diff --git a/uniform_marked_polygon_bridge_v2.txt b/uniform_marked_polygon_bridge_v2.txt
index a5bedf4..1c49515 100644
--- a/uniform_marked_polygon_bridge_v2.txt
+++ b/uniform_marked_polygon_bridge_v2.txt
@@ -135,16 +135,16 @@ edges. A boundary port of a tiled domain is a midpoint of a
 boundary edge; paths enter and exit normally at ports. The auxiliary
 constants L=pi/8, b=3L, C=2-sqrt2=4cos^2 b are eigenvalues
 of the row transfer matrix. For a triangular lattice vertex y and
-Rin d_0mathbb Z_bbZ_{>0} with d_0=sqrt3/2, H(y;R) is the
+Rin d_0Z_{>0} with d_0=sqrt3/2, H(y;R) is the
 regular pure-sided hexagon of inradius R centred at y.
 
 Introduction.: 
 
 The honeycomb lattice and its critical weight.: 
 
-Every closed walk of mathbb HbbH drawn without self-intersection
+Every closed walk of H drawn without self-intersection
 encloses a union of triangles. A simple polygon
-Lsubsetmathbb HbbH is a simple closed curve of the honeycomb
+LsubsetH is a simple closed curve of the honeycomb
 graph; it may be unbounded on a cylinder
 (Section~sec:intro-cylinder), in which case it is
 essential, or bounded, in which case it is
@@ -177,14 +177,14 @@ edges plus 1, for a path with nonempty interior.
 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 mathcal B_hcalB_h be the set of bridges of
-mathbb H:bbH: paths from a fixed bottom port to any port on the
+integer hge1, let B_h be the set of bridges of
+H:bbH: 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_{gammainmathcal B_h}calB_h kappa^{ell(gamma)},
-B_h(E) = sum_{gammainmathcal B_h}calB_h mathbf 1_E(bf1_E(gamma)kappa^{ell(gamma)}.
+B_h = sum_{gammainB_h} kappa^{ell(gamma)},
+B_h(E) = sum_{gammainB_h} mathbf 1_E(bf1_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
@@ -198,7 +198,7 @@ There exist constants c,C>0 such that for every sufficiently
 large integer h,
 
 c h^{-1/4} &le B_h,le, C h^{-1/4},
-sum_{gammainmathcal B_h}calB_h ell(gamma)kappa^{ell(gamma)}
+sum_{gammainB_h} ell(gamma)kappa^{ell(gamma)}
 &le C h^{13/12},
 B_h{ellge c h^{4/3}} &ge c h^{-1/4}.
 
@@ -217,8 +217,8 @@ The second theorem uses two points, not one.
 
 [Two axial marks]
 Let e_1,e_2 be unit lattice vectors at angle 60^circ. Fix
-P_1,P_2inmathbb Z_bbZ_{>0}, P=P_1e_1+P_2e_2, d=P_1+P_2. Let
-Esubset mathbb HbbH be a honeycomb link crossing a tiling edge
+P_1,P_2inZ_{>0}, P=P_1e_1+P_2e_2, d=P_1+P_2. Let
+Esubset H be a honeycomb link crossing a tiling edge
 in either axial direction, and let E+P be its translate by
 P. Suppose d is even. For an integer kge2, quotient by
 translation by kP; the two marks remain distinct and their
@@ -235,7 +235,7 @@ choice of axial edge direction.
 
 
 [Nesting and free boundary length]
-For a triangular lattice vertex y and Rin d_0mathbb Z_bbZ_{>0}
+For a triangular lattice vertex y and Rin d_0Z_{>0}
 with Rge1, let H(y;R) be the pure-sided regular hexagon of
 Euclidean inradius R centred at y; the lattice restriction on
 R and the centre is verified in
@@ -386,7 +386,7 @@ and sin(4L-2L)=sin(2L)=1/sqrt2.
 The local R-matrix.: 
 
 
-Let V=mathbb C^3bbC^3 have basis {|+>,|0>,|->},
+Let V=C^3 have basis {|+>,|0>,|->},
 interpreted as spins +1,0,-1. Set s = e^{2iL} = e^{ipi/4} so
 s^4=-1 and s^2+s^{-2}=0. Put Q(u)=sin(2L+u)sin(b+u) and
 define
@@ -469,7 +469,7 @@ reduces to the (pm,pm) check.
 Pi-periodicity. Q(u+pi)=-Q(u); a(u+pi)=-a(u),
 h(u+pi)=-h(u) (numerator flips sign), c(u+pi)=c(u),
 d(u+pi)=d(u), v(u+pi)=v(u). Thus on (pm,pm), R is
-pi-periodic; on (+,0)-type, R is antisymmetric under umapsto
+pi-periodic; on (+,0)-type, R is antisymmetric under u|->
 u+pi, matching the J insertion on either side; on (+-,00,-+),
 the off-diagonal terms h/s flip sign, and the J_1 on either side
 supplies the same flip. This identifies both sides.
@@ -488,9 +488,9 @@ Each entry of (eq:R-crossing) follows.
 
 Special values at 0, b, 2L. At u=0: a=h=d=0,
 v=c=1; R reduces to permutation P. At u=b: Q(b)=sin(4L)sin(2b)
-= 1 cdot sin(6L) = sin(pi/4+pi/2) = sin(3pi/4) = S;
+= 1 · sin(6L) = sin(pi/4+pi/2) = sin(3pi/4) = S;
 direct check gives R(b)=ΩbarbarΩ^T.Ω^T. At u=2L:
-Q(2L)=sin(4L)sin(2L+3L)=sin(4L)sin(5L)=1cdot S; direct check
+Q(2L)=sin(4L)sin(2L+3L)=sin(4L)sin(5L)=1· S; direct check
 gives R(2L)=WZ.
 
 
@@ -528,7 +528,7 @@ and (+,-) and (+,0) and (0,0) and (+,f)-type with fin{+,0,-}.
 Each case reduces to sine-product identities in t_j=sin(u+jL). The
 specific list, writing primes for argument L-u, Q'=Q(L-u),
 Q_+=Q(u+L), and using QQ'/Q_+=-t_{-3}t_2 (a direct computation from
-sin(cdot) products), is:
+sin(·) products), is:
 
 On (+,+)-input:
 
@@ -657,7 +657,7 @@ cos(Lpm y)=sin(bmp y).
 Fusion below. We sketch. Transposing the second fusion gives the
 fourth (using Fbarbar D_eD_e F = D_{e/s}^T at formal real e). For the
 third, (eq:K-fusion-below), evaluate at u=2t and use
-Q(2t)=sin(4L)sin(2t+3L)cdot...; on the three columns of
+Q(2t)=sin(4L)sin(2t+3L)·...; on the three columns of
 D_{e,2}R(2t)D_{e,1}barbar WW the constant term in e reduces to
 v k_+ + a = k and the linear term to k_-(a k_+ + v)=k and
 k_-(h+kappa c k_+)=kappa. Writing
@@ -731,7 +731,7 @@ closes the contraction, giving identity.
 Commutation of T(t) and T(y).: 
 
 [Commuting double rows]
-For all t,y in mathbb CbbC away from poles,
+For all t,y in C away from poles,
 
 T(t)T(y) = T(y)T(t).
 
@@ -855,7 +855,7 @@ d(L)&=(sin(L)sin(0))/(sin(3L))=0.
 
 These are exactly the honeycomb path weights: passing through a
 triangle centre costs kappa, and pairing two arcs at a shared centre
-costs kappa^2=kappacdotkappa.
+costs kappa^2=kappa·kappa.
 
 The mto∞ limit.: 
 Entries of T(gamma)^m are sums over configurations in an m-long,
@@ -884,7 +884,7 @@ site's rapidity to imaginary infinity: the local R-matrices become
 block-triangular with respect to that site's spin, and the traced sector
 reduces to a smaller T acting on N-1 sites at the same parameter.
 By induction on N, plus the base case N=1 (which is one-site, and
-where the diagonal R's give 1+xi s^{-6p}+xi^{-1}s^{6p}neq 1),
+where the diagonal R's give 1+xi s^{-6p}+xi^{-1}s^{6p}!= 1),
 simplicity persists at every N.
 
 Hence T(gamma) has a joint generalized-eigenspace decomposition; on
@@ -1023,36 +1023,36 @@ u=0 and order N at u=L.
 We will use the following functional-analytic representation in the
 next section. For u in the horizontal strip |Re u|le 5L/2,
 
-p(u)=mathcal LcalL!((C_u)/(D)),
+p(u)=L!((C_u)/(D)),
 
 where
 [
 C_u(x)=cosh(usqrt x), D(x)=2C_L(x)-1,
 ]
-mathcal LcalL is a fixed linear functional, and x=z^2.
+L is a fixed linear functional, and x=z^2.
 
-The value 2B_N=2cdot(1)/(2)q(gamma)=q(gamma)=p(0) becomes
+The value 2B_N=2·(1)/(2)q(gamma)=q(gamma)=p(0) becomes
 
-2B_N=p(0)=mathcal L(1/D).calL(1/D).
+2B_N=p(0)=L(1/D).calL(1/D).
 
 
 The finite functional.: 
 
 
-We now pin down mathcal L.calL. Set mu_0(x)=C_{L/2}(x),
+We now pin down L. Set mu_0(x)=C_{L/2}(x),
 mu_1(x)=C_{3L/2}(x), and
 M=span{1,x,...,x^{N-1};∂_m^j C_m(x)|_{m=L}}_{j=0}^{N-1}.
 Then:
 
 [Finite functional]
-There is a unique linear functional mathcal LcalL on the span of
+There is a unique linear functional L on the span of
 M, C_{L/2}, C_{3L/2}/D such that:
 
-mathcal L(f)calL(f) &= 0  for every fin M,
-mathcal L(C_calL(C_{L/2}) &= 0,
-mathcal L(C_calL(C_{3L/2}-C_{L/2}/D) &= 0.
+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=mathcal L(1/D)calL(1/D) for every N.
+For this functional, 2B_N=L(1/D)calL(1/D) for every N.
 
 
 Proof. [Outline]
@@ -1066,7 +1066,7 @@ int_0^∞(cosh(uz)sinh(mz))/(sinh(pi z/2)),dz
 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 mathcal 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.calL(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,
@@ -1085,8 +1085,8 @@ sums to the displayed tan-combination.
 The positive interpolation estimate.: 
 
 Section~c6:strip-transfer reduced the strip bridge mass to a
-single linear functional mathcal LcalL on a (2N+2)-dimensional space.
-Here we evaluate mathcal L(1/D)calL(1/D) up to an absolute constant, obtaining
+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
 B_Nasymp N^{-1/4}. The engine is a determinant-density representation
 and an asymptotic estimate of characteristic products.
 
@@ -1095,16 +1095,16 @@ Spaces, measure, determinant density.:
 Retain from Proposition~prop:finite-functional:
  measure dmu(x)=zD(x),dz/sinh(4Lz), x=z^2;
  M_N=span{1,x,...,x^{N-1};∂_m^jC_m(x)|_{m=L}}_{j=0}^{N-1};
- mathcal L(M_N)=calL(M_N)=mathcal L(C_calL(C_{L/2})=
-mathcal L(C_calL(C_{3L/2}-C_{L/2}/D)=0;
- 2B_N=mathcal L(1/D).calL(1/D).
+ 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).
 
 
 Set
 
 c_1&=C_{L/2}, c_3=C_{3L/2},
 h_0=1/D, h_1=c_1/D, g(x)=sinh(Lsqrt x)/sqrt x,
-F_0&=M_N+mathbb Cc_1,bbCc_1, F=F_0+mathbb Cc_3,bbCc_3, G=M_N+mathbb Cg.bbCg.
+F_0&=M_N+Cc_1, F=F_0+Cc_3, G=M_N+Cg.
 
 Note g(x)=int_0^L C_t(x),dt.
 
@@ -1138,12 +1138,12 @@ positive integral of C_t rows, 0<t<L.
 
 
 [Determinant density]
-For H with int|Hf|dmu<∞ (fin M_N+mathbb Cg):bbCg):
+For H with int|Hf|dmu<∞ (fin M_N+Cg):bbCg):
 
-(mathcal L(H))/(4a_*)=calL(H))/(4a_*)=nu[e_{-4}H]+s_*,nu[e_0H],
+(L(H))/(4a_*)=calL(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 mathcal L.calL.
+at nodes X, and s_*=rho/(4a_*) the scalar in L.
 
 
 Proof. 
@@ -1151,7 +1151,7 @@ Apply Andr'eief's identity
 intdet F_0det Gprod dmu=det(int F_0G^Tdmu) row by row: the
 integral int Hf,dmu against the density reduces to a projective
 nu-average; point evaluations at -4 and 0 enter as residual
-evaluation vectors because H is fixed on the discrete mathcal LcalL
+evaluation vectors because H is fixed on the discrete L
 component. See Tracy--Widom [cite] for the identity
 itself.
 
@@ -1197,7 +1197,7 @@ Deform all beta-labels to 0 continuously; interlacing is preserved.
 Reduction to polynomial. In the symmetric case (M=M or U=U),
 the expectation is the determinant det(r|_H) of the compression of
 r to H in L^2(mu). By L^2(mu) approximation, H lies between
-the polynomial spaces mathcal P_calP_{<N} and mathcal P_calP_{<CN} for
+the polynomial spaces P_{<N} and P_{<CN} for
 a large absolute C, with operator-norm error e^{-N}. Hence the
 determinant changes by 1+o(1).
 
@@ -1255,13 +1255,13 @@ D_J(0) are positive with derivative ratio bounded by
 theta_N=(sumbeta_i)/(2(2N+1)); for large N the minors remain
 positive after translating by a.
 
-Laplace structure. Write h_j(w-a)=h_j(-a)mathbb EbbE e^{-wZ_j},
+Laplace structure. Write h_j(w-a)=h_j(-a)E e^{-wZ_j},
 where Z_jge0 is a random variable. Explicitly, the product formula
 for D(x)=2C_L(x)-1 (via cosh-product) gives
 D(x)propto(x+lambda_*^2)prod_{kge2}(x+s_k) with
 s_k=((2pi k+pi/3)/L)^2; similarly c_1 and (c_3-c_1) have
 product form. Hence
-mathbb P(Z_jbbP(Z_jge t)asymp e^{-lambda_*^2t} (for h_1, one pairs
+P(Z_j t)asymp e^{-lambda_*^2t} (for h_1, one pairs
 each numerator factor with a distinct smaller s_k, splitting off
 the smallest rate by a change-of-measure uniform bound).
 
@@ -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:
 [
-mathbb E[e^bbE[e^{-(lambda_*^2-a)Y}]=prod_{i}(v_i+a)/(v_i+lambda_*^2)
+E[e^bbE[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.
 
@@ -1361,22 +1361,22 @@ Assembly.:
 Combine (eq:schur-identity), (eq:P-7-estimates), and
 Lemma~lem:weight-comp. Write
 A_y=nu[e_yc_3], H_{j,y}=nu[e_yh_j]
-(yin{0,-4}). The constraint mathcal L(c_3-h_1)=0calL(c_3-h_1)=0 determines
+(yin{0,-4}). The constraint L(c_3-h_1)=0calL(c_3-h_1)=0 determines
 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 mathcal L(h_0)/(4a_*)calL(h_0)/(4a_*) is then
+The exact form of L(h_0)/(4a_*)calL(h_0)/(4a_*) is then
 
-(mathcal L(h_0))/(4a_*)calL(h_0))/(4a_*)
+(L(h_0))/(4a_*)calL(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}}.
 
-The numerator has leading term A_0nu_3[E_{-4}h_0]asymp N^{-1/4}cdot
+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
-mathcal L(h_0)/(4a_*)calL(h_0)/(4a_*)asymp N^{-1/4}, hence
+L(h_0)/(4a_*)calL(h_0)/(4a_*)asymp N^{-1/4}, hence
 B_Nasymp N^{-1/4}.
 
 With this, Theorem~thm:finite-bridges items (eq:bridge-mass)
@@ -1399,7 +1399,7 @@ The cylinder and the finite formula.:
 
 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 mathcal Z_N(y)calZ_N(y) sum systems of
+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
 in a system separates A from B; the empty system is included.
@@ -1434,12 +1434,12 @@ The denominator is a Pfaffian. Put
 
 B(u)=(sin(u+b)sin(u-b))/(sin u), h(u)=(cos u)/(B(u)).
 
-Let mathsf h_Nsfh_N be the skew matrix with upper entries
-(mathsf h_N)_sfh_N)_{jk}=h(v_k-v_j), j<k; for odd N append a last
+Let h_Nsfh_N be the skew matrix with upper entries
+(h_N)_sfh_N)_{jk}=h(v_k-v_j), j<k; for odd N append a last
 column of ones and a row of minus ones with bottom-right entry 0.
 Set a_{2l}=2^l/sqrt 2, a_{2l+1}=(-2)^l, and
 
-f_N(v)=a_Nprod_{j<k}B(v_k-v_j),Pf,mathsf h_N.sfh_N.
+f_N(v)=a_Nprod_{j<k}B(v_k-v_j),Pf,h_N.sfh_N.
 
 Explicitly, f_0=1/sqrt2 and f_1=1, and for even size
 
@@ -1451,7 +1451,7 @@ The apparent singularities in~(eq:f_N) are removable
 [Homogeneous physical pairing]
 For y>0,
 
-mathcal Z_N(y)=calZ_N(y)=left.(G_N(v))/(2f_N(v)^2)|_{v_1=...=v_N=0}.
+Z_N(y)=/(2f_N(v)^2)|_{v_1=...=v_N=0}.
 
 The denominator is nonzero there, and the identity holds also as a
 Laurent-polynomial identity in y.
@@ -1467,7 +1467,7 @@ Let K(xi)_{pp}=xi^p denote the spin twist (pin{+1,0,-1})
 and T_xi(t) the single-row transfer matrix
 
 T_xi(t)=Tr_a,K_a(xi)R_{a1}(t+v_1)... R_{aN}(t+v_N),
-qquadxi=i or -i,
+ xi=i or -i,
 
 acting on total-spin-zero site states. The empty site vector is
 denoted ∅; site order is site 1 first even though operator
@@ -1481,13 +1481,13 @@ is a trigonometric polynomial of degree le N-1 in each v_j,
 with the following properties:
 
 itemParity. On a component with site spin p_j, Psi_N is
-(-1)^{N-1+p_j}-parity in v_jmapsto v_j+pi.
+(-1)^{N-1+p_j}-parity in v_j|-> v_j+pi.
 itemFusion. With remaining list U,
 
-Psi_N(u,u+2L,U)=prod_{zin U}cos(z-u-L)cdot W_{12}Psi_{N-1}(u+L,U),
+Psi_N(u,u+2L,U)=prod_{zin U}cos(z-u-L)· W_{12}Psi_{N-1}(u+L,U),
 
 
-Psi_N(u,u+b,U)=2cos bprod_{zin U}cos(z-u-L)cos(z-u-2L)cdotΩ_{12}Psi_{N-2}(U).
+Psi_N(u,u+b,U)=2cos bprod_{zin U}cos(z-u-L)cos(z-u-2L)·Ω_{12}Psi_{N-2}(U).
 
 itemInfinity. As |v_N|to i∞,
 Psi_N(v)/prod_{j<N}cos(v_N-v_j)tosqrt2Psi_{N-1}(v_1,...,v_{N-1}) ⊗ |0>_N.
@@ -1518,15 +1518,15 @@ Degree bound. For each row T_xi(t) on the vacuum line,
 Fourier-Laurent expansion gives
 [
 T_xi(t)=I+sum_k[A_kcot(t+v_k+2L)+B'_kcot(t+v_k+3L)],
-qquadsum_k(A_k+B'_k)=0.
+ sum_k(A_k+B'_k)=0.
 ]
 Applying the fusion equations to T_j=T_xi(-v_p+jL) gives commuting-operator
-annihilator rows mathcal A_calA_{p,0}=c_0-c_1+c_2+X_0c_2 and
-mathcal A_calA_{p,1}=c_1-c_2+c_3+X_1c_3. Collecting into operator
-matrix mathcal A,calA, one has
-detmathcal A=(calA=(det H_{mathcal A}calA)P^xi, where H_{mathcal A}calA
+annihilator rows A_{p,0}=c_0-c_1+c_2+X_0c_2 and
+A_{p,1}=c_1-c_2+c_3+X_1c_3. Collecting into operator
+matrix A, one has
+detA=( H_{A}, where H_{A}
 is the cotangent kernel r(u)=cot(u+3L)-cot(u+4L)+cot(u+5L).
-The scalar Pfaffian J_N=Pf(H_{mathcal A}calA)prod_{j<k}p(v_k-v_j),
+The scalar Pfaffian J_N=Pf(H_{A}{j<k}p(v_k-v_j),
 
 p(u)=frac{sin(u+2L)sin(u-2L)cos u [sin(u+b)sin(u-b)]^2}
 {sin^2u,sin(u+L)sin(u-L)},
@@ -1569,8 +1569,8 @@ divergence at v_2-v_1in{b} in the double fusion~(eq:psi-fusion-b):
 the wheel factor there forces d_N^ige N-1 and d_N^{-i}ge 2(N-1).
 Matching (eq:f_N) gives the exact scalar fusion identities
 
-Psi_N(u,u+2L,U) &= c_1prod_{zin U}cos(z-u-L)cdot W_{12}Psi_{N-1}(u+L,U),
-Psi_N(u,u+b,U) &= c_2prod_{zin U}cos(z-u-L)cos(z-u-2L)cdotΩ_{12}Psi_{N-2}(U),
+Psi_N(u,u+2L,U) &= c_1prod_{zin U}cos(z-u-L)· W_{12}Psi_{N-1}(u+L,U),
+Psi_N(u,u+b,U) &= c_2prod_{zin U}cos(z-u-L)cos(z-u-2L)·Ω_{12}Psi_{N-2}(U),
 
 with c_1=1 (after normalization) and c_2=2cos b
 (cross-checked at N=2,3 and by induction).
@@ -1598,7 +1598,7 @@ degree after dividing by the fusion-denominator product and vanishes
 at each step. The base cases N=0,1 match.
 
 Evaluating at v_1=...=v_N=0 with the sandwiched K_j(delta)'s:
- mathcal Z_NcalZ_N at homogeneous v has a natural expansion as an
+ Z_N at homogeneous v has a natural expansion as an
 oriented sum over polygon systems; the contractible and essential
 sectors get phases i and pm i (reversing conjugates), and the
 `left-side switch' factor (-1)^{1_{Ain D'}} plus
@@ -1610,7 +1610,7 @@ weights.
  The empty system gives 1, matching (∅,∅)
 entry.
 
-For positive y, exhaustion proves mathcal Z_N=G_N/(2f_N^2)|_calZ_N=G_N/(2f_N^2)|_{v=0},
+For positive y, exhaustion proves Z_N=G_N/(2f_N^2)|_calZ_N=G_N/(2f_N^2)|_{v=0},
 and the equality extends to a Laurent polynomial identity.
 
 
@@ -1634,7 +1634,7 @@ 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),
 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_{mathbb R}bbRV(s)rho(s),ds.
+I=int_{R},ds.
 
 [Homogeneous gas upper bound]
 As mto∞, locally uniformly in |Im r|<.54pi:
@@ -1757,7 +1757,7 @@ P_{m,T}(e^r) equals
 
 sum_{{nu_{g,i}:,(eq:winding-neutrality)}}
 frac{1}{prod_{g,i}nu_{g,i}!}
-,e^{ccdot 2pi iTr,sigma^tnu/(4h_*)-S_{rm bulk}(T)-S_{rm gas}[nu;r]},
+,e^{c· 2pi iTr,sigma^tnu/(4h_*)-S_{rm bulk}(T)-S_{rm gas}[nu;r]},
 
 with:
  S_{rm bulk}(T)=(1)/(2T)sum_{pne0}widehatwidehat J^tK(p)J^tK(p)widehatwidehat JJ,widehatrho(p)^2
@@ -1807,7 +1807,7 @@ log P_m(i)=Im^2+frac5{24}log m+o(log m).
 
 Equivalently, the nesting ratio satisfies
 
-mathcal Z_calZ_{m,m}(1)=(P_m(1))/(P_m(i))=m^{1/6+o(1)}.
+Z_{m,m}(1)=(P_m(1))/(P_m(i))=m^{1/6+o(1)}.
 
 
 
@@ -1889,7 +1889,7 @@ log P_m(1)=Im^2+frac38log m+o(log m).
 
 Combining (eq:P-m-i-final) and (eq:saturation-at-1):
 
-mathcal Z_calZ_{m,m}(1)=P_m(1)/P_m(i)=m^{1/6+o(1)},
+Z_{m,m}(1)=P_m(1)/P_m(i)=m^{1/6+o(1)},
 
 as claimed.
 
@@ -1952,7 +1952,7 @@ using the flux rule with an artificial arc below the base line. Combined
 with the diameter bound over a window of length asymp l: K_llesssim l^{-5/4}.
 
 Deep exits. For an exit from T(a,n_0,t) through the side with
-outward normal n_1, at depth q=n_0cdot(p_1-a),
+outward normal n_1, at depth q=n_0·(p_1-a),
 
 mathop{average}_{tin[alpha R,AR]}sum_{qgedelta R}k(t,q)
 ge c_{alpha,A,delta}R^{-1/4},
@@ -2022,25 +2022,25 @@ confined arches and averaging over their shared base level gives
 Positivity of the negative polymer polynomial.: 
 
 Let D be the finite cylinder between horizontal row-boundary caps,
-with height H=h d_0 and quotient by N steps. Let mathcal PcalP be
+with height H=h d_0 and quotient by N steps. Let P be
 its set of simple polygons. For 0lebeta'le3/8, put
 alpha=pibeta'/3, x=(2cosalpha)^{-1}, and
 
-omega(L)=x^{|L|}cdot
+omega(L)=x^{|L|}·
 -2cos(4alpha),&L contractible,
 -2cos(2alpha),&L essential.
 
 
-For mathcal A⊆calA⊆mathcal P:calP:
+For A⊆calA⊆P:calP:
 
-Q(mathcal A)=calA)=sum_{substack{mathcal F⊆calF⊆mathcal AcalA
-polygons in mathcal FcalF disjoint}}
-prod_{Linmathcal F}calFomega(L).
+Q(A)={substack{F⊆calF⊆A
+polygons in F disjoint}}
+prod_{LinF}
 
 
 [Positivity]
-For every Nge1, hge1, 0lebeta'le3/8, mathcal A⊆calA⊆mathcal P:calP:
-Q(mathcal A)>0.calA)>0.
+For every Nge1, hge1, 0lebeta'le3/8, A⊆calA⊆P:calP:
+Q(A)>0.calA)>0.
 
 
 Proof. [Sketch]
@@ -2050,12 +2050,12 @@ interior lattice points) form a connected overlap family meeting every
 triangle centre, so the incompatibility graph is connected. The signed
 exit identity gives
 
-Q(mathcal P)=calP)=sum_{gamma:pto∂ D}x^{|gamma|}
+Q(P)={gamma:pto∂ D}x^{|gamma|}
 e^{ibeta'W(gamma)}Q({L:Lcapgamma=∅}).
 
 Continue from beta'=0, x=1/2 where defect terms are nonnegative,
-to beta'=3/8. A minimal-zero subset mathcal AcalA must be all of
-mathcal PcalP by connectedness; but a fixed bridge through every level
+to beta'=3/8. A minimal-zero subset A must be all of
+P by connectedness; but a fixed bridge through every level
 gives a positive residual polynomial, contradiction.
 
 
@@ -2070,7 +2070,7 @@ H_H(gamma;t)geexpBig(t!sum_{ell essential, ellcapgammane∅}!kappa^{|ell|}Big).
 Combining (eq:cylinder-deletion-ratio) with
 (eq:slab-essential-mass) and (eq:strip-upper) gives
 
--log Z_H(sqrt2)le CH/Nqquad(Hge N),
+-log Z_H(sqrt2)le CH/N (Hge N),
 
 and hence
 
@@ -2119,7 +2119,7 @@ cost by (eq:cylinder-deletion-ratio). Contractible separators
 lift to plane polygons around the marked point; the barrier bounds
 (eq:cylinder-barriers) discard those with excessive vertical or
 horizontal span. Systems inject. Hence
-mathcal ZcalZlesssim Z(CN^{1+2xi})^2 for any xi>0, and combining
+Z Z(CN^{1+2xi})^2 for any xi>0, and combining
 with (eq:nesting-ratio) and taking xi↓0 gives
 (eq:Z-R).
 
@@ -2178,7 +2178,7 @@ the deleted set K, for 0le tlesqrt2.
 Proof. 
 For a single deleted polygon v:
 
-Q_P/Q_{P∖ v}=1-tkappa^{|v|}cdot Q_{P∖ N[v]}/Q_{P∖ v},
+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
@@ -2216,7 +2216,7 @@ 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=-(mathcal Y(y)/Q_D-1)/calY(y)/Q_D-1)/sqrt2.
+Prop.~prop:occupancy: J=-(Y(y)/Q_D-1)/calY(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
@@ -2301,15 +2301,15 @@ Classwise convergence of the normalization.:
 
 [Classwise convergence]
 For rin{0,ipi/2} put sigma(r)=3/8+2r^2/(3pi^2) and
-mathcal A_m(r)=e^calA_m(r)=e^{-Im^2}m^{-sigma(r)}P_m(e^r). There exist a fixed
+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
 
-mathcal A_m(r)=calA_m(r)=mathfrak a_fraka_{mbmod Q}(r)+O(m^{-delta}).
+A_m(r)= a_fraka_{mbmod Q}(r)+O(m^{-delta}).
 
 Consequently
 
-mathcal Z_calZ_{m,m}(1)=P_m(1)/P_m(i)asymp m^{1/6}.
+Z_{m,m}(1)=P_m(1)/P_m(i)asymp m^{1/6}.
 
 
 
@@ -2318,12 +2318,12 @@ Work with the periodic grand sum at period
 T=2a_0^{-1}log m+g+itheta, |theta|lepi/2, for a suitably large
 fixed g. The stability estimate (eq:Z-coercive) and
 localization (eq:gas-localization) continue to hold with the
-coercivity error O(1/g)mathcal QcalQ absorbed by g large; the period
+coercivity error O(1/g)Q absorbed by g large; the period
 graph is fixed-mod-g independent of m on half intervals.
 
 Multiplying the lifted polynomial by
 e^{-2Am^2e^{-2T}}e^{-Im^2}m^{-sigma(r)} produces an
-F_m(theta,r) whose circle mean is exactly mathcal A_m(r).calA_m(r). We show
+F_m(theta,r) whose circle mean is exactly A_m(r). We show
 F_m is Cauchy at exponential speed along each class modulo Q.
 
 Scalar factor. The quadratic bulk contribution gives
@@ -2382,24 +2382,24 @@ 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},mathcal J_calJ_{rm ext}!!
+Y=Q_{lh},J_{rm ext}!!
 sum_{substack{a_jin{-1,0,1}
 sum_Aa_j-sum_Ba_j=h-l}}
-(i/delta)^{sum_Aa_j},mathcal W(a)calW(a),mathcal M(a),calM(a),
+(i/delta)^{sum_Aa_j},W(a),M(a),
 
 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 mathcal J_calJ_{rm ext} is a
+ J(u)=cos(u+L)cos(u+2L), and J_{rm ext} is a
 product of J-factors between extra slots in each list.
- mathcal W(a)=calW(a)=prod_jc_{a_j}prod_{j<j'}P_{a_ja_{j'}}(v_{j'}-v_j)
+ 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
 Section~c6:periodic-polynomial.
  M^1_+(u)=(1,cos(u+L),J(u)), M^1_0=1 extracted by
 convention, and M^2_a(u)=M^1_{-a}(-u).
- mathcal M(a)=calM(a)=prod_{jin Acup B}prod_{einextras}
+ M(a)={jin Acup B}prod_{einextras}
 M^{d(j,e)}_{a_j}(x_e-v_j) with d(j,e)in{1,2} determined by list
 and spin type.
 
@@ -2576,7 +2576,7 @@ test) are
 
 2pi i,sigma_i!!!sum_{e:,g_e=g_i}!!!
 int_0^{s+i(v_e+delta_i L)}!!!
-sum_{nuinmathbb Z}bbZrho(z+nu T),dz,
+sum_{nuinZ}+nu T),dz,
 sigma=(0,1,1,-1,0).
 
 
@@ -2723,14 +2723,14 @@ energies.
 
 [Homogeneous interval operators]
 On the ambient cell space with the Fock fibers above, let
-mathcal T_calT_pm be homogeneous cell operators with all five color-A
+T_ be homogeneous cell operators with all five color-A
 species and only S,F of color B, other species having zero
 outgoing point weight, no magnetic functions or potentials
 (eq:legacy-P-25), constant tilt a_0'=pm6Li, r=-2a_0', and
 weights consisting of the pair term G, the phases
 (eq:legacy-P-27), and the exponent (eq:legacy-P-29).
 Then both are Hilbert--Schmidt and
-spr(mathcal T_calT_pm)<exp(S_0/2).
+spr(T_<exp(S_0/2).
 
 Proof. 
 The Hilbert--Schmidt assertion follows from the Fock trace-norm bound.
@@ -2744,7 +2744,7 @@ recovers precisely (eq:legacy-P-28) at positive separations of at
 least two cells and nothing else.
 
 Spectral radius via powers.  For a compact Hilbert--Schmidt
-operator, spr(mathcal T)=limsup_mcalT)=limsup_m|mathcal T^mcalT^m|^{1/m};
+operator, spr(T)=limsup_m|T^m|^{1/m};
 splitting off finite-dimensional outer eigenvalues leaves a strictly
 smaller radius on a complementary invariant space whose power traces
 are trace-class.  It suffices to bound the homogeneous log-trace per
@@ -2819,7 +2819,7 @@ either sign of the tilt.
 
 For later use, the spectral-radius formula gives a fixed block length
 q and ε>0 with
-|mathcal T_calT_pm^q|leexp((1/2-ε)qS_0) for both tilts.
+|T_|leexp((1/2-ε)qS_0) for both tilts.
 Continuity of a finite ordered product in operator norm then preserves
 exp(qS_0/2) for sufficiently close q-blocks in the same state
 space and gauge.  Section~u:core-product proves this closeness
@@ -3085,7 +3085,7 @@ tfrac12sum_{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_{mathbb R}bbR!!
+I_{gh}(v)=2q_gq_hlog2+int_{R}!!
 [K_{gh}(p)widehatrho(p)^2cosh(vp)-4q_gq_h/p^2],(dp)/(2pi).
 ]
 The subtracted pole all-mode sum is q_gq_hT/3.  Contour shifting
@@ -3139,7 +3139,7 @@ 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
 [
-2sum_{nge1}mathfrak d(2frakd(2pi n/T)=sum_{ninmathbb Z}bbZ
+2sum_{nge1}mathfrak d(2frakd(2pi n/T)=sum_{ninZ}
 mathfrak d(2frakd(2pi n/T)-mathfrak d(0).frakd(0).
 ]
 Since mathfrak dfrakd is even, regular at zero, and exponentially
@@ -3335,7 +3335,7 @@ just specified, the average overlineoverline EE over the level and support
 randomness satisfies, with constants uniform in the event and the
 admissible levels:
  If it requires the use of two fixed internal links, then
-overlineoverline EEle Cl^{3/4}mathcal K,calK, where mathcal KcalK is the
+overlineoverline EEle Cl^{3/4}K, where K is the
 single-polygon mass using those links on a cylinder of period vector
 in any lattice direction chosen for the marks and length large
 compared with R.
@@ -3420,7 +3420,7 @@ 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
 prescribed corridors at those levels, and put
-Y(eta)=mathbb E_bbE_{mathbf u}bfumathbf1_bf1_{mathbf u}bfu(eta) and
+Y(eta)=E_{mathbf u}bfumathbf1_bf1_{mathbf u}bfu(eta) and
 P=prod_{j=1}^ks_j^{-1/4}.  Concatenating the confined crossings
 gives
 [
@@ -3470,7 +3470,7 @@ exactly two boundary crossings.  Let Ω be the six
 phase--slope pairs with the product law and mu critical weight on
 target polygons through the prescribed links (case 1) or target
 bridges satisfying the common deterministic event (case 2), with
-total mass lemathcal KcalK or B^*.  In a level window I of size
+total mass leK or B^*.  In a level window I of size
 asympε l, let b_h(Gamma,Ω) sum the recovered
 ordered decompositions of Gamma at ∂ D_h of the
 chord-event indicator times the exterior trimming factor, so
@@ -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 ⊗ mathbb P_Ω)bbP_Ω)asymp Foverlineoverline E,E,
+int X,d(mu ⊗ P_Ω)bbP_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 ⊗ mathbb P_Ω)bbP_Ω)le CFoverlineoverline EE,Xi.
+int X^2,d(mu ⊗ P_Ω)bbP_Ω)le CFoverlineoverline EE,Xi.
 
 
 
@@ -3558,7 +3558,7 @@ C(1+min(g,t))^{-1/2}min{1,t/(1+g)}le C(1+g)^{-1/2}.
 Conditional on the gap, Xile Cl^{-1/2} on each window.  Applying
 Cauchy--Schwarz to (eq:sewing-first-moment) and
 (eq:sewing-second-moment) yields
-overlineoverline EEle Cmathcal KcalK,Xi/F and
+overlineoverline EEle CK,Xi/F and
 B^*ge cFoverlineoverline E/E/Xi, proving both assertions.  Averaging over
 windows recombines them.
 
@@ -3583,7 +3583,7 @@ for any 0<eta<1/2; bounded-gap atoms satisfy the same after
 increasing C_eta.  Keeping this gain in the first sewing estimate
 gives
 
-overlineoverline EEle C_eta l^{3/4}(t/l)^etamathcal K.calK.
+overlineoverline EEle C_eta l^{3/4}(t/l)^etaK.
 
 
 Indeed, the first moment on the target polygon measure is at least
@@ -3623,12 +3623,12 @@ d=P_1+P_2 is even.  Choose an integer period multiple kP of length
 comparable to a large constant times R so the sewn polygon projects
 injectively; then Nasymp R and
 [
-mathcal KcalKle CR^{-4/3}(R/d)^{15/8}.
+K CR^{-4/3}(R/d)^{15/8}.
 ]
 With O(R^2) choices of the first centre and O(d) displacements with
 P_1+P_2=d,
 
-sum_{eligible pairs}mathcal KcalK
+sum_{eligible pairs}K
 le CR^2R^{-4/3}R^{15/8}sum_{1le dle CR}d^{1-15/8}
 le CR^{8/3}.
 
@@ -3707,7 +3707,7 @@ so the square of the strip partition is at most that quantity.
 
 Summing over the O(h) link representatives,
 [
-sum_{gammainmathcal B_h}calB_h(ell(gamma)-1)w(gamma)le Ch^{13/12}.
+sum_{gammainB_h}-1)w(gamma)le Ch^{13/12}.
 ]
 The one missing centre per bridge adds only B_hle Ch^{-1/4},
 completing (eq:bridge-first-moment).  The strip mass
@@ -3764,7 +3764,7 @@ Journal of Statistical Mechanics: Theory and Experiment (2017), 053102.
 
 
 A.~Glazman and I.~Manolescu.
-Self-avoiding walk on mathbb Z^2bbZ^2 with Yang--Baxter weights:
+Self-avoiding walk on Z^2 with Yang--Baxter weights:
 universality of critical fugacity and 2-point function.
 Annales de l'Institut Henri Poincar'e, Probabilit'es et Statistiques
 56 (2020), 2281--2300.