diff --git a/uniform_marked_polygon_bridge_v2.txt b/uniform_marked_polygon_bridge_v2.txt
index 1c49515..1215b8c 100644
--- a/uniform_marked_polygon_bridge_v2.txt
+++ b/uniform_marked_polygon_bridge_v2.txt
@@ -178,13 +178,13 @@ A pure lattice cut is a line parallel to one of the three
 edge directions of the triangular tiling. A pure-sided
 domain is one whose sides lie on pure lattice cuts. For an
 integer hge1, let B_h be the set of bridges of
-H:bbH: paths from a fixed bottom port to any port on the
+H:H: paths from a fixed bottom port to any port on the
 opposite horizontal cut at Euclidean distance h d_0 (where
 d_0=sqrt3/2 is the row spacing of the triangular lattice),
 staying strictly between the cuts. Put
 
 B_h = sum_{gammainB_h} kappa^{ell(gamma)},
-B_h(E) = sum_{gammainB_h} mathbf 1_E(bf1_E(gamma)kappa^{ell(gamma)}.
+B_h(E) = sum_{gammainB_h} 1_E(1_E(gamma)kappa^{ell(gamma)}.
 
 By lattice symmetry B_h does not depend on the chosen bottom
 port or on the choice of pure lattice orientation. Diameters
@@ -684,7 +684,7 @@ The double-row operator T(t).:
 
 For an integer Nge1, let V_{sites}=V^{ ⊗  N} with
 
-U_alpha(t) = big(D_e(t),R(t+zeta_{a_1})... R(t+zeta_{a_N})big)_{a_1... a_N;b_1... b_N}
+U_alpha(t) = (D_e(t),R(t+zeta_{a_1})... R(t+zeta_{a_N})big)_{a_1... a_N;b_1... b_N}
 
 where alphain{1,2} labels the two rows and zeta_j are the
 N fixed rapidities. Concretely: each row uses N local R-matrices
@@ -849,7 +849,7 @@ At t=gamma we have k(gamma)=sin(gamma-gamma)/sin(2gamma)=0:
 the boundary is closed. The sites sit at xi_j=-L/2 so the tile
 argument is gamma+xi_j=L. Substituting:
 
-a(L)&=(Ssin(2L))/(sin(3L))=frac{1}{√(2+sqrt2)}=kappa,
+a(L)&=(Ssin(2L))/(sin(3L))=1/√(2+sqrt2)=kappa,
 h(L)&=v(L)=c(L)=(Ssin(L))/(sin(3L))=(1)/(2+sqrt2)=kappa^2,
 d(L)&=(sin(L)sin(0))/(sin(3L))=0.
 
@@ -895,7 +895,7 @@ Now let e_ell,e_r be formal variables (with e^{-2}=0). The
 eigenvalue of T(t) near the physical point, expanded to first order
 in e_ell e_r, is
 
-T(t) on the eigenvalue-1 linequad↝
+T(t) on the eigenvalue-1 lineuad↝
 1+e_ell,e_r,q(t;xi)+O((e_ell e_r)^2)
 
 where q(t;xi) is a scalar function. Equivalently, in the mixed
@@ -931,31 +931,31 @@ Analytic structure of q(t;xi).:
 
 We now show that q(t;xi), viewed as a function of t, has
 enough structure to be pinned down by finitely many linear conditions.
-Denote mathbf X(t)=bfX(t)=cos(2t).
+Denote X(t)=X(t)=cos(2t).
 
 [Rational structure of q]
 There exists a fixed scalar rho (depending only on a_*,...)
-and a polynomial p(t) of mathbf X(t)bfX(t) of degree at most N+1
+and a polynomial p(t) of X(t)X(t) of degree at most N+1
 such that
 
-q(t;xi)=frac{p(t)}{(mathbf X+1)^bfX+1)^{N}(mathbf X+bfX+cos 2L)^{N+1}},
+q(t;xi)=frac{p(t)}{(X+1)^X+1)^{N}(X+X+cos 2L)^{N+1}},
 
-and the polynomial part of q as mathbf XbfXto∞ is 4mathbf X+bfX+rho.
+and the polynomial part of q as Xto∞ is 4X+X+rho.
 
 
 Proof. 
 The mto∞ limit and (1) column-by-column: the only pole sources
 in T(t) as a function of t are the zeros of Q(u). For each U
 (row transfer), we have N local R-matrices plus two boundary
-K's; setting mathbf X=bfX=cos(2t) converts all t-dependences to
-rational in mathbf XbfX since sin(2t) etc. are polynomial in
-cos(2t). The two boundary rows give poles at mathbf X=-1bfX=-1 and
-mathbf X=-bfX=-cos(2L); the N local poles pile at
-(mathbf X+1)^N(bfX+1)^N(mathbf X+bfX+cos 2L)^{N+1}. (The extra cos 2L
+K's; setting X=X=cos(2t) converts all t-dependences to
+rational in X since sin(2t) etc. are polynomial in
+cos(2t). The two boundary rows give poles at X=-1bfX=-1 and
+X=-X=-cos(2L); the N local poles pile at
+(X+1)^N(X+1)^N(X+X+cos 2L)^{N+1}. (The extra cos 2L
 power comes from combining the left and right boundary K's.)
 
-For the polynomial part: as mathbf XbfXto∞, T(t) tends to
-I+ single-t-matrix terms by unitarity, contributing 4mathbf XbfX
+For the polynomial part: as Xto∞, T(t) tends to
+I+ single-t-matrix terms by unitarity, contributing 4bfX
 from the leading coefficient; a fixed scalar rho traces the linear
 correction.
 
@@ -1003,18 +1003,18 @@ The scalar crossing function.:
 At homogeneous xi_j (i.e., at the physical rapidities with correction
 -L/2 already absorbed), define
 
-p(u)=q(gamma+u;xi), mathbf X(u)=bfX(u)=cos(2(gamma+u)).
+p(u)=q(gamma+u;xi), X(u)=X(u)=cos(2(gamma+u)).
 
 Then p is even: p(u)=p(-u) (from T(t)=T(r), r=b-t,
 so p(u)=q(gamma+u)=q(b-gamma-u)=q(gamma-u)=p(-u)). And p
-is pi-periodic since T is. So p is a function of mathbf X,bfX, and
+is pi-periodic since T is. So p is a function of X,X, and
 (eq:q-rational) with a change of variable gives
 
-p(u)=frac{P(mathbf X(u))}bfX(u)){(mathbf X+1)^N(bfX+1)^N(mathbf X+bfX+cos 2L)^{N+1}}.
+p(u)=frac{P(X(u))}X(u)){(X+1)^N(X+1)^N(X+X+cos 2L)^{N+1}}.
 
 
 The conditions (eq:q-zeros-1), (eq:q-zeros-2) translate to:
- P(mathbf X(3L/2))=P(bfX(3L/2))=P(mathbf X(5L/2))=0,bfX(5L/2))=0, from
+ P(X(3L/2))=P(X(3L/2))=P(X(5L/2))=0,X(5L/2))=0, from
 p(3L/2)=p(5L/2)=0 via (eq:q-zeros-1).
  The combination p(u-L)+p(u+L)-p(u) vanishes to order 2N at
 u=0 and order N at u=L.
@@ -1033,7 +1033,7 @@ L is a fixed linear functional, and x=z^2.
 
 The value 2B_N=2·(1)/(2)q(gamma)=q(gamma)=p(0) becomes
 
-2B_N=p(0)=L(1/D).calL(1/D).
+2B_N=p(0)=L(1/D).L(1/D).
 
 
 The finite functional.: 
@@ -1052,26 +1052,26 @@ L(f) &= 0  for every fin M,
 L(C_{L/2}) &= 0,
 L(C_{3L/2}-C_{L/2}/D) &= 0.
 
-For this functional, 2B_N=L(1/D)calL(1/D) for every N.
+For this functional, 2B_N=L(1/D)L(1/D) for every N.
 
 
 Proof. [Outline]
-Apply partial fractions to p(u) in mathbf X.bfX. Standard identities
+Apply partial fractions to p(u) in X.X. Standard identities
 
 int_0^∞(cosh(uz)sinh(mz))/(sinh(pi z/2)),dz
-=(1)/(2)[tan(m+u)+tan(m-u)]=(Ssin(2m))/(mathbf X+bfX+cos 2m)
+=(1)/(2)[tan(m+u)+tan(m-u)]=(Ssin(2m))/(X+X+cos 2m)
 
 (at z real, x=z^2) give the interpolation basis {1,x,...,x^{N-1},
 ∂_m^j C_m(x)|_{m=L}}_j. Conditions (eq:q-zeros-2)
 of Section~sec:strip-q-structure translate to
 (eq:L-M); (eq:q-zeros-1) translate to
 (eq:L-C-L-2) and (eq:L-C-3L-2). The remaining free
-coefficient of 1/D is L(1/D)=p(0)=2B_N.calL(1/D)=p(0)=2B_N.
+coefficient of 1/D is L(1/D)=p(0)=2B_N.L(1/D)=p(0)=2B_N.
 
 The interpolation basis {1,x,...,x^{N-1}}cup
 {∂_m^j C_m(x)|_{m=L}}_{j=0}^{N-1} is 2N-dimensional,
 matching the space of rational functions with pole structure
-(mathbf X+1)^bfX+1)^{-N}(mathbf X+bfX+cos 2L)^{-(N+1)} and polynomial
+(X+1)^X+1)^{-N}(X+X+cos 2L)^{-(N+1)} and polynomial
 degree le 2N+1 subject to the two homogeneous constraints
 (eq:q-zeros-1) and (eq:q-zeros-2). The identity
 (eq:cheb-integral) is a standard residue computation:
@@ -1086,7 +1086,7 @@ The positive interpolation estimate.:
 
 Section~c6:strip-transfer reduced the strip bridge mass to a
 single linear functional L on a (2N+2)-dimensional space.
-Here we evaluate L(1/D)calL(1/D) up to an absolute constant, obtaining
+Here we evaluate L(1/D)L(1/D) up to an absolute constant, obtaining
 B_Nasymp N^{-1/4}. The engine is a determinant-density representation
 and an asymptotic estimate of characteristic products.
 
@@ -1097,7 +1097,7 @@ Retain from Proposition~prop:finite-functional:
  M_N=span{1,x,...,x^{N-1};∂_m^jC_m(x)|_{m=L}}_{j=0}^{N-1};
  L(M_N)= L(C_calL(C_{L/2})=
 L(C_{3L/2}-C_{L/2}/D)=0;
- 2B_N=L(1/D).calL(1/D).
+ 2B_N=L(1/D).L(1/D).
 
 
 Set
@@ -1138,9 +1138,9 @@ positive integral of C_t rows, 0<t<L.
 
 
 [Determinant density]
-For H with int|Hf|dmu<∞ (fin M_N+Cg):bbCg):
+For H with int|Hf|dmu<∞ (fin M_N+Cg):Cg):
 
-(L(H))/(4a_*)=calL(H))/(4a_*)=nu[e_{-4}H]+s_*,nu[e_0H],
+(L(H))/(4a_*)=L(H))/(4a_*)=nu[e_{-4}H]+s_*,nu[e_0H],
 
 with e_{-a}H=H(-a)-I_{F_0,X}H(-a) the residual of F_0-interpolation
 at nodes X, and s_*=rho/(4a_*) the scalar in L.
@@ -1272,7 +1272,7 @@ old interpolant's coefficient at each node has sign (-1)^i; a
 single-power increase changes the residual by a sign-consistent
 amount. Reduce to the pure-exponential case:
 [
-E[e^bbE[e^{-(lambda_*^2-a)Y}]=prod_{i}(v_i+a)/(v_i+lambda_*^2)
+E[e^E[e^{-(lambda_*^2-a)Y}]=prod_{i}(v_i+a)/(v_i+lambda_*^2)
 ]
 since Y is a sum of exponential variables with rates v_i+a.
 
@@ -1317,7 +1317,7 @@ Step 1: both measures are MTP_2. For measures whose density
 has the determinant-product form w=det(F,G)prod_i dmu(x_i),
 MTP_2 holds whenever the base mu is MTP_2; the proof
 (Bogso cite[Theorem 2.27]{Bogso2014}, following Karlin--McGregor)
-reduces the MTP_2 inequality to Pl"ucker's 2times 2 identity on
+reduces the MTP_2 ineuality to Pl"ucker's 2times 2 identity on
 the alternating minors det(F,G) and then to
 log-supermodularity of the single-variable base density. Both
 nu and nu_b, built over the same jet spaces from the base
@@ -1326,12 +1326,12 @@ measures dmu and dmu_b, are therefore MTP_2.
 Step 2: FKG. For an MTP_2 measure eta and any
 coordinatewise nondecreasing bounded f_1,...,f_m,
 eta[prod_jf_j]geprod_jeta[f_j] (Harris). Indeed one-coordinate
-heat-bath smoothing preserves the FKG inequality: conditional
+heat-bath smoothing preserves the FKG ineuality: conditional
 densities of one variable given the rest have monotone likelihood
 ratio in the other coordinates by Pl"ucker's quadratic relation, so
-the FKG product inequality passes to the full d-variable
+the FKG product ineuality passes to the full d-variable
 distribution; the common unit-quantile coupling then extends the
-inequality from indicators of increasing cylinders to all bounded
+ineuality from indicators of increasing cylinders to all bounded
 increasing integrands.
 
 Step 3: the comparison. From the lemma computation,
@@ -1366,9 +1366,9 @@ s_*=-frac{A_{-4}-H_{1,-4}}{A_0-H_{1,0}}; the denominator is
 nonzero for large N since |A_0|gtrsim N^{-1/2} and
 |H_{1,0}|le N^{-1+o(1)}.
 
-The exact form of L(h_0)/(4a_*)calL(h_0)/(4a_*) is then
+The exact form of L(h_0)/(4a_*)L(h_0)/(4a_*) is then
 
-(L(h_0))/(4a_*)calL(h_0))/(4a_*)
+(L(h_0))/(4a_*)L(h_0))/(4a_*)
 =frac{A_0,nu_3[E_{-4}h_0]
 +H_{1,-4}H_{0,0}-H_{1,0}H_{0,-4}}
 {A_0-H_{1,0}}.
@@ -1376,7 +1376,7 @@ The exact form of L(h_0)/(4a_*)calL(h_0)/(4a_*) is then
 The numerator has leading term A_0nu_3[E_{-4}h_0]asymp N^{-1/4}·
 N^{-1/2}=N^{-3/4}; the remaining quotient contributes
 O(N^{-1/2+o(1)}), negligible. So
-L(h_0)/(4a_*)calL(h_0)/(4a_*)asymp N^{-1/4}, hence
+L(h_0)/(4a_*)L(h_0)/(4a_*)asymp N^{-1/4}, hence
 B_Nasymp N^{-1/4}.
 
 With this, Theorem~thm:finite-bridges items (eq:bridge-mass)
@@ -1401,7 +1401,7 @@ Take the lozenge-cylinder of period N. Put corners A,B of tiles
 on a common horizontal line, A just before column 1 and B just
 after column m, with m+n=N. Let Z_N(y) sum systems of
 finitely many disjoint simple polygons in the cylinder, each with
-weight kappa^{rm length} and a factor y+y^{-1}. Every polygon
+weight kappa^{length} and a factor y+y^{-1}. Every polygon
 in a system separates A from B; the empty system is included.
 
 Introduce spectral variables v_1,...,v_N, one per column. Put
@@ -1547,7 +1547,7 @@ in the order 1,2,3 with remaining list Y. Perturbing
 v_2=u+2L+z, v_3=u+5L+z' and using~(eq:psi-fusion-2L):
 at z=0, Psiinim(W_{12}); on z'=z and z'=0 we get
 specific columns b_0,c_0 proportional to Ω_{23} and to
-K_3 J^{rm spin}_1 Ω_{31} (composed with site-3 factors). The
+K_3 J^{spin}_1 Ω_{31} (composed with site-3 factors). The
 column is in im(W_{12}) only at specific spin coefficient
 identities; matching at spin pm1 then at spin 0 forces both
 gamma,alpha=0 (for xi=-i), giving vanishing of order ge2.
@@ -1631,7 +1631,7 @@ Put
 
 P_m(e^r)=frac{G(e^r)}{C^mD(0)^{m(m-1)}}.
 
-With L=pi/8, b=3L, C=2-sqrt2, widehatrho(p)=1/(2cosh Lp-1),
+With L=pi/8, b=3L, C=2-sqrt2, rhohat(p)=1/(2cosh Lp-1),
 and V(s) the real log of R_s(is)R_c(is) for
 R_s(z)=-1/[sin(z-L)sin(z+L)], R_c(z)=cos^2z, put
 I=int_{R},ds.
@@ -1698,23 +1698,23 @@ Let H_*=H(0;0), A_0=sum_{ain(2)}H(a;1), A_1=sum_{ain(2,3)}H(a;-1),
 A_2=1, A_3=-1, A_4=A_1+A_3. For 0le i,j<4, set
 
 F^{(c)}_{ij}(z)=P_{ij}(z)+frac{A_i(z)A_j(z^{-1})}{H_*(z)}
-+(-1+2c)sgn(i-j){mathbf 1}bf1_{{i,j}={1,3}},
++(-1+2c)sgn(i-j){1}1_{{i,j}={1,3}},
 
-F_{i4}=F_{i1}+F_{i3}+{mathbf 1}bf1_{i=3}-{mathbf 1}bf1_{i=1},
+F_{i4}=F_{i1}+F_{i3}+{1}1_{i=3}-{1}1_{i=1},
 F_{4i}(z)=-F_{i4}(z^{-1}), F_{44}=F_{11}+F_{13}+F_{31}+F_{33},
 where P_{ij} collects lower-derivative residue rules
 P_{ij}(z)=sum_{ain b_i,kin b_j}H(a-k;sgn(j-i)) for i,j<2
 and sgn(j-i) for i,jge2, and 0 otherwise.
 
 On the two R-lines (before fusion), the matrix symbol has a clean form.
-Let K=K_{RR} and widehatrho(p)=1/(2cosh Lp-1); then
+Let K=K_{RR} and rhohat(p)=1/(2cosh Lp-1); then
 
 K=(1-e(p))K_0,
 e(p)=(sinh(3Lp)-sinh(2Lp))/(sinh(4Lp))in(0,1),
 
 
 K_0=(2pi)/(p)coth(bp)&-csch(bp)
--csch(bp)&coth(bp), V_R=K{mathbf 1}bf1widehatrho.
+-csch(bp)&coth(bp), V_R=K{1}1rhohat.
 
 (Here V_R denotes the potential contribution.)
 
@@ -1726,7 +1726,7 @@ The key analytic input is a positive-definite bound on Z.
 Put h_*=3pi b/4 and alpha=2pi^2/h_*, and let W=(w,w)^t be the
 weight column. Then for every real pne0 there is eta>0 with
 
-Z_{rm s}(p)succcurlyeq(etaalpha)/(p^2)WW^t+ε_0(1+p^2)^{-1}{bf I},
+Z_{s}(p)succcurlyeq(etaalpha)/(p^2)WW^t+ε_0(1+p^2)^{-1}{ I},
 
 for some ε_0>0 independent of T and m, where Z_s=(Z+Z^t)/2.
 
@@ -1734,10 +1734,10 @@ for some ε_0>0 independent of T and m, where Z_s=(Z+Z^t)/2.
 Proof. [Sketch]
 On each R-line, K=(1-e(p))K_0 with e(p)in(0,1) and K_0
 positive definite for pne0. The leading behavior at small p is
-K_0sim2pi/(p)(coth bp{bf I}-cschbp,P) where
+K_0sim2pi/(p)(coth bp{ I}-cschbp,P) where
 P is the color-swap permutation matrix; the eigen-decomposition is
 common and gives the WW^t/p^2 leading term. Away from p=0, the
-residual is bounded below by ε_0(1+p^2)^{-1}{bf I} after
+residual is bounded below by ε_0(1+p^2)^{-1}{ I} after
 partial-fraction expansion of the explicit F-rules.
 
 
@@ -1746,7 +1746,7 @@ Z(p)=alpha WW^t/p^2+M_*/p+O(1)(pto0),
  M_* real skew.
 
 
-Z(p)=(2pi)/(|p|){mathbf I}bfI+2ff^t+O(e^{-ε|p|})(|p|to∞), f={mathbf e}bfe_4.
+Z(p)=(2pi)/(|p|){I}I+2ff^t+O(e^{-ε|p|})(|p|to∞), f={e}e_4.
 
 
 The grand sum.: 
@@ -1757,13 +1757,13 @@ P_{m,T}(e^r) equals
 
 sum_{{nu_{g,i}:,(eq:winding-neutrality)}}
 frac{1}{prod_{g,i}nu_{g,i}!}
-,e^{c· 2pi iTr,sigma^tnu/(4h_*)-S_{rm bulk}(T)-S_{rm gas}[nu;r]},
+,e^{c· 2pi iTr,sigma^tnu/(4h_*)-S_{bulk}(T)-S_{gas}[nu;r]},
 
 with:
- S_{rm bulk}(T)=(1)/(2T)sum_{pne0}widehatwidehat J^tK(p)J^tK(p)widehatwidehat JJ,widehatrho(p)^2
+ S_{bulk}(T)=(1)/(2T)sum_{pne0}Jhat^tK(p)J^tK(p)JJhat,rhohat(p)^2
 -(1)/(2)(T/3-2log2)(M_A-M_B)^2,
 a real quadratic form;
- S_{rm gas}[nu;r]=-(1)/(2T)sum_{pne0}sum_{d,e}Z_{g_de_d}(p)e^{-ip(s_d-s_e)}-(1)/(T)sum_{linear}...,
+ S_{gas}[nu;r]=-(1)/(2T)sum_{pne0}sum_{d,e}Z_{g_de_d}(p)e^{-ip(s_d-s_e)}-(1)/(T)sum_{linear}...,
 the pairwise and single-particle interactions against real positions
 s_din(-T/2,T/2);
  ε_nuin{pm1} is a fixed phase depending only on the multiplicity pattern;
@@ -1783,7 +1783,7 @@ from below by a positive quadratic form. After regularizing the diagonal,
 the sum over nu_{g,i} is bounded by a geometric series in
 sum_{g,i}nu_{g,i} weighted by a Gaussian-decaying kernel in the
 positions s_d and a bounded activity from the chemical potential. For any configuration with a particle at |Re s_d|>log m+const,
-the one-particle suppression from S_{rm bulk} dominates the local
+the one-particle suppression from S_{bulk} dominates the local
 gain, and the sum over such configurations is at most e^{-ε m}.
 Consequently the whole sum is dominated by the localized region
 (eq:gas-localization).
@@ -1879,7 +1879,7 @@ uniformly in |Im r|<.54pi. From (eq:P-m-i-final) at
 r=ipi/2 (i.e. y=i), f_m(ipi/2)to0.
 
 The function f_m is subharmonic. Apply the subharmonic area
-mean inequality on a finite chain of disks starting at r=ipi/2 and
+mean ineuality on a finite chain of disks starting at r=ipi/2 and
 reaching any compact subset of |Im r|<.54pi: the two ends of the
 chain give f_mto0 in local L^1 throughout the strip. In
 particular at r=0:
@@ -2031,16 +2031,16 @@ omega(L)=x^{|L|}·
 -2cos(2alpha),&L essential.
 
 
-For A⊆calA⊆P:calP:
+For A⊆A⊆P:P:
 
-Q(A)={substack{F⊆calF⊆A
+Q(A)={substack{F⊆F⊆A
 polygons in F disjoint}}
 prod_{LinF}
 
 
 [Positivity]
-For every Nge1, hge1, 0lebeta'le3/8, A⊆calA⊆P:calP:
-Q(A)>0.calA)>0.
+For every Nge1, hge1, 0lebeta'le3/8, A⊆A⊆P:P:
+Q(A)>0.A)>0.
 
 
 Proof. [Sketch]
@@ -2183,7 +2183,7 @@ Q_P/Q_{P∖ v}=1-tkappa^{|v|}· Q_{P∖ N[v]}/Q_{P∖ v},
 where N[v] is the closed conflict neighbourhood. By positivity
 (eq:Q-def) the ratio is ge1. Induction on |P| (deleting
 neighbours successively in the smaller set) gives the multi-deletion
-inequality; extra neighbours only increase the ratio.
+ineuality; extra neighbours only increase the ratio.
 
 
 For a simple polygon Lsubset D and a boundary source a, let
@@ -2216,12 +2216,12 @@ The essential sum is absent in the plane.
 Proof. [Sketch]
 Evaluate the same prefix current J two ways.
 (i) Telescoping along Gamma via the cut-link identity of
-Prop.~prop:occupancy: J=-(Y(y)/Q_D-1)/calY(y)/Q_D-1)/sqrt2.
+Prop.~prop:occupancy: J=-(Y(y)/Q_D-1)/Y(y)/Q_D-1)/sqrt2.
 (ii) Continuing each prefix to its first exit gives
 J=I+2A+E, where A is the contractible enclosing sum and |E|le
 sqrt2sum_{essential}kappa^{|L|}H_D(L)sum_aP(a,L) (turn
 increments pm4pi/3 versus pm2pi/3). Comparing gives the
-inequality.
+ineuality.
 
 
 The annular bound.: 
@@ -2303,7 +2303,7 @@ Classwise convergence of the normalization.:
 For rin{0,ipi/2} put sigma(r)=3/8+2r^2/(3pi^2) and
 A_m(r)=e^{-Im^2}m^{-sigma(r)}P_m(e^r). There exist a fixed
 integer Qge1, a delta>0, and positive constants
-mathfrak a_b(r),fraka_b(r), 0le b<Q, such that
+a_b(r),a_b(r), 0le b<Q, such that
 
 A_m(r)= a_fraka_{mbmod Q}(r)+O(m^{-delta}).
 
@@ -2354,7 +2354,7 @@ We prove the third main theorem: the two-edge marked-polygon bound
 in the ratio N/d. The argument has three parts: (a) identify the
 two-edge observable as a contraction of polynomial columns, (b) in the
 balanced (antipodal) case reduce it to a two-line gas and estimate,
-(c) for unequal intervals control the two long columns outside a small
+(c) for uneual intervals control the two long columns outside a small
 core.
 
 Setup.: 
@@ -2382,7 +2382,7 @@ Y=big<Psi_bbig|,delta^{sum_A p_j}big|Psi_tbig>,
 of the columns (below with physical rapidities, above with negated
 rapidities, spin-fixed extra lists -,+,+,-) satisfies
 
-Y=Q_{lh},J_{rm ext}!!
+Y=Q_{lh},J_{ext}!!
 sum_{substack{a_jin{-1,0,1}
 sum_Aa_j-sum_Ba_j=h-l}}
 (i/delta)^{sum_Aa_j},W(a),M(a),
@@ -2391,8 +2391,8 @@ where the right-hand data are specified below.
 
 
 Here:
- Q_{lh}=tfrac12C^{k/2}s^{-k}(-s^{-1})^ls^{-h}.
- J(u)=cos(u+L)cos(u+2L), and J_{rm ext} is a
+ Q_{lh}=1/2C^{k/2}s^{-k}(-s^{-1})^ls^{-h}.
+ J(u)=cos(u+L)cos(u+2L), and J_{ext} is a
 product of J-factors between extra slots in each list.
  W(a)={a_j}prod_{j<j'}P_{a_ja_{j'}}(v_{j'}-v_j)
 prod_{j<j',,same list}D(v_{j'}-v_j), with P, D from
@@ -2557,16 +2557,16 @@ winding.  Phases are retained in this step: the interval estimate
 below uses the resulting transfer operator rather than the modulus
 of an unidentified periodic partition.
 
-The extended mode matrix.:  Put E(p)=sum_e e^{-p a_e}{bf e}_{g_e}.
+The extended mode matrix.:  Put E(p)=sum_e e^{-p a_e}{ e}_{g_e}.
 Include extra background slots E_e of count-mode 1 in the mode
-matrix, with pair entries from the potentials ({cal M}_{R,E}n_E=-Kwidehatrho E)
+matrix, with pair entries from the potentials ({cal M}_{R,E}n_E=-Krhohat E)
 and inverse couplings initially zero; their mutual block is zero
 except at Laurent zero, where we assign 4q_eq_f/p^2+4q_eq_f(a_e-a_f)/p
 with zero finite part.  Then:
  The bulk log term is
 
-S(T)=frac1{2T}sum_p E(-p)^tK(p)E(p)widehatrho(p)^2
--tfrac12(T/3-2log2)(M_A-M_B)^2.
+S(T)=frac1{2T}sum_p E(-p)^tK(p)E(p)rhohat(p)^2
+-1/2(T/3-2log2)(M_A-M_B)^2.
 
 
 Determinants and measure conventions of (eq:legacy-P-star) are
@@ -2585,15 +2585,15 @@ Fugacity still brings Tr_j^2/(4h_*)-r_j(u_*+barbar L).L).
 
 Extended zero modes.:   At zero, take the raw right count
 vector with left and inverse modes as before, put the background
-modes at 1, and prescribe q^tgamma'_{rm raw}
+modes at 1, and prescribe q^tgamma'_{raw}
 =A_L+sum_e q_ea_e+pi l/2.  The extended matrix has the same
-charge/offset/winding poles, and the saddle term widehatrho E
+charge/offset/winding poles, and the saddle term rhohat E
 neutralizes the appended charges at 0 and contributes exactly
 sum_e q_ea_e to the derivative condition, while its count value
 (M_A,M_B) leaves the free integer and Gaussian shift unchanged.  The
 assigned background block has zero contracted finite part at zero, and
 the linear symbol at a particle against a same-color site is
--2pisigma_i e^{(v_e+delta_i L)p}widehatrho(p)/p; the primitive
+-2pisigma_i e^{(v_e+delta_i L)p}rhohat(p)/p; the primitive
 of (eq:legacy-P-25) follows.
 
 Ordered-position form.:   For particles in increasing real
@@ -2601,7 +2601,7 @@ contour order write H(s)=j+sum_{Re s_d>Re s}W_d (j the Poisson
 integer), with mid-value H_m at jumps; winding neutrality closes the
 integer path.  Set c^t by
 [
-{bf1}^t(J(p)-K(p)ell)=2pi W^t/p-2h_*c^t+O(p),
+{1}^t(J(p)-K(p)ell)=2pi W^t/p-2h_*c^t+O(p),
 ]
 so u_*+barbar L=c^tn-(L=c^tn-(pi i/h_*)sum W_ds_d.  (Here J is the
 coupling block of the transform, not the site function of
@@ -2683,7 +2683,7 @@ pi^2/h_*int H^2) plus the real adjacent pair energy from
 (eq:legacy-P-28) (cyclically omitting separations of at least
 two cells).  By (eq:legacy-P-P3), for large S_0,
 
-sum A(x_i,x_{i+1})ge4ε_Ssum|x_i|_0^2-C_Sn_{rm cells},
+sum A(x_i,x_{i+1})ge4ε_Ssum|x_i|_0^2-C_Sn_{cells},
 |x|_0^2=N_x^2+H_{x,rm in}^2,
 
 
@@ -2713,7 +2713,7 @@ with V=sum_{bin x_{i+1}}v_bD_{d_b^L},
 U=sum_{ain x_i}D_{ell'+d_a^R}u_a, symmetric tensor powers, and
 exponential-vector normalization.  Each block is rescaled so
 |U|+|V|le Ce^{-ε S_0}(N_x+N_y); splitting
-Gamma(D_{ell'})=Gamma(tfrac12I)Gamma(4D_{ell'})Gamma(tfrac12I)
+Gamma(D_{ell'})=Gamma(1/2I)Gamma(4D_{ell'})Gamma(1/2I)
 and using < a^dagger(U)^k>le|U|^k√((m+k)!/m!)
 bounds the trace norm by exp(C|U|^2+C|V|^2), within the
 Gaussian margins.  The resulting integral operators on L^2 with the
@@ -2789,8 +2789,8 @@ u_2=beta(beta^2-1)(beta^2-4)/120,
 and the lower polynomial has first two coefficients
 U_k=sum_ju_kB_j (k=0,1) and last
 [
-U_2=sum_ju_2B_j-tfrac12sum_{12nmid j}
-(u_2B_j+|u_2|diag(|B_j|_{rm ent}{bf1})).
+U_2=sum_ju_2B_j-1/2sum_{12nmid j}
+(u_2B_j+|u_2|diag(|B_j|_{ent}{1})).
 ]
 A finite integer check (1000 times each U_k is integer-valued with
 consecutive leading determinants positive after subtracting 3I)
@@ -2842,8 +2842,8 @@ the left retain direct U_T factors with the mean subtracted; on the
 right insert a-barbar a.a.  Completing Cartan nonzero modes as in the
 balanced case yields an extra external scalar
 
-S_a(T)=frac1Tsum_pwidehatrho(p){bf1}^tE(-p)widehatwidehat a(p)a(p)
-+frac1Tsum_{pne0}(Q_+(p)-1/h_*)widehatwidehat a(-p)a(-p)widehatwidehat a(p),a(p),
+S_a(T)=frac1Tsum_prhohat(p){1}^tE(-p)ahat(p)a(p)
++frac1Tsum_{pne0}(Q_+(p)-1/h_*)ahat(-p)a(-p)ahat(p),a(p),
 
 
 and the local height-square becomes
@@ -2852,7 +2852,7 @@ and the local height-square becomes
 
 
 The added particle-source nonzero-mode multiplier is
--J(-p)^t{bf1}Q_+(p); its simple pole +2pi W/(ph_*) acting on
+-J(-p)^t{1}Q_+(p); its simple pole +2pi W/(ph_*) acting on
 a-barbar aa gives the mean-zero primitive of -i(a-barbar a),a), and
 summing the jumps by parts against H recovers the quadratic in
 (eq:legacy-P-26) together with the mean and test-variance
@@ -2866,7 +2866,7 @@ exponent, beyond (eq:legacy-P-27), is
 
 where the b_{i'} are the physical integer offsets and b_i' is the
 Fourier action on a' of the regular symbol
-[-J(-p)^t{bf1}Q_+(p)-2pi W/(ph_*)-(2c-ell_Σ)]_i/(-ip).
+[-J(-p)^t{1}Q_+(p)-2pi W/(ph_*)-(2c-ell_Σ)]_i/(-ip).
 Here Q_+=1/h_*+O(p^2) fixes the constant 2c-ell_Σ, whose
 mean times the counts matches the terms in (eq:legacy-P-31) and
 (eq:legacy-P-32) via int(H-j)=sum W_ds_d.
@@ -3005,7 +3005,7 @@ p_i(x)=delta,depth_iN_x with fixed small delta>0, and
 p_i=0 on exterior states at the two core interfaces.  If K_i(x,y)
 is the outgoing edge kernel, replace it by
 [
-K_i^{rm sym}(x,y)=exp!Bigl[frac{p_i(x)-p_{i+1}(y)}2Bigr]K_i(x,y).
+K_i^{sym}(x,y)=exp!Bigl[frac{p_i(x)-p_{i+1}(y)}2Bigr]K_i(x,y).
 ]
 The reserved exponent on such an edge is
 -p_i(x)/2-p_{i+1}(y)/2.  In a product through the core the multipliers
@@ -3065,7 +3065,7 @@ the core scalar but excluding (eq:legacy-P-24),
 (eq:legacy-P-31) and the determinant, the overall log modulus is
 at most
 
--2R+tfrac52(R-2u_0)+O(1).
+-2R+5/2(R-2u_0)+O(1).
 
 
 
@@ -3075,18 +3075,18 @@ Matching the scalar and the Pfaffian normalization.:
 The interval bound controls the particle integral.  To recover the
 polygon observable we must still match its scalar factors with the
 normalizing polynomial f_N^2, even when the staircase uses only two
-rapidity values, because those values may have arbitrarily unequal
+rapidity values, because those values may have arbitrarily uneual
 multiplicities.
 
 External scalars.:   Uniformly on the circle,
 (eq:legacy-P-24) equals
 [
-tfrac12sum_{e,f}I_{g_eg_f}(v_e-v_f)+C_2e^{-2T}+O(1)
+1/2sum_{e,f}I_{g_eg_f}(v_e-v_f)+C_2e^{-2T}+O(1)
 ]
 for a site-data constant C_2, where
 [
 I_{gh}(v)=2q_gq_hlog2+int_{R}!!
-[K_{gh}(p)widehatrho(p)^2cosh(vp)-4q_gq_h/p^2],(dp)/(2pi).
+[K_{gh}(p)rhohat(p)^2cosh(vp)-4q_gq_h/p^2],(dp)/(2pi).
 ]
 The subtracted pole all-mode sum is q_gq_hT/3.  Contour shifting
 with exponentially bounded tails gives error O(N^2e^{-a_0Re T}),
@@ -3099,14 +3099,14 @@ e^{I_{gh}(v)}=B(v)^2tan^2(lambda_bv)
 
 
 Verification: for real v the multiplier
-(p/2pi)K_{gh}widehatrho^2 on p>0 is
+(p/2pi)K_{gh}rhohat^2 on p>0 is
 t^2/[(1-t+t^2)(1-t^8)] times 1+t^6 (same color) or -2t^3
 (opposite), with t=e^{-Lp}.  Thus it expands as
 sum_{kge1}c_ke^{-kLp} with
 [
-c_k=-2{bf1}_{k=pm3(8)}+2{bf1}_{k=0(8)}
-+2{bf1}_{k=3(6)}-2{bf1}_{k=0(6)}
-+{bf1}_{g=h}{bf1}_{k=pm2,pm3(8)}.
+c_k=-2{1}_{k=pm3(8)}+2{1}_{k=0(8)}
++2{1}_{k=3(6)}-2{1}_{k=0(6)}
++{1}_{g=h}{1}_{k=pm2,pm3(8)}.
 ]
 Integrating gives sum_k-c_klog(1-v^2/(kL)^2), matching
 (eq:legacy-P-35) by the sine product; the constant follows by
@@ -3126,7 +3126,7 @@ costs O(1+Ne^{-a_0Re T/2}).  The principal per-background integral
 is the whole-line core-log convolution shifted by i(x_e-v_e).  Its
 derivative has transform
 [
-2pi iwidehatrho(p)(z-z^7)/(1-z^8)
+2pi irhohat(p)(z-z^7)/(1-z^8)
 =2pi i(z^2+z^3-z^5-z^6)/(1-z^8), z=e^{-Lp},
 ]
 obtained from the localized coth difference; hence the convolution is
@@ -3137,14 +3137,14 @@ separated).
 Determinant scalar.:   Remove C_2e^{-2T}+C_1e^{-T} by an
 exponential analytic multiplier and use maximum modulus.  The
 two-color transform determinant adds (3/8)log N+O(1).  For the
-remaining determinant put mathfrak d(p)=-frakd(p)=-log(1-e(p)) and use
+remaining determinant put d(p)=-d(p)=-log(1-e(p)) and use
 [
-2sum_{nge1}mathfrak d(2frakd(2pi n/T)=sum_{ninZ}
-mathfrak d(2frakd(2pi n/T)-mathfrak d(0).frakd(0).
+2sum_{nge1}d(2frakd(2pi n/T)=sum_{ninZ}
+d(2frakd(2pi n/T)-d(0).d(0).
 ]
-Since mathfrak dfrakd is even, regular at zero, and exponentially
+Since d is even, regular at zero, and exponentially
 decaying, the periodization contributes
-pi T/(16L)-mathfrak d(0)+O(e^frakd(0)+O(e^{-cRe T}) as in the two-line case.
+pi T/(16L)-d(0)+O(e^d(0)+O(e^{-cRe T}) as in the two-line case.
 The two square-root factors have constant product.  With
 Re T=(3/4)log N+O(1) this gives the stated uniform bound on the
 nome circle.  Formula (eq:legacy-P-35), with
@@ -3225,11 +3225,11 @@ with a bounded additive error.
 Assembly.:   The log of the normalized marked-polygon mass
 is at most
 [
--2R+tfrac52(R-2u_0)+(frac38-frac5{24})log N+O(1).
+-2R+5/2(R-2u_0)+(frac38-frac5{24})log N+O(1).
 ]
 Choosing R=(3/4)log N+O(1) and 2u_0=(3/4)log d+O(1) makes this
 [
--tfrac43log N+(15)/(8)log(N/d)+O(1).
+-4/3log N+(15)/(8)log(N/d)+O(1).
 ]
 Exponentiation proves the uniform two-edge bound
 (eq:marked-uniform).  If the shorter core has only a bounded
@@ -3297,7 +3297,7 @@ Two exterior sewing estimates.:
 
 
 Take a triangular-lattice vertex as origin and pure-support domains
-D=D(0;mathbf b)bfb) with |b_j-R|leε R for a fixed
+D=D(0;b)b) with |b_j-R|leε R for a fixed
 0<εle1/20; these are convex hexagons with minimum side
 asymp R, and Section~sec:hexagon-completion supplies uniform
 nesting and central-occupancy estimates for every such domain,
@@ -3417,10 +3417,10 @@ for case 1 shared at both ends.  Let nu be critical weight on all
 exterior objects with the stated confinement and depth (paths
 pto q in case 1, or ordered pairs of disjoint paths to the two
 assigned caps in case 2), counting all centres in exterior paths.
-For a cut list mathbf u,bfu, let
-mathbf1_bf1_{mathbf u}bfu(eta) indicate that eta follows the
+For a cut list u,u, let
+mathbf1_bf1_{u}u(eta) indicate that eta follows the
 prescribed corridors at those levels, and put
-Y(eta)=E_{mathbf u}bfumathbf1_bf1_{mathbf u}bfu(eta) and
+Y(eta)=E_{u}umathbf1_bf1_{u}u(eta) and
 P=prod_{j=1}^ks_j^{-1/4}.  Concatenating the confined crossings
 gives
 [
@@ -3478,7 +3478,7 @@ chord-event indicator times the exterior trimming factor, so
 X(Gamma,Ω)=average_{hin I}b_h(Gamma,Ω).
 Then
 
-int X,d(mu ⊗ P_Ω)bbP_Ω)asymp Foverlineoverline E,E,
+int X,d(mu ⊗ P_Ω)P_Ω)asymp Foverlineoverline E,E,
 
 
 where the left side includes the level average and support
@@ -3499,7 +3499,7 @@ at r alone suffices.  Bounding the averaged cost Xi of the two
 band bridges uniformly conditional on the inner level and phases
 gives
 
-int X^2,d(mu ⊗ P_Ω)bbP_Ω)le CFoverlineoverline EE,Xi.
+int X^2,d(mu ⊗ P_Ω)P_Ω)le CFoverlineoverline EE,Xi.