diff --git a/spherical_perceptron_bi_orthogonal_v2.txt b/spherical_perceptron_bi_orthogonal_v2.txt
index 8073d32..328ba8a 100644
--- a/spherical_perceptron_bi_orthogonal_v2.txt
+++ b/spherical_perceptron_bi_orthogonal_v2.txt
@@ -1,6 +1,6 @@
 Let A be a real Mtimes N matrix, and let sigma_N denote uniform
-probability measure on the sphere S^N={xinmathbb R^N:bbR^N:|x|^2=N}.
-For a bounded continuous f:mathbb RbbRtomathbb R,bbR, the normalized
+probability measure on the sphere S^N={xinR^N:bbR^N:|x|^2=N}.
+For a bounded continuous f:R, the normalized
 log-partition function of the spherical perceptron is
 
 F_N(A,f)=frac1Nlogint_{S^N}exp{sum_{r=1}^M f((Ax)_r)}dsigma_N(x).
@@ -130,7 +130,7 @@ Panchenko and Talagrand and Panchenko's scalar ultrametricity theorem
 enter where the matrix symmetry of the overlap is proved. The argument
 keeps an independent residual Gaussian noise on every visit
 to a cascade leaf and verifies the marking rules for the fractional
-moments and zero restrictions. ./refs/ghirlanda-guerra-1998.pdf
+moments and zero restrictions. ./pdf/ghirlanda-guerra-1998.pdf
 ./pdf/panchenko-2013.pdf
 
 The singular values of A describe the strength and rank of the
@@ -140,7 +140,7 @@ when rows or columns are null. Let delta_a denote the Dirac mass at
 a, and let supp,nu denote the support of a probability
 measure nu on [0,∞).
 
-Fix alpha>0, M=lflooralpha Nrfloor, and fin C_b(mathbb R).bbR).
+Fix alpha>0, M=floor(alpha N), and fin C_b(R).
 Suppose the law of A_N is unchanged under left and right orthogonal
 multiplication, so A_Noverset{d}{=}UA_NV for any independent
 uniform orthogonal Uin O(M), Vin O(N), and suppose the empirical
@@ -161,13 +161,13 @@ Approximate the singular slots in operator norm by finitely many
 positive values s_1,...,s_h with multiplicities n_j satisfying
 n_j/Mto p_j>0. Then sum_jp_j=min(1,1/alpha). Put
 v_j=alpha p_j, and when alpha<1 add an input null block with
-v_0=1-alpha. Let mathcal I=calI={1,...,h} when alphage 1 and
-mathcal I=calI={0,1,...,h} when alpha<1. For each block
-iinmathcal IcalI let rho_i denote the fraction of the input squared
+v_0=1-alpha. Let I={1,...,h} when alphage 1 and
+I={0,1,...,h} when alpha<1. For each block
+iinI let rho_i denote the fraction of the input squared
 norm assigned to block i, with rho_i>0 and sum_irho_i=1. The
 value for this finite approximation is
 
-sup_{rho}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)},
+sup_{rho}{frac12sum_{iinI}/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)},
 
 where P_p^f is the angular pressure characterised by the Gaussian theorem and
 p=(p_j)_j fixes every block proportion. The limit of these
@@ -176,10 +176,10 @@ approximations as the spectral mesh tends to zero is F_{alpha,nu}(f).
 The term frac12sum v_ilog(rho_i/v_i) is the logarithmic Dirichlet
 density, at rate per input coordinate, for distributing the squared
 length of a spherically uniform input across blocks of dimensions
-v_iN. The factor alpha converts the per-mathbb R^MbbR^M angular
-pressure into a per-mathbb R^NbbR^N pressure.
+v_iN. The factor alpha converts the per-R^M angular
+pressure into a per-R^N pressure.
 
-Let E_1,...,E_hsubsetmathbb R^MbbR^M be mutually orthogonal Haar
+Let E_1,...,E_hsubsetR^M be mutually orthogonal Haar
 subspaces with dim E_j=n_j and n_j/Mto p_j>0, so p=sum_jp_jle 1.
 For a_jge 0, let sigma_{E_j,Ma_j} denote uniform probability
 measure on the sphere {yin E_j:|y|^2=Ma_j} of squared radius
@@ -194,7 +194,7 @@ P_M^f(a) converges in probability, uniformly on compact sets of
 ain[0,∞)^h, to a deterministic continuous function P_p^f(a).
 For p<1 and positive a, the limit equals
 
-P_p^f(a)=J_p-H_p(a)+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta),
+P_p^f(a)=J_p-H_p(a)+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}B_{p,a}(kappa,lambda,B,theta),
 
 for every sufficiently large fixed kappa, with
 
@@ -210,7 +210,7 @@ P_p^f(a)=lim_{eta↓ 0}P_{(1-eta)p}^f(a) with a fixed.
 
 Let d_j=h-j+1. For each j, let R_j be an upper triangular
 d_jtimes d_j matrix with positive diagonal and every column norm at
-most kappa. For a row vector xi_jinmathbb R^bbR^{d_j}, label the
+most kappa. For a row vector xi_jinR^{d_j}, label the
 entries of xi_j^TR_j by x,j and z_i,j for jle i<h. Write
 
 y=sum_j(xi_j^TR_j)_{x,j}, u_i=sum_{jle i}(xi_j^TR_j)_{z_i,j}.
@@ -222,9 +222,9 @@ g_1=y^2, c_1=s_a; g_2=-min(y^2,B), c_2=-s_a; g_{i+2}=(y-u_i)^2, c_{i+2}=sum_{j>i
 
 Define
 
-mathcal B_calB_{p,a}(kappa,lambda,B,theta)=sup_{R_1,...,R_h}[sum_jp_j(log(kappa√((2pi e)/(p_j)))+logfrac{R_{j,11}}kappa)+thetasum_{ell=2}^{d_j}p_jlogfrac{R_{j,ellell}}kappa+thetamin_{0le t_elllelambda/theta}{tcdot c+mathcal G_calG_{theta,p}((f(y))/(theta)-tcdot g)}],
+B_{p,a}(kappa,lambda,B,theta)=sup_{R_1,...,R_h}[sum_jp_j(log(kappa√((2pi e)/(p_j)))+logfrac{R_{j,11}}kappa)+thetasum_{ell=2}^{d_j}p_jlogfrac{R_{j,ellell}}kappa+thetamin_{0le t_elllelambda/theta}{t· c+G_{theta,p}((f(y))/(theta)-t· g)}],
 
-where mathcal G_calG_{theta,p} is the Gaussian functional obtained from
+where G_{theta,p} is the Gaussian functional obtained from
 the powered Stiefel integral. The first column of
 R_j has weight p_j and fires the outer entropy. The remaining
 columns have weight p_jtheta and the entropy of those columns
@@ -237,19 +237,19 @@ sought from the vectors that witness its projection constraints. Fix
 arbitrary column counts d_jge 1 and a positive exponent
 0<theta<1. Let n_j/Mto p_j>0 and let
 
-X_jinSt(n_j,d_j):={Xinmathbb R^bbR^{n_jtimes d_j}:X^TX=n_jI_{d_j}}
+X_jinSt(n_j,d_j):={XinR^{n_jtimes d_j}:X^TX=n_jI_{d_j}}
 
 be uniform on the Stiefel manifold. The first columns, jointly across
 species, are outer variables; the remaining columns have the
 conditional Stiefel law and are inner variables. For independent
-standard Gaussian vectors g_{r,j}inmathbb R^bbR^{n_j}, define the row
+standard Gaussian vectors g_{r,j}inR^{n_j}, define the row
 fields
 
 xi_{r,j}(X)=frac{X_j^Tg_{r,j}}{√(n_j)}, 1le rle M.
 
 For a Hamiltonian H define
 
-mathcal Z_calZ_{M,theta}(H)=[int(int e^H,dmu_{in})^theta dmu_{out}]^{1/theta}.
+Z_{M,theta}(H)=[int(int e^H,dmu_{in})^theta dmu_{out}]^{1/theta}.
 
 The natural tilted path law samples the outer variables with density
 proportional to the inner partition function to power theta, and then
@@ -273,36 +273,36 @@ containing theta. A trial in block j consists of symmetric matrices
 
 with the pinning condition q_{j,i}e_1=e_1 whenever zeta_igetheta.
 Write q_j(u)=q_{j,i} on [zeta_i,zeta_{i+1}). Let
-F_{j,i}=mathbb R^bbR^{d_j} before the sharing level and
+F_{j,i}=R^{d_j} before the sharing level and
 F_{j,i}=e_1^perp afterwards, and put F_{j,k}={0}. The
 susceptibility and precision are
 
-mathsf K_sfK_{j,i}=sum_{r=i+1}^kzeta_r(q_{j,r}-q_{j,r-1}^{phantom{}}), mathsf P_sfP_{j,i}=(mathsf K_sfK_{j,i}|_{F_{j,i}})^{-1}.
+K_{j,i}=sum_{r=i+1}^kzeta_r(q_{j,r}-q_{j,r-1}^{phantom{}}), P_{j,i}=(K_{j,i}|_{F_{j,i}})^{-1}.
 
 An eligible trial has strictly positive susceptibility on every
 nonempty active space. The precision extends by zero on the orthogonal
 complement. Let q_j^0 be zero on active coordinates and the identity
 on shared coordinates, with q_{j,k}^0=I, and define
-mathsf K_sfK_{j,i}^0 analogously. The block entropy is
+K_{j,i}^0 analogously. The block entropy is
 
-2S_theta(q_j)=tr(mathsf P_sfP_{j,0}q_{j,0})+sum_{i=1}^kfrac1{zeta_i}log(frac{detmathsf K_sfK_{j,i-1}}{detmathsf K_sfK_{j,i}}frac{detmathsf K_sfK_{j,i}^0}{detmathsf K_sfK_{j,i-1}^0}),
+2S_theta(q_j)=tr(P_{j,0}q_{j,0})+sum_{i=1}^kfrac1{zeta_i}log(frac{detK_{j,i-1}}{detK_{j,i}}frac{detK_{j,i}^0}{detK_{j,i-1}^0}),
 
 where each determinant is taken on the active space at its own index.
 The normalisation compensates for the dimension change at theta.
 
-For a continuous terminal D:prod_jmathbb R^bbR^{d_j}tomathbb RbbR
+For a continuous terminal D:prod_jR^{d_j}toR
 satisfying -C(1+|z|^2)le D(z)le C for some C<∞, let
 Z_0,...,Z_k be independent centred Gaussian vectors with
 block-diagonal covariances
 diag_j(q_{j,i}-q_{j,i-1}). Set U_k=D and define backwards,
 for i=k,k-1,...,1,
 
-U_{i-1}(z)=frac1{zeta_i}logmathbb EbbEexp{zeta_iU_i(z+Z_i)}.
+U_{i-1}(z)=frac1{zeta_i}logE{zeta_iU_i(z+Z_i)}.
 
-The single-pattern value is V_D(q)=mathbb EU_0(Z_0);bbEU_0(Z_0); the outer
+The single-pattern value is V_D(q)=EU_0(Z_0); the outer
 expectation averages the root Gaussian data. Define
 
-mathcal G_calG_{theta,p}(D)=inf_q{sum_{j=1}^hp_jS_theta(q_j)+V_D(q)},
+G_{theta,p}(D)=inf_q{sum_{j=1}^hp_jS_theta(q_j)+V_D(q)},
 
 over all eligible finite grids and trials. Refinements use a common
 grid for all blocks. The diagonal increment I-q_{j,k-1} is integrated
@@ -314,18 +314,18 @@ The Gaussian theorem states that, with the Stiefel model, fixed
 of a bounded continuous function and a nonpositive quadratic form, the
 normalized log integral satisfies
 
-frac1Mlogmathcal Z_calZ_{M,theta}(sum_{r=1}^MD(xi_r))→mathcal G_calG_{theta,p}(D)
+frac1MlogZ_{M,theta}(sum_{r=1}^MD(xi_r))→G_{theta,p}(D)
 
 in probability and in expectation. More generally, let each g_ell
 be bounded continuous or a nonnegative quadratic form, let
-barbar g=M^g=M^{-1}sum_rg(xi_r), and fix d>0 and cinmathbb R^s.bbR^s.
+barbar g=M^g=M^{-1}sum_rg(xi_r), and fix d>0 and cinR^s.
 The pressure for the soft-penalised Hamiltonian
 
 H=sum_rD(xi_r)-Mdsum_{ell=1}^s(barbar g_g_ell-c_ell)_+
 
 converges in probability to
 
-min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.
+min_{tin[0,d]^s}{t· c+G_{theta,p}(D-t· g)}.
 
 No restriction on sum_jp_j is needed.
 
@@ -335,9 +335,9 @@ construction. The base bounded case is trivial for the estimates. The
 negative quadratic form is the shape produced when a shrinkage penalty
 enforces bounded squared norm. A positive quadratic would increase
 exponentially with the field and break the variational formula, so the
-sign restriction excludes exactly that case. In mathcal B_calB_{p,a} the
+sign restriction excludes exactly that case. In B_{p,a} the
 row tests g_ell are nonnegative quadratics, and the contributions
--tcdot g fall in the negative-quadratic class.
+-t· g fall in the negative-quadratic class.
 
 Reparametrise the powered integral as the partition function of a
 branching marked cascade. Choose a Poisson point process on the
@@ -359,10 +359,10 @@ limits at each mark are as in the single-root case: spatial frames
 converge to uniform Stiefel points, and read covariances are the
 paths q.
 
-Because mathcal Z_calZ_{M,theta}(H) is a normalised log of a partition
+Because Z_{M,theta}(H) is a normalised log of a partition
 function over independent patterns and independent disorder, standard
 subadditivity and Gaussian concentration bound the variance of
-M^{-1}logmathcal Z_calZ_{M,theta}(H) by O(M^{-1}). Consequently the
+M^{-1}logZ_{M,theta}(H) by O(M^{-1}). Consequently the
 normalised log concentrates at its mean, and convergence in probability
 and convergence in expectation are equivalent.
 
@@ -370,11 +370,11 @@ Overlaps and matrix synchronization. Write replicas as superscripts
 a,b. The scalar and matrix overlaps of a configuration (X,A) at
 the disorder level are
 
-mathcal R_j^calR_j^{ab}=((X_j^a)^TX_j^b)/(n_j).
+R_j^{ab}=((X_j^a)^TX_j^b)/(n_j).
 
 The array indexed jointly by replica and column is Gram and the self
 blocks are fixed at I. The perturbing scalarizations are
-v^T(mathcal R_j^calR_j^{ab})^{circell}v for ell=1,2 and a countable
+v^T(R_j^{ab})^{circell}v for ell=1,2 and a countable
 dense set of vectors v, together with the numerical label overlap.
 Each such scalarization is a Gram kernel: take tensor products of the
 column vectors for the entrywise powers. Products, nonnegative
@@ -388,9 +388,9 @@ hold in the limit. Precisely, for every bounded test F of the
 complete arrays on the first r replicas and every continuous test
 psi of the generated scalarization-and-label vector on one edge,
 
-mathbb E<bbE< Fpsi(T^{1,r+1})>=frac1rmathbb E<bbE< F>,mathbb E<bbE<psi(T^{12})>+frac1rsum_{a=2}^rmathbb E<bbE< Fpsi(T^{1a})>,
+E<bbE< Fpsi(T^{1,r+1})>=frac1rE<bbE< F>,E<bbE<psi(T^{12})>+frac1rsum_{a=2}^rE<bbE< Fpsi(T^{1a})>,
 
-where brackets denote Gibbs expectation and mathbb EbbE averages all
+where brackets denote Gibbs expectation and E averages all
 randomness. Monomial GG identities give all continuous tests of this
 generated vector by polynomial approximation on its compact range.
 At this stage the identities do not yet identify a nonsymmetric
@@ -402,9 +402,9 @@ identities and Panchenko's ultrametricity theorem for scalarized
 coordinates. A countable separating family of first-power quadratic
 forms is comonotone on an edge; summing over a countable family with
 positive weights gives a single order coordinate. If the matrix
-overlap mathcal R_j^calR_j^{12} were not symmetric in the limit, dense
+overlap R_j^{12} were not symmetric in the limit, dense
 first-power forms alpha and antisymmetric perturbations beta of
-mathcal R_j^calR_j^{12} would generate crossed values on edges 12 and
+R_j^{12} would generate crossed values on edges 12 and
 13 that the GG and ultrametricity structure forbid. Ultrametricity
 of each of two crossed coordinates forces edge 23 to have the lower
 value in each coordinate, contradicting ultrametricity of the sum.
@@ -424,14 +424,14 @@ deterministic matrix quantiles q_j(u), nondecreasing in Loewner
 order. Gram positivity and the fixed self overlap I imply
 0preceq q_jpreceq I. The label coordinate retains its prescribed
 atom masses, so its sharing threshold sits at quantile theta. When
-the first column is shared, mathcal R_j^calR_j^{ab}e_1=e_1; hence the
+the first column is shared, R_j^{ab}e_1=e_1; hence the
 matrix quantiles satisfy the pinning condition at that threshold.
 
 Variation of the single-pattern value. For smooth D, the functional
 V_D is Lipschitz in the sum of the L^1 path distances. Along a
 step variation,
 
-left.(d)/(dt)V_D(q+tdot q)|_{t=0}=-frac12sum_{j,i}w_i,tr(dot q_{j,i}mathbb E[M_bbE[M_{j,i}M_{j,i}^T]),
+left.(d)/(dt)V_D(q+tdot q)|_{t=0}=-frac12sum_{j,i}w_i,tr(dot q_{j,i}E[M_bbE[M_{j,i}M_{j,i}^T]),
 
 where M_{j,i} is the conditional mean of the terminal j-gradient
 under the successive tilted kernels. Gaussian covariance
@@ -439,7 +439,7 @@ differentiation inserts half the Hessian and the step coefficient
 times the gradient square; propagating these through the cascade makes
 the Hessian terms telescope, and consecutive coefficients differ by
 w_i. The conditional gradients form a martingale, so the second
-moment matrices G_{j,i}=mathbb E[M_bbE[M_{j,i}M_{j,i}^T] are nondecreasing
+moment matrices G_{j,i}=E[M_bbE[M_{j,i}M_{j,i}^T] are nondecreasing
 and uniformly bounded. Put T_{j,i}=sum_{r=i}^{k-1}w_r(widetilde
 q_{j,r}-q_{j,r}) and T_{j,k}=0. Summation by parts gives
 
@@ -479,7 +479,7 @@ kernels unchanged. Write gamma_r=N(0,I_r) and Q_1,Q_{Fmid1} for
 the marginal and conditional laws of this trial. The chain rule for
 relative entropy gives
 
-theta^{-1}Ent(Q_1midgamma_1)+mathbb E_bbE_{Q_1}Ent(Q_{Fmid1}midgamma_{d-1})letheta^{-1}Ent(N(0,Q)midgamma_d),
+theta^{-1}Ent(Q_1midgamma_1)+E_{Q_1}Ent(Q_{Fmid1}midgamma_{d-1})letheta^{-1}Ent(N(0,Q)midgamma_d),
 
 and this permits correlated trial coordinates. Hence
 
@@ -489,30 +489,30 @@ Choosing Q=I+tfrac12sign(C) yields
 
 Psi(C,b)-tfrac12tr,Cgetfrac14|C|_*-c_d/theta,
 
-with |cdot|_* the trace norm. When |b_{k-1}|le B_0, the
+with |·|_* the trace norm. When |b_{k-1}|le B_0, the
 comparison constant C=-(B_0+1)I gives Psi(C,b)le 0. Nested
 exponents dominate 1 and Jensen's bound controls the ordinary
 Gaussian exponential integral. Thus minimisers of
 Psi(C,b)-tr,C/2 lie in a compact set depending only on
 d,theta,B_0. Lower semicontinuity gives existence; finiteness
 forces strict precision inequalities, and differentiating in C
-gives the tilted self-covariance identity mathbb Ess^T=I.bbEss^T=I.
+gives the tilted self-covariance identity Ess^T=I.
 
-Under the tilted kernels, let mathcal H_icalH_i be the sigma-field
+Under the tilted kernels, let H_i be the sigma-field
 generated by the field increments up to zeta_i plus the shared spin
-coordinates when zeta_igetheta, and let M_i=mathbb E[sbbE[smid
-mathcal H_i].calH_i]. Let mathsf K_i^*sfK_i^* denote the Hessian of U_i in its
+coordinates when zeta_igetheta, and let M_i=E[sbbE[smid
+H_i].calH_i]. Let K_i^*sfK_i^* denote the Hessian of U_i in its
 field argument, supported on the active space F_i. Differentiating a
 single log-exponential integration gives the recursion
 
-mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_iCov(M_imidmathcal H_calH_{i-1}),
+K_{i-1}^*=K_i^*+sfK_i^*+zeta_iCov(M_imidH_{i-1}),
 
 with deterministic conditional covariances. Averaging and the
 martingale property yield
 
-q_i^*=mathbb E[M_iM_i^T],bbE[M_iM_i^T], mathsf K_k^*=0,sfK_k^*=0, mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_i(q_i^*-q_{i-1}^*), q_0^*=mathsf K_0^*b_0sfK_0^*b_0mathsf K_0^*.sfK_0^*.
+q_i^*=E[M_iM_i^T],bbE[M_iM_i^T], K_k^*=0,sfK_k^*=0, K_{i-1}^*=K_i^*+sfK_i^*+zeta_i(q_i^*-q_{i-1}^*), q_0^*=K_0^*b_0sfK_0^*b_0K_0^*.sfK_0^*.
 
-The inverse identity mathsf K_isfK_imathsf P_isfP_imathsf K_i=sfK_i=mathsf K_isfK_i
+The inverse identity K_i P_isfP_iK_i= K_isfK_i
 on the active space, combined with the pinning across theta,
 identifies q_0^* with the precision condition. The final statement
 is the canonical duality: the minimiser C(b) of
@@ -530,16 +530,16 @@ S_theta is strictly convex in the path in the
 2S=trPq+... sense: expanding A_{i-1}=A_i+zeta_i(e_{i-1}-e_i)
 successively cancels cross terms and leaves a sum of squares. In
 addition S_theta(q^0)=0 and the gradient at q^0 vanishes; hence
-S_thetage 0. At the canonical path, mathsf P_0q_0^*sfP_0q_0^*mathsf P_0=b_0.sfP_0=b_0.
+S_thetage 0. At the canonical path, P_0q_0^*sfP_0q_0^*P_0=b_0.
 Compress the inverse identity to each nested F_i and telescope; the
 scaled entropy gradient in the rightmost expression becomes
 b_i|_{F_i}. Uniqueness of the minimiser then holds.
 
 The dual values agree. Vary the field levels along the segment from
 zero to b. Covariance differentiation of Psi before subtracting
-the diagonal shift inserts tfrac12(mathsf K_i^*+sfK_i^*+zeta_imathbb
+the diagonal shift inserts tfrac12(K_i^*+sfK_i^*+zeta_imathbb
 EM_iM_i^T)bbEM_iM_i^T) for an increment at level i. Telescoping and
-mathbb Ess^T=IbbEss^T=I give
+Ess^T=I give
 
 dot J=-tfrac12sum_iw_itr(q_i^*dot b_i),
 
@@ -563,8 +563,8 @@ covariance comparison and the compact multiplier bound give local
 uniformity in b.
 
 Upper bound for the Gaussian theorem. Fix a trial path q in the
-Gaussian functional mathcal G_calG_{theta,p}. The proof produces an
-expected pressure upper bound that matches mathcal G_calG_{theta,p} at
+Gaussian functional G_{theta,p}. The proof produces an
+expected pressure upper bound that matches G_{theta,p} at
 this trial, uniformly in q; taking the infimum over q gives the
 theorem. The comparison follows Mourrat's supersolution method for
 mean-field spin glasses with vector spins, adapted to the
@@ -584,9 +584,9 @@ the spin columns.
 
 Define the contact functional
 
-A_{M,r}(b,u)=-mathbb EbbEmathcal P_calP_{M,r}(b,u)+sum_jp_jS_theta(tildetilde q_q_{j})+frac r{2}sum_iw_itr(tildetilde q_ib_i)+q_ib_i)+Delta(r,u),
+A_{M,r}(b,u)=-E P_calP_{M,r}(b,u)+sum_jp_jS_theta(tildetilde q_q_{j})+frac r{2}sum_iw_itr(tildetilde q_ib_i)+q_ib_i)+Delta(r,u),
 
-where mathcal P_calP_{M,r} is the expected (perturbed, enriched)
+where P_{M,r} is the expected (perturbed, enriched)
 pressure, tildetilde q_jq_j is the covariance of the linear field, b is
 the trial linear-field covariance path, u collects the perturbation
 parameters, and Delta is a small penalty that keeps the
@@ -594,7 +594,7 @@ perturbation sizes and b in a compact window.
 
 Contact-point argument. Suppose the trial q violates the desired
 bound, so that the true r=1 expected pressure exceeds
-mathcal G_calG_{theta,p} at that trial by 3delta for some
+G_{theta,p} at that trial by 3delta for some
 delta>0. Then A_{M,0}(0,u) is O(e_M^2) and A_{M,1}(b_{r=1},u)
 is negative. A large perturbation penalty is nonnegative. There is
 thus an interior minimiser of A with r>0, small perturbation
@@ -621,7 +621,7 @@ minimising perturbation parameters, the joint Ghirlanda-Guerra
 identities hold in the limit Mto∞, and the matrix overlap
 limits form an eligible matrix path. Supply of the GG identities uses
 the monomial perturbations. Concentration of
-M^{-1}logmathcal Z_calZ_{M,theta} on a bounded window of the
+M^{-1}logZ_{M,theta} on a bounded window of the
 perturbation parameters, combined with Gaussian interpolation on the
 perturbing fields, bounds the expected first derivative of the
 perturbed pressure by the amplitude of the perturbation. Markov's
@@ -635,7 +635,7 @@ valued function of the label quantile, and the pinning and Loewner
 order properties of the path quantile are exactly the properties
 required for the field at the minimiser to match an eligible trial
 path. Uniqueness of the path identifes tildetilde qq with the trial
-q, so the contact-point upper bound matches mathcal G_calG_{theta,p}
+q, so the contact-point upper bound matches G_{theta,p}
 at the trial q. Because the trial was arbitrary, the upper bound
 follows uniformly over q by the compactness of the trial paths.
 
@@ -658,7 +658,7 @@ with
 
 c_q=etamin_{r:F_rne{0}}{w_rlambda_{min}((I-q_r)|_{F_r})}>0,
 
-where strict positivity holds because I-q_rsucceqmathsf K_rsfK_rsucc 0
+where strict positivity holds because I-q_rsucceqK_r 0
 on F_r. Applying the Stiefel restriction bound with q', bounded
 pattern energy, and the O(e_M^2) perturbation cost, the contact
 functional is positive at a sufficiently large fixed trace cap. Thus a
@@ -708,7 +708,7 @@ C_a,s,s_M^2/M+C/(ssqrt M)
 for the integrated absolute fluctuation of the first derivative. At
 s=M^{-1/8} this implies
 
-intmathbb E<|Y_a-bbE<|Y_a-mathbb E<bbE< Y_a>|>/s_M,dpito 0.
+intE<|Y_a-bbE<|Y_a-E<bbE< Y_a>|>/s_M,dpito 0.
 
 The estimates hold on a slightly larger coupling interval so endpoint
 difference quotients cause no difficulty, and a diagonal selection
@@ -739,7 +739,7 @@ Gaussian columns proves the conditional version.
 Fix small ε>0 and restrict the enlarged partition function to
 the thick shell
 
-mathcal I=calI={|Z_j^TZ_j/r_j-I|leεhbox{ for every }j}.
+I={|Z_j^TZ_j/r_j-I|leεhbox{ for every }j}.
 
 This only lowers its logarithm. On the shell, replace the larger
 perturbation by the base perturbation at the same u. For fixed
@@ -799,18 +799,18 @@ canonical path q^*.
 
 Summing the pieces. Let V_D(q) denote the limiting expected log
 increment of one independent pattern at a GG limit; the entropy S_theta
-and the Gaussian functional mathcal G_calG_{theta,p} already entered
+and the Gaussian functional G_{theta,p} already entered
 via the canonical lemma. The cavity comparison identifies the
 enlarged pressure with ell times the canonical value minus o(ell)
 error, so
 
-frac1ellmathbb E[bbE[logmathcal Z_calZ_{M+ell}-logmathcal Z_M]calZ_M]gemathcal G_calG_{theta,p}(D)-o_{tol}(1).
+frac1ellE[bbE[logZ_{M+ell}-logZ_M]calZ_M]geG_{theta,p}(D)-o_{tol}(1).
 
 Uniformity in the base size. For every delta>0 there is ell_0
 such that for each integer ellgeell_0, every sequence of base
 sizes with n_j/Mto p_j and increments r_j=p_jell+O(1) satisfies
 
-liminf_{Mto∞}frac1ellintDelta_{M,ell}(u),dpi(u)gemathcal G_calG_{theta,p}(D)-delta,
+liminf_{Mto∞}frac1ellintDelta_{M,ell}(u),dpi(u)geG_{theta,p}(D)-delta,
 
 where Delta_{M,ell} is the log-increment of the enlarged pressure
 beyond the base at parameter u. The constant in O(1) is fixed
@@ -822,7 +822,7 @@ have pi-measure zero and use the fixed-ell uniform lower
 increment bound.
 
 Telescoping. Along the triangular chain M' with M'equiv M
-modulo ell and block dimensions lfloor M'n_j/Mrfloor, divide by
+modulo ell and block dimensions floor( M'n_j/M), divide by
 M' and send M'to∞, then ellto∞ and delta↓ 0.
 The same parameter law pi at both endpoints makes log increments
 telescope exactly; the o(M) perturbation error is removable. This
@@ -833,7 +833,7 @@ the pressure at the larger argument up to o(M), uniformly on shells.
 
 Maximal sub-sequential identification. Combining the upper and lower
 bounds, every subsequential limit of the normalised expected log
-pressure equals mathcal G_calG_{theta,p}(D). Concentration of the
+pressure equals G_{theta,p}(D). Concentration of the
 normalised log on a bounded window transfers this
 to convergence in probability. Boundedness of D and uniform
 integrability bound the pressure, giving convergence in expectation.
@@ -846,7 +846,7 @@ sum of a bounded continuous function and a nonpositive quadratic form
 path is separate from the approximation, and both are needed.
 
 Row tails. For every trial path q and every R<∞ the
-Gaussian row fields satisfy mathbb EbbEexp{c|Z_j|^2}le C_{c,R} for
+Gaussian row fields satisfy E{c|Z_j|^2}le C_{c,R} for
 |c| small enough and all rows rle M; the constants are uniform in
 the finite grid and the trial path. Bounded smooth approximation of
 D on compacts, together with a Markov tail bound on the residual
@@ -859,13 +859,13 @@ H=sum_rD(xi_r)-Mdsum_ell(barbar g_g_ell-c_ell)_+ is exactly the
 shape in which a nonpositive quadratic enters. Rewrite the soft
 penalty using the convex dual of x_+:
 
--Mdsum_ell(barbar g_g_ell-c_ell)_+=min_{0le t_ellle d}M(tcdot c-sum_ell t_ellbarbar g_g_ell).
+-Mdsum_ell(barbar g_g_ell-c_ell)_+=min_{0le t_ellle d}M(t· c-sum_ell t_ellbarbar g_g_ell).
 
 The minimum is attained because the constraint set is compact and the
 objective is linear in t. Substituting yields a Hamiltonian of the
-form H=sum_r(D(xi_r)-tcdot g(xi_r)) with bounded linear-t
+form H=sum_r(D(xi_r)-t· g(xi_r)) with bounded linear-t
 coefficients. The unpenalised Gaussian theorem with terminal
-D-tcdot g applies uniformly in tin[0,d]^s because negative
+D-t· g applies uniformly in tin[0,d]^s because negative
 quadratic coefficients remain negative, so it covers every
 t-terminal in one stroke.
 
@@ -876,7 +876,7 @@ v(t), obtained as a limit of barbar gg along compactly supported
 clippings. For each direction a into the compact box [0,d]^s, the
 one-sided derivative of the pressure with respect to t is
 
-∂_ap(t)=-acdot v(t).
+∂_ap(t)=-a· v(t).
 
 Passage of slopes through clipping: interpolate only the increment in
 the coupling. Increasing a nonnegative quadratic coupling adds a
@@ -894,9 +894,9 @@ Duality in closed form. Once the empirical average concentrates and
 the derivative formula holds, standard convex duality in finite
 dimensions gives the soft-penalty pressure as
 
-min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.
+min_{tin[0,d]^s}{t· c+G_{theta,p}(D-t· g)}.
 
-The unpenalised functional mathcal G_calG_{theta,p}(D-tcdot g) is
+The unpenalised functional G_{theta,p}(D-t· g) is
 jointly convex in t by direct differentiation and continuity of the
 unpenalised pressure in h under the terminal class; no separate
 lower-semicontinuity hypothesis is needed, because the compact box and
@@ -930,7 +930,7 @@ inequality n/Mto p<1 makes the small-ball exponent dominate the
 net size.
 ./pdf/arxiv-mezzadri-0609050.pdf
 
-The volume Jacobian of xmapsto Gx on the coefficient sphere is
+The volume Jacobian of x|-> Gx on the coefficient sphere is
 √(det(G^TG)). Its logarithm divided by M converges in
 probability to
 
@@ -945,7 +945,7 @@ Coefficient integral. Write A=Udiag(s)V^T and put
 x=Vw so that x is uniform on S^N. The column blocks of A
 multiply x block by block: Ax=sum_jU_jdiag(s_j)V_j^Tx.
 On the Haar flag the blocks V_j^Tx of a uniform coefficient decompose
-as independent uniform directions in mathbb R^bbR^{n_j} up to the
+as independent uniform directions in R^{n_j} up to the
 Jacobian. Thus the angular pressure satisfies
 
 P_M^f(a)=frac1Mlogint_{S^n}e^{sum_rf((Gx)_r)}|det(G^TG)|^{1/2}dsigma_n(x)+o(1).
@@ -974,7 +974,7 @@ S^{n_j}, and the Gaussian column blocks G_j act on each x_j
 independently. Encoding the constraint |y-u_i|^2/Mle
 sum_{j>i}a_j via the soft penalty at multiplier t_ell and sending
 the cap lambda/theta to infinity gives exactly the functional
-mathcal B_calB_{p,a}(kappa,lambda,B,theta) after taking a supremum
+B_{p,a}(kappa,lambda,B,theta) after taking a supremum
 over the trial coefficient frames R_j that parametrise the inner
 witness block-by-block.
 
@@ -1065,7 +1065,7 @@ sufficiently large B, independently of theta. Finally take
 theta small enough that the witness entropy is below ε. A
 union bound over the candidate rotations and the concentration
 estimate turns the witness lower bound into the corresponding
-deterministic bound for J_p+mathcal B_calB_{p,a} with one additional
+deterministic bound for J_p+B_{p,a} with one additional
 ε loss. Thresholds may depend on the fixed lambda; no
 uniformity in its later limit is required.
 
@@ -1076,7 +1076,7 @@ lambda. These monotonicities pass to the deterministic limits. The
 upper bound and the witness lower bound establish the successive
 limits and
 
-Phi(a)=J_p+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta).
+Phi(a)=J_p+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}B_{p,a}(kappa,lambda,B,theta).
 
 Using the shell-volume identity for Phi proves the angular formula
 and identifies every subsequential P_∞. Equicontinuity extends
@@ -1092,14 +1092,14 @@ probability is uniformly small, so Fubini and Markov apply after
 arbitrary bounded activation reweighting. Zero-radius blocks are
 ignored. Letting eta↓ 0 recovers the p=1 case.
 
-Continuity in f and a. For fin C_b(mathbb R)bbR) choose bounded
+Continuity in f and a. For fin C_b(R) choose bounded
 Lipschitz approximants with a common bound and uniform convergence on
 compact intervals. Every angular configuration satisfies
 M^{-1}sum_ry_r^2=sum_ja_j. On a compact radius set the fraction of
 entries outside [-R,R] is uniformly O(R^{-2}). Compact
 approximation controls the normalised energy uniformly and transfers
 all angular conclusions. The same estimate applies to
-mathcal B_calB_{p,a} on a bounded operator-norm event, independently of
+B_{p,a} on a bounded operator-norm event, independently of
 lambda,B,theta, so its limiting formula has the same continuity.
 This completes the angular theorem.
 
@@ -1127,8 +1127,8 @@ parameters (n_j/2) (including the null block when it is present).
 
 Under this split, the measurement is
 Ax=sum_jU_j(s_j√(Nrho_j/alpha)),G_jtheta_j plus a null
-contribution. Block j has image subspace E_j=U_jmathbb R^bbR^{n_j} of
-Haar dimension n_j in mathbb R^M,bbR^M, and the vector
+contribution. Block j has image subspace E_j=U_jR^{n_j} of
+Haar dimension n_j in R^M, and the vector
 y_j=s_j√(Nrho_j)G_jtheta_j is uniform on the sphere of radius
 √(Ma_j) in E_j with a_j=s_j^2rho_j/alpha. The y_j are
 mutually independent once rho is fixed.
@@ -1138,11 +1138,11 @@ the independence. On the coefficient sphere the factor
 |det(G^TG)|^{1/2} contributes the Jacobian rate J_p. The
 normalised log of the integral over shell radii equals
 
-sup_{substack{rho_i>0sum_{iinmathcal I}calIrho_i=1}}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)}+o(1),
+sup_{substack{rho_i>0sum_{iinI}}}{frac12sum_{iinI}/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)}+o(1),
 
 where v_j=alpha p_j, v_0=1-alpha when alpha<1, and
-mathcal I=calI={1,...,h} when alphage 1 and
-mathcal I=calI={0,1,...,h} when alpha<1. The first term is the
+I={1,...,h} when alphage 1 and
+I={0,1,...,h} when alpha<1. The first term is the
 Dirichlet entropy of the split of squared norm, whose density near
 rho is rho_1^{n_1/2-1}..., contributing
 frac12sum_jp_jlog(rho_j/p_j) plus a normalisation giving
@@ -1184,10 +1184,10 @@ Dirichlet arguments use only limiting proportions. The additional case
 is alpha=1 with N>M and k=N-M=o(N) structural right-null
 coordinates. Let B be the restriction of A to the M visible
 input coordinates, and let Z_M(r) be the angular partition function
-of B at radius r. If b=|f|_∞, the sphere in mathbb R^NbbR^N
+of B at radius r. If b=|f|_∞, the sphere in R^N
 decomposes as
 
-x=√(v),x'+u, x'inmathbb R^M,bbR^M, uinmathbb R^k,bbR^k,
+x=√(v),x'+u, x'inR^M, uinR^k,
 
 with v=|x|^2/NsimBeta(M/2,k/2) and x' uniform on
 S^{M-1} conditionally. The measured field at radius √(Nv) has
@@ -1241,7 +1241,7 @@ For kle M/2 and lambda_xgelambda_0, this is at most
 (C_0/lambda_0)^k. If a measurable set B has Lebesgue measure at
 most delta, then with k=lceilgamma Mrceil and 0<gamma<1/2,
 
-mathbb PbbP{#{r:(Ax)_rin B}ge k}lebinom{M}{k}(C_0delta/lambda_0)^k.
+P{#{r:(Ax)_rin B}ge k}lebinom{M}{k}(C_0delta/lambda_0)^k.
 
 Choosing R with Λ^2/R^2legamma and approximating f in
 Lebesgue measure on [-R,R] by continuous g with |g|le b gives
@@ -1304,8 +1304,8 @@ and Montanari gives the same conclusion with explicit constants.
 
 Counterexample: unbounded limiting spectrum. Let d_k=exp(8^k) and
 nu=sum_{kge 1}2^{-k}delta_{d_k} with unbounded support. For
-large N choose K=lfloorlog_2log Nrfloor and multiplicities
-n_j=lfloor N2^{-j}rfloor, with the top block about N/2. Let f
+large N choose K=floor(log_2log N) and multiplicities
+n_j=floor( N2^{-j}), with the top block about N/2. Let f
 be continuous on [0,1] equal to c_Kin{0,1} on
 [d_K^{1/2},d_K^2], alternating in c_K with K, interpolating in
 the gaps. The top block has singular value d_K=exp(8^K) and its