diff --git a/low_temp_sk_glauber_typical_v2.txt b/low_temp_sk_glauber_typical_v2.txt
index ccef313..dc4bb14 100644
--- a/low_temp_sk_glauber_typical_v2.txt
+++ b/low_temp_sk_glauber_typical_v2.txt
@@ -73,7 +73,7 @@ Two conventions are essential and we fix them now.
 [label=(roman*)]
  The start is random and then held fixed. We first draw the
   disorder J, then one configuration sigma_0simpi^J_{n,beta},
-  and only then look at the law P_t^{J,beta}(sigma_0,cdot) of the chain
+  and only then look at the law P_t^{J,beta}(sigma_0,·) of the chain
   started at that single sigma_0. The total-variation distance in the
   mixing time is between this fixed-start law and pi^J_{n,beta}. If one
   instead averaged sigma_0 into the starting distribution, stationarity
@@ -102,24 +102,24 @@ field. The local field at site i is
 When the clock at site i rings, the new value equals sin{-1,1}
 with probability
 
- mathbb PbbP{new spin=smidsigma}
+ P{new spin=smidsigma}
  =frac{exp(beta s,h_i(sigma))}
         {2cosh(beta h_i(sigma))} .
 
 The update may leave sigma_i unchanged. Let
-P_t^{J,beta}(sigma,cdot) denote the law at time t of this
+P_t^{J,beta}(sigma,·) denote the law at time t of this
 continuous-time chain started from the fixed configuration sigma, and
 let
 
  T^J_{n,beta}(sigma)
  =infBigl{tge0:
- bigl|P_t^{J,beta}(sigma,cdot)-pi^J_{n,beta}bigr|_{TV}
+ bigl|P_t^{J,beta}(sigma,·)-pi^J_{n,beta}bigr|_{TV}
  letfrac14Bigr}
 
 be the mixing time from that start. The joint law of interest samples
 the disorder and one Gibbs configuration,
-mathbb P_bbP_{J,,sigma_0simpi^J_{n,beta}}
-=mathbb E_JbbE_J,pi^J_{n,beta}.
+P_{J,,sigma_0simpi^J_{n,beta}}
+=E_J,pi^J_{n,beta}.
 
 ## Main results
 
@@ -127,7 +127,7 @@ mathbb P_bbP_{J,,sigma_0simpi^J_{n,beta}}
 For every fixed beta>1 there is a constant c_betain(0,1) such
 that
 [
- mathbb P_bbP_{J,,sigma_0simpi^J_{n,beta}}Bigl{
+ P_{J,,sigma_0simpi^J_{n,beta}}Bigl{
  T^J_{n,beta}(sigma_0)leexp(n^{1-c_beta})Bigr}
  → 1
   (nto∞).
@@ -150,7 +150,7 @@ produces a two-sided statement.
 For every fixed beta>1, in the model and continuous-time convention
 above,
 [
- mathbb P_bbP_{J,,sigma_0simpi^J_{n,beta}}Bigl{
+ P_{J,,sigma_0simpi^J_{n,beta}}Bigl{
  exp(n^{1/10000})<T^J_{n,beta}(sigma_0)
  leexp(n^{1-1/40000000})Bigr}
  → 1 .
@@ -160,7 +160,7 @@ above,
 Proof. 
 The continuous-time form of cite[Theorem~1.1]{OpenAISKBarriers2026}
 states that, with joint probability tending to one,
-|P^{J,beta}_{exp(n^{1/10000})}(sigma_0,cdot)-pi^J_{n,beta}|_{TV}>1/4
+|P^{J,beta}_{exp(n^{1/10000})}(sigma_0,·)-pi^J_{n,beta}|_{TV}>1/4
 for sigma_0simpi^J_{n,beta}. By (eq:T) and continuity of the
 finite-state semigroup this forces the strict lower bound
 T^J_{n,beta}(sigma_0)>exp(n^{1/10000}). The upper bound is
@@ -261,14 +261,14 @@ for finite Markov chains cite[Lemmas~2.5,~2.7]{DiaconisSaloffCoste1996}.
 %======================================================================
 
 Fix a finite beta>1. Constants may depend on this fixed parameter;
-when that dependence matters we write C_beta or O_beta(cdot). All
+when that dependence matters we write C_beta or O_beta(·). All
 logarithms are natural.
 
 Distances and inner products.:  For probability measures
 mullnu on a common space, set
 {rm KL}(muVertnu)=intlog(dmu/dnu),dmu. On the spin cube,
 total variation is one half of the ell^1 distance. For vectors
-v,winmathbb R^nbbR^n write
+v,winR^n write
 [
  < v,w>_n=frac1nsum_i v_i w_i,
   |v|_n^2=< v,v>_n .
@@ -284,18 +284,18 @@ diagonal entries are independent N(0,2/n). Write G for the vector of
 independent standard Gaussian drivers of all these entries and
 gamma for its law. Since x_i^2=1 for every spin,
 [
- tfrac12,x^{mathsf T}sfTdiag(W),x
+ tfrac12,x^{T},x
  =tfrac12sum_i W_{ii}
 ]
 is the same random constant for every configuration. Hence the diagonal
 contributes the same additive constant to every Gibbs weight; it changes
 neither pi^J_{n,beta} nor any heat-bath rate nor the transition
-kernel. Its effect on mathbb EbbE F(G) is a deterministic constant that
-we absorb into the pressure functional mathcal PcalP below.
+kernel. Its effect on E F(G) is a deterministic constant that
+we absorb into the pressure functional P below.
 
 Gibbs weights on the completed matrix.:  Define, with W=W(G),
 [
- widehatwidehat H_G(x)=H_G(x)=tfrac12 x^{mathsf T}sfTW(G),x,
+ widehatwidehat H_G(x)=H_G(x)=tfrac12 x^{T},x,
  
  F(G)=logsum_{xin{-1,1}^n}e^{betawidehatwidehat H_G(x)}H_G(x),
  
@@ -333,8 +333,8 @@ following external inputs.
 
 [leftmargin=*]
  Pressure. Guerra's finite-volume interpolation bound
-  cite[Theorem~3]{Guerra2003}: mathbb EbbE F(G)le nmathcal P,calP, where
-  mathcal PcalP is the Parisi pressure functional, whose exact
+  cite[Theorem~3]{Guerra2003}: E F(G)le nP, where
+  P is the Parisi pressure functional, whose exact
   normalization we give below.
  Regularity and stability. Auffinger--Chen
   cite[Propositions~1--2]{AC2015}: the scalar Parisi equation has
@@ -375,11 +375,11 @@ this scalar normalization agrees with our completed Hamiltonian.
 Set v_*=beta^2 q_*. The pressure functional and the finite-volume
 upper bound read
 
- mathcal P=calP=Phi(0,0)-frac1{2beta^2}int_0^{beta^2}valpha_v,dv,
+ P=,0)-frac1{2beta^2}int_0^{beta^2}valpha_v,dv,
  
- mathbb EbbE F(G)le nmathcal PcalP .
+ E F(G)le nP .
 
-The centered diagonal shifts neither mathcal PcalP nor mathbb EbbE F(G)
+The centered diagonal shifts neither P nor E F(G)
 in any way that matters below. Since alpha_v=1 above v_*, the
 Gaussian recursion gives, at v_*,
 
@@ -414,7 +414,7 @@ a_v=a(v,Y_v). The stationarity identity, the differentiated PDE, and
 It^o's formula give the key scalar facts
 
  m_0=0, dm_v=a_v,dZ_v,
- mathbb EbbE m_v^2=(v)/(beta^2)(0le vle v_*),
+ E m_v^2=(v)/(beta^2)(0le vle v_*),
 
 and this martingale equation continues past v_* by direct It^o
 calculus with the explicit functions (eq:phi-extension). We use,
@@ -427,16 +427,16 @@ Why (eq:a-m-bound) holds. We derive it in the form that
 behaves well under step approximation. One Parisi step of CDF height
 lambdain[0,1] and variance delta>0 maps a profile f to
 [
- (Tf)(y)=frac1lambdalogmathbb EbbE,
+ (Tf)(y)=frac1lambdalogE,
  e^{lambda f(y+sqrtdelta U)}, Usim N(0,1),
 ]
-with Tf(y)=mathbb EbbE f(y+sqrtdelta U) when lambda=0. Write
-mathbb E_bbE_lambda for the Gaussian expectation tilted by
+with Tf(y)=E f(y+sqrtdelta U) when lambda=0. Write
+E_ for the Gaussian expectation tilted by
 e^{lambda f(y+sqrtdelta U)} and renormalized. Differentiating gives
 the exact formulas
 [
- (Tf)'=mathbb E_bbE_lambda f',
- (Tf)''=mathbb E_bbE_lambda f''+lambdaVar_lambda(f') .
+ (Tf)'=E_ f',
+ (Tf)''=E_ f''+lambdaVar_lambda(f') .
 ]
 Suppose 0<f''le1-(f')^2. Then
 [
@@ -455,7 +455,7 @@ tanh directly.
 
 Define the three deterministic functions
 
- p(v)=mathbb EbbE a_v, r(v)=mathbb EbbE a_v^2,
+ p(v)=E a_v, r(v)=E a_v^2,
   rho(v)=beta√(r(v)) .
 
 The martingale second-moment identity (eq:martingale) and
@@ -484,10 +484,10 @@ cosh y is symmetric and nonincreasing in |y|. It suffices to check
 this for step CDFs and pass to the limit; stability
 cite[Propositions~1--2]{AC2015} justifies the passage. For a step CDF
 with masses alpha^{(j)} on intervals [v_j,v_{j+1}], and
-varphi_t the centered Gaussian density of variance t, the transition
+phi_t the centered Gaussian density of variance t, the transition
 density on one interval is
 [
- K_j(z,y)=varphi_{v_{j+1}-v_j}(y-z)
+ K_j(z,y)=phi_{v_{j+1}-v_j}(y-z)
  expbigl{alpha^{(j)}[Phi(v_{j+1},y)-Phi(v_j,z)]bigr} .
 ]
 The Gaussian recursion normalizes K_j; its Doob transform has drift
@@ -511,16 +511,16 @@ decreasing profile can only increase the expectation of
 sech^2y, since two nonincreasing functions of |y| have
 nonnegative covariance. Integration by parts gives
 [
- frac{int_{mathbb R}bbRsech^{k+2}y,dy}
-      {int_{mathbb R}bbRsech^{k}y,dy}
+ frac{int_{R}{k+2}y,dy}
+      {int_{R}{k}y,dy}
  =(k)/(k+1)(k>0).
 ]
-Consequently r/pge1/2 and mathbb EbbE a_v^3/r(v)ge3/4.
+Consequently r/pge1/2 and E a_v^3/r(v)ge3/4.
 
 Differential identities. On the extension a_v=1-m_v^2, and
 direct It^o calculations yield
 [
- p'=-r, r'=4r-6,mathbb EbbE a_v^3 .
+ p'=-r, r'=4r-6,E a_v^3 .
 ]
 The two moment ratios imply p'le-p/2 and r'le0. Since
 0<a_vle1, the second identity also gives r'ge-2r. Integrating
@@ -530,9 +530,9 @@ r(v_*)=1/beta^2 and monotonicity of r imply rhole1.
 
 ## The entropy--energy identity
 
-Let mathcal H(z)calH(z) be the entropy of a sign with mean zin(-1,1):
+Let H(z) be the entropy of a sign with mean zin(-1,1):
 [
- mathcal H(z)=-(1+z)/(2)calH(z)=-(1+z)/(2)log(1+z)/(2)
+ H(z)=-(1+z)/(2)calH(z)=-(1+z)/(2)log(1+z)/(2)
               -(1-z)/(2)log(1-z)/(2),
 ]
 with continuous values at z=pm1. For V>v_* define the two scalars
@@ -542,37 +542,37 @@ matrix proposal:
  e(V)=int_0^V√(r(v)),p(v),rho(v),dv,
  
  j(V)=frac12int_0^Vrho(v)^2,
-        mathbb E(a_v^2)/(1-m_v^2)bbE(a_v^2)/(1-m_v^2),dv .
+        E(a_v^2)/(1-m_v^2)bbE(a_v^2)/(1-m_v^2),dv .
 
 The factor rho(v)^2 inside j accounts for the independent noise we
 introduce in the proposal. Below the support endpoint, It^o's formula
 together with the PDE and stationarity gives
 [
- mathbb E[Y_bbE[Y_{v_*}m_{v_*}]=int_0^{v_*}(p(v)+alpha_v v/beta^2),dv,
+ E[Y_bbE[Y_{v_*}m_{v_*}]=int_0^{v_*}(p(v)+alpha_v v/beta^2),dv,
  
- mathbb EbbEPhi(v_*,Y_{v_*})=Phi(0,0)
+ E*,Y_{v_*})=Phi(0,0)
    +frac12int_0^{v_*}alpha_v v/beta^2,dv .
 ]
 These also follow by step approximation with bounded first and second
 spatial derivatives. Substituting the endpoint value (eq:phi-at-vstar)
-and using mathcal H(tanhcalH(tanh y)=log(2cosh y)-ytanh y gives
+and using H(tanh y)=log(2cosh y)-ytanh y gives
 
- int_0^{v_*}p(v),dv+mathbb EbbE,mathcal H(m_calH(m_{v_*})
- =mathcal P-(calP-(beta^2)/(4)(1-q_*)^2 .
+ int_0^{v_*}p(v),dv+E,H(m_{v_*})
+ =P-(calP-(beta^2)/(4)(1-q_*)^2 .
 
 
 [Entropy--energy identity]
 For every V>v_*,
 
- log2-j(V)+beta e(V)=mathcal P-(calP-(beta^2)/(4),p(V)^2 .
+ log2-j(V)+beta e(V)=P-(calP-(beta^2)/(4),p(V)^2 .
 
 
 
 Proof. 
 The same martingale calculation applied to binary entropy yields
 [
- frac12int_0^{v_*}mathbb E(a_v^2)/(1-m_v^2)bbE(a_v^2)/(1-m_v^2),dv
- =log2-mathbb EbbE,mathcal H(m_calH(m_{v_*}).
+ frac12int_0^{v_*}E(a_v^2)/(1-m_v^2)bbE(a_v^2)/(1-m_v^2),dv
+ =log2-E,H(m_{v_*}).
 ]
 Localization justifies It^o's formula near z=pm1; the integrand is
 bounded because ale1-m^2. Since rho=1 and betasqrt r=1 on
@@ -754,10 +754,10 @@ expectation, with exceptional probability at most exp(-n^{1-8l}).
 
 The population identities are
 
- mathbb E[T_jT_k]_bbE[T_jT_k]_{rm row}=mathbf1_bf1_{j=k},
- mathbb E[T_jg_k]_bbE[T_jg_k]_{rm row}=0 (jle k),
- mathbb E[T_kx]_bbE[T_kx]_{rm row}=s_k,
- mathbb E[g_kx]_bbE[g_kx]_{rm row}=b_k,
+ E[T_jT_k]_bbE[T_jT_k]_{rm row}=mathbf1_bf1_{j=k},
+ E[T_jg_k]_bbE[T_jg_k]_{rm row}=0 (jle k),
+ E[T_kx]_bbE[T_kx]_{rm row}=s_k,
+ E[g_kx]_bbE[g_kx]_{rm row}=b_k,
 
 where
 b_k=h^{-1/2}int_{t_k}^{t_{k+1}}p(v)rho(v),dv. These follow from the
@@ -775,10 +775,10 @@ sets, so retaining that scale is contractive; summing over scales proves
 the bound.
 
 Let A be a Gram matrix with |A-I|leeta<1/2, and let
-R^{mathsf T}sfTR=A be its upper Cholesky factor. Along
+R^{T} be its upper Cholesky factor. Along
 A(t)=I+t(A-I), differentiation shows that dot RR^{-1} is the upper
 triangular part, with half the diagonal, of
-R^{-mathsf T}sfTdot AR^{-1}. The norms of R and R^{-1} are uniformly
+R^{-T} AR^{-1}. The norms of R and R^{-1} are uniformly
 bounded; the preceding triangular estimate plus integration give
 [
  |R-I|le Cetalog(2m).
@@ -798,9 +798,9 @@ writing c_k=< u_k,x>_n, the empirical event gives
  
 ]
 The middle estimate uses that left multiplication of the population cross
-matrix by the lower-triangular R^{-mathsf T}sfT preserves its zero upper
+matrix by the lower-triangular R^{-T} preserves its zero upper
 triangle, including the diagonal. Also |c|le1, because the query
-directions are orthonormal in |cdot|_n.
+directions are orthonormal in |·|_n.
 
 Energy of the final spin.:  Let H_0=Pi_{>K-1}. Revealing the
 columns in order decomposes the quadratic form exactly as
@@ -815,10 +815,10 @@ CKexp(-cneta^2): the diagonal estimate follows from the conditional
 Gaussian law and sum c_k^2le1, while the residual term uses that the
 spin has not seen the unrevealed block. Removing the projections in the
 first term costs at most Cetalog(2K) by the triangular bound. Its
-remaining value is scdot b up to the same error. The bound on
+remaining value is s· b up to the same error. The bound on
 (log r)' in (eq:1) implies s_k/sqrt h differs from
 √(r(v)) on [t_k,t_{k+1}] by a relative O(h). The missing initial
-interval contributes O(h), and e(V) is bounded. Hence scdot b=e(V)+O_beta(h),
+interval contributes O(h), and e(V) is bounded. Hence s· b=e(V)+O_beta(h),
 and altogether
 
  widehatwidehat H_G(x)/n=e(V)+O_H_G(x)/n=e(V)+O_beta(h+etalog(2K)).
@@ -845,12 +845,12 @@ linked experiment. Summing the linking density over the uniform
 spin shows the path and disorder marginal is unchanged.
 
 For a continuous local martingale N starting at zero write
-mathcal E(N)_V=calE(N)_V=exp(N_V-< N>_V/2). The linking density
+E(N)_V=-< N>_V/2). The linking density
 factors as E_BE_D with
 [
  
- E_B&=mathcal E(calE(sum_iint_0^{cdot}xi_irho,dB_i)_V,
- E_D&=mathcal E(calE(sum_iint_0^{cdot}xi_i√(1-rho^2),dD_i)_V,
+ E_B&=E({·}xi_irho,dB_i)_V,
+ E_D&=E({·}xi_i√(1-rho^2),dD_i)_V,
  
 ]
 using the zero quadratic covariation of B and D. Define the
@@ -860,30 +860,30 @@ disorder and uniform spin. The conditional mean-one assertion behind this
 fact needs a filtration check, since the matrix-query endpoints depend on
 the past of D.
 
-Filtration check.:  Let mathcal AcalA contain all primitive
+Filtration check.:  Let A contain all primitive
 auxiliary inputs except D, and under the unlinked law set
 [
- mathcal F_v=calF_v=sigma(x,B_{[0,v]},D_{[0,v]}),
- mathcal G_v=calG_v=sigma(x,G,mathcal A,D_calA,D_{[0,v]}).
+ F_v=,B_{[0,v]},D_{[0,v]}),
+ G_v=,G,A,D_calA,D_{[0,v]}).
 ]
 Interval by interval, every query direction is determined by past D and
-by information in mathcal G_0.calG_0. Each new endpoint and its entire bridge
+by information in G_0. Each new endpoint and its entire bridge
 are known before using the current D-increment. Deterministic-coefficient
 bridge integrals are linear Gaussian integrals with the endpoint term
-fixed by that past. Thus mathcal F_v⊆calF_v⊆mathcal G_v.calG_v. One can
+fixed by that past. Thus F_v⊆calF_v⊆G_v. One can
 also see this pathwise: rho=1 before v_*, and (eq:1) gives
 -rholerho'le0 afterward, so rho has bounded variation and the
 identity intrho,dB=rho B-int B,drho, applied on successive
 intervals, uses no future D.
 
-The pair (B,D) is Brownian in mathcal F,calF, and D is Brownian in
-mathcal GcalG because its time-zero information is independent of D. The
+The pair (B,D) is Brownian in F, and D is Brownian in
+G because its time-zero information is independent of D. The
 bounded continuous D-integrand has the same stochastic integral in
 these two filtrations: identical left-endpoint approximations converge in
 L^2 by the It^o isometry. The bounded-integrand exponential-martingale
-theorem in mathcal GcalG now gives conditional mean one given
-(x,G,mathcal A),calA), proving the claimed marginal. All later Brownian
-changes of measure use mathcal F;calF; no Brownian property for B in the
+theorem in G now gives conditional mean one given
+(x,G,A), proving the claimed marginal. All later Brownian
+changes of measure use F; no Brownian property for B in the
 enlarged filtration is required.
 
 Information cost and likelihood.:  The density of the linked
@@ -902,7 +902,7 @@ across the i.i.d. coordinate streams and uniform spins; conditionally on
 the streams through time v, the linked spin has probabilities
 (1pm m(v,Y_{v,i}))/2. We obtain
 
- mathbb EbbE L=nj(V), L=nj(V)+O(neta),
+ E L=nj(V), L=nj(V)+O(neta),
 
 outside an event of probability at most exp(-n^{1-4l}) for large n.
 The centered integral is controlled by exponential Markov; the
@@ -924,7 +924,7 @@ constant at most C_betasqrt n. The Gaussian concentration inequality
 cite[Section~5.4, Theorem~5.6]{BLM2013} together with the pressure bound
 (eq:pressure) therefore implies
 [
- F(G)le nmathcal P+ncalP+neta
+ F(G)le nP+ncalP+neta
 ]
 outside an event of probability at most 2exp(-c_beta neta^2). This
 uses the ordinary Gaussian marginal of G, which is unchanged after
@@ -982,8 +982,8 @@ connecting sequence in Section~sec:connections: its exceptional-event
 estimate exp(-n^{1-10b}) is far stronger, even though its entropy bound
 (eq:5) is weaker.
 
-[Smoothing map mathcal S]calS]
-For any joint law of disorder and spin, let mathcal ScalS retain the spin
+[Smoothing map S]calS]
+For any joint law of disorder and spin, let S retain the spin
 and replace its disorder by
 
  G^circ=√(1-delta),G+sqrtdelta,barbar G,G,
@@ -991,9 +991,9 @@ and replace its disorder by
 where barbar GGsimgamma is independent of everything.
 
 
-Both mathcal SPcalSP and mathcal SQ_fcalSQ_f have standard Gaussian disorder
-marginal. To generate the conditional spin kernel nu(cdotmid G^circ)
-of mathcal SQ_fcalSQ_f given the smoothed disorder, use Gaussian regression:
+Both SP and SQ_f have standard Gaussian disorder
+marginal. To generate the conditional spin kernel nu(·mid G^circ)
+of SQ_f given the smoothed disorder, use Gaussian regression:
 given G^circ, run the l=f proposal on
 [
  √(1-delta),G^circ+sqrtdelta,G'
@@ -1008,7 +1008,7 @@ The central reverse inequality is the following.
 
 [Smoothing reverses KL]
 
- {rm KL}(mathcal SPcalSPVertmathcal SQ_f)calSQ_f)
+ {rm KL}(SP SQ_f)calSQ_f)
  ledelta^{-1},{rm KL}(Q_fVert P) .
 
 
@@ -1042,7 +1042,7 @@ product is its Jacobian determinant. The entropy formula is
  =int(U(T(z))-U(z)-logdetnabla T(z)),dmu(z).
 ]
 Strong convexity bounds the first difference below by
-nabla U(z)cdot(T(z)-z)+tfrac12|T(z)-z|^2. Integration by parts
+nabla U(z)·(T(z)-z)+tfrac12|T(z)-z|^2. Integration by parts
 replaces the linear term by div(T-id). There is
 no boundary term: on a face where the i-th coordinate is an endpoint of
 the box, the conditional quantile map fixes that endpoint, so the normal
@@ -1062,7 +1062,7 @@ entropies converge, and tightness plus lower semicontinuity pass the
 coupling bound to the original laws.
 
 Applying to P and Q_f. Apply the transport bound with
-mu=P(cdotmid x) and lambda=Q_f(cdotmid x). For any coupling of
+mu=P(·mid x) and lambda=Q_f(·mid x). For any coupling of
 two disorder values g,widetilde g, the Gaussian smoothing kernels
 satisfy
 [
@@ -1088,15 +1088,15 @@ coverage consequence used in the next section.
 [Coverage at a target disorder]
 Let G,barbar GG be independent standard Gaussian disorder vectors, set
 G^circ=√(1-delta),G+sqrtdelta,barbar G,G, and let
-nu(cdotmid G^circ) be the conditional spin kernel of
-mathcal SQ_fcalSQ_f (eq:smoothing). Then
+nu(·mid G^circ) be the conditional spin kernel of
+SQ_f (eq:smoothing). Then
 
- mathbb EbbE,{rm KL}(pi_GVertnu(cdotmid G^circ))
+ E,{rm KL}(pi_GVertnu(·mid G^circ))
  le O_beta(nsqrtdelta)+2n^{1-f/2}/delta .
 
 With probability tending to one over the target/base pair,
 
- {rm KL}(pi_GVertnu(cdotmid G^circ))le n^{1-4d} .
+ {rm KL}(pi_GVertnu(·mid G^circ))le n^{1-4d} .
 
 On the same event, simultaneously for every set
 A⊆{-1,1}^n with pi_G(A)ge n^{-d},
@@ -1107,22 +1107,22 @@ The set A may be chosen after the target and base are realized.
 
 
 Proof. 
-Let overlinepi(cdotmid G^circ) be the conditional spin law of
-mathcal SP,calSP, the conditional mixture of pi_G over the target that
+Let overlinepi(·mid G^circ) be the conditional spin law of
+SP, the conditional mixture of pi_G over the target that
 produced the base G^circ. Inserting this law into the logarithmic ratio
 gives the exact decomposition
 [
  
- mathbb EbbE,{rm KL}(pi_GVertnu(cdotmid G^circ))
- &=mathbb EbbE,{rm KL}(pi_GVert
-                 overlinepi(cdotmid G^circ))
- &+{rm KL}(mathcal SPcalSPVertmathcal SQ_f).calSQ_f).
+ E,{rm KL}(pi_GVertnu(·mid G^circ))
+ &=E,{rm KL}(pi_GVert
+                 overlinepi(·mid G^circ))
+ &+{rm KL}(SP SQ_f).calSQ_f).
  
 ]
 Both expectations are over the target/base pair (G,G^circ) of the
 statement. The conditional mixture minimizes averaged divergence among
 spin laws depending only on the base, so the first term on the right is
-at most mathbb EbbE,{rm KL}(pi_GVertpi_{G^circ}).
+at most E,{rm KL}(pi_GVertpi_{G^circ}).
 
 For any two Hamiltonians, the Gibbs divergence is bounded by twice beta
 times the supremum of their absolute energy difference. Here those
@@ -1197,7 +1197,7 @@ make this sequence impossible.
 For a sufficiently large fixed C, Gaussian quadratic-form tails on a
 1/4-net give |W(G)|le C except with probability O(e^{-cn}). On this
 event every one-spin energy difference is O(sqrt n), because flipping a
-single spin changes x^{mathsf T}sfTWx/2 by an O(|W|sqrt n) amount. The
+single spin changes x^{T}/2 by an O(|W|sqrt n) amount. The
 heat-bath rate across any one-spin edge is therefore at least
 exp(-O_beta(sqrt n)).
 
@@ -1214,10 +1214,10 @@ Set R_n=lceil n^{1-b/4}rceil and enlarge ∂ S by Hamming
 distance R_n. For two spins at Hamming distance at most R_n and with
 |W(G)|le C, the logarithmic Gibbs weight ratio is bounded by
 O_beta(n^{1-b/8}): the change in widehatwidehat H_GH_G along any path of
-R_n flips is at most Csqrt ncdot R_n/sqrt ncdot n^{1/2}=O_beta(n^{1-b/8}),
+R_n flips is at most Csqrt n· R_n/sqrt n· n^{1/2}=O_beta(n^{1-b/8}),
 and a Hamming ball of radius R_n has logarithmic cardinality
 O(n^{1-b/4}log n). The enlarged boundary
-mathcal D_R(G)calD_R(G) therefore has target Gibbs mass at most
+D_R(G) therefore has target Gibbs mass at most
 
  exp(-tfrac12 n^{1-b/20})
 
@@ -1225,13 +1225,13 @@ for n large, since 1-b/20<1-b/8 and b/20<b/4.
 
 Among cuts violating (eq:11), select the first one in a fixed ordering
 of the subsets of the cube whenever such a cut exists and |W(G)|le C.
-Let mathcal D(G)calD(G) be its enlarged boundary, and set
-mathcal D(G)=∅calD(G)=∅ if no such cut exists or the norm bound fails.
+Let D(G) be its enlarged boundary, and set
+D(G)=∅calD(G)=∅ if no such cut exists or the norm bound fails.
 This is measurable because the cube has finitely many subsets. Let
-mathcal EcalE be the event that a witness cut exists, the target-norm bound
+E be the event that a witness cut exists, the target-norm bound
 holds, and the coverage statement (eq:10) holds for the sampled
 base G^circ. The norm and coverage estimates reduce
-(eq:11) to showing Pr(mathcal E)=o(1).calE)=o(1).
+(eq:11) to showing Pr(E)=o(1).
 
 ## A sequence with independent fine endpoints
 
@@ -1239,7 +1239,7 @@ The fine horizon l=f and the coarse horizon l=b serve distinct
 purposes.
 
 [leftmargin=*]
- Fine horizon supplies coverage. On mathcal E,calE, the coverage
+ Fine horizon supplies coverage. On E, the coverage
   statement (eq:cov) gives each side of the cut fine-proposal
   probability at least exp(-n^{1-2d}). Two conditionally independent
   fine endpoints therefore cross the cut with probability at least
@@ -1284,7 +1284,7 @@ and draw a fine spin. Coarse spin draws are not used in any
 continuation. All final spin draws may use fresh independent randomness.
 
 Conditional structure.:  The two fine endpoints are independent
-conditional on (G,G^circ) and each has law nu(cdotmid G^circ).
+conditional on (G,G^circ) and each has law nu(·mid G^circ).
 For each individual grid point, averaging over the disorder and
 auxiliary inputs on the right side of (eq:rotation) gives the
 appropriate proposal law; its primitive auxiliary collection is
@@ -1296,7 +1296,7 @@ continuity along the grid.
 We now prove that, outside an event of probability at most
 exp(-n^{1-d/2}), the following two properties both hold:
 [label=(alph*)]
- every coarse spin avoids mathcal D(G);calD(G);
+ every coarse spin avoids D(G);
  the sequence consisting of the first fine spin, all coarse spins in
   grid order, and the last fine spin has consecutive Hamming distances at
   most R_n.
@@ -1326,7 +1326,7 @@ The primitive auxiliary inputs and the final spin draw for this run are
 independent of all the matrix drivers. Thus, after fixing
 G(theta)=g and adjoining (G,G^circ), this remains the conditional
 good-output kernel given the past. In particular, the inequality applies
-to the random set mathcal D(G)calD(G) once those matrices have been fixed.
+to the random set D(G) once those matrices have been fixed.
 The separate energy-change event is used only to compare Gibbs weights;
 it is not built into the proposal good event.
 
@@ -1379,7 +1379,7 @@ angles at most a polynomial factor times |theta-theta'|.
 
 Interval induction.:  At a Gram--Schmidt step, the difference of
 the two projection operators is bounded by twice the sum of preceding
-direction differences in |cdot|_n. Normalizing residuals of norm at
+direction differences in |·|_n. Normalizing residuals of norm at
 least 1/2 costs only a fixed factor. Using 1/s_k, the new direction
 difference is controlled by earlier magnetization and direction
 differences at polynomial cost. The same is true of the new g_k, by its
@@ -1402,14 +1402,14 @@ magnetization difference negligible relative to n^{-b/3}.
 For either a coarse spin or a fine endpoint spin compared with its own
 coarse magnetization, the normalized squared distance has mean
 [
- 1-mathbb EbbE m_{V_b}^2=p(V_b)le C_beta n^{-b/2},
+ 1-E m_{V_b}^2=p(V_b)le C_beta n^{-b/2},
 ]
 by p(V_b)le C_beta e^{-V_b} and V_b=blog n. For a fine spin this
 follows from the martingale conditional mean through the continuation.
 Under the unconditioned single-run law, each squared distance is an average
 of bounded i.i.d. coordinate variables; consequently it is at most
 n^{-b/3} except with probability O(exp(-n^{1-b})). The triangle
-inequality in |cdot|_n, along with the negligible magnetization
+inequality in |·|_n, along with the negligible magnetization
 increments above, then gives Hamming step sizes at most R_n for large n,
 since n^{-b/3}ll R_n^2/n=n^{1-b/2}/n=n^{-b/2}.
 
@@ -1422,7 +1422,7 @@ sufficiently large n.
 
 ## The crossing contradiction
 
-On mathcal E,calE, the coverage statement (eq:cov) gives each side of
+On E, the coverage statement (eq:cov) gives each side of
 the chosen cut fine-endpoint probability at least exp(-n^{1-2d}).
 Conditional independence therefore puts the two fine endpoints on opposite
 sides of the cut with probability at least exp(-2n^{1-2d}).
@@ -1431,13 +1431,13 @@ But if both sequence properties (a) and (b) hold and the fine endpoints
 lie on opposite sides, then some consecutive pair in the sequence must
 cross the cut. Each such pair contains a coarse spin. A shortest one-spin
 path between the pair meets ∂ S and has length at most R_n,
-placing that coarse spin in mathcal D(G).calD(G). This contradicts property (a).
+placing that coarse spin in D(G). This contradicts property (a).
 
 Consequently
 
- Pr(mathcal E)calE),exp(-2n^{1-2d})leexp(-n^{1-d/2}) .
+ Pr(E),exp(-2n^{1-2d})leexp(-n^{1-d/2}) .
 
-Since d/2<2d, this implies Pr(mathcal E)=o(1)calE)=o(1) and proves
+Since d/2<2d, this implies Pr(E)=o(1) and proves
 (eq:11). The auxiliary diagonal and smoothing base are used
 only in this proof; the resulting flow bound concerns the original heat
 bath.
@@ -1475,7 +1475,7 @@ centering
   node[below=1mm] at (crossright.south) {v};
   node[fill=white,inner sep=1.5pt] at (7.3,.48) {d_H(u,v)le R_n};
   node at (6.9,-1.15)
-    {Every step has a coarse endpoint; coarse outputs avoid mathcal D(G).}calD(G).;
+    {Every step has a coarse endpoint; coarse outputs avoid D(G).};
   node at (7.3,-1.9)
     {Expand the crossing step into single-spin flips: length ellle R_n.};
   draw[densely dotted,black!55] (7.3,-2.25) -- (7.3,-3.35);
@@ -1495,7 +1495,7 @@ centering
   node at (12,-2.85) {S^{mathrm c}rmc};
   node[red!65!black,align=center] at (7.3,-3.8)
     {d_H(u,∂ S),,d_H(v,∂ S)leellle R_n
-     ;⇒; u,vinmathcal D(G).}calD(G).;
+     ;⇒; u,vinD(G).};
 
 
 
@@ -1532,7 +1532,7 @@ Let Ω be a finite set, pi a strictly positive probability on
 Ω, and P_t a continuous-time Markov semigroup reversible with
 respect to pi with jump rates c(x,y). Assume that for some R>0
 [
- sum_{yne x}c(x,y)le Rqquad(xinΩ).
+ sum_{yne x}c(x,y)le R (xinΩ).
 ]
 Let 0<ε<1/3 and B>0. Suppose that
 [
@@ -1541,7 +1541,7 @@ Let 0<ε<1/3 and B>0. Suppose that
 ]
 Then for every T>0,
 
- sum_{xinΩ}pi(x)bigl|P_T(x,cdot)-pibigr|_{rm TV}
+ sum_{xinΩ}pi(x)bigl|P_T(x,·)-pibigr|_{rm TV}
  lefrac{√(2R,H(pi))}{Bsqrt T}+2ε,
   H(pi)=-sum_xpi(x)logpi(x).
 
@@ -1665,7 +1665,7 @@ The last choice is valid because there are exactly n rate-one update
 clocks, so the total jump rate at any vertex is n. Also
 H(pi_G)le nlog2, since the state space has 2^n elements. Hence
 [
- sum_xpi_G(x)bigl|P_T^G(x,cdot)-pi_Gbigr|_{rm TV}
+ sum_xpi_G(x)bigl|P_T^G(x,·)-pi_Gbigr|_{rm TV}
  lefrac{√(2log2);n}{Bsqrt T}+2n^{-d}.
 ]
 Take