diff --git a/goe_rrg_bulk_universality_v2.txt b/goe_rrg_bulk_universality_v2.txt
index 9fc175e..953d4ce 100644
--- a/goe_rrg_bulk_universality_v2.txt
+++ b/goe_rrg_bulk_universality_v2.txt
@@ -64,14 +64,14 @@ frac1nsum_{i=1}^ndelta_{lambda_i} converges weakly, in probability,
 to the Kesten--McKay density
 
   rho_d(t);=;frac{d,√(4(d-1)-t^2)}{2pi,(d^2-t^2)}
-  qquadfor |t|<2√(d-1),
+   for |t|<2√(d-1),
 
 and rho_d(t)=0 otherwise. It integrates to 1. Its support is the
 compact interval [-2√(d-1),,2√(d-1)], the spectrum of the
 d-regular tree.
 
 For example, when d=3 the support is [-2sqrt2,2sqrt2] and
-rho_3(0)=3sqrt8/(2picdot9)approx 0.150, so the mean gap between
+rho_3(0)=3sqrt8/(2pi·9)approx 0.150, so the mean gap between
 neighbouring eigenvalues near 0 is about 1/0.150approx 6.7 in the
 variable lambda.
 
@@ -393,14 +393,14 @@ vertex; there are binom d2 candidate pairs at each of n vertices,
 each realized with probability 1/(nd-1). So the expected number of
 loops is
 [
-  n,(d(d-1))/(2)cdotfrac1{nd-1}→ (d-1)/(2).
+  n,(d(d-1))/(2)·frac1{nd-1}→ (d-1)/(2).
 ]
 A double edge is a pair of matched stub pairs between the same
 two distinct vertices. Choose the two vertices, two stubs at each, and a
 pairing of the four stubs into two pairs: the expected number of double
 edges is
 [
-  binom n2cdot ((binom d2)^2cdot 2)/((nd-1)(nd-3))
+  binom n2· ((binom d2)^2· 2)/((nd-1)(nd-3))
   → ((d-1)^2)/(4).
 ]
 A prescribed family of e disjoint stub pairs is realized with
@@ -422,7 +422,7 @@ converges in distribution to a Poisson variable with parameter
 limit; in particular it is positive.
 
 For a triangle through a fixed vertex, sum over the other two vertices
-and their stub choices: O(n^2)cdot O(n^{-3})=O(n^{-1}) in the
+and their stub choices: O(n^2)· O(n^{-3})=O(n^{-1}) in the
 matching model, and conditioning on simplicity multiplies this by a
 bounded factor.
 
@@ -558,7 +558,7 @@ Each child subtree is again such a branch, so a Schur complement at the
 root gives
 [
   F=frac1{zeta-(d-1)F},
-  qquadi.e. zeta=F^{-1}+(d-1)F,
+   i.e. zeta=F^{-1}+(d-1)F,
 ]
 and Fto0 at infinity; thus F=q. At the root of the full tree, a
 Schur complement over the d branches gives
@@ -579,7 +579,7 @@ vertices by homogeneity. The resolvent equation (zeta-A)G=I at the
 root gives
 [
   zeta g-1=d g_1 ,
-  qquadsoqquad g_1=(1+zeta m_d(zeta))/(d),
+   so  g_1=(1+zeta m_d(zeta))/(d),
 ]
 and applied to a neighbour column gives
 [
@@ -632,7 +632,7 @@ by evaluating the quadratic at zeta=E. Substituting into
 |q|=1/sqrt2, |1-q^2|=3/2, and the right side is
 log(sqrt3/2).)
 
-Finally, tmapstolog|t-E| is bounded near E because rho_d is,
+Finally, t|->log|t-E| is bounded near E because rho_d is,
 and
 [
   intlog|1-(mathrm irmieta)/(t-E)|rho_d(t),dd t
@@ -679,7 +679,7 @@ depends only on the graph. Independently of that choice, sample k
 uniform oriented edges (v_1,v'_1),...,(v_k,v'_k). Set
 [
   b_j=(u_{i(j)}(v_1),...,u_{i(j)}(v_k),
-           u_{i(j)}(v'_1),...,u_{i(j)}(v'_k))^{mathsf T}sfT
+           u_{i(j)}(v'_1),...,u_{i(j)}(v'_k))^{T}
   inR^{2k},
    j=1,...,ell .
 ]
@@ -724,11 +724,11 @@ the graph is conditioned on.
 
 We need two notions of entropy. For a random variable on a finite set we
 write H for the discrete entropy; for a random vector with a density
-we write mathcal HcalH for the differential entropy; both use natural
+we write H for the differential entropy; both use natural
 logarithms. The only fact about H we use besides its definition is
 that the number of labelings with a given empirical distribution of
 local observations is at most exponential in n times the entropy of
-that distribution; the only facts about mathcal HcalH are the Gaussian
+that distribution; the only facts about H are the Gaussian
 maximum-entropy bound and the continuity estimates for binning, both
 spelled out below.
 
@@ -852,13 +852,13 @@ each fixed moment.
 On events of graphs of probability tending to one, uniformly over all
 choices of V and all 0leεle1,
 
-  mathcal H(F_calH(F_{rm star})-frac d2mathcal H(F_calH(F_{rm edge})
-   ge frac1nmathcal H(Y_calH(Y_{rm global})-O(n^{-c}),
+  H(F_{rm star})-frac d2H(F_{rm edge})
+   ge frac1nH(Y_{rm global})-O(n^{-c}),
 
 where Y_{rm global} is the full nell-dimensional Gaussian array.
 Moreover
 
-  frac1nmathcal H(Y_calH(Y_{rm global})
+  frac1nH(Y_{rm global})
   =ell(frac12logfrac{2pi{rm e}eta}{pirho}-U(E))
    +O(n^{-c}),
 
@@ -876,15 +876,15 @@ alphabet has
 
 so by 2ell(d+1)gamma<1 the hypothesis
 (eq:type-count-hypothesis) holds with room to spare. Let
-mathcal LcalL be the resulting random quantized labeling of the vertices,
+L be the resulting random quantized labeling of the vertices,
 and let T,Q be its star and edge histograms. Entropy of the type
-index. The total number of labelings is exp(nH(mathcal L)),calL)), the
+index. The total number of labelings is exp(nH(L)), the
 number of possible type indices is at most
 exp(O(M^{d+1}log n)), and for a given type index the count is at
 most (eq:ent-type-count). Hence, by concavity of H in its first
 argument,
 
-  frac1nH(mathcal L)calL)
+  frac1nH(L)
   le H(EE T)-frac d2,EE H(Q)+O(n^{-c}) .
 
 
@@ -916,7 +916,7 @@ the diagonal term being smaller. There are M^2=n^{4ellgamma+o(1)}
 cells, so Cauchy--Schwarz bounds the mean ell^1 deviation
 [
   EE|Q-EE Q|_1
-  le (M^2cdot Cn^{-1/2+2alpha})^{1/2}
+  le (M^2· Cn^{-1/2+2alpha})^{1/2}
   =n^{2ellgamma-1/4+alpha+o(1)}=O(n^{-c}),
 ]
 because 4ellgamma+2alpha<tfrac14. For two probability vectors on a
@@ -933,7 +933,7 @@ spectrum in [ceta,C/eta], because Gamma does and because a
 deterministic shift does not change covariance. Unbounded binning of a
 Gaussian vector and the entropy bound on a cube give
 [
-  frac1nH(mathcal L)calL) ge frac1nmathcal H(Y_calH(Y_{rm global})
+  frac1nH(L) ge frac1nH(Y_{rm global})
   -elllog b-O(n^{-c});
 ]
 the clipping error is in fact superpolynomially small, since
@@ -943,7 +943,7 @@ superpolynomially small, uniformly in the shift. For either local
 observation, of dimension ell(d+1) for the star and 2ell for the
 edge, we also have the two-sided expansion
 
-  H(binned X)=mathcal H(X)-(calH(X)-(dim X)log b+O(n^{-c}).
+  H(binned X)=H(X)-(calH(X)-(dim X)log b+O(n^{-c}).
 
 Here is the justification, since everything downstream rests on it.
 Bin averaging changes a Gaussian density, or a mixture of Gaussians, in
@@ -968,16 +968,16 @@ deterministic shift of Y_{rm global} we obtain
 Step 4: the global entropy. The determinant formula for a
 Gaussian gives
 [
-  frac1nmathcal H(Y_calH(Y_{rm global})
+  frac1nH(Y_{rm global})
   =(ell)/(2)logfrac{2pi{rm e}eta}{pirho}
    -(ell)/(n)sum_{i=1}^nlog|lambda_i-E-mathrm irmieta|,
 ]
 because the eigenvalues of Gamma are
 etabigl/(pirho,((lambda_i-E)^2+eta^2)), each with
-multiplicity ell. The function tmapstolog|t-E-mathrm irmieta| is
+multiplicity ell. The function t|->log|t-E-mathrm irmieta| is
 Lipschitz with constant at most eta^{-1}. By rigidity we may replace
 the eigenvalues by Kesten--McKay quantiles at a total error of at most
-ncdoteta^{-1}cdot n^{-c_0}; the quantile sum differs from its
+n·eta^{-1}· n^{-c_0}; the quantile sum differs from its
 integral against rho_d by O((neta)^{-1}), by summing the
 lengths of the quantile intervals; an omitted or extra bounded
 eigenvalue costs O(log n/n). The choice alpha<c_0 makes the first
@@ -1003,12 +1003,12 @@ the d neighbours). Let P be the orthogonal projection of R^d onto
 define the linear map to the ``clean'' coordinates, and the residual,
 by
 [
-  mathcal B(Z,X)=(Z,(PX)/(calB(Z,X)=(Z,(PX)/(sqrt a)),
+  B(Z,X)=(Z,(PX)/(calB(Z,X)=(Z,(PX)/(sqrt a)),
   R_{rm res}(Z,X)=sum_{i=1}^dX_i-EZ .
 ]
 Everything is tensored with the identity on the ell colours, so
 D:=(R ⊕ {bf 1}^{perp}) ⊗ R^{ell} is the space in which
-mathcal BcalB takes its values. Let w denote the unit vector along the
+B takes its values. Let w denote the unit vector along the
 central direction R inside R ⊕ {bf 1}^{perp}, and put
 [
   v_i=p,w+sqrt a,Pe_i,
@@ -1024,7 +1024,7 @@ they are linearly independent and the coordinate map on U_i is
 invertible.
 
 Let P_i be the orthogonal projection of D onto U_i, put
-B_V=mathcal B(V_calB(V_{rm star}) where V_{rm star} is the true
+B_V=B(V_{rm star}) where V_{rm star} is the true
 eigenvector star, and let G be an independent standard Gaussian vector
 in D. All entropies on D and on U_i are computed with their
 Euclidean volume.
@@ -1033,7 +1033,7 @@ Euclidean volume.
 On the graph events of Lemma~ent:star-edge, uniformly for
 0leεle1 and for each i,
 
-  mathcal H(εcalH(ε B_V+G)-frac d2mathcal H(P_i(εcalH(P_i(ε
+  H(εcalH(ε B_V+G)-frac d2H(P_i(εcalH(P_i(ε
   B_V+G)) ge -O(n^{-c}).
 
 Moreover B_V has mean zero and second-moment matrix
@@ -1044,12 +1044,12 @@ interchanges w and v_i and acts trivially on the colours.
 
 Proof. 
 The Jacobian. Describe the map
-(Z,X)mapsto(mathcal B(Z,X),R_calB(Z,X),R_{rm res}(Z,X)) in the orthonormal
+(Z,X)|->(B(Z,X),R_calB(Z,X),R_{rm res}(Z,X)) in the orthonormal
 coordinates of R^{d+1}=R ⊕ R{bf 1} ⊕ {bf 1}^{perp}, one
 colour at a time. On the central line it is the identity. On the
 all-ones line, R_{rm res} reads sqrt d,t-EZ where X=t{bf
 1}/sqrt d+..., so the diagonal entry is sqrt d. On each of the
-d-1 remaining directions, mathcal BcalB divides by sqrt a, so the
+d-1 remaining directions, B divides by sqrt a, so the
 diagonal entry is a^{-1/2}. The map is triangular in these
 coordinates, hence its absolute determinant, per colour, is
 sqrt d,a^{-(d-1)/2}, and for ell colours it is
@@ -1082,7 +1082,7 @@ terms vanish, and because |Delta|le S^2/n entrywise. The maximum
 entropy bound by the covariance, together with the fact that
 conditioning decreases entropy, therefore gives
 [
-  mathcal H(R_calH(R_{rm res}(F_{rm star})mid mathcal B(F_calB(F_{rm star}))
+  H(R_{rm res}(F_{rm star})mid B(F_{rm star}))
   le(ell)/(2)logfrac{2pi{rm e}eta}{pirho}+O(n^{-c}).
 ]
 The relative correction inside the logarithm is
@@ -1097,7 +1097,7 @@ The noise is standard. By the typical-star estimate of
 Proposition~prop:spectral-input at the height E+mathrm irmieta,
 together with the tree covariance of Lemma~lem:tree-identities,
 the covariance matrix of the smoothed star noise
-mathcal B(Y_calB(Y_{rm star}) differs from Id_D by a polynomial error
+B(Y_{rm star}) differs from Id_D by a polynomial error
 outside a polynomially small fraction of centres. Two centred Gaussian
 vectors whose covariance matrices differ by delta in operator norm
 have laws at total variation distance at most Cdelta after
@@ -1114,16 +1114,16 @@ absolute determinant (1-p^2)^{ell/2}.
 Bookkeeping: the constants cancel. For a linear bijection Phi
 with Jacobian J_Phi
 [
-  mathcal H(calH(Phi(X))=mathcal H(X)+calH(X)+log|det J_Phi| ,
+  H( H(X)+calH(X)+log|det J_Phi| ,
 ]
 since the density of Phi(X) is the density of X divided by
 |det J_Phi|. Applied to the triangular map
-Phi=(Z,X)mapsto(mathcal B(Z,X),R_calB(Z,X),R_{rm res}(Z,X)), whose determinant
+Phi=(Z,X)|->(B(Z,X),R_calB(Z,X),R_{rm res}(Z,X)), whose determinant
 per colour is sqrt d,a^{-(d-1)/2}, and combining with the
 conditional bound on the residual, we get
 [
-  mathcal H(F_calH(F_{rm star})
-  lemathcal H(calH(mathcal B(F_calB(F_{rm star}))
+  H(F_{rm star})
+  leH( B(F_calB(F_{rm star}))
    +(ell)/(2)logfrac{2pi{rm e}eta}{pirho}
    -ell(frac12log d-(d-1)/(2)log a)+O(n^{-c}).
 ]
@@ -1131,8 +1131,8 @@ The edge observation is the image of P_iF_{rm star} under the
 coordinate map of U_i, which has determinant
 (1-p^2)^{ell/2}, so
 [
-  mathcal H(F_calH(F_{rm edge})
-  =mathcal H(P_iF_calH(P_iF_{rm star})+(ell)/(2)log(1-p^2)+O(n^{-c}).
+  H(F_{rm edge})
+  =H(P_iF_{rm star})+(ell)/(2)log(1-p^2)+O(n^{-c}).
 ]
 Substituting both formulas into (eq:ent-star-edge), and using
 (eq:ent-global-entropy) for the right-hand side, the terms with
@@ -1144,13 +1144,13 @@ log(2pi{rm e}eta/pirho) cancel, and the remaining constant is
 which is exactly zero by the potential identity
 (eq:tree-potential). Hence
 [
-  mathcal H(calH(mathcal B(F_calB(F_{rm star}))
-  -frac d2,mathcal H(P_iF_calH(P_iF_{rm star}) ge -O(n^{-c}).
+  H( B(F_calB(F_{rm star}))
+  -frac d2,H(P_iF_{rm star}) ge -O(n^{-c}).
 ]
 Replacing the smoothed eigenvector star F_{rm star} by
 ε B_V+G costs polynomial entropy in exactly the same way as in
 the noise-replacement step above: the covariance of
-mathcal B(F_calB(F_{rm star}) is Id_D+O(n^{-1+o(1)}) uniformly in
+B(F_{rm star}) is Id_D+O(n^{-1+o(1)}) uniformly in
 ε, and the entropy is a continuous function of the covariance
 on the fixed finite-dimensional space D. This proves
 (eq:ent-projected-entropy).
@@ -1161,16 +1161,16 @@ eigenvector relations, orthogonality of distinct eigenvectors, and
 uniform permutation of the d neighbours give the exact second-moment
 identities
 [
-  EE ZZ^{mathsf T}sfT=EE X_iX_i^{mathsf T}sfT=Id_{ell},
-  EE ZX_i^{mathsf T}sfT=(Λ)/(d),
-  EE X_iX_j^{mathsf T}sfT=frac{Λ^2-d,Id_{ell}}{d(d-1)}
+  EE ZZ^{T} X_iX_i^{T}{ell},
+  EE ZX_i^{T}Λ)/(d),
+  EE X_iX_j^{T}{Λ^2-d,Id_{ell}}{d(d-1)}
   (ine j).
 ]
 The mean is zero because every selected eigenvector is orthogonal to the
 constant vector. Since Λ=E,Id_ell+O(n^{-1+o(1)}) on the graph
 event, these identities give, using 1-(E^2-d)/(d(d-1))=a,
 [
-  EE B_VB_V^{mathsf T}sfT=Id_D+O(n^{-1+o(1)}).
+  EE B_VB_V^{T}+O(n^{-1+o(1)}).
 ]
 Finally, a uniform oriented edge is invariant in law under reversal. Its
 two coordinate vectors differ from the observations of B_V along w
@@ -1189,11 +1189,11 @@ Two facts do the work: a formal identity expressing the entropy deficit
 as a difference of relative entropies, and a strict inequality for
 symmetric tensors that forbids a nonzero first Hermite coefficient.
 
-Write varphi_L for the standard Gaussian density on a Euclidean space
+Write phi_L for the standard Gaussian density on a Euclidean space
 L, and for a random vector Z with density f on L define the
 relative entropy
 [
-  Ent(Law(Z)midvarphi_L)=int_L flog(f)/(varphi_L) .
+  Ent(Law(Z)midphi_L)=int_L flog(f)/(phi_L) .
 ]
 Relative entropy is nonnegative, and it vanishes only for Z standard
 Gaussian.
@@ -1203,17 +1203,17 @@ Let Z be a centred random vector in D with identity covariance
 matrix and finite differential entropies for Z and for each P_iZ.
 Then
 
-  mathcal H(Z)-calH(Z)-frac12sum_{i=1}^dmathcal H(P_iZ)calH(P_iZ)
-  =-Ent(Law(Z)midvarphi_D)
-   +frac12sum_{i=1}^dEnt(Law(P_iZ)midvarphi_{U_i}).
+  H(Z)-calH(Z)-frac12sum_{i=1}^dH(P_iZ)
+  =-Ent(Law(Z)midphi_D)
+   +frac12sum_{i=1}^dEnt(Law(P_iZ)midphi_{U_i}).
 
 
 
 Proof. 
 For any X in a Euclidean space L with a density,
-mathcal H(X)=calH(X)=mathcal H(calH(varphi_L)-Ent(Law(X)midvarphi_L); this is
-the definition of relative entropy together with intvarphilogvarphi
-=-mathcal H(calH(varphi). Also mathcal H(calH(varphi_L)=(dim L)/(2)log(2pi{rm e}).
+H(X)= H(calH(phi_L)-Ent(Law(X)midphi_L); this is
+the definition of relative entropy together with intphilogphi
+=-H( Also H( L)/(2)log(2pi{rm e}).
 Substituting into the left side of (eq:ent-deficit-sign), the
 Gaussian entropies contribute
 (dim D)/(2)-frac12sum_i(dim U_i)/(2) times
@@ -1238,7 +1238,7 @@ For every fixed jge3 there is c_j>0 such that every symmetric
 j-tensor H on D satisfying
 [
   sigma_i^{ ⊗  j}P_i^{ ⊗  j}H=P_i^{ ⊗  j}H
-  qquadfor all i=1,...,d
+   for all i=1,...,d
 ]
 obeys
 
@@ -1281,7 +1281,7 @@ particular the two numbers
 ]
 are positive and sum to one. With these weights the frame identity
 
-  sum_{i=1}^d(b_s,s_is_i^{mathsf T}sfT+b_t,t_it_i^{mathsf T}sfT)
+  sum_{i=1}^d(b_s,s_is_i^{T}+b_t,t_it_i^{T}
   =Id_{R ⊕ {bf 1}^{perp}}
 
 holds in R ⊕ {bf 1}^{perp}, tensored with the identity on the
@@ -1294,7 +1294,7 @@ parallel central and Pe_i components with i-independent
 coefficients and sum_iPe_i=0; and the {bf 1}^{perp} block is
 cP with
 c=b_s(1-q_0)/(2)+b_t(1+q_0)/(2)=1, because
-sum_i(Pe_i)(Pe_i)^{mathsf T}sfT=P.
+sum_i(Pe_i)(Pe_i)^{T}
 
 Nonnegativity. Let R^{(1)}_{i,s} denote the projection of the
 first tensor slot onto s_i, tensored with the identity on the other
@@ -1335,7 +1335,7 @@ for t. Adding the two identities over the orthonormal pair
 s_i,t_i, and using symmetry of H to move a projection from slot one
 to any other slot, gives
 
-  P_i^{(a)}H=P_i^{ ⊗  j}Hqquadfor every slot a.
+  P_i^{(a)}H=P_i^{ ⊗  j}H for every slot a.
 
 Now take ine k and three distinct slots, which exist because jge3:
 [
@@ -1348,7 +1348,7 @@ all pairs ine k.
 Let a_i'in{bf 1}^{perp} be the unit vector along Pe_i. On
 R ⊕ {bf 1}^{perp} the spatial part of this commutator is
 [
-  ip{a_i'}{a_k'}(a_i'(a_k')^{mathsf T}sfT-a_k'(a_i')^{mathsf T}sfT),
+  ip{a_i'}{a_k'}(a_i'(a_k')^{T}-a_k'(a_i')^{T},
    ip{a_i'}{a_k'}=-frac1{d-1}ne0  (dge3),
 ]
 which is an invertible rotation generator on
@@ -1381,7 +1381,7 @@ Fix an integer qge2. Let B_n be random vectors in D with
 |B_n|le K_n almost surely, where K_n=n^{o(1)} is deterministic.
 Suppose that for some c_*>0
 [
-  |EE B_n|+bigl|EE B_nB_n^{mathsf T}sfT-Id_Dbigr|=O(n^{-c_*}),
+  |EE B_n|+bigl|EE B_nB_n^{T}-Id_Dbigr|=O(n^{-c_*}),
 ]
 and that for every 1le jle q and every i
 [
@@ -1391,8 +1391,8 @@ and that for every 1le jle q and every i
 Let G be an independent standard Gaussian in D and suppose in
 addition that, uniformly for 0leεle1,
 [
-  mathcal H(εcalH(ε B_n+G)-frac12sum_{i=1}^d
-    mathcal H(P_i(εcalH(P_i(ε B_n+G)) ge -O(n^{-c_*}).
+  H(εcalH(ε B_n+G)-frac12sum_{i=1}^d
+    H(P_i(εcalH(P_i(ε B_n+G)) ge -O(n^{-c_*}).
 ]
 Then every fixed polynomial moment of B_n of degree at most q
 differs from its standard Gaussian value by O(n^{-c}) for some c>0.
@@ -1433,13 +1433,13 @@ whose second-moment matrix is Id_D+u^2O(n^{-c_*}). Feeding this into
 (eq:ent-deficit-sign), and absorbing the second-moment error into
 the Gaussian entropies, gives
 
-  Ent(Law(Z_u)midvarphi_D)
-  -frac12sum_{i=1}^dEnt(Law(P_iZ_u)midvarphi_{U_i})
+  Ent(Law(Z_u)midphi_D)
+  -frac12sum_{i=1}^dEnt(Law(P_iZ_u)midphi_{U_i})
   le O(n^{-c_*}).
 
 
 The relative entropy is essentially the j-th Hermite moment.
-Let f_u be the density ratio of Z_u to varphi_D. The coefficient
+Let f_u be the density ratio of Z_u to phi_D. The coefficient
 of u^k in the Taylor expansion of f_u is
 [
   frac1{k!}ip{H_k}{He_k(y)} .
@@ -1450,11 +1450,11 @@ EEexp(uip{B_n}{t}-u^2|t|^2/2), and pairing two generating
 exponentials under Gaussian measure gives Hermite orthogonality
 together with
 [
-  Bigl|frac{ip{H_k}{He_k}}{k!}Bigr|_{L^2(varphi_D)}^2
+  Bigl|frac{ip{H_k}{He_k}}{k!}Bigr|_{L^2(phi_D)}^2
   =(|H_k|^2)/(k!) .
 ]
 The Taylor remainder through degree j is
-O(u^{j+1}n^{o(1)}) in L^3(varphi_D). At a fixed source value b_0
+O(u^{j+1}n^{o(1)}) in L^3(phi_D). At a fixed source value b_0
 the ratio at parameter sin[0,u] is
 [
   (1-s^2)^{-dim D/2}
@@ -1463,22 +1463,22 @@ the ratio at parameter sin[0,u] is
 whose required derivatives are bounded by polynomials in
 |y|,|b_0| times Cexp(Cs|b_0||y|); since
 |b_0|le K_n=n^{o(1)} and u=n^{-tau+o(1)}, their
-L^3(varphi_D) norms are n^{o(1)}, uniformly over the source
+L^3(phi_D) norms are n^{o(1)}, uniformly over the source
 values. Consequently
 [
-  int_D|f_u-1|^3,ddvarphi_Dle Cu^3n^{o(1)} ,
+  int_D|f_u-1|^3,ddphi_Dle Cu^3n^{o(1)} ,
 ]
 and, since |(1+h)log(1+h)-h-h^2/2|le C|h|^3 for |h|letfrac12,
-while int(f_u-1)ddvarphi_D=0, we get
+while int(f_u-1)ddphi_D=0, we get
 
-  Ent(Law(Z_u)midvarphi_D)
-  =frac12int_D(f_u-1)^2ddvarphi_D+O(u^3n^{o(1)}).
+  Ent(Law(Z_u)midphi_D)
+  =frac12int_D(f_u-1)^2ddphi_D+O(u^3n^{o(1)}).
 
 
 Extracting the j-th Hermite coefficient. Orthogonality of the
 Hermite tensors gives, for the squared norm above,
 [
-  int_D(f_u-1)^2ddvarphi_D
+  int_D(f_u-1)^2ddphi_D
   =u^{2j}(|H_j|^2)/(j!)
    +sum_{kne j}u^{2k}(|H_k|^2)/(k!) .
 ]
@@ -1493,7 +1493,7 @@ with mu=ub_0/(1-u^2); a Gaussian integral then gives
 chi^2=O(u^2(dim D+|b_0|^2)) for small u, whose mean is
 O(u^2), because EE|B_n|^2=dim D+o(1). Therefore
 
-  Ent(Law(Z_u)midvarphi_D)
+  Ent(Law(Z_u)midphi_D)
   =frac{u^{2j}}{2}(|H_j|^2)/(j!)+O(n^{-2tau+o(1)}),
 
 and the same identity holds on U_i with H_j replaced by
@@ -1559,7 +1559,7 @@ projected star field of the projection step. Complete
 delocalization gives the deterministic bound |B_n|le n^{o(1)}.
 Lemma~ent:projected-entropy supplies the entropy hypothesis, the
 first two moments EE B_n=0 and
-EE B_nB_n^{mathsf T}sfT=Id_D+O(n^{-1+o(1)}), and the moment
+EE B_nB_n^{T}+O(n^{-1+o(1)}), and the moment
 symmetries (sigma_i^{ ⊗  j}-Id)EE(P_iB_n)^{ ⊗  j}=O(n^{-1+o(1)})
 for every fixed j, all uniformly over the selected eigenindices and
 in 0leεle1. Averaging (eq:ent-projected-entropy) over
@@ -1667,14 +1667,14 @@ The defining convenience of the switch is that Z has rank two in the
 following explicit sense. Put
 [
   u=e_v-e_y, v=e_w-e_x,
-  Z=uv^{mathsf T}sfT+vu^{mathsf T}sfT.
+  Z=uv^{T}+vu^{T}
 ]
 Expanding,
 [
-  Z=e_ve_w^{mathsf T}sfT+e_we_v^{mathsf T}sfT+e_xe_y^{mathsf T}sfT
-    +e_ye_x^{mathsf T}sfT
-    -e_ve_x^{mathsf T}sfT-e_xe_v^{mathsf T}sfT
-    -e_we_y^{mathsf T}sfT-e_ye_w^{mathsf T}sfT,
+  Z=e_ve_w^{T}+e_we_v^{T}+e_xe_y^{T}
+    +e_ye_x^{T}
+    -e_ve_x^{T}-e_xe_v^{T}
+    -e_we_y^{T}-e_ye_w^{T},
 ]
 so Z deletes (v,x) and (w,y) and inserts (v,w) and (x,y),
 as required. Because Z has rank at most two, the resolvent equation
@@ -1683,8 +1683,8 @@ has an explicit elementary solution.
 [Rank-two resolvent identity]
 Let W=A-zeta with Imzeta>0, G=W^{-1}, G'=(W+Z)^{-1}, and
 [
-  A_{uv}=v^{mathsf T}sfTGu, A_{uu}=u^{mathsf T}sfTGu,
-  A_{vv}=v^{mathsf T}sfTGv,
+  A_{uv}=v^{T}, A_{uu}=u^{T},
+  A_{vv}=v^{T},
   D=(1+A_{uv})^2-A_{uu}A_{vv}.
 ]
 Then Dneq0 and
@@ -1702,17 +1702,17 @@ G'_{vw}.
 
 
 Proof. 
-From (W+uv^{mathsf T}sfT+vu^{mathsf T}sfT)G'=I we get
+From (W+uv^{T}+vu^{T}'=I we get
 [
-  G'=G-Guv^{mathsf T}sfTG'-Gvu^{mathsf T}sfTG' .
+  G'=G-Guv^{T}'-Gvu^{T}' .
 ]
-Set X=v^{mathsf T}sfTG' and Y=u^{mathsf T}sfTG', row vectors. Multiplying
-the displayed identity on the left by v^{mathsf T}sfT and by
-u^{mathsf T}sfT gives the linear system
+Set X=v^{T}' and Y=u^{T}', row vectors. Multiplying
+the displayed identity on the left by v^{T} and by
+u^{T} gives the linear system
 [
   1+A_{uv}&A_{vv} A_{uu}&1+A_{uv}
   X Y
-  =v^{mathsf T}sfTG u^{mathsf T}sfTG,
+  =v^{T} u^{T},
 ]
 whose determinant is D. Since Im G is negative definite and Z is
 symmetric with zero exponential type, W+Z has its spectrum in the open
@@ -1720,16 +1720,16 @@ upper half-plane, so G' exists; consistency of the system gives
 Dne0 (otherwise W+Z would have a null vector with real
 eigenvalue). Cramer's rule gives
 [
-  X=D^{-1}[(1+A_{uv})v^{mathsf T}sfTG-A_{vv}u^{mathsf T}sfTG],
-  Y=D^{-1}[(1+A_{uv})u^{mathsf T}sfTG-A_{uu}v^{mathsf T}sfTG].
+  X=D^{-1}[(1+A_{uv})v^{T}-A_{vv}u^{T}],
+  Y=D^{-1}[(1+A_{uv})u^{T}-A_{uu}v^{T}].
 ]
-Now take the (x,y) entry of G'=G-Guv^{mathsf T}sfTG'-Gvu^{mathsf T}sfTG':
+Now take the (x,y) entry of G'=G-Guv^{T}'-Gvu^{T}':
 [
-  G'_{xy}=G_{xy}-(Gu)_x(e_y^{mathsf T}sfTv^{mathsf T}sfTG')
-                  -(Gv)_x(e_y^{mathsf T}sfTu^{mathsf T}sfTG'),
+  G'_{xy}=G_{xy}-(Gu)_x(e_y^{T}{T}')
+                  -(Gv)_x(e_y^{T}{T}'),
 ]
-and e_y^{mathsf T}sfTv^{mathsf T}sfTG'=(G'v)_y=(X)_y,
-e_y^{mathsf T}sfTu^{mathsf T}sfTG'=(Y)_y, while
+and e_y^{T}{T}'=(G'v)_y=(X)_y,
+e_y^{T}{T}'=(Y)_y, while
 (Gu)_x=G_{xv}-G_{xy} and (Gv)_x=G_{xw}-G_{xx} because
 u=e_v-e_y and v=e_w-e_x. Substituting the expressions for X,Y
 proves (eq:switch-entries).
@@ -1752,9 +1752,9 @@ uniform oriented edges. For fixed kge1, let W_{n,k} be the ntimes2k
 matrix whose columns are the coordinate vectors of the first k
 starting vertices followed by their k ending vertices. Define
 
- M_{n,k}(z)=W_{n,k}^{mathsf T}sfT(A-E-frac zn)^{-1}W_{n,k}
- =sum_ifrac{b_ib_i^{mathsf T}sfT}{x_i-z},
- b_i=W_{n,k}^{mathsf T}sfTu_i, Im z>0 .
+ M_{n,k}(z)=W_{n,k}^{T}-E-frac zn)^{-1}W_{n,k}
+ =sum_ifrac{b_ib_i^{T}}{x_i-z},
+ b_i=W_{n,k}^{T}, Im z>0 .
 
 The normalization |u_i|^2=n accounts for the missing factor n^{-1}.
 We show that, at any fixed microscopic height, the scalar m_n and the
@@ -1793,10 +1793,10 @@ uniform vertices, and within one edge the typical-star estimate gives the
 adjacent entry (1+Em_*)/d. Also #{|x_i|le S^2}=O(S^2)=n^{o(1)}.
 
 Concentration under a trace bound. Descend from height S to y
-by halving the imaginary part at fixed real part s; let mathcal Z_ncalZ_n
+by halving the imaginary part at fixed real part s; let Z_n
 be the O(log S) dyadic arguments. For every fixed C<∞, outside
 an event of limiting probability O_{k,D,C}(y^{-D}), simultaneously for
-zinmathcal Z_n,calZ_n,
+zinZ_n,
 
  |m_n(z)|le C ⇒ M_{n,k}(z)=K_k(m_n(z))+O((Im z)^{-c_2}),
 
@@ -1808,7 +1808,7 @@ matrix bound immediate). Inside the band, with
 w_i(z)=(x_i-z)^{-1}-(x_i-z_0)^{-1}, a bounded trace gives
 sum_{|x_i|le S^2}|w_i(z)|^2le 2Im m_n(z)/Im z+2Im m_n(z_0)/Sle C'/Im z.
 Replacing the marks by independent N(0,C_k) vectors, each entry of
-sum(b_ib_i^{mathsf T}sfT-C_k)w_i(z) has 2j-th absolute moment
+sum(b_ib_i^{T}-C_k)w_i(z) has 2j-th absolute moment
 O_{j,k,C}((Im z)^{-j}) (any spectral index occurring once has zero
 expectation; the remaining partitions are bounded by Gaussian moments and
 the weight squares). Conditionally on a graph satisfying the fixed-order
@@ -1833,8 +1833,8 @@ function by a bounded amount, so
 Write the first two pairs as (v_1,w_1),(v_2,w_2) and set
 f_1=(e_{v_1}-e_{v_2})/sqrt2, f_2=(e_{w_1}-e_{w_2})/sqrt2,
 F=(f_1,f_2), B=0&-2-2&0; the switch adds
-FBF^{mathsf T}sfT, and the resolvent identity reads
-(I+UB)widehatwidehat U=UU=U with U=F^{mathsf T}sfT(A-E-z/n)^{-1}F. Assuming
+FBF^{T}, and the resolvent identity reads
+(I+UB)widehatwidehat U=UU=U with U=F^{T}-E-z/n)^{-1}F. Assuming
 |m_n(z)|le C and applying (eq:lim-concentration-implication) and
 (eq:lim-rank-trace) to both data sets,
 
@@ -1900,7 +1900,7 @@ there are standard Gaussian vectors g_jinR^{2k}, independent over
 j conditionally on nu, such that
 
  m(z)=m_{rm r}+PVsum_jfrac1{x_j-z},
- M_k(z)=K_k(m_{rm r})+PVsum_jfrac{C_k^{1/2}g_jg_j^{mathsf T}sfTC_k^{1/2}}{x_j-z}.
+ M_k(z)=K_k(m_{rm r})+PVsum_jfrac{C_k^{1/2}g_jg_j^{T}{1/2}}{x_j-z}.
 
 These representations are consistent as k grows by keeping the first
 k channels, a separate mark is assigned to each occurrence of a
@@ -1914,11 +1914,11 @@ Tightness. For fixed L,y>0, positivity gives
 nu_n([-L,L])le(L^2+y^2)y^{-1}Im m_n(mathrm iy);rmiy); using
 Proposition~prop:endpoint-control with large fixed y (chosen per
 tolerance) gives tightness of the local counts. For compact-open
-tightness, fix a compact mathcal KcalKsubsetC_+ and y>0. For a resolvent
+tightness, fix a compact K+ and y>0. For a resolvent
 sum f with nonnegative residues, comparing kernels on the real line
-gives Im f(z)le C_{mathcal K,y}calK,yIm f(mathrm iy)rmiy) and
-|f(z)-f(mathrm iy)|rmiy)|le C_{mathcal K,y}calK,yIm f(mathrm iy)rmiy) for
-zinmathcal K;calK; applying these to m_n and, by polarization, to the
+gives Im f(z)le C_{K,y}calK,yIm f(mathrm iy)rmiy) and
+|f(z)-f(mathrm iy)|rmiy)|le C_{K,y}calK,yIm f(mathrm iy)rmiy) for
+zinK; applying these to m_n and, by polarization, to the
 real quadratic forms of M_{n,k} bounds each fixed compact, and Cauchy's
 estimates give equicontinuity on smaller compacts. Extracting diagonally
 over growing compacts and channel dimensions gives the joint limit;
@@ -1959,8 +1959,8 @@ compact marked sums converge, so
  
  Im m(z)&=sum_jImfrac1{x_j-z},&
  m(z)-m(mathrm i)&=rmi)&=sum_j(frac1{x_j-z}-frac1{x_j-mathrm i}rmi),
- Im M_k(z)&=sum_jb_jb_j^{mathsf T}sfTImfrac1{x_j-z},&
- M_k(z)-M_k(mathrm i)&=rmi)&=sum_jb_jb_j^{mathsf T}sfT(frac1{x_j-z}-frac1{x_j-mathrm i}rmi).
+ Im M_k(z)&=sum_jb_jb_j^{T}{x_j-z},&
+ M_k(z)-M_k(mathrm i)&=rmi)&=sum_jb_jb_j^{T}{x_j-z}-frac1{x_j-mathrm i}rmi).
  
 
 No imaginary spectral mass is lost at infinity; real additive constants
@@ -2002,13 +2002,13 @@ S_nu(mathrm iR)=rmiR)=mathrm irmipirho+o(1). By
 imaginary part; the two displays identify it as m_{rm r}.
 
 Marked principal values. Fix k, condition on nu, and set
-Z_j=b_jb_j^{mathsf T}sfT-C_k, independent centered with bounded second
+Z_j=b_jb_j^{T}-C_k, independent centered with bounded second
 moments. Ordered by increasing |x_j|, sum_jZ_j/(x_j-mathrm i)rmi)
 converges almost surely entrywise, and
 sum_{|x_j|>1}|Z_j|x_j^{-2}<∞ conditionally almost surely, so
 the difference series between z and mathrm irmi converges absolutely
 and locally uniformly; with the scalar PV this gives local uniform
-convergence of S_{k,nu}(z)=PVsum_jb_jb_j^{mathsf T}sfT(x_j-z)^{-1}.
+convergence of S_{k,nu}(z)=PVsum_jb_jb_j^{T}-z)^{-1}.
 The linear count gives sum_j|x_j-mathrm iR|^rmiR|^{-2}=O(R^{-1}), so each
 entry of the centered marked series at mathrm iRrmiR has conditional
 second moment O(R^{-1}), whence
@@ -2036,7 +2036,7 @@ The microscopic-limit step leaves us with the microscopic limiting law and
 its Gaussian endpoint marks. Those marks give a symmetry that a general
 law would not have: conjugating M_k by an orthogonal Q_U preserves
 the law. Composing many small conjugations then produces an honest GOE
-perturbation: the natural graph trace mathcal S_r(z)calS_r(z) is approximated,
+perturbation: the natural graph trace S_r(z) is approximated,
 in law and up to o(1), by m_{rm r} + Tr(H_r-z)^{-1}, where H_r
 is a fixed diagonal plus independent GOE noise.
 
@@ -2044,11 +2044,11 @@ is a fixed diagonal plus independent GOE noise.
 
 Fix kge2, write M=M_k, and for Oin O(k) set
 D_O=I_k&00&O and
-B_O=D_OJ_kD_O^{mathsf T}sfT-J_k, which is symmetric. For an analytic
+B_O=D_OJ_kD_O^{T}-J_k, which is symmetric. For an analytic
 symmetric matrix function M with positive definite imaginary part, put
 
- mathcal T_O(m,M)=(m-∂_zcalT_O(m,M)=(m-∂_zlogdet(I_{2k}+B_OM),,
- D_O^{mathsf T}sfT(M^{-1}+B_O)^{-1}D_O).
+ T_O(m,M)=(m-∂_zcalT_O(m,M)=(m-∂_zlogdet(I_{2k}+B_OM),,
+ D_O^{T}{-1}+B_O)^{-1}D_O).
 
 This is well defined: M is invertible and
 Im(M^{-1})=-M^{-*}(Im M)M^{-1}<0, so M^{-1}+B_O is invertible and
@@ -2057,19 +2057,19 @@ derivative is used, so no branch of the logarithm is chosen.
 
 [Continuous switching symmetry]
 For every fixed kge2 and every Oin O(k), the joint law of (m,M_k)
-under nu is invariant under mathcal T_O.calT_O.
+under nu is invariant under T_O.
 
 
 Proof. 
 Let O be a permutation matrix. On the event, of probability
 1-O_k(n^{-1}), that the 2k sampled endpoints are distinct and induce
-exactly the sampled matching, the map Amapsto A+WB_OW^{mathsf T}sfT,
-Wmapsto WD_O permutes the pairing between the two endpoint lists, is a
+exactly the sampled matching, the map A|-> A+WB_OW^{T},
+W|-> WD_O permutes the pairing between the two endpoint lists, is a
 bijection of the restricted graph/list space preserving all weights, and
 is inverted by the inverse permutation. The determinant and inverse
 identities for finite-rank perturbations give
 widehatwidehat m_n=m_n-∂_zm_n=m_n-∂_zlogdet(I_{2k}+B_OM_{n,k}) and
-widehatwidehat M_M_{n,k}=D_O^{mathsf T}sfT(M_{n,k}^{-1}+B_O)^{-1}D_O (the factor
+widehatwidehat M_M_{n,k}=D_O^{T}{n,k}^{-1}+B_O)^{-1}D_O (the factor
 n^{-1} in m_n=n^{-1}Tr R supplies the derivative). Compact-open
 convergence of analytic functions gives convergence of derivatives, and
 positive-definite imaginary parts make the inverse regular, so passing to
@@ -2078,13 +2078,13 @@ the limit proves the claim for permutation matrices.
 For Uin O(k) put Q_U=diag(U,U). As C_k and K_k(m_{rm r})
 commute with Q_U, the Gaussian representation
 (eq:marked-representation) shows that
-mathcal R_U(m,M)=(m,Q_U^calR_U(m,M)=(m,Q_U^{mathsf T}sfTMQ_U) preserves the joint law. Since
-Q_UD_OQ_U^{mathsf T}sfT=D_{UOU^{mathsf T}sfT} and
-Q_UB_OQ_U^{mathsf T}sfT=B_{UOU^{mathsf T}sfT}, we get
-mathcal R_U^calR_U^{-1}mathcal T_OcalT_Omathcal R_U=calR_U=mathcal T_calT_{UOU^{mathsf T}sfT};
+R_U(m,M)=(m,Q_U^calR_U(m,M)=(m,Q_U^{T} preserves the joint law. Since
+Q_UD_OQ_U^{T}{UOU^{T}} and
+Q_UB_OQ_U^{T}{UOU^{T}}, we get
+R_U^{-1}T_O R_U=calR_U=T_{UOU^{T}};
 so the law is invariant under every orthogonal conjugate of a permutation
-switch. The maps satisfy B_{OO'}=B_O+D_OB_{O'}D_O^{mathsf T}sfT and
-D_{OO'}=D_OD_{O'}, so mathcal T_OcalT_Omathcal T_calT_{O'}=mathcal T_calT_{OO'}
+switch. The maps satisfy B_{OO'}=B_O+D_OB_{O'}D_O^{T} and
+D_{OO'}=D_OD_{O'}, so T_O T_calT_{O'}=T_{OO'}
 (the inverse updates compose and the determinant ratios multiply before
 differentiation). A transposition is a hyperplane reflection; its
 orthogonal conjugates are all hyperplane reflections, and finite products
@@ -2115,11 +2115,11 @@ beta be a diagonal matrix of eigenvalues of Gamma_r and set
 
 Index the limiting positions with multiplicity, put
 I={j:|x_j|le K}, q_*=|I| and D_I=diag(x_j:jin I). Conditionally
-on nu, let mathcal WcalW be a raw GOE matrix of size q_* (variances 1
+on nu, let W be a raw GOE matrix of size q_* (variances 1
 off the diagonal and 2 on it), independent of nu apart from its
 dimension, and set
 
- H_r=D_I+s_1Id_{q_*}+√(s_2),mathcal WcalW .
+ H_r=D_I+s_1Id_{q_*}+√(s_2),W .
 
 
 [Gaussian insertion]
@@ -2135,10 +2135,10 @@ jointly at every fixed finite list of zinC_+, and the unscaled
 eigenvalue point measure of H_r converges locally vaguely in law to
 nu. More precisely, generating the Gaussian marks of M_k
 conditionally on nu separately for each r, and writing
-mathcal S_r(z)=m(z)-∂_zcalS_r(z)=m(z)-∂_zlogdet(I_{2k}+BM_k(z)), expectations of
+S_r(z)=m(z)-∂_zcalS_r(z)=m(z)-∂_zlogdet(I_{2k}+BM_k(z)), expectations of
 every bounded smooth function with bounded derivatives of finitely many
 real and imaginary trace values differ by o(1) when
-m_{rm r}+Tr(H_r-z)^{-1} is replaced by mathcal S_r(z).calS_r(z).
+m_{rm r}+Tr(H_r-z)^{-1} is replaced by S_r(z).
 
 
 First cut at L. Elementary consequences of (eq:discrepancy)
@@ -2167,16 +2167,16 @@ gives
 with probability tending to one (using Lgg r^4). As B^{-1} is real
 symmetric, this bounds |(B^{-1}+M)^{-1}| and |(B^{-1}+M_L)^{-1}| by
 r^5, so cutting both scalar and marked sums in
-mathcal S_r(z)=m(z)-calS_r(z)=m(z)-Tr(B^{-1}+M(z))^{-1}M'(z) costs o_{PP}(1): the
-marked error is O_{PP}(rcdot r^5cdot r^{-98})+O_{PP}(rcdot r^{10}
-cdot r^{-98}cdot r^2)=O_{PP}(r^{-85}), and the scalar tail is
+S_r(z)=m(z)-calS_r(z)=m(z)-Tr(B^{-1}+M(z))^{-1}M'(z) costs o_{PP}(1): the
+marked error is O_{PP}(r· r^5· r^{-98})+O_{PP}(r· r^{10}
+· r^{-98}· r^2)=O_{PP}(r^{-85}), and the scalar tail is
 O(L^{-chi}). Diagonalizing Gamma_r and rotating the independent
-Gaussian row vectors in channel space gives a matrix mathcal GcalG of
+Gaussian row vectors in channel space gives a matrix G of
 independent standard Gaussian entries with 4r columns, and the
 finite-rank inverse identity (Woodbury) yields
 
- mathcal S_r(z)=m_calS_r(z)=m_{rm r}+Tr(D+A'-z)^{-1}+o_{PP}(1),
- A'=mathcal GcalGbetamathcal G^calG^top, D=diag(x_j:|x_j|le L).
+ S_r(z)=m_{rm r}+Tr(D+A'-z)^{-1}+o_{PP}(1),
+ A'=G G^calG^top, D=diag(x_j:|x_j|le L).
 
 The coefficient bounds follow from
 (B^{-1}+M^0)^{-1}=(Id+BM^0)^{-1}B=B+O(theta^2): every singular value
@@ -2195,11 +2195,11 @@ In the estimates below, polylog(r) denotes a fixed power
 of log r whose exponent may rise from one occurrence to the next.
 
 [Weighted block estimates]
-For mathcal AcalAin{I,F} put a_I=h, a_F=K. With probability tending
+For A{I,F} put a_I=h, a_F=K. With probability tending
 to one,
 
- bigl|U_{mathcal A}calA(A'_{mathcal AcalAmathcal A}calA-s_1Id)U_{mathcal A}calAbigr|
- lepolylog(r)(sqrt{T/a_{mathcal A}calA}+theta),
+ bigl|U_{A}'_{A A}calA-s_1Id)U_{A}|
+ lepolylog(r)(sqrt{T/a_{A}}+theta),
 
 and, for P_0=U_IA'_{IF}U_F,
 
@@ -2210,15 +2210,15 @@ and, for P_0=U_IA'_{IF}U_F,
 Proof. 
 The centered matrix in (eq:sw-bernstein) is a sum of independent
 self-adjoint sum_{alpha=1}^{4r}beta_{alphaalpha}(v_alpha
-v_alpha^top-P) with P=U_{mathcal A}calA^2 and v_alpha Gaussian of
+v_alpha^top-P) with P=U_{A} and v_alpha Gaussian of
 covariance P. Gaussian fourth moments give
 EE[(v_alpha v_alpha^top)^2]=(Tr P)P+2P^2. Truncating each Gaussian
 column at |entry|lelog r (a superpolynomially small event, with
 superpolynomially small centering correction) leaves summands of norm at
 most C|beta|(log r)^2Tr Plethetapolylog(r), and
 bounds the matrix variance parameter by Cs_2(Tr P)|P|le
-(T/a_{mathcal A}calA)polylog(r). The bounded matrix Bernstein
-inequality gives a two-sided tail 2|mathcal A|calA|exp(-ct^2/
+(T/a_{A} The bounded matrix Bernstein
+inequality gives a two-sided tail 2|A|calA|exp(-ct^2/
 (sigma_{rm mat}^2+bt)), and a large power of log r in t
 proves (eq:sw-bernstein). For the cross block, independence of
 distinct Gaussian rows gives EE|P_0|_{HS}^2=s_2(Tr U_I^2)(Tr U_F^2),
@@ -2253,9 +2253,9 @@ Q_F=U_F^{-1}G_FU_F^{-1} has bounded norm. Also
 For Σ, replacing G_F by D_F^{-1} costs at most
 |P_0|_{HS}^2delta_Fle Tpolylog(r)delta_F in the
 U_I-weighted norm. For A'_{IF}D_F^{-1}A'_{FI}, condition on the inner
-block and set Q=mathcal G_IcalG_Ibeta^2mathcal G_I^calG_I^top; the columns a_j
+block and set Q=G_I G_I^calG_I^top; the columns a_j
 of A'_{IF}, jin F, are independent centered Gaussian with covariance
-Q, so EE[sum_{jin F}a_ja_j^top/x_jmidcdot]=Qsum_{jin F}x_j^{-1}
+Q, so EE[sum_{jin F}a_ja_j^top/x_jmid·]=Qsum_{jin F}x_j^{-1}
 and the entrywise variance is (Q_{aa}Q_{bb}+Q_{ab}^2)sum_{jin F}x_j^{-2}.
 With EE[Q_{aa}^2]=s_2^2+2Trbeta^4=O(T^2) and |Q_{ab}|^2le Q_{aa}Q_{bb},
 (eq:sw-tail-sums) and (eq:sw-weight-bounds) give
@@ -2309,7 +2309,7 @@ most |U_I^{-1}widetilde G_IU_I^{-1}||P_0|_{HS}^2|Q_FU_F^2Q_F|
 Conditional on almost every nu, bounded smooth tests with bounded
 derivatives of finitely many trace values have expectations differing by
 o(1) between D_I+A'_{II} and
-H_r=D_I+s_1Id_{q_*}+√(s_2),mathcal W.calW.
+H_r=D_I+s_1Id_{q_*}+√(s_2),W.
 
 
 Proof. 
@@ -2318,8 +2318,8 @@ Write A'_{II}-s_1Id_{q_*}=sum_{alpha=1}^{4r}beta_{alphaalpha}
 g_alpha. Since Cov(g_ag_b,g_cg_d)=delta_{ac}delta_{bd}+delta_{ad}
 delta_{bc}, each g_alpha g_alpha^top-Id and a raw GOE matrix have
 equal mean (zero) and entry covariances; replace the summands one at a time
-by beta_{alphaalpha}mathcal W_calW_alpha with independent raw GOE
-mathcal W_calW_alpha (negative coefficients allowed, both covariances scaling
+by beta_{alphaalpha}W_ with independent raw GOE
+W_ (negative coefficients allowed, both covariances scaling
 by beta_{alphaalpha}^2). For either kind of summand X,
 EE|X|^3le Ctheta^3q_*^3 (Gaussian moments of |g|^2; Hilbert--
 Schmidt norm for GOE). Differentiating the resolvent in a symmetric
@@ -2329,16 +2329,16 @@ C_{z,Phi}q_*^3|Delta|^3 at every real symmetric intermediate matrix.
 Taylor expansion to second order cancels means and covariances in each
 replacement, and the summed third-order remainder is
 C_{z,Phi}q_*^6rtheta^3=O(r^{-0.44+(3/2)tau})=o(1). Finally
-sum_alphabeta_{alphaalpha}mathcal W_calW_alpha has the law
-√(s_2),mathcal W.calW.
+sum_alphabeta_{alphaalpha}W_ has the law
+√(s_2),W.
 
 
 Proof. [Completion of Proposition~prop:gaussian-insertion]
 The coefficient bounds were proved after (eq:sw-finite-matrix). That
 equation, Lemma~sw:trace-cut and Lemma~sw:lindeberg prove the
-conditional comparison with mathcal S_r.calS_r. Bounded convergence averages
+conditional comparison with S_r. Bounded convergence averages
 the conditional error, and by Proposition~prop:continuous-switching
-the unconditional law of mathcal S_rcalS_r is the law of m for every r;
+the unconditional law of S_r is the law of m for every r;
 this proves (eq:sw-trace-limit).
 
 For the point measures, let nu^{(r)} be the eigenvalue counting measure
@@ -2429,9 +2429,9 @@ conditional matrices of Proposition~prop:gaussian-insertion:
 K=r^{1/100}, T=r^{tau} with 0<tau<chi/100000,
 I={j:|x_j|le K}, q_*=|I|, and
 
- H_r=D_I+s_1Id+√(s_2),mathcal W_r,calW_r, |s_1|le CT, cTle s_2le CT,
+ H_r=D_I+s_1Id+√(s_2),W_r, |s_1|le CT, cTle s_2le CT,
 
-with mathcal W_rcalW_r raw GOE of dimension q_*. By (eq:discrepancy),
+with W_r raw GOE of dimension q_*. By (eq:discrepancy),
 
  q_*=2rho K+O(K^{1-chi}), T=q_*^{100tau+o(1)}.
 
@@ -2505,22 +2505,22 @@ preserve (eq:dbm-time-window) for c_r^2t_*.
 By (eq:dbm-normalized-matrix) the rescaled eigenvalues supplied by
 Theorem~thm:dbm-input are
 
- (q_*)/(pi)cdot(c_ralpha_*)/(q_*)lambda_j(H_r)=c_rrho,lambda_j(H_r),
+ (q_*)/(pi)·(c_ralpha_*)/(q_*)lambda_j(H_r)=c_rrho,lambda_j(H_r),
 
 with c_r=1 at exceptional initial values.
 
 
-Let mathcal G_m=calG_m=sum_{j=1}^mdelta_{(m/pi)lambda_j(W_m)}. These
+Let G_m={j=1}^mdelta_{(m/pi)lambda_j(W_m)}. These
 converge vaguely in law along all dimensions to a common locally finite
-process mathcal S_calS_{GOE}: the eigenvalues of sqrt m,W_m have
+process S_{GOE}: the eigenvalues of sqrt m,W_m have
 density propto e^{-frac14sum y_j^2}prod_{i<j}|y_i-y_j|, and
 Theorem~1 of Valk'o--Vir'ag (for beta=1, center zero) gives vague
 convergence after multiplying by 2sqrt m; dividing by 2pi gives
-mathcal G_m.calG_m.
+G_m.
 
 
 For every compact JsubsetR and every b>0,
-sup_{mge1}EEexp(b,mathcal G_m(J))<∞.calG_m(J))<∞.
+sup_{mge1}EEexp(b,G_m(J))<∞.calG_m(J))<∞.
 
 
 Proof. 
@@ -2528,7 +2528,7 @@ Bound the GOE count by twice a Gaussian-unitary count plus one via the
 Gaussian decimation identity of Forrester--Rains. Let mathrm O_mrmO_m be the
 decreasingly ordered points with density
 proptoDelta(y)prod_i b_0(y_i), b_0=e^{-y^2/2}, the law of
-sqrt m,W_m/sqrt2; then mathcal G_m(J)calG_m(J) is the mathrm O_mrmO_m count in
+sqrt m,W_m/sqrt2; then G_m(J) is the mathrm O_mrmO_m count in
 J_m=pi J/√(2m), so |J_m|=O(m^{-1/2}). Superposing independent
 mathrm O_mrmO_m and mathrm O_rmO_{m+1} gives density
 P(z)=sum_{|S|=m}Delta(z_S)Delta(z_{S^c}), which by the
@@ -2558,7 +2558,7 @@ EE(N_m^{even})_ale(2√(2m),|J_m|)^ale C_J^a, so
 EE e^{vN_m^{even}}leexp(C_J(e^v-1)). As the
 superposition count is at most twice its even count plus one and dominates
 the mathrm O_mrmO_m count,
-EE e^{bmathcal G_m(J)}calG_m(J)le e^bEE e^{2bN_m^{even}}.
+EE e^{bG_m(J)} e^bEE e^{2bN_m^{even}}.
 
 
 
@@ -2578,29 +2578,29 @@ seen by Bernoulli variables of success probabilities f(y_i). Conditional
 on almost every nu, let zeta_r have coordinates c_rrho,lambda_j(H_r);
 Lemma~dbm:regularity-density and Theorem~thm:dbm-input with
 test prod_j f(alpha_j) give, for each fixed ell,
-EE[B_ell(zeta_r)midnu]-EE B_ell(mathcal G_calG_{q_*})to0. To close
+EE[B_ell(zeta_r)midnu]-EE B_ell(G_{q_*})to0. To close
 the brackets uniformly in q_*, take Jsupsetsupp h; since
-A_a(mathcal G_m)calG_m)lebinom{mathcal G_m(J)}calG_m(J)ale2^{-a}3^{mathcal G_m(J)}calG_m(J)
+A_a(G_m){G_m(J)}{-a}3^{G_m(J)}
 (from sum_a2^abinom Na=3^N), Lemma~lem:goe-counts gives
 
- sup_mEE[B_{2ell}(mathcal G_m)-B_calG_m)-B_{2ell+1}(mathcal G_m)]calG_m)]le C_J2^{-(2ell+1)}.
+ sup_mEE[B_{2ell}(G_m)-B_calG_m)-B_{2ell+1}(G_m)]calG_m)]le C_J2^{-(2ell+1)}.
 
 Using (eq:dbm-bonferroni) for both models, sending rto∞ at
 fixed ell and then ellto∞, gives
 
  EEBigl[exp(-int h,ddzeta_r)midnuBigr]to
- EEexp(-int h,ddmathcal S_calS_{GOE}).
+ EEexp(-int h,ddS_{GOE}).
 
 Both sides lie in [0,1], so bounded convergence removes the conditioning.
-For c>0 let mathcal D_ccalD_czeta scale positions by c; this map is
+For c>0 let D_c scale positions by c; this map is
 continuous in (c,zeta) for c near a positive number and zeta
 locally vaguely convergent. Since c_rto1 almost surely,
-zeta_r=mathcal D_calD_{c_rrho}nu^{(r)}⇒mathcal D_calD_rhonu, and
+zeta_r=D_{c_rrho}nu^{(r)}⇒D_, and
 bounded continuity of the Laplace functional of a compactly supported
 continuous nonnegative test gives
 
- EEexp(-int h,ddmathcal D_calD_rhonu)
- =EEexp(-int h,ddmathcal S_calS_{GOE})
+ EEexp(-int h,ddD_
+ =EEexp(-int h,ddS_{GOE})
 
 for 0le hin C_c^∞, extended to 0le hin C_c by convolution with
 a nonnegative approximate identity and bounded convergence.
@@ -2608,9 +2608,9 @@ a nonnegative approximate identity and bounded convergence.
 Proof. [Proof of Theorem~thm:main]
 Start with any sequence of admissible sizes; Proposition~prop:microscopic-limit
 supplies a further subsequence on which nu_n⇒nu as above. The
-graph point measure is mathcal D_calD_rhonu_n, and by bounded continuity of
+graph point measure is D_, and by bounded continuity of
 its Laplace functional and (eq:dbm-limit-identification) its Laplace
-expectation along this subsequence tends to that of mathcal S_calS_{GOE};
+expectation along this subsequence tends to that of S_{GOE};
 the reference GOE expectations at dimension n tend to the same value along
 all sizes. Every original subsequence thus has a further subsequence on
 which the difference tends to zero, proving the full-sequence limit. All