diff --git a/coordinate_sweep_contraction_v2.txt b/coordinate_sweep_contraction_v2.txt
index 87ec941..6d12f79 100644
--- a/coordinate_sweep_contraction_v2.txt
+++ b/coordinate_sweep_contraction_v2.txt
@@ -131,14 +131,14 @@ 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_{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
 [
  Biglceil(2)/(N)log_2(3N!)/(4)Bigrceil
-  le t_{rm mix}(d) le C,d .
+  le t_{mix}(d) le C,d .
 ]
-In particular t_{rm mix}(d)=Θ(log N).
+In particular t_{mix}(d)=Θ(log N).
 
 
 The upper bound is the hard direction and occupies Parts
@@ -173,7 +173,7 @@ 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
-route, whose overlap estimate (Theorem~thm:occupied) is an inequality
+route, whose overlap estimate (Theorem~thm:occupied) is an ineuality
 about a board with deleted cells. It is the most self-contained. The other
 two routes ("deletion frames" and "coherent states") prove the same kind
 of bound by other techniques; we mention them in
@@ -254,8 +254,8 @@ to quote.
   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
+  Ind_{S_{s_e}times S_{s_o}}^{S_s}(V_gamma ⊗  V_{xi^{tr}}),
+  where xi^{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.)
@@ -322,7 +322,7 @@ 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).
+ D_lambda ge binom{u+ell}{ell},exp!(-(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
@@ -386,8 +386,8 @@ By (T5)--(T7) the standard signed realization carries V_a inside
 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
+ V_{xi^{tr}}). There is a sector containing V_a: take gamma as the
+first h rows of a and xi^{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.
 
@@ -396,7 +396,7 @@ 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 leexp(O(h^2log(N+2))): given
-(gamma,xi) contributing to chi, the skew shape chi/xi^{rm tr}
+(gamma,xi) contributing to chi, the skew shape chi/xi^{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|)
@@ -405,10 +405,10 @@ u=|gamma|+|xi|)
  
  H(gamma,xi)=sum_igamma_ilog(u/gamma_i)+sum_jxi_jlog(u/xi_j).
 
-The first inequality is D_chilebinom{u}{|gamma|}D_gamma D_xi (from
+The first ineuality 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 |gammacapxi^{rm tr}|le h^2.
+together with |gammacapxi^{tr}|le h^2.
 
 smallskipnoindentTurning overlap into a trace.
 Let P_sigma,P_theta be the row- and column-isotypic projections inside
@@ -463,7 +463,7 @@ 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,
+integral for L and the triangle ineuality,
 [
  Tr_{L} le exp{C h^2log(N+2)+t},exp(-H(gamma,xi)),
 ]
@@ -502,7 +502,7 @@ within columns; both are products of smaller independent sweeps.
 We assume for the estimate that the current lambda is not sparse, in the
 sense
 
- k=N-lambda_1>N^{1-eps}, lambda^{rm tr}_1le N/2,
+ k=N-lambda_1>N^{1-eps}, lambda^{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
@@ -528,7 +528,7 @@ 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, delta=tfrac1{16}, b_1=(b_0delta)/(4),
+ b_0=1/4, delta=1/{16}, b_1=(b_0delta)/(4),
  A=10, eta=(b_1)/(2A), eps=(eta)/(8),
 ]
 and the penalty function
@@ -543,7 +543,7 @@ For a diagram lambda⊢ L and a subdiagram a⊆lambda
 
 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
+0le J(lambda)le1/4log D_lambda; hence
 
  D_lambda^{3/4} le W_lambda le D_lambda.
 
@@ -620,7 +620,7 @@ Proof.
 For fixed l, B|->< 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
+sum_s c_sle|A_l|_op. The triangle ineuality in ell^2(r), then
 squaring and summing in l, finishes.
 
 
@@ -689,16 +689,16 @@ part{Schatten interpolation}
 
 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 B(M,M')calB(M,M') on V_lambda be the product of the
+smaller sweep. Let B(M,M')B(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 2le p<∞,
 
- W_lambda|B(M,M')calB(M,M')|_p^p
+ W_lambda|B(M,M')B(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),
+ (sum_mu W_mu|M_mu|_p^pBig)(sum_nu W_nu|M'_nu|_p^pBig),
 
 with the constant in the exponent independent of p.
 
@@ -706,21 +706,21 @@ 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|B(M,M')calB(M,M')|_2^2le e^{E}(sum_mu W_mu|M_mu|_2^2)
+W_lambda|B(M,M')B(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 |B(M,M')calB(M,M')|_oplemax_mu|M_mu|_opmax_nu|M'_nu|_op
+Also |B(M,M')B(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
-B(M,M')calB(M,M') with output weighted trace-norm. It has norm le e^{E/2}
+B(M,M')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.
 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
+p'=p/(p-1). Boundary norms are le1; the three-lines ineuality 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.
@@ -883,7 +883,7 @@ By Theorem~thm:main and W_lambdage D_lambda^{3/4},
 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{lambdane(N) lambda^{rm tr}_1le N/2}} D_lambda^{-2}.
+ 4|Pr_j-pi|_{TV}^2 le sum_{substack{lambdane(N) lambda^{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},
@@ -893,7 +893,7 @@ 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
 (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_{mix}(d)le Cd. This is
 the upper bound of Corollary~cor:mix.
 
 %=========================================================================
@@ -921,7 +921,7 @@ permutation and every non-reachable point in the complement),
 
 
 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_{mix}(d)=Θ(d)=Θ(log N).
 
 %=========================================================================
 {99}
@@ -956,7 +956,7 @@ T.~M.~Liggett, Negative dependence and the geometry of polynomials, J. Amer.
 Math. Soc. 22 (2009), 521--567.
 bibitem[Carlen--Cordero-Erausquin(2009)]{CCE} E.~A.~Carlen,
 D.~Cordero-Erausquin, Subadditivity of the entropy and Brascamp--Lieb type
-inequalities, GAFA 19 (2009), 373--405.
+ineualities, GAFA 19 (2009), 373--405.
 bibitem[Vershik--Okounkov(2005)]{VO} A.~M.~Vershik, A.~Yu.~Okounkov, A new
 approach to the representation theory of the symmetric groups II, J. Math.
 Sci. 131 (2005), 5471--5494.