diff --git a/spherical_perceptron_bi_orthogonal_v2.txt b/spherical_perceptron_bi_orthogonal_v2.txt
index 328ba8a..6b65ac7 100644
--- a/spherical_perceptron_bi_orthogonal_v2.txt
+++ b/spherical_perceptron_bi_orthogonal_v2.txt
@@ -1,5 +1,5 @@
 Let A be a real Mtimes N matrix, and let sigma_N denote uniform
-probability measure on the sphere S^N={xinR^N:bbR^N:|x|^2=N}.
+probability measure on the sphere S^N={xinR^N:R^N:|x|^2=N}.
 For a bounded continuous f:R, the normalized
 log-partition function of the spherical perceptron is
 
@@ -349,7 +349,7 @@ mark. Reads on inner visits are independent at every occurrence,
 including repeated reads from the same leaf.
 
 Under this construction, the expected cascade logarithm conditioned on
-disorder equals (1/theta)logint(int e^Hdmu_{rm in})^theta dmu_{rm out}.
+disorder equals (1/theta)logint(int e^Hdmu_{in})^theta dmu_{out}.
 When H has no further label dependence, adding redundant cascade
 levels leaves this value unchanged. Merged masses at a branching
 exponent z follow normalised stable jumps at exponent z; at the
@@ -388,7 +388,7 @@ 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,
 
-E<bbE< Fpsi(T^{1,r+1})>=frac1rE<bbE< F>,E<bbE<psi(T^{12})>+frac1rsum_{a=2}^rE<bbE< Fpsi(T^{1a})>,
+E<E< Fpsi(T^{1,r+1})>=frac1rE<E< F>,E<E<psi(T^{12})>+frac1rsum_{a=2}^rE<E< Fpsi(T^{1a})>,
 
 where brackets denote Gibbs expectation and E averages all
 randomness. Monomial GG identities give all continuous tests of this
@@ -473,7 +473,7 @@ for the tilted spin density N(0,Q) and the tilted kernels attached
 to C. The exact expression is not needed; only its structure
 matters. This Psi(C,b) is lower semicontinuous in C with value
 +∞ outside its finite domain, and has a coercivity bound.
-Indeed for any covariance Q with tfrac12Ipreceq Qpreceqtfrac32 I,
+Indeed for any covariance Q with 1/2Ipreceq Qpreceq3/2 I,
 use N(0,Q) as trial in the spin integration leaving the field
 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
@@ -483,11 +483,11 @@ theta^{-1}Ent(Q_1midgamma_1)+E_{Q_1}Ent(Q_{Fmid1}midgamma_{d-1})letheta^{-1}Ent(
 
 and this permits correlated trial coordinates. Hence
 
-Psi(C,b)-tfrac12tr,Cgetfrac12tr(C(Q-I))-frac1{2theta}(tr,Q-logdet Q-d).
+Psi(C,b)-1/2tr,Cge1/2tr(C(Q-I))-frac1{2theta}(tr,Q-logdet Q-d).
 
-Choosing Q=I+tfrac12sign(C) yields
+Choosing Q=I+1/2sign(C) yields
 
-Psi(C,b)-tfrac12tr,Cgetfrac14|C|_*-c_d/theta,
+Psi(C,b)-1/2tr,Cge1/4|C|_*-c_d/theta,
 
 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
@@ -495,13 +495,13 @@ 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
+forces strict precision ineualities, and differentiating in C
 gives the tilted self-covariance identity Ess^T=I.
 
 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=E[sbbE[smid
-H_i].calH_i]. Let K_i^*sfK_i^* denote the Hessian of U_i in its
+H_i].H_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
 
@@ -510,7 +510,7 @@ 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^*=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^*.
+q_i^*=E[M_iM_i^T],E[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 K_i P_isfP_iK_i= K_isfK_i
 on the active space, combined with the pinning across theta,
@@ -518,7 +518,7 @@ identifies q_0^* with the precision condition. The final statement
 is the canonical duality: the minimiser C(b) of
 Psi(C,b)-tr,C/2 yields
 
-J(b):=Psi(C(b),b)-tfrac12tr(C(b)+b_{k-1})=S_theta(q^*)-frac12sum_iw_itr(q_i^*b_i)=inf_q{S_theta(q)-frac12sum_iw_itr(q_ib_i)},
+J(b):=Psi(C(b),b)-1/2tr(C(b)+b_{k-1})=S_theta(q^*)-frac12sum_iw_itr(q_i^*b_i)=inf_q{S_theta(q)-frac12sum_iw_itr(q_ib_i)},
 
 and q^* is the unique minimiser of the right side. The path q^* is
 eligible. For fixed theta,d and bounded b_{k-1}, all such
@@ -537,11 +537,11 @@ 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(K_i^*+sfK_i^*+zeta_imathbb
-EM_iM_i^T)bbEM_iM_i^T) for an increment at level i. Telescoping and
+the diagonal shift inserts 1/2(K_i^*+sfK_i^*+zeta_imathbb
+EM_iM_i^T)EM_iM_i^T) for an increment at level i. Telescoping and
 Ess^T=I give
 
-dot J=-tfrac12sum_iw_itr(q_i^*dot b_i),
+dot J=-1/2sum_iw_itr(q_i^*dot b_i),
 
 and the entropy infimum has the same derivative by its unique
 minimiser. Envelope differentiation is justified by the compact
@@ -606,14 +606,14 @@ fresh pattern with the negative of r∂ A/∂ r at the
 minimiser. The latter is bounded below by -V_D(q^*)-delta/2, and
 this is the upper bound on the fresh pattern log-pressure.
 
-Differentiating in b gives the tail contact inequality: for every
+Differentiating in b gives the tail contact ineuality: for every
 nondecreasing variation tildetilde bbsucceq b of the linear-field path
 with compact support, the covariance comparison V_D(tildetilde q)q)le
 V_D(q) (from the path-order implication) implies
 q^*preceq q in the Loewner order path sense. The gradient of the
 supersolution with respect to b is a signed measure whose negative
 part is supported where q^* stays below q. Summation by parts and
-the contact inequality at the minimiser identify the distribution of
+the contact ineuality at the minimiser identify the distribution of
 q^* with a path of the same type.
 
 Ghirlanda-Guerra identities. The critical step is that, at the
@@ -625,7 +625,7 @@ 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
-inequality gives that the perturbed first derivative concentrates.
+ineuality gives that the perturbed first derivative concentrates.
 Passing to a subsequence in which the perturbation parameters converge
 then yields the GG identities.
 
@@ -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
 
-intE<|Y_a-bbE<|Y_a-E<bbE< Y_a>|>/s_M,dpito 0.
+intE<|Y_a-E<|Y_a-E<E< 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
@@ -804,7 +804,7 @@ via the canonical lemma. The cavity comparison identifies the
 enlarged pressure with ell times the canonical value minus o(ell)
 error, so
 
-frac1ellE[bbE[logZ_{M+ell}-logZ_M]calZ_M]geG_{theta,p}(D)-o_{tol}(1).
+frac1ellE[E[logZ_{M+ell}-logZ_M]Z_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
@@ -817,7 +817,7 @@ beyond the base at parameter u. The constant in O(1) is fixed
 across these choices. Begin with any good-parameter sequence and any
 compact array limit. The tolerance, shell, and rounding choices apply
 to it with uniform canonical constants; failure of the eventual bound
-would contradict the cavity inequality. The bad parameter sets
+would contradict the cavity ineuality. The bad parameter sets
 have pi-measure zero and use the fixed-ell uniform lower
 increment bound.
 
@@ -926,7 +926,7 @@ at most (1+2/delta)^r; fixed-vector Gaussian norms have arbitrarily
 strong exponential upper tails at a large constant, with small-ball
 probability at most (Cε)^M. Take a fixed mesh for the upper
 bound and a mesh of order ε for the lower bound; the strict
-inequality n/Mto p<1 makes the small-ball exponent dominate the
+ineuality n/Mto p<1 makes the small-ball exponent dominate the
 net size.
 ./pdf/arxiv-mezzadri-0609050.pdf
 
@@ -955,7 +955,7 @@ the Gaussian matrix G with its Jacobian. The error o(1) comes from
 a bounded activation and the good-norm event.
 
 Witnesses. Introduce auxiliary unit-block vectors z_{i,j}inmathbb
-R^bbR^{n_j} for jle i<h and put
+R^R^{n_j} for jle i<h and put
 
 u_i=sum_{jle i}G_jz_{i,j}.
 
@@ -1016,7 +1016,7 @@ cumulative dominance give
 
 sum_ic_ia_i=int_0^pCAleint_0^pC^*Aleint_0^pC^*B^*=max_lambdasum_{i,j}c_ilambda_{ij}b_j/p_j.
 
-The second inequality is summation by parts; equal total mass permits
+The second ineuality is summation by parts; equal total mass permits
 arbitrary, including negative, prices. Finite-dimensional separation
 gives a table attaining a. Set T_{ij}=lambda_{ij}/p_j; then
 
@@ -1196,7 +1196,7 @@ singular value. The endpoints v small and v near one contribute
 negligibly because the Beta density makes such values exponentially rare at
 the field-energy scale. Integration by parts over v converts the
 sphere integral into a ball integral of Z_M(r) over
-r^2in[alpha s_-^2v_{rm lo},alpha s_+^2v_{rm hi}] times a
+r^2in[alpha s_-^2v_{lo},alpha s_+^2v_{hi}] times a
 Jacobian. The ball integral differs from the sphere integral by
 frac12log of an explicit Beta-type factor, which is exactly the
 Dirichlet entropy of a null block of mass 1-alpha. The same radial
@@ -1284,12 +1284,12 @@ inf_u(1/M)sum_rmin(M(Uu)_r^2,1)ge 1/80 with probability at least
 1-e^{-c_0M}. So the even-N pressure is at most -calpha/80,
 which is -12.5c for alpha=1000. On odd N one input direction
 maps to zero; a uniform input decomposes as
-sqrt Vu_{rm live}+√(1-V)u_{rm null} with
+sqrt Vu_{live}+√(1-V)u_{null} with
 VsimBeta((N-1)/2,1/2). The event Vle v has
 log-probability per N tending to frac12log v, and on this event
 the energy is at least -cv, giving
 
-{rm pressure}ge tfrac12log v -cv = -0.5log 160-6.25capprox -2.5-6.25c.
+{pressure}ge 1/2log v -cv = -0.5log 160-6.25capprox -2.5-6.25c.
 
 For c=1 the even-N pressure is at most -12.5 while the odd-N
 pressure is at least -8.75, so the limits differ. The rectangular
@@ -1338,7 +1338,7 @@ sum_r(A_Nx)_r^8ge N^2/256,
 
 and therefore
 
-{rm pressure}ge frac1N[(N^2)/(256)+log(frac18 2^{-(N-3)/2})]=frac N{256}-frac12log 2-(3log 2)/(2N)to+∞.
+{pressure}ge frac1N[(N^2)/(256)+log(frac18 2^{-(N-3)/2})]=frac N{256}-frac12log 2-(3log 2)/(2N)to+∞.
 
 The pressure diverges on odd N while it is zero on even N, and both
 spectra converge to delta_0 with no outliers. Continuous activation