^{ ⊗  L}$ the groups $U(h)$ and $S_L$ act and commute,
   and
-  $(\mathbb C^h)^{ ⊗  L}\cong⊕_{\gamma⊢ L,\ \ell(\gamma)\le h}
-  W_\gamma ⊗  V_\gamma$ with $W_\gamma$ the polynomial
-  $U(h)$-representation of highest weight $\gamma$. A positive matrix $P$
-  acting identically on all $L$ factors has, on the $U(h)$-type $\gamma$,
+  $(mathbb C^h)^bbC^h)^{ ⊗  L}cong⊕_{gamma⊢ L, ell(gamma)le h}
+  W_gamma ⊗  V_gamma$ with $W_gamma$ the polynomial
+  $U(h)$-representation of highest weight $gamma$. A positive matrix $P$
+  acting identically on all $L$ factors has, on the $U(h)$-type $gamma$,
   largest eigenvalue given by the highest-weight monomial
-  $\prod_i x_i^{\gamma_i}$ in the eigenvalues of $P$ sorted decreasingly;
-  this is also a lower bound for $\Tr(P^\gamma)$. (Etingof et al.,
+  $prod_i x_i^{gamma_i}$ in the eigenvalues of $P$ sorted decreasingly;
+  this is also a lower bound for $Tr(P^gamma)$. (Etingof et al.,
   Thm 4.57 and Cor 4.59; Schur characters in Stanley EC2 Thm 7.10.3.)
 
-\item[(T6)] Signed tensor space $(\mathbb C^h ⊕ \mathbb C^h)^{ ⊗  s}$
+item[(T6)] Signed tensor space $(mathbb C^hbbC^h ⊕ mathbb C^h)^bbC^h)^{ ⊗  s}$
   (the "even" block and the "odd" block). The group $S_s$ acts by
   permuting factors and inserting a sign equal to the sign of the
   reordering of the odd factors. This $S_s$-action commutes with a diagonal
-  $U(h)\times U(h)$ action, one $U(h)$ per block. Fixing the number $s_e$ of
-  even slots and $s_o$ of odd, a $U(h)\times U(h)$-sector with top weights
-  $(\gamma,\xi)$ has multiplicity space, as an $S_s$-representation,
-  $\Ind_{S_{s_e}\times S_{s_o}}^{S_s}(V_\gamma ⊗  V_{\xi^{\rm tr}})$,
-  where $\xi^{\rm tr}$ means the transpose because the odd block contributes
+  $U(h)times U(h)$ action, one $U(h)$ per block. Fixing the number $s_e$ of
+  even slots and $s_o$ of odd, a $U(h)times U(h)$-sector with top weights
+  $(gamma,xi)$ has multiplicity space, as an $S_s$-representation,
+  $Ind_{S_{s_e}times S_{s_o}}^{S_s}(V_gamma ⊗  V_{xi^{rm tr}})$,
+  where $xi^{rm tr}$ means the transpose because the odd block contributes
   a sign. This is the standard "sign twist and induction" combination.
   (Etingof et al., with James 1978 Thm 6.7 for the sign and Thm 16.4
   for the induction.)
 
-\item[(T7)] Group-algebra matrix actions. Given a matrix $M$ on an
-  irreducible $V_\mu$ of a subgroup $R⊆ S_N$, form the element of
-  $\mathbb CR$ with matrices $M$ on the $V_\mu$-block and $0$ elsewhere.
+item[(T7)] Group-algebra matrix actions. Given a matrix $M$ on an
+  irreducible $V_mu$ of a subgroup $R⊆ S_N$, form the element of
+  $mathbb CR$bbCR$ with matrices $M$ on the $V_mu$-block and $0$ elsewhere.
   Viewed in any larger representation, its restriction to $R$ acts as
   "the chosen matrix on one copy of the carrier, and identity on every
-  multiplicity copy". Its coefficient at $g\in R$ is
-  $(D_\mu)/(|R|)\Tr(M\rho_\mu(g^{-1}))$. (Orthogonality and
+  multiplicity copy". Its coefficient at $gin R$ is
+  $(D_mu)/(|R|)Tr(Mrho_mu(g^{-1}))$. (Orthogonality and
   Plancherel; Etingof et al.) We call this a matrix action
-  supported on $\mu$. When $M$ is the rank-one projector onto a unit
+  supported on $mu$. When $M$ is the rank-one projector onto a unit
   vector $e$, it is an orthogonal projection on the $R$-part and is written
-  $P_{\mu,e}$; it uses one direction of the carrier and is the identity on
+  $P_{mu,e}$; it uses one direction of the carrier and is the identity on
   multiplicity copies.
 
-\item[(T8)] Cauchy--Schwarz for sums of projections. When a positive
+item[(T8)] Cauchy--Schwarz for sums of projections. When a positive
   operator is a sum of orthogonal projections onto orthogonal subspaces,
   comparison between two such decompositions is controlled by
   Cauchy--Schwarz, and nonnegativity of each irreducible's contribution
   lets one take traces representation by representation. We use this
   without further comment.
 
-\item[(T9)] Entropy of an occupation law. For a probability vector
-  $(\omega_{ij})$ supported on the cells of a board and marginals
-  $(p_i),(q_j)$, the relative entropy of $\omega$ from the product
-  $(p_iq_j)$ is at most $\log(mn/s)$ when $\omega$ is uniform on $s$
-  occupied cells of an $m\times n$ board. (Standard; the argument is
+item[(T9)] Entropy of an occupation law. For a probability vector
+  $(omega_{ij})$ supported on the cells of a board and marginals
+  $(p_i),(q_j)$, the relative entropy of $omega$ from the product
+  $(p_iq_j)$ is at most $log(mn/s)$ when $omega$ is uniform on $s$
+  occupied cells of an $mtimes n$ board. (Standard; the argument is
   the entropy/subadditivity duality of Carlen--Cordero-Erausquin 2009,
   Thm 2.1.)
 
-\item[(T10)] Interpolation of a bilinear Schatten map. If a bilinear
-  map has norm $\le e^{E/2}$ at the $2$-Schatten endpoint and $\le1$ at the
-  operator-norm endpoint, then it has norm $\le e^{E/p}$ at the
-  $p$-Schatten norm for $2\le p<∞$, with $E$ independent of $p$.
+item[(T10)] Interpolation of a bilinear Schatten map. If a bilinear
+  map has norm $le e^{E/2}$ at the $2$-Schatten endpoint and $le1$ at the
+  operator-norm endpoint, then it has norm $le e^{E/p}$ at the
+  $p$-Schatten norm for $2le p<∞$, with $E$ independent of $p$.
   (Complex interpolation of Schatten classes / three-lines; see the
   computation in Part~part:interp. We spell it out.)
 
 
 %=========================================================================
-\part{A dimension lower bound}
+part{A dimension lower bound}
 %=========================================================================
 
 # Every nontrivial type that survives is large
 
 One matching layer kills every partition of length $>N/2$: averaging a
 swap along one coordinate projects onto the invariants of a Young subgroup
-$S_2^{N/2}$, and only partitions of length $\le N/2$ survive. So we only
-need to bound types $\lambda$ with at most $N/2$ rows. Below, $k=N-\lambda_1$
+$S_2^{N/2}$, and only partitions of length $le N/2$ survive. So we only
+need to bound types $lambda$ with at most $N/2$ rows. Below, $k=N-lambda_1$
 is the number of boxes below the first row.
 
 [Dimension away from the first row]
-There is an absolute $c>0$ such that, for every $N\ge2$ and every
-nontrivial partition $\lambda⊢ N$ with at most $N/2$ rows,
-\[
- \log D_\lambda \ \ge\ c\,k,\qquad k=N-\lambda_1.
-\]
+There is an absolute $c>0$ such that, for every $Nge2$ and every
+nontrivial partition $lambda⊢ N$ with at most $N/2$ rows,
+[
+ log D_lambda  ge c,k,qquad k=N-lambda_1.
+]
 
 Proof. 
-If both width and height are $<N/10$, every hook length is $\le N/5$, so
-the hook formula gives $\log D_\lambda\ge cN\ge ck$.
+If both width and height are $<N/10$, every hook length is $le N/5$, so
+the hook formula gives $log D_lambdage cNge ck$.
 
-Otherwise pass to $\lambda$ or its transpose so the first row has length
-$u\ge N/10$; let $v=N-u$ be the remaining boxes. Keep that whole first
-row and a subdiagram of size $\ell=\min(v,\lfloor u/2\rfloor)$ below it.
-By branching (T1) $D_\lambda$ is at least the dimension of that
+Otherwise pass to $lambda$ or its transpose so the first row has length
+$uge N/10$; let $v=N-u$ be the remaining boxes. Keep that whole first
+row and a subdiagram of size $ell=min(v,lfloor u/2rfloor)$ below it.
+By branching (T1) $D_lambda$ is at least the dimension of that
 subdiagram. For a diagram with first row $u$ and any fixed lower part of
-size $\ell$, the hook formula plus (T4) bounds
-\[
- D_\lambda\ \ge\ \binom{u+\ell}{\ell}\,\exp\!\Big(-(\ell)/(u-\ell+1)\Big).
-\]
-The lower part has width $\le\ell$, so it adds at most $\ell$ to each top-row
+size $ell$, the hook formula plus (T4) bounds
+[
+ D_lambda ge binom{u+ell}{ell},exp!Big(-(ell)/(u-ell+1)Big).
+]
+The lower part has width $leell$, so it adds at most $ell$ to each top-row
 hook; the correction is the displayed exponential. In the original case
-$\ell\ge c_1 k$ (large $N$); in the transpose case the height hypothesis
-gives $v\ge N/2\ge k$, so again $\ell\ge c_1 k$. With $\ell\le u/2$ the
-logarithm is $\ge(\log3-1)\ell\ge c_2 k$. Finite small $N$ are covered by
+$ellge c_1 k$ (large $N$); in the transpose case the height hypothesis
+gives $vge N/2ge k$, so again $ellge c_1 k$. With $ellle u/2$ the
+logarithm is $ge(log3-1)ellge c_2 k$. Finite small $N$ are covered by
 decreasing $c$; the only one-dimensional irreducibles are trivial and
 sign, both excluded here.
 
 
-In particular, if $k>N^{1-\eps}$ for a small fixed $\eps>0$ then
+In particular, if $k>N^{1-eps}$ for a small fixed $eps>0$ then
 
- \log D_\lambda\ \ge\ c_0\,N^{1-\eps}.
+ log D_lambda ge c_0,N^{1-eps}.
 
 
 %=========================================================================
-\part{The overlap theorem}
+part{The overlap theorem}
 %=========================================================================
 
 # A board with deleted cells
 
-This is the technical heart. Picture an $m\times n$ board, $mn=N$. Colour
+This is the technical heart. Picture an $mtimes n$ board, $mn=N$. Colour
 some cells occupied (there are $s$ of them) and the rest
 deleted (there are $t=N-s$). Put the symmetric group $G_s$ on the
 occupied cells. Restricting a representation $V_a$ of $G_s$ to the
@@ -366,123 +366,123 @@ gives two decompositions that do not commute. The theorem bounds how a
 row-type direction and a column-type direction can overlap.
 
 [Overlap on an arbitrary occupied board]
-Let $a⊢ s$ be a partition lying in the $(h,h)$-hook, i.e.\ no box of
-$a$ has both row index and column index greater than $h$; here $1\le h\le N$.
+Let $a⊢ s$ be a partition lying in the $(h,h)$-hook, i.e. no box of
+$a$ has both row index and column index greater than $h$; here $1le hle N$.
 Decompose $V_a$ under the row group and, separately, under the column
-group. For row-types $\sigma$ (a tuple of partitions) and column-types
-$\theta$, let $I_{\sigma,u}\colon V_\sigma\to V_a$ and
-$I_{\theta,v}\colon V_\theta\to V_a$ be isometries onto orthogonal copies,
+group. For row-types $sigma$ (a tuple of partitions) and column-types
+$theta$, let $I_{sigma,u}colon V_sigmato V_a$ and
+$I_{theta,v}colon V_thetato V_a$ be isometries onto orthogonal copies,
 over all copies $u$ and $v$, chosen so that their images span the
 corresponding isotypic parts. Then
 
- \sum_{u,v}\|I_{\sigma,u}^* I_{\theta,v}\|_{\op}^2
- \ \le\ \exp\!\big\{C[(m+n+1)h^2\log(N+2)+t]\big\}\,
-      \min\Big\{1,\ (D_\sigma D_\theta)/(D_a)\Big\}.
+ sum_{u,v}|I_{sigma,u}^* I_{theta,v}|_{op}^2
+  le exp!big{C[(m+n+1)h^2log(N+2)+t]big},
+      minBig{1, (D_sigma D_theta)/(D_a)Big}.
 
 Here $C$ is absolute. When $s=0$ the empty representation has dimension $1$.
 
 
 In plain words. The overlaps between the two decompositions are
 tiny: the total squared overlap is bounded by an explicit factor (the
-exponential) times the "quantum" ratio $D_\sigma D_\theta/D_a$, never
+exponential) times the "quantum" ratio $D_sigma D_theta/D_a$, never
 more than $1$. The exponential depends on the hook width $h$ and on the
 number $t$ of deleted cells, which is exactly the cost of having a
 non-rectangular occupied region.
 
 Proof. [Proof of Theorem~thm:occupied]
 For $s=0$ there is nothing to prove. Throughout we use the signed tensor
-space $\mathcal L=(\mathbb C^h ⊕ \mathbb C^h)^{ ⊗  s}$ and the
+space $mathcal L=(calL=(mathbb C^hbbC^h ⊕ mathbb C^h)^bbC^h)^{ ⊗  s}$ and the
 signed $S_s$-action (T6).
 
-\smallskip\noindentRealisation.
+smallskipnoindentRealisation.
 By (T5)--(T7) the standard signed realization carries $V_a$ inside
-$\mathcal L$; concretely, at fixed parity counts $s_e,s_o$ and sector top
-weights $(\gamma,\xi)$ (both of length $\le h$, degrees $s_e,s_o$), the
-sector's multiplicity space is $\Ind_{S_{s_e}\times S_{s_o}}^{S_s}(V_\gamma ⊗ 
- V_{\xi^{\rm tr}})$. There is a sector containing $V_a$: take $\gamma$ as the
-first $h$ rows of $a$ and $\xi^{\rm tr}$ as the rest (a diagram of width
-$\le h$), so that the relevant Littlewood--Richardson coefficient is
+$mathcal L$;calL$; concretely, at fixed parity counts $s_e,s_o$ and sector top
+weights $(gamma,xi)$ (both of length $le h$, degrees $s_e,s_o$), the
+sector's multiplicity space is $Ind_{S_{s_e}times S_{s_o}}^{S_s}(V_gamma ⊗ 
+ V_{xi^{rm tr}})$. There is a sector containing $V_a$: take $gamma$ as the
+first $h$ rows of $a$ and $xi^{rm tr}$ as the rest (a diagram of width
+$le h$), so that the relevant Littlewood--Richardson coefficient is
 positive. This sector is the arena for the rest of the proof.
 
-\smallskip\noindentSize and entropy of sectors.
-The dimensions of the $U(h)\times U(h)$ carriers, and the number of
-sectors, are at most $\exp(O(h^2\log(N+2)))$ (count symbol occupations row
+smallskipnoindentSize and entropy of sectors.
+The dimensions of the $U(h)times U(h)$ carriers, and the number of
+sectors, are at most $exp(O(h^2log(N+2)))$ (count symbol occupations row
 by row in a semistandard tableau). The multiplicity of any $S_s$-type
-$\chi$ in the signed action is also $\le\exp(O(h^2\log(N+2)))$: given
-$(\gamma,\xi)$ contributing to $\chi$, the skew shape $\chi/\xi^{\rm tr}$
-has content $\gamma$ and its part in the first $h$ columns has at most
+$chi$ in the signed action is also $leexp(O(h^2log(N+2)))$: given
+$(gamma,xi)$ contributing to $chi$, the skew shape $chi/xi^{rm tr}$
+has content $gamma$ and its part in the first $h$ columns has at most
 $h^2$ boxes, beyond which columns are determined. Hence (with
-$u=|\gamma|+|\xi|$)
+$u=|gamma|+|xi|$)
 
- D_\chi\ \le\ \exp(H(\gamma,\xi))\ \le\ D_\chi\exp(O(h^2\log(N+2))),
- \qquad
- H(\gamma,\xi)=\sum_i\gamma_i\log(u/\gamma_i)+\sum_j\xi_j\log(u/\xi_j).
+ D_chi le exp(H(gamma,xi)) le D_chiexp(O(h^2log(N+2))),
+ qquad
+ H(gamma,xi)=sum_igamma_ilog(u/gamma_i)+sum_jxi_jlog(u/xi_j).
 
-The first inequality is $D_\chi\le\binom{u}{|\gamma|}D_\gamma D_\xi$ (from
+The first inequality is $D_chilebinom{u}{|gamma|}D_gamma D_xi$ (from
 T2 and transposition-invariance of dimension) with multinomial bounds; the
 second is (T4) applied to the hooks in the first $h$ rows and columns
-together with $|\gamma\cap\xi^{\rm tr}|\le h^2$.
+together with $|gammacapxi^{rm tr}|le h^2$.
 
-\smallskip\noindentTurning overlap into a trace.
-Let $P_\sigma,P_\theta$ be the row- and column-isotypic projections inside
+smallskipnoindentTurning overlap into a trace.
+Let $P_sigma,P_theta$ be the row- and column-isotypic projections inside
 $V_a$, and expand the embeddings: the sum of squared Hilbert--Schmidt norms
-$\sum_{u,v}\|I_{\sigma,u}^*I_{\theta,v}\|_2^2=\Tr_{V_a}(P_\sigma P_\theta)$,
+$sum_{u,v}|I_{sigma,u}^*I_{theta,v}|_2^2=Tr_{V_a}(P_sigma P_theta)$,
 which dominates the required sum of squared operator norms. Lift to a
-sector projector $P$ of $\mathcal L$ and subgroup isotypic projectors
-$Q_\sigma,Q_\theta$. Since $P$ commutes with the signed permutations and
+sector projector $P$ of $mathcal L$calL$ and subgroup isotypic projectors
+$Q_sigma,Q_theta$. Since $P$ commutes with the signed permutations and
 each irreducible contributes nonnegatively,
-\[
- \Tr_{V_a}(P_\sigma P_\theta)\ \le\ \Tr_{\mathcal L}(Q_\sigma P Q_\theta P).
-\]
+[
+ Tr_{V_a}(P_sigma P_theta) le Tr_{mathcal L}calL(Q_sigma P Q_theta P).
+]
 
-\smallskip\noindentMajorising by product states.
-We bound $Q_\sigma$ in the positive semidefinite order by a probability
+smallskipnoindentMajorising by product states.
+We bound $Q_sigma$ in the positive semidefinite order by a probability
 mixture of tensor products $S=⊗_{cells} S_i$, with the same
 even state $S_i$ in every cell of row $i$. The prefactor is
-$D_\sigma\exp(Cm\,h^2\log(N+2))$. On one row of length $l$, bound the
-projection for a partition $\chi$ by the sum over compatible sector top
-weights $(\gamma_0,\xi_0)$; in each sector, average by
-$U(h)\times U(h)$ the identical-cell product of the diagonal state with
-entries $(\gamma_{0,i}/l,\xi_{0,j}/l)$. After averaging this product acts
-as a scalar $\ge\exp(-H(\gamma_0,\xi_0))/\dim(carrier)$ on that
+$D_sigmaexp(Cm,h^2log(N+2))$. On one row of length $l$, bound the
+projection for a partition $chi$ by the sum over compatible sector top
+weights $(gamma_0,xi_0)$; in each sector, average by
+$U(h)times U(h)$ the identical-cell product of the diagonal state with
+entries $(gamma_{0,i}/l,xi_{0,j}/l)$. After averaging this product acts
+as a scalar $geexp(-H(gamma_0,xi_0))/dim(carrier)$ on that
 sector (highest-weight eigenvalue via T5), and the dimensions are folded
-into the prefactor. Do the same for $Q_\theta$ with column states $T_j$ and
-prefactor $D_\theta\exp(Cn\,h^2\log(N+2))$. Positivity then reduces us to
-bounding $\Tr_{\mathcal L}(S P T P)$.
+into the prefactor. Do the same for $Q_theta$ with column states $T_j$ and
+prefactor $D_thetaexp(Cn,h^2log(N+2))$. Positivity then reduces us to
+bounding $Tr_{mathcal L}calL(S P T P)$.
 
-\smallskip\noindentThe quarter-root filter.
+smallskipnoindentThe quarter-root filter.
 Two normalisations are needed at once: the global sector must pay
-$\exp(-H(\gamma,\xi))$ in the squared norm, and the per-cell overlaps must
-average to $\le1$ under the row- and column-marginals. Let $\bar S,\bar T$
+$exp(-H(gamma,xi))$ in the squared norm, and the per-cell overlaps must
+average to $le1$ under the row- and column-marginals. Let $barbar S,S,barbar T$T$
 be the average cell-states over the $s$ occupied cells, put
-$X=\bar S^{1/2}$, $Y=\bar T^{1/2}$, and set
+$X=barbar S^S^{1/2}$, $Y=barbar T^T^{1/2}$, and set
 $S_i'=X^{-1/2}S_iX^{-1/2}$, $T_j'=Y^{-1/2}T_jY^{-1/2}$ (inverses on
 supports). With $L=(X^{1/2})^{ ⊗  s}P(Y^{1/2})^{ ⊗  s}$ we have
-$\Tr_{\mathcal L}(SPTP)=\|(S')^{1/2}L(T')^{1/2}\|_2^2$. The matrix $L$ acts
-on the sector with operator norm $\le\exp(-H(\gamma,\xi)/2)$: indeed
-$(X^{1/2})^{ ⊗  s}=(\bar S^{1/4})^{ ⊗  s}$ has norm
-$\le\exp(-H(\gamma,\xi)/4)$ on that sector by the largest-weight monomial
-in the eigenvalues of $\bar S$ (whose sum is $1$); the same for $\bar T$.
+$Tr_{mathcal L}calL(SPTP)=|(S')^{1/2}L(T')^{1/2}|_2^2$. The matrix $L$ acts
+on the sector with operator norm $leexp(-H(gamma,xi)/2)$: indeed
+$(X^{1/2})^{ ⊗  s}=(barbar S^S^{1/4})^{ ⊗  s}$ has norm
+$leexp(-H(gamma,xi)/4)$ on that sector by the largest-weight monomial
+in the eigenvalues of $barbar S$S$ (whose sum is $1$); the same for $barbar T$.T$.
 By matrix-coefficient orthogonality for the block-unitary group, $L$ is an
 integral of $U^{ ⊗  s}$ over block unitaries with total coefficient
-$\le \exp(-H(\gamma,\xi)/2)\cdot(squared carrier dimension)$.
+$le exp(-H(gamma,xi)/2)cdot(squared carrier dimension)$.
 
-\smallskip\noindentThe cell product and an entropy bound.
-For a block unitary $U$ let $z_{ij}=\Tr(S_i'UT_j'U^*)\ge0$. The weighted
+smallskipnoindentThe cell product and an entropy bound.
+For a block unitary $U$ let $z_{ij}=Tr(S_i'UT_j'U^*)ge0$. The weighted
 squared Hilbert--Schmidt norm of $U^{ ⊗  s}$ on the occupied cells is
-$\prod_{cells}z_{ij}$. Under the product of row-marginals $p_i$ and
+$prod_{cells}z_{ij}$. Under the product of row-marginals $p_i$ and
 column-marginals $q_j$ (uniform occupation of the $s$ cells), the mean is
-$\Tr(\bar S^{1/2}U\bar T^{1/2}U^*)\le1$ by Cauchy--Schwarz. So the
-$\omega$-average of $\log z_{ij}$ is at most the relative entropy of the
-occupation law $\omega$ from $p_iq_j$, which is $\le\log(mn/s)\le\log(N/s)$
-(T9). Hence $\prod z_{ij}\le\exp(s\log(N/s))\le\exp(t)$. Inserting the
+$Tr(barbar S^S^{1/2}Ubarbar T^T^{1/2}U^*)le1$ by Cauchy--Schwarz. So the
+$omega$-average of $log z_{ij}$ is at most the relative entropy of the
+occupation law $omega$ from $p_iq_j$, which is $lelog(mn/s)lelog(N/s)$
+(T9). Hence $prod z_{ij}leexp(slog(N/s))leexp(t)$. Inserting the
 integral for $L$ and the triangle inequality,
-\[
- \Tr_{\mathcal L}(SPTP)\ \le\ \exp\{C h^2\log(N+2)+t\}\,\exp(-H(\gamma,\xi)),
-\]
-and $D_a\le\exp(H(\gamma,\xi))$ by (eq:sector-entropy). Multiplying by
-the mixture prefactors $D_\sigma D_\theta$ and dividing by $D_a$ gives
-(eq:occupied), with $\min\{1,D_\sigma D_\theta/D_a\}$ recording that
+[
+ Tr_{mathcal L}calL(SPTP) le exp{C h^2log(N+2)+t},exp(-H(gamma,xi)),
+]
+and $D_aleexp(H(gamma,xi))$ by (eq:sector-entropy). Multiplying by
+the mixture prefactors $D_sigma D_theta$ and dividing by $D_a$ gives
+(eq:occupied), with $min{1,D_sigma D_theta/D_a}$ recording that
 the overlap can never exceed $1$.
 
 
@@ -498,37 +498,37 @@ A third proof, on coherent states, keeps the deleted positions as an
 ordered list and compares row and column projections by averaging
 highest-weight coherent vectors and matching their one-cell densities by a
 minimum-norm (Kempf--Ness type) argument. It yields an explicit restoration
-factor $e^{3t}\binom{N}{t}$. These routes are not needed below.
+factor $e^{3t}binom{N}{t}$. These routes are not needed below.
 
 
 %=========================================================================
-\part{From overlap to the row-column Fourier estimate}
+part{From overlap to the row-column Fourier estimate}
 %=========================================================================
 
 # The weighted overlap
 
 Now fix the split of the sweep. Write $N=2^d$ as $mn$ with $m,n$ both of
-order $\sqrt N$ (they are the sizes of a column and a row for a balanced
+order $sqrt N$ (they are the sizes of a column and a row for a balanced
 split of the $d$ bits). The row part acts within rows, the column part
 within columns; both are products of smaller independent sweeps.
 
-We assume for the estimate that the current $\lambda$ is not sparse, in the
+We assume for the estimate that the current $lambda$ is not sparse, in the
 sense
 
- k=N-\lambda_1>N^{1-\eps},\qquad \lambda^{\rm tr}_1\le N/2,
+ k=N-lambda_1>N^{1-eps},qquad lambda^{rm tr}_1le N/2,
 
-with the small $\eps$ of (eq:dim-large); the sparse range is handled
-in Part~part:sparse. The weights $W_\lambda$ are built in
+with the small $eps$ of (eq:dim-large); the sparse range is handled
+in Part~part:sparse. The weights $W_lambda$ are built in
 (eq:Jdef) below; for the moment only their two properties matter,
-$D_\lambda^{3/4}\le W_\lambda\le D_\lambda$, and the exact minimisation
+$D_lambda^{3/4}le W_lambdale D_lambda$, and the exact minimisation
 formula (eq:Jdef).
 
 [Weighted overlap]
-Under (eq:not-sparse), for irreducibles $\mu$ of the row group and
-$\nu$ of the column group and unit carrier vectors $e,f$,
+Under (eq:not-sparse), for irreducibles $mu$ of the row group and
+$nu$ of the column group and unit carrier vectors $e,f$,
 
- W_\lambda\,\Tr_{V_\lambda}(P_{\mu,e}P_{\nu,f})
- \ \le\ W_\mu W_\nu\,\exp\!\big\{O((\log D_\lambda)/d)\big\},
+ W_lambda,Tr_{V_lambda}(P_{mu,e}P_{nu,f})
+  le W_mu W_nu,exp!big{O((log D_lambda)/d)big},
 
 with an absolute implied constant independent of the types and of $d$.
 
@@ -540,178 +540,178 @@ The proof needs the weight formula, so we state it now.
 The weight is a small ``tax'' on each diagram designed so that the tax of
 a parent equals the tax of its children plus exactly enough to pay
 for restoring deleted cells. Fix absolute constants
-\[
- b_0=\tfrac14,\quad \delta=\tfrac1{16},\quad b_1=(b_0\delta)/(4),\quad
- A=10,\quad \eta=(b_1)/(2A),\quad \eps=(\eta)/(8),
-\]
+[
+ b_0=tfrac14,quad delta=tfrac1{16},quad b_1=(b_0delta)/(4),quad
+ A=10,quad eta=(b_1)/(2A),quad eps=(eta)/(8),
+]
 and the penalty function
-\[
- F_L(r)=r\min\{\log(L/r),\ \eta\log L\}\qquad(1\le r\le L),\qquad F_L(0)=0.
-\]
-For a diagram $\lambda⊢ L$ and a subdiagram $a⊆\lambda$
+[
+ F_L(r)=rmin{log(L/r), etalog L}qquad(1le rle L),qquad F_L(0)=0.
+]
+For a diagram $lambda⊢ L$ and a subdiagram $a⊆lambda$
 (itself a Young diagram) let $r=L-|a|$ and define
 
- J(\lambda)=\min_{a⊆\lambda}\big\{b_0\log D_a+b_1 r\log L-AF_L(r)\big\},
- \qquad W_\lambda=D_\lambda\,e^{-J(\lambda)}.
+ J(lambda)=min_{a⊆lambda}big{b_0log D_a+b_1 rlog L-AF_L(r)big},
+ qquad W_lambda=D_lambda,e^{-J(lambda)}.
 
-Empty diagrams have $J=0$, $W=1$. Since $AF_L(r)\le(b_1)/(2)r\log L$,
-the minimised expression is nonnegative, and $a=\lambda$ shows
-$0\le J(\lambda)\le\tfrac14\log D_\lambda$; hence
+Empty diagrams have $J=0$, $W=1$. Since $AF_L(r)le(b_1)/(2)rlog L$,
+the minimised expression is nonnegative, and $a=lambda$ shows
+$0le J(lambda)letfrac14log D_lambda$; hence
 
- D_\lambda^{3/4}\ \le\ W_\lambda\ \le\ D_\lambda.
+ D_lambda^{3/4} le W_lambda le D_lambda.
 
 On tuples (product groups) $J$ is summed and $W$ multiplied.
 
 The point of $J$ is this trimming lemma: the minimising $a$ can be
-taken inside the $(h,h)$-hook, $h=\lfloor N^\delta\rfloor$, at additive
+taken inside the $(h,h)$-hook, $h=lfloor N^deltarfloor$, at additive
 cost $O(h^2)$ in the minimum of (eq:Jdef). Indeed, trimming the boxes
 with both indices $>h$ from any minimiser $a_1$ to get $a$ changes
-$\log D_{a_1}-\log D_a$ by at least $x(\log h-C)-Ch^2$ where $x$ is the
-number trimmed, while the penalty changes by at most $(b_1\log N+A)x$; as
-$b_1<b_0\delta$ the trim is favorable. Consequently, taking $a$ in the
+$log D_{a_1}-log D_a$ by at least $x(log h-C)-Ch^2$ where $x$ is the
+number trimmed, while the penalty changes by at most $(b_1log N+A)x$; as
+$b_1<b_0delta$ the trim is favorable. Consequently, taking $a$ in the
 hook with $t=N-|a|$,
 
- b_0\log D_a+b_1t\log N-AF_N(t)=J(\lambda)+O(h^2),
- \qquad t\log N\le C(\log D_\lambda+h^2).
+ b_0log D_a+b_1tlog N-AF_N(t)=J(lambda)+O(h^2),
+ qquad tlog Nle C(log D_lambda+h^2).
 
 
 # Proof of the weighted overlap
 
 Proof. [Proof of Proposition~prop:weighted-overlap]
 Take the hook subdiagram $a$ from (eq:trim), and choose a diagram
-$b⊢ t$ so that $V_\lambda$ occurs in
-$\mathcal I=\Ind_{G_{N-t}\times G_t}^{G_N}(V_a ⊗  V_b)$ (possible by
-branching and decomposing the restriction). The trace on $\mathcal I$ of
-$XY$, for $X=P_{\mu,e}$ and $Y=P_{\nu,f}$, bounds the wanted trace on
-$V_\lambda$, since each $G_N$-type contributes nonnegatively.
-
-\smallskip\noindentFibers at deleted sets.
-$\mathcal I$ is a direct sum over the $t$-sets $Z$ of the board; at $Z$, the
+$b⊢ t$ so that $V_lambda$ occurs in
+$mathcal I=calI=Ind_{G_{N-t}times G_t}^{G_N}(V_a ⊗  V_b)$ (possible by
+branching and decomposing the restriction). The trace on $mathcal I$calI$ of
+$XY$, for $X=P_{mu,e}$ and $Y=P_{nu,f}$, bounds the wanted trace on
+$V_lambda$, since each $G_N$-type contributes nonnegatively.
+
+smallskipnoindentFibers at deleted sets.
+$mathcal I$calI$ is a direct sum over the $t$-sets $Z$ of the board; at $Z$, the
 stabiliser acts as $V_a$ on the complement of $Z$ and $V_b$ on $Z$. Let
 $v_i$ count $Z$ in row $i$ and $u_j$ in column $j$, and set
-\[
- L_R=\prod_i\binom{n}{v_i},\quad
- L_C=\prod_j\binom{m}{u_j},\quad
- L_0=\binom{N}{t},\quad
+[
+ L_R=prod_ibinom{n}{v_i},quad
+ L_C=prod_jbinom{m}{u_j},quad
+ L_0=binom{N}{t},quad
  E_0=H_R+H_C-H_0,
-\]
-$H_R=\sum_i v_i\log(n/v_i)$, $H_C=\sum_j u_j\log(m/u_j)$,
-$H_0=t\log(N/t)$ (zero terms are $0$). For a nonempty profile $E_0\ge0$ (it
+]
+$H_R=sum_i v_ilog(n/v_i)$, $H_C=sum_j u_jlog(m/u_j)$,
+$H_0=tlog(N/t)$ (zero terms are $0$). For a nonempty profile $E_0ge0$ (it
 is $t$ times a relative entropy). The layout count is controlled by
 
- \log\frac{L_{RC}^2L_0}{L_RL_C}\ \le\ E_0+O(t),
+ logfrac{L_{RC}^2L_0}{L_RL_C} le E_0+O(t),
 
 where $L_{RC}$ is the number of $t$-sets with the given row- and
-column-profiles: $\log L_R\ge H_R$, $\log L_C\ge H_C$,
-$\log L_0\le H_0+t$ by binomial bounds, and subadditivity of entropy gives
-$\log L_{RC}\le E_0+2t$ (take a uniformly random occupied cell with
+column-profiles: $log L_Rge H_R$, $log L_Cge H_C$,
+$log L_0le H_0+t$ by binomial bounds, and subadditivity of entropy gives
+$log L_{RC}le E_0+2t$ (take a uniformly random occupied cell with
 independent row index $I$ and column index $J$, and use
-$H(V\mid I,J)\le H(V\mid I)+H(V\mid J)-H(V)$).
+$H(Vmid I,J)le H(Vmid I)+H(Vmid J)-H(V)$).
 
-\smallskip\noindentDiagonal blocks.
+smallskipnoindentDiagonal blocks.
 The block trace of $XY$ pairs $Z,Z'$ with equal profiles; off-diagonal
 blocks are bounded by the geometric mean of diagonal overlaps (Cauchy--
 Schwarz for $X^{1/2}UX'^{1/2}$). For a diagonal block at $Z$, the row
-stabiliser $R_Z$ acts separately off and on $Z$; if $\alpha=(\sigma,\tau)$
+stabiliser $R_Z$ acts separately off and on $Z$; if $alpha=(sigma,tau)$
 denotes the two parts, then (T7, orthogonality, $|R|=L_R|R_Z|$)
-\[
- X_{ZZ}=(D_\mu)/(L_R)\sum_\alpha(1)/(D_\alpha)\mathcal P_{\alpha,M_\alpha},
-\]
-the $M_\alpha\succeq0$ with $\sum_\alpha\Tr M_\alpha=1$ obtained by tracing
-out the multiplicity space. Similarly $Y_{ZZ}$ with $D_\nu/L_C$ and column
-stabiliser types $\zeta=(\theta,\pi)$. The only subtlety is that $M_\alpha$
+[
+ X_{ZZ}=(D_mu)/(L_R)sum_alpha(1)/(D_alpha)mathcal P_calP_{alpha,M_alpha},
+]
+the $M_alphasucceq0$ with $sum_alphaTr M_alpha=1$ obtained by tracing
+out the multiplicity space. Similarly $Y_{ZZ}$ with $D_nu/L_C$ and column
+stabiliser types $zeta=(theta,pi)$. The only subtlety is that $M_alpha$
 need not be a product of a surviving-cell state and a deleted-cell state;
 the next lemma (which needs no independence) removes this worry.
 
 [Arbitrary carrier vectors]
-Let $A_l\colon E'\to E$ and $B_r\colon F'\to F$ be finite families with
-$\sum_r|< x,B_r y>|^2\le C\|x\|^2\|y\|^2$. Then for unit
-$w\in E ⊗  F$ and $w'\in E' ⊗  F'$,
-$\sum_{l,r}|< w,(A_l ⊗  B_r)w'>|^2\le C\sum_l\|A_l\|_\op^2$.
+Let $A_lcolon E'to E$ and $B_rcolon F'to F$ be finite families with
+$sum_r|< x,B_r y>|^2le C|x|^2|y|^2$. Then for unit
+$win E ⊗  F$ and $w'in E' ⊗  F'$,
+$sum_{l,r}|< w,(A_l ⊗  B_r)w'>|^2le Csum_l|A_l|_op^2$.
 
 Proof. 
-For fixed $l$, $B\mapsto< w,(A_l ⊗  B)w'>$ has operator
-norm $\le\|A_l\|_\op$; by trace-norm duality it is
-$\sum_s c_s< x_s,By_s>$ with unit $x_s,y_s$, $c_s\ge0$,
-$\sum_s c_s\le\|A_l\|_\op$. The triangle inequality in $\ell^2(r)$, then
+For fixed $l$, $Bmapsto< w,(A_l ⊗  B)w'>$ has operator
+norm $le|A_l|_op$; by trace-norm duality it is
+$sum_s c_s< x_s,By_s>$ with unit $x_s,y_s$, $c_sge0$,
+$sum_s c_sle|A_l|_op$. The triangle inequality in $ell^2(r)$, then
 squaring and summing in $l$, finishes.
 
 
 The fiber overlap for arbitrary pure carrier vectors $w,w'$ therefore has
-the form $\sum_{l,r}|< w,(A_l ⊗  B_r)w'>|^2$ with $A_l$ the
+the form $sum_{l,r}|< w,(A_l ⊗  B_r)w'>|^2$ with $A_l$ the
 row-decomposition overlaps and $B_r$ the column ones. On $Z$ itself the
 row and column subgroups intersect trivially, so the pairwise products of
 their group elements use unique factorisations; Plancherel in $G_t$ gives
 for unit $x,y$
 
- \sum_r|< x,B_ry>|^2\ \le\ (D_\tau D_\pi)/(D_b)
- (t!)/(\prod_i v_i!\prod_j u_j!)\ \le\ (D_\tau D_\pi)/(D_b)e^{E_0+O(t)}
+ sum_r|< x,B_ry>|^2 le (D_tau D_pi)/(D_b)
+ (t!)/(prod_i v_i!prod_j u_j!) le (D_tau D_pi)/(D_b)e^{E_0+O(t)}
 
-(using $E_0=t\log t-\sum v_i\log v_i-\sum u_j\log u_j$ and factorial
+(using $E_0=tlog t-sum v_ilog v_i-sum u_jlog u_j$ and factorial
 bounds). Lemma~lem:entangled upgrades this to arbitrary carrier
-vectors, and Theorem~thm:occupied bounds $\sum_l\|A_l\|_\op^2$.
+vectors, and Theorem~thm:occupied bounds $sum_l|A_l|_op^2$.
 
-\smallskip\noindentPutting the pieces together.
-Write $S=\max\{0,\log(D_\sigma D_\theta/D_a)\}$. Combining
+smallskipnoindentPutting the pieces together.
+Write $S=max{0,log(D_sigma D_theta/D_a)}$. Combining
 (eq:occupied), the fiber bound via Lemma~lem:entangled, and
-(eq:on-Z), and dividing by $D_\alpha D_\zeta$ from the diagonal-block
+(eq:on-Z), and dividing by $D_alpha D_zeta$ from the diagonal-block
 formulas, the pure-state bound becomes
-\[
- (1)/(D_aD_b)\exp\{E_0-S+O(t+(m+n+1)h^2\log(N+2))\}.
-\]
+[
+ (1)/(D_aD_b)exp{E_0-S+O(t+(m+n+1)h^2log(N+2))}.
+]
 The positive mixtures have total weight $1$, so no extra count appears.
-Multiply by $D_\mu D_\nu L_{RC}^2/(L_RL_C)$ and use
-$D_\lambda\le L_0D_aD_b$ and (eq:layout):
+Multiply by $D_mu D_nu L_{RC}^2/(L_RL_C)$ and use
+$D_lambdale L_0D_aD_b$ and (eq:layout):
 
- (D_\mu D_\nu)/(D_\lambda)
- \exp\{2E_0-S+O(t+(m+n+1)h^2\log(N+2))\}.
+ (D_mu D_nu)/(D_lambda)
+ exp{2E_0-S+O(t+(m+n+1)h^2log(N+2))}.
 
 
-\smallskip\noindentThe weight budget pays $2E_0-S$.
-Restrict $\mu_i$ to its complement row of size $n-v_i$: the surviving type
-$\sigma_i$ is a subpartition of $\mu_i$ of that size; likewise $\theta_j$
-is a subpartition of $\nu_j$ of size $m-u_j$. So
-\[
- J(\mu)+J(\nu)\le b_0\log(D_\sigma D_\theta)+b_1t\log N
- -A\Big(\sum_i F_n(v_i)+\sum_j F_m(u_j)\Big).
-\]
+smallskipnoindentThe weight budget pays $2E_0-S$.
+Restrict $mu_i$ to its complement row of size $n-v_i$: the surviving type
+$sigma_i$ is a subpartition of $mu_i$ of that size; likewise $theta_j$
+is a subpartition of $nu_j$ of size $m-u_j$. So
+[
+ J(mu)+J(nu)le b_0log(D_sigma D_theta)+b_1tlog N
+ -ABig(sum_i F_n(v_i)+sum_j F_m(u_j)Big).
+]
 Together with (eq:trim), this gives
 
- J(\lambda)-J(\mu)-J(\nu)\ \ge\ -b_0 S+AE_0
- -O\big(h^2+(mn^{1-\eta}+nm^{1-\eta})\log(N+2)\big).
-
-Therefore, with $\mathcal E$ the error in (eq:budget),
-\[
- 2E_0-S-[J(\lambda)-J(\mu)-J(\nu)]
- \le (2-A)E_0-(1-b_0)S+\mathcal E\le\mathcal E,
-\]
-since $A\ge2$, $b_0\le1$, $E_0,S\ge0$. This is exactly the statement that
-replacing $D_\mu D_\nu/D_\lambda$ by $W_\mu W_\nu/W_\lambda$ (i.e.\
-dividing (eq:pref-final) by $D_\lambda e^{-J}$ and multiplying by the
-child weights) leaves only the error $\mathcal E$. All error terms and the
-count of profiles are $O((\log D_\lambda)/d)$ by (eq:dim-large), and
-$h\le N^{1/16}$, $m,n\asymp\sqrt N$, $\eps<\eta/2$. This is
+ J(lambda)-J(mu)-J(nu) ge -b_0 S+AE_0
+ -Obig(h^2+(mn^{1-eta}+nm^{1-eta})log(N+2)big).
+
+Therefore, with $mathcal E$calE$ the error in (eq:budget),
+[
+ 2E_0-S-[J(lambda)-J(mu)-J(nu)]
+ le (2-A)E_0-(1-b_0)S+mathcal EcalElemathcal E,calE,
+]
+since $Age2$, $b_0le1$, $E_0,Sge0$. This is exactly the statement that
+replacing $D_mu D_nu/D_lambda$ by $W_mu W_nu/W_lambda$ (i.e.
+dividing (eq:pref-final) by $D_lambda e^{-J}$ and multiplying by the
+child weights) leaves only the error $mathcal E$.calE$. All error terms and the
+count of profiles are $O((log D_lambda)/d)$ by (eq:dim-large), and
+$hle N^{1/16}$, $m,nasympsqrt N$, $eps<eta/2$. This is
 (eq:weighted-overlap).
 
 
 %=========================================================================
-\part{Schatten interpolation}
+part{Schatten interpolation}
 %=========================================================================
 
 # A $p$-moment bilinear estimate
 
 The row part and the column part of the sweep multiply (in the scan order)
 to give $K_d$. Each part, inside a row or a column, is an independent
-smaller sweep. Let $\mathcal B(M,M')$ on $V_\lambda$ be the product of the
-two subgroup matrix actions, using arbitrary matrices $M_\mu$, $M'_\nu$ on
+smaller sweep. Let $mathcal B(M,M')$calB(M,M')$ on $V_lambda$ be the product of the
+two subgroup matrix actions, using arbitrary matrices $M_mu$, $M'_nu$ on
 their irreducibles.
 
 
-Under (eq:not-sparse), for $2\le p<∞$,
+Under (eq:not-sparse), for $2le p<∞$,
 
- W_\lambda\|\mathcal B(M,M')\|_p^p
- \le e^{O((\log D_\lambda)/d)}
- \Big(\sum_\mu W_\mu\|M_\mu\|_p^p\Big)\Big(\sum_\nu W_\nu\|M'_\nu\|_p^p\Big),
+ W_lambda|mathcal B(M,M')calB(M,M')|_p^p
+ le e^{O((log D_lambda)/d)}
+ Big(sum_mu W_mu|M_mu|_p^pBig)Big(sum_nu W_nu|M'_nu|_p^pBig),
 
 with the constant in the exponent independent of $p$.
 
@@ -719,78 +719,78 @@ Proof.
 For a single pair of types, the squared $2$-norm is a trace of a product of
 positive matrix actions; diagonalising and using (eq:weighted-overlap)
 gives
-$W_\lambda\|\mathcal B(M,M')\|_2^2\le e^{E}(\sum_\mu W_\mu\|M_\mu\|_2^2)
-(\sum_\nu W_\nu\|M'_\nu\|_2^2)$ with $E=O((\log D_\lambda)/d)$. Summing over
+$W_lambda|mathcal B(M,M')calB(M,M')|_2^2le e^{E}(sum_mu W_mu|M_mu|_2^2)
+(sum_nu W_nu|M'_nu|_2^2)$ with $E=O((log D_lambda)/d)$. Summing over
 type tuples with triangle and Cauchy--Schwarz absorbs the log counts
-(which are $O(N^{3/4}\log N)$, covered by (eq:dim-large)) into $E$.
-Also $\|\mathcal B(M,M')\|_\op\le\max_\mu\|M_\mu\|_\op\max_\nu\|M'_\nu\|_\op$
+(which are $O(N^{3/4}log N)$, covered by (eq:dim-large)) into $E$.
+Also $|mathcal B(M,M')calB(M,M')|_oplemax_mu|M_mu|_opmax_nu|M'_nu|_op$
 by unitary restriction.
 
-Now interpolate. Let the bilinear map $\Phi$ carry $(M,M')$ to
-$\mathcal B(M,M')$ with output weighted trace-norm. It has norm $\le e^{E/2}$
-at the $2$-Schatten endpoint and $\le1$ at the operator-norm endpoint.
+Now interpolate. Let the bilinear map $Phi$ carry $(M,M')$ to
+$mathcal B(M,M')$calB(M,M')$ with output weighted trace-norm. It has norm $le e^{E/2}$
+at the $2$-Schatten endpoint and $le1$ at the operator-norm endpoint.
 Normalise inputs to $p$-norm $1$; test the output against a dual-norm-one
 matrix. On the strip with value at $z=2/p$, extend the inputs blockwise by
 polar factors with exponent $pz/2$ on absolute values (zero singular values
 kept zero) and the test matrix with exponent $p'(1-z/2)$,
-$p'=p/(p-1)$. Boundary norms are $\le1$; the three-lines inequality gives
-factor $e^{(E/2)\theta}=e^{E/p}$ at $\theta=2/p$. Raising to the $p$-th
+$p'=p/(p-1)$. Boundary norms are $le1$; the three-lines inequality gives
+factor $e^{(E/2)theta}=e^{E/p}$ at $theta=2/p$. Raising to the $p$-th
 power gives (eq:interp), and the exponent coefficient does not grow
 with $p$.
 
 
 %=========================================================================
-\part{The sparse shapes}
+part{The sparse shapes}
 %=========================================================================
 
 # Representations whose first row is almost the whole diagram
 
 
-When $k=N-\lambda_1$ is small, the general machinery is overkill; a direct
+When $k=N-lambda_1$ is small, the general machinery is overkill; a direct
 combinatorial estimate is better. This is the range that seeds the
 induction.
 
 
-For $k=N-\lambda_1\ge1$,
+For $k=N-lambda_1ge1$,
 
- \|K_d(\lambda)\|_\op\ \le\ (Cd)^k\,(k/N)^{k/8}
+ |K_d(lambda)|_op le (Cd)^k,(k/N)^{k/8}
 
 for an absolute $C$.
 
 
 Proof. 
-Work on ordered injective $k$-tuples of positions $x\mapsto y$. Let $K$ be
+Work on ordered injective $k$-tuples of positions $xmapsto y$. Let $K$ be
 the sweep transition kernel. On functions of $k$-tuples that sum to zero
 over any one coordinate (all others fixed), $K$ acts; this subspace contains
-$V_\lambda$ because $V_\lambda$ occurs on $k$-tuples (invariants upon
+$V_lambda$ because $V_lambda$ occurs on $k$-tuples (invariants upon
 restriction to $G_{N-k}$, branching T1) but not on $(k-1)$-tuples.
 
-\smallskip\noindentThe pair-factor formula.
+smallskipnoindentThe pair-factor formula.
 A sweep sends each card along a path; the path of a card is fixed by its
 start and end, since the successive end coordinates replace the start ones.
-For two cards, let $\ell$ be the last coordinate where they differ at the
+For two cards, let $ell$ be the last coordinate where they differ at the
 start and $r$ the first coordinate where they differ at the end. Give the
-pair weight $w=0$ if $r>\ell$, $w=2$ if $r=\ell$, $w=1$ otherwise. Then
-$K(x,y)=N^{-k}\prod_{pairs}w$. (Paths do not collide iff
-$r\le\ell$ for all pairs; a switch used by two paths rather than one
-contributes the factor $2$ exactly when a pair has $r=\ell$.) This works for
+pair weight $w=0$ if $r>ell$, $w=2$ if $r=ell$, $w=1$ otherwise. Then
+$K(x,y)=N^{-k}prod_{pairs}w$. (Paths do not collide iff
+$rleell$ for all pairs; a switch used by two paths rather than one
+contributes the factor $2$ exactly when a pair has $r=ell$.) This works for
 any subset of cards.
 
-\smallskip\noindentCancelled kernel.
+smallskipnoindentCancelled kernel.
 Because the zero-sum subspace forces all proper-coordinate marginals to
 vanish, we may replace $K$ by the fully cancelled
-\[
- D(x,y)=N^{-k}\sum_{B⊆[k]}(-1)^{k-|B|}\prod_{pairs in B}w .
-\]
-This vanishes unless the contact graph (pairs with $w\ne1$) has no isolated
-vertex, and its absolute column sums are $\le2^k$ (each subset term in
+[
+ D(x,y)=N^{-k}sum_{B⊆[k]}(-1)^{k-|B|}prod_{pairs in B}w .
+]
+This vanishes unless the contact graph (pairs with $wne1$) has no isolated
+vertex, and its absolute column sums are $le2^k$ (each subset term in
 absolute value is the nonnegative doubly stochastic sweep kernel).
 
-\smallskip\noindentInclusion of occupied sets.
+smallskipnoindentInclusion of occupied sets.
 In $K^*K$ we sample $x$ from the row $z$ of $K^*$ (a reverse scan). The
 occupied set $S$ of $x$ satisfies, for every fixed $F$,
 
- \Pr(S⊇ F)\le(k/N)^{|F|}.
+ Pr(S⊇ F)le(k/N)^{|F|}.
 
 This is the upper product-of-marginals property: true at the reverse scan
 start, and preserved by a random switch (test one switched site, average
@@ -798,186 +798,186 @@ the two bounds; test both, and the product of marginals can only grow under
 averaging). The final marginals are $k/N$ (one position is uniform after
 all coordinates).
 
-\smallskip\noindentA spanning forest.
-A no-isolated-vertex graph on $S$ contains a spanning forest with $c\le k/2$
-rooted trees of size $\ge2$. There are $\le C_0^k$ forest shapes, choose
-roots in $\binom Nc$ ways and path types in $2^k$ ways. Conditioned on the
+smallskipnoindentA spanning forest.
+A no-isolated-vertex graph on $S$ contains a spanning forest with $cle k/2$
+rooted trees of size $ge2$. There are $le C_0^k$ forest shapes, choose
+roots in $binom Nc$ ways and path types in $2^k$ ways. Conditioned on the
 common routing, expose children after parents: a common-routing child has
 at most two possible start positions per meeting switch, and an
 independent-path child summed over fresh positions meets a given parent at
-stage $i$ with expectation $\le2^i2^{-(i-1)}$. Each embedding uses $k$ sites,
+stage $i$ with expectation $le2^i2^{-(i-1)}$. Each embedding uses $k$ sites,
 so multiply by $(k/N)^k$ via (eq:inclusion). Summing gives, for the
-signed kernel $K^*D$, absolute row sum $\le(C_1d)^k(k/N)^{k/2}$ using
-$\binom Nc\le(C_2N/k)^{k/2}$ for $c\le k/2$. The $\ell^2$-operator (Schur)
+signed kernel $K^*D$, absolute row sum $le(C_1d)^k(k/N)^{k/2}$ using
+$binom Ncle(C_2N/k)^{k/2}$ for $cle k/2$. The $ell^2$-operator (Schur)
 bound on this subspace bounds $K^*K$ there, proving (eq:sparse).
 
 
 %=========================================================================
-\part{The induction that proves the main moment bound}
+part{The induction that proves the main moment bound}
 %=========================================================================
 
 # Closing the induction
 
 Proof. [Proof of Theorem~thm:main]
-All $K_d(\lambda)$ are contractions. Partitions of length $>N/2$ are killed
+All $K_d(lambda)$ are contractions. Partitions of length $>N/2$ are killed
 by one matching layer; the trivial one satisfies (eq:main-bound) at any
 exponent.
 
-\smallskip\noindentSparse range. For $1\le k\le N^{1-\eps}$,
-(eq:sparse) with $W_\lambda\le D_\lambda\le N^k$ gives, for all large
+smallskipnoindentSparse range. For $1le kle N^{1-eps}$,
+(eq:sparse) with $W_lambdale D_lambdale N^k$ gives, for all large
 $d$ and any sufficiently large absolute exponent $p$,
-\[
- W_\lambda\|K_d(\lambda)\|_p^p
- \le N^k\cdot N^{k}\,N^{-\eps k p/8}\cdot(\log-factor)^k\le1 .
-\]
-
-\smallskip\noindentThe recursive step. Suppose (eq:main-bound)
-holds at the two smaller bit sizes $d_1,d_2\le\lceil d/2\rceil$ with their
-exponents. Apply (eq:interp) at $p=\max(p_{d_1},p_{d_2})\ge2$, feeding
+[
+ W_lambda|K_d(lambda)|_p^p
+ le N^kcdot N^{k},N^{-eps k p/8}cdot(log-factor)^kle1 .
+]
+
+smallskipnoindentThe recursive step. Suppose (eq:main-bound)
+holds at the two smaller bit sizes $d_1,d_2lelceil d/2rceil$ with their
+exponents. Apply (eq:interp) at $p=max(p_{d_1},p_{d_2})ge2$, feeding
 in the two subgroup sweep parts. Each input on a tuple is a tensor product of
-smaller sweeps, so each weighted $p$-th power term of the inputs is $\le1$
+smaller sweeps, so each weighted $p$-th power term of the inputs is $le1$
 by the inductive hypothesis, and the sums over type tuples use only the
 count bounds. Thus
-\[
- W_\lambda\|K_d(\lambda)\|_p^p\le\exp\{C(\log D_\lambda)/d\}
-\]
-with $C$ absolute. Since $W_\lambda\ge D_\lambda^{3/4}$,
-\[
- \|K_d(\lambda)\|_p^p\ \le\ D_\lambda^{-3/4}e^{C(\log D_\lambda)/d}
- \ \le\ D_\lambda^{-1/2}
-\]
-for large $d$ (so $C/d\le1/4$), which gives $\|K\|_\op^p\le\|K\|_p^p\le D_\lambda^{-1/2}$.
-
-Push the exponent by a small factor $1+\theta$. Interpolation of Schatten norms
-gives $\|K\|_{p(1+\theta)}^{p(1+\theta)}\le\|K\|_\op^{\theta p}\|K\|_p^p$ for
-$\theta\ge0$. Write $E=C(\log D_\lambda)/d$, so that $W_\lambda\|K\|_p^p\le e^{E}$
-and $\|K\|_\op^{\theta p}\le(D_\lambda^{-3/4}e^{E})^{\theta}$ (using
-$W_\lambda\ge D_\lambda^{3/4}$). Then
-\[
- W_\lambda\|K\|_{p(1+\theta)}^{p(1+\theta)}
- \le D_\lambda^{-3\theta/4}\,e^{E(1+\theta)}\le1
-\]
-once $\theta\ge C'/d$ with an absolute $C'$ (take $C'=2C$). Take
+[
+ W_lambda|K_d(lambda)|_p^pleexp{C(log D_lambda)/d}
+]
+with $C$ absolute. Since $W_lambdage D_lambda^{3/4}$,
+[
+ |K_d(lambda)|_p^p le D_lambda^{-3/4}e^{C(log D_lambda)/d}
+  le D_lambda^{-1/2}
+]
+for large $d$ (so $C/dle1/4$), which gives $|K|_op^ple|K|_p^ple D_lambda^{-1/2}$.
+
+Push the exponent by a small factor $1+theta$. Interpolation of Schatten norms
+gives $|K|_{p(1+theta)}^{p(1+theta)}le|K|_op^{theta p}|K|_p^p$ for
+$thetage0$. Write $E=C(log D_lambda)/d$, so that $W_lambda|K|_p^ple e^{E}$
+and $|K|_op^{theta p}le(D_lambda^{-3/4}e^{E})^{theta}$ (using
+$W_lambdage D_lambda^{3/4}$). Then
+[
+ W_lambda|K|_{p(1+theta)}^{p(1+theta)}
+ le D_lambda^{-3theta/4},e^{E(1+theta)}le1
+]
+once $thetage C'/d$ with an absolute $C'$ (take $C'=2C$). Take
 $p_d=p(1+C'/d)$; then (eq:main-bound) holds at $p_d$.
 
-\smallskip\noindentBase case. For each fixed small $d$, a nontrivial
+smallskipnoindentBase case. For each fixed small $d$, a nontrivial
 sweep matrix is a product of orthogonal projections that together generate
 $S_N$, so its operator norm is strictly below $1$ on every nontrivial
-$V_\lambda$ (equality would need a common invariant vector for all matching
+$V_lambda$ (equality would need a common invariant vector for all matching
 switches, whose generated group is all of $S_N$). Hence a finite absolute
-base exponent $p_0$ exists covering the finitely many $d\le d_0$, chosen also
+base exponent $p_0$ exists covering the finitely many $dle d_0$, chosen also
 large enough for the sparse range.
 
-\smallskip\noindentBounded exponent. The recurrence multiplies the
+smallskipnoindentBounded exponent. The recurrence multiplies the
 two child exponents by $1+C'/d$ and then takes a maximum. Along repeated
-halvings $d\mapsto\lceil d/2\rceil$ the sum of $1/d$ is a convergent
-geometric series, so $p_d\le p_*<∞$ with $p_*$ absolute. This proves
+halvings $dmapstolceil d/2rceil$ the sum of $1/d$ is a convergent
+geometric series, so $p_dle p_*<∞$ with $p_*$ absolute. This proves
 (eq:main-bound), hence Theorem~thm:main.
 
 
 %=========================================================================
-\part{From the moment bound to mixing}
+part{From the moment bound to mixing}
 %=========================================================================
 
 # Plancherel and Cauchy--Schwarz
 
-Let $\pi$ be the uniform permutation distribution, and let $\Pr_j$ be the law
+Let $pi$ be the uniform permutation distribution, and let $Pr_j$ be the law
 of $j$ independent forward sweeps. Finite-group Plancherel and
 Cauchy--Schwarz (Diaconis--Shahshahani 1981, Sections 2--3) give
 
- 4\,\|\Pr_j-\pi\|_{\TV}^2\ \le\ \sum_{\lambda\ne(N)} D_\lambda\|K_d(\lambda)^j\|_{\HS}^2 .
+ 4,|Pr_j-pi|_{TV}^2 le sum_{lambdane(N)} D_lambda|K_d(lambda)^j|_{HS}^2 .
 
-We bound $\|K_d(\lambda)^j\|_{\HS}^2\le D_\lambda\|K_d(\lambda)\|_\op^{2j}$.
-By Theorem~thm:main and $W_\lambda\ge D_\lambda^{3/4}$,
-$\|K_d(\lambda)\|_\op\le D_\lambda^{-3/(4p_*)}$. Hence
-\[
- D_\lambda\|K_d(\lambda)^j\|_{\HS}^2\ \le\ D_\lambda^2\,D_\lambda^{-3j/(2p_*)}.
-\]
+We bound $|K_d(lambda)^j|_{HS}^2le D_lambda|K_d(lambda)|_op^{2j}$.
+By Theorem~thm:main and $W_lambdage D_lambda^{3/4}$,
+$|K_d(lambda)|_ople D_lambda^{-3/(4p_*)}$. Hence
+[
+ D_lambda|K_d(lambda)^j|_{HS}^2 le D_lambda^2,D_lambda^{-3j/(2p_*)}.
+]
 Taking $j=8p_*/3$ (an absolute number of sweeps) gives
-$D_\lambda^2D_\lambda^{-4}=D_\lambda^{-2}$. So
-\[
- 4\|\Pr_j-\pi\|_{\TV}^2\ \le\ \sum_{\substack{\lambda\ne(N)\\ \lambda^{\rm tr}_1\le N/2}} D_\lambda^{-2}.
-\]
-This sum tends to $0$ as $d\to∞$: for $k>N^{1-\eps}$ it is bounded by
-(eq:dim-large) and the partition count; for $1\le k\le N^{1-\eps}$,
-$D_\lambda\ge e^{-1}(N/k)^k$ (top-row hook estimate as in Lemma~lem:dim)
+$D_lambda^2D_lambda^{-4}=D_lambda^{-2}$. So
+[
+ 4|Pr_j-pi|_{TV}^2 le sum_{substack{lambdane(N) lambda^{rm tr}_1le N/2}} D_lambda^{-2}.
+]
+This sum tends to $0$ as $dto∞$: for $k>N^{1-eps}$ it is bounded by
+(eq:dim-large) and the partition count; for $1le kle N^{1-eps}$,
+$D_lambdage e^{-1}(N/k)^k$ (top-row hook estimate as in Lemma~lem:dim)
 and there are at most $2^k$ such diagrams. Since the product law determines
 the deck from any fixed starting deck, this is a worst-case bound. Thus an
-absolute number $j$ of sweeps gives TV distance $\le1/4$ uniformly in $d$
+absolute number $j$ of sweeps gives TV distance $le1/4$ uniformly in $d$
 (large $d$; the finitely many small $d$ converge by the strict Fourier gap
 and only increase the absolute sweep count, and convolution cannot increase
-TV). One sweep is $d$ physical shuffles, so $t_{\rm mix}(d)\le Cd$. This is
+TV). One sweep is $d$ physical shuffles, so $t_{rm mix}(d)le Cd$. This is
 the upper bound of Corollary~cor:mix.
 
 %=========================================================================
-\part{The matching lower bound}
+part{The matching lower bound}
 %=========================================================================
 
 # Counting reachable permutations
 
 [Support obstruction]
-Let $q_d$ be one physical shuffle. For every integer $t\ge0$,
-\[
- \|q_d^{*t}-U_{S_N}\|_{\TV}\ \ge\ 1-\frac{2^{tN/2}}{N!}.
-\]
-Thus TV distance $\le1/4$ needs at least
-$\lceil(2/N)\log_2(3N!/4)\rceil=2d-O(1)$ physical shuffles.
+Let $q_d$ be one physical shuffle. For every integer $tge0$,
+[
+ |q_d^{*t}-U_{S_N}|_{TV} ge 1-frac{2^{tN/2}}{N!}.
+]
+Thus TV distance $le1/4$ needs at least
+$lceil(2/N)log_2(3N!/4)rceil=2d-O(1)$ physical shuffles.
 
 Proof. 
 One physical shuffle uses $N/2$ independent fair bits (one per pair), so
 $t$ shuffles use $tN/2$ fair bits and can produce at most $2^{tN/2}$
 permutations. Testing against the uniform measure (which weighs every
 permutation and every non-reachable point in the complement),
-$\|q_d^{*t}-U_{S_N}\|_{\TV}\ge1-2^{tN/2}/N!$. The threshold
-$2^{tN/2}/N!\le3/4$ rearranges to the stated lower bound; Stirling gives the
+$|q_d^{*t}-U_{S_N}|_{TV}ge1-2^{tN/2}/N!$. The threshold
+$2^{tN/2}/N!le3/4$ rearranges to the stated lower bound; Stirling gives the
 $2d-O(1)$ form.
 
 
 This is the lower bound of Corollary~cor:mix. Combined with the upper
-bound, $t_{\rm mix}(d)=Θ(d)=Θ(\log N)$.
+bound, $t_{rm mix}(d)=Θ(d)=Θ(log N)$.
 
 %=========================================================================
 {99}
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