diff --git a/o4_model_mass_bounds_v2.txt b/o4_model_mass_bounds_v2.txt
index e84dfed..8e8cf3b 100644
--- a/o4_model_mass_bounds_v2.txt
+++ b/o4_model_mass_bounds_v2.txt
@@ -338,9 +338,9 @@ bound survives any subsequential local limit of the cylinder states as
 w=2^knto∞.
 
 Now identify cylinder limits with square-torus limits. Let
-widehatwidehat K_w=K_w/K_w=K_w/lambda_1(w). For w=n_k and rle w/2, the
+Khat_w=K_w/K_w=K_w/lambda_1(w). For w=n_k and rle w/2, the
 square-torus expectation equals
-Tr(A_Fwidehatwidehat K_w^K_w^{w-r})/Trwidehatwidehat K_w^w.K_w^w. Splitting off the ground
+Tr(A_FKhat_w^K_w^{w-r})/TrKhat_w^w.K_w^w. Splitting off the ground
 projection in each trace shows the difference from the cylinder
 expectation is at most 2norm F_∞ s_{w/2}(w). But
 p(w/2,w)leDelta_beta(w/2) and s_{w/2}(w)le e^{Delta_beta(w/2)}-1to0.
@@ -463,7 +463,7 @@ Use quaternion algebra: writing a unit quaternion as a pair of circle
 valued fields and then decomposing further into Ising variables lets us
 reduce any local Lipschitz observable to a sum of charged
 observables in ferromagnetic subsystems. Positive (FKG/Ginibre)
-inequalities upgrade the two-point bounds to full mixing for all
+ineualities upgrade the two-point bounds to full mixing for all
 observables.
 
 Finally the difference of the mean bond energies on n- and 2n-tori
@@ -499,7 +499,7 @@ where z_e is the edge midpoint and mu(e) the edge direction. Its
 second variation under infinitesimal spin twists gives the stiffness
 session
 
-  H_Q(F,G)=S^calH_Q(F,G)=S^{-2}Esum_e b_e t_e,barbar F_eF_e· G_e
+  H_Q(F,G)=S^H_Q(F,G)=S^{-2}Esum_e b_e t_e,barbar F_eF_e· G_e
   -Cov(X_Q(F),X_Q(G)).
 
 This is ``direct quadratic cost minus current covariance''. The
@@ -565,7 +565,7 @@ bounded in absolute value by bsum_e(f_y-f_x)^2, so Haar invariance
 gives
 
   Eexp(tD_fA)
-  le exp!Bigl[tfrac12 bt^2sum_e(f_y-f_x)^2Bigr].
+  le exp!Bigl[1/2 bt^2sum_e(f_y-f_x)^2Bigr].
 
 This is the exponential-tail estimate for a compact score. All fixed
 moments of the likelihood ratio follow up to a fixed power of
@@ -575,7 +575,7 @@ Next extract a transverse current on a buffered interior patch. Take a
 scalar test h on a patch of side d, pick a rotation axis a
 perpendicular to the current colour to extract; let θ be an
 affine compact potential with slope 1/d. Rotate the score by opposite
-profiles f=tfrac12arcsin(h/N) with N=C√(1+b),H and take the
+profiles f=1/2arcsin(h/N) with N=C√(1+b),H and take the
 difference. This gives the transverse current multiplied by
 2sin(f_x+f_y), and multiplication by N extracts h up to CH/d.
 The moment cost is O_p((1+b)H), and the deterministic total error over
@@ -613,8 +613,8 @@ constants, not on R,L,S. Then ((eq:pre:2-2) and
 (eq:pre:2-1))
 
   |E_c X_c(F)|&le P(B)N_c(F),
-  |H_c(F,G)|calH_c(F,G)|le P(B)N_c(F)N_c(G),
-  |H_c(F,G)-calH_c(F,G)-barbar F_cF_c H_c G_c|
+  |H_c(F,G)|H_c(F,G)|le P(B)N_c(F)N_c(G),
+  |H_c(F,G)-H_c(F,G)-barbar F_cF_c H_c G_c|
   &le P(B)[D_c(F)N_c(G)+N_c(F)D_c(G)].
 
 The second equation says: ``up to slowly varying corrections, the cell
@@ -668,8 +668,8 @@ discrete gradient Gram matrix. Substitute p into the density by an
 exact change of variables; the new density is proportional to e^{-V(p)}
 with
 
-  V(p) = tfrac12 c_0 p^TG_0 p + S_g(e^{atheta}omega^0)
-  +tfrac12sum_c|n_c^0+√(C_c^0),a,nablatheta(z_c)/L|^2.
+  V(p) = 1/2 c_0 p^TG_0 p + S_g(e^{atheta}omega^0)
+  +1/2sum_c|n_c^0+√(C_c^0),a,nablatheta(z_c)/L|^2.
 
 The change of variables is exact: the ring rotation preserves product
 Haar, the n_0 substitution is Gaussian, and the n_c substitution is
@@ -683,7 +683,7 @@ so we can integrate by parts normally. Then
   = Esum_{ij}∂_j Y_i,∂_i Y_j
   +Esum_{ij}Y_i∂_i∂_j V,Y_j.
 
-This is our basic inequality: it expresses the ``V'' Hessian and
+This is our basic ineuality: it expresses the ``V'' Hessian and
 ``crossed derivative'' cost of a long-run Gaussian in terms of two
 tractable sums. Everything from here down is bookkeeping to estimate
 these two sums.
@@ -750,7 +750,7 @@ noise drift, and non-anchor square-root derivative. Equation
 Same-frequency Ward bound.
 For the uncentred defect
 [
-  D(F,G)=vL^calD(F,G)=vL^{-2}sum_cbarbar F_cF_c· G_c-EJ(F)barJ(G),
+  D(F,G)=vL^D(F,G)=vL^{-2}sum_cbarbar F_cF_c· G_c-EJ(F)barJ(G),
 ]
 and for a compact eta of band O(K) with a fixed Fourier symbol,
 apply the raw Ward identity (eq:pre:1-5) before replacing the
@@ -768,7 +768,7 @@ Close a two-colour estimate in two steps. For perpendicular colours
 r,r', and for frequencies 2Mle|k_p|le4M, put
 b_p=|k_p|^{-2} and G_p^r=r,d_S(psitildephi_p/|k_p|). The
 Ward bound (eq:pre:5-14) applied to the two annuli in the two
-colours produces a closed inequality
+colours produces a closed ineuality
 [
   sum_p b_pE D_p^2
   le CvU^r+CT_wU^{r'}+O(B^{-N_*}),
@@ -787,10 +787,10 @@ using vge B^{-D},
 
 At a fixed total complex frequency n=k_p+k_i, |n|le4K_1, we
 take w(n-k_p)ge c(M+|n|)^{-2} uniformly in the annulus. Write
-psi_mu^+(z)=tfrac12[psi(z+e_mu/(2S))+psi(z-e_mu/(2S))]. The
+psi_mu^+(z)=1/2[psi(z+e_mu/(2S))+psi(z-e_mu/(2S))]. The
 exact gradient factor in spatial direction mu is
 [
-  widehatwidehat k_k_{pmu}sinc(k_{pmu}/(2S))psi_mu^+
+  khat_k_{pmu}sinc(k_{pmu}/(2S))psi_mu^+
   +frac{Scos(k_{pmu}/(2S))}{i|k_p|}
   [psi(z+e_mu/(2S))-psi(z-e_mu/(2S))].
 ]
@@ -843,7 +843,7 @@ hge Alog B and a shallow regime hle Clog B.
 
 noindentDepletion regime. For hge Alog B, set
 [
-  v=h+1, c_0=tfrac12,
+  v=h+1, c_0=1/2,
   L=bigllceil e^{√(hlog B)}bigrrceil,
 ]
 with A large compared with J^2 and the constants in
@@ -865,9 +865,9 @@ This is what produces the pibeta leading term of Xi(beta) in
 
 
 noindentShallow regime. Stop at hle Clog B and take a
-small fixed delta<c_{rm sh}/4 with
+small fixed delta<c_{sh}/4 with
 [
-  v=h+deltamin(h,1), c_0=tfrac12deltamin(h,1),
+  v=h+deltamin(h,1), c_0=1/2deltamin(h,1),
   L=lceil B^{11J}rceil.
 ]
 The same bookkeeping bounds the shallow remainder by an additional
@@ -903,9 +903,9 @@ A union bound over such chains gives the following uniform bound: for
 every bounded original observable O on a finite set A whose average
 under a common left rotation is identically zero,
 
-  |E O|le C|O|_∞|A|,e^{-cd_{rm pin}/s},
+  |E O|le C|O|_∞|A|,e^{-cd_{pin}/s},
 
-with d_{rm pin} the distance from A to the pinned boundary and s
+with d_{pin} the distance from A to the pinned boundary and s
 a microscopic constant. Extending to pairs O(q_x)O'(q_y) in
 sufficiently large unpinned tori gives
 
@@ -939,7 +939,7 @@ variables. The two resulting XY subsystems are independent ferromagnetic
 XY models. Condition further on the XY-spin magnitudes
 and phases; the remaining sign degrees of freedom form independent
 ferromagnetic Ising models. At each conditioning step, positive
-correlation (association / FKG) inequalities apply.
+correlation (association / FKG) ineualities apply.
 
 
 noindentAssociation + Ginibre. For a finite collection of
@@ -954,7 +954,7 @@ functions Upm F,Vpm G are increasing, so their covariance is
 non-negative. Summing the two matching-sign bounds gives the lower bound;
 summing the opposite-sign bounds gives the upper bound.)
 
-The O(4) plane-rotator form of Ginibre's inequality
+The O(4) plane-rotator form of Ginibre's ineuality
 ([cite]) applies here. Two-replica substitution
 theta=phi+psi,theta'=phi-psi is an onto homomorphism of
 the product tori that pushes product Haar to product Haar; expanded in
@@ -1139,9 +1139,9 @@ covariance localization.
 The conditional mean gives an exact algebraic identity for the free
 energy. For real ambient fields u,V on the two lattices,
 
-  tfrac12inner{u}{K_hu}+tfrac a2|V-Qu|^2
-  =tfrac12inner{V}{K_{h+1}V}
-  +tfrac12inner{u-P_hV}{C_h^{-1}(u-P_hV)}.
+  1/2inner{u}{K_hu}+tfrac a2|V-Qu|^2
+  =1/2inner{V}{K_{h+1}V}
+  +1/2inner{u-P_hV}{C_h^{-1}(u-P_hV)}.
 
 Indeed C_h^{-1}P_h=aQ^* and a(I-QP_h)=K_{h+1} make the cross and
 constant terms agree after expansion. Although K_h vanishes on
@@ -1166,14 +1166,14 @@ There is a coarse-edge to fine-edge map T satisfying
 
 whose aliases are bounded by C_JL^{-1}< l>^{-J}, and whose
 rows, columns, and fine-difference bounds depend only on L. One can
-find a positive scalar B_2 with an inequality
+find a positive scalar B_2 with an ineuality
 [
   widetilde B'U_*widetilde B'=c'^*B_2sfc'^*B_2c'sfc'
 ]
 such that the Schur complement U_* is nonnegative; consequently there
 exists an ambient-field identity
 
-  U_*+M^*calM^*M+calM+N^*calN^*N=I,
+  U_*+M^*M^*M+M+N^*N^*N=I,
 
 with M,N defined in terms of T,B',A_s
 and S'=A'(sfA'(widetilde B')^{-1}(A')^*sfA')^* an orthogonal
@@ -1211,15 +1211,15 @@ ell=(widetilde B-c_0I)^{1/2}. Taking square roots: all remain
 uniformly analytic on a smaller strip. Define
 
   S'&=A'(sfA'(widetilde B')^{-1}(A')^*,sfA')^*,
-  Mz&=(calMz&=(A B^{-1}T(A')^*z,sfA')^*z,
+  Mz&=(Mz&=(A B^{-1}T(A')^*z,sfA')^*z,
                    a^{-1/2}d'^*(A')^*z),sfA')^*z),
-  Nz&=((I-S')z,calNz&=((I-S')z,
+  Nz&=((I-S')z,Nz&=((I-S')z,
      B_2^{1/2}c'(sfc'(widetilde B')^{-1}(A')^*z).sfA')^*z).
 
 Substitution of (eq:free:Tidentities) proves
 (eq:free:ambient) and the cokernel property; c'd'=0sfc'd'=0
 proves that N'scalN's second factor has the requested form. Using
-the definition of U_* gives M^*calM^*M+calM+N^*calN^*N=I.
+the definition of U_* gives M^*M^*M+M+N^*N^*N=I.
 The extra factors are uniformly summable convolution kernels, so they
 preserve the proved bounds for T. At depth zero take
 widetilde B=I instead of the fine curl correction; this deletes a
@@ -1287,11 +1287,11 @@ what fixes the sqrtbeta e^{-pibeta} scale. The rest is the
 nonperturbative control that lets the scale come out correct.
 
 Let me present the step in four parts.
-{bf (a)} Setup: kinetic energy, observation kernel, positivity.
-{bf (b)} Class of densities retained at every scale.
-{bf (c)} The exact integration and the Schur-type identities that
+{ (a)} Setup: kinetic energy, observation kernel, positivity.
+{ (b)} Class of densities retained at every scale.
+{ (c)} The exact integration and the Schur-type identities that
 reorganise it into the retained class.
-{bf (d)} Statement of the closure theorem, and matching to the
+{ (d)} Statement of the closure theorem, and matching to the
 two-loop canonical expansion.
 
 The technical proofs are distributed through
@@ -1322,7 +1322,7 @@ noindentKinetic energy.
 Given a fine configuration q=(q_x)_{x}, define
 
   E_{h,t}(q) = sum_e S_t(r_e)
-    + tfrac12sum_e m_e(q),|(ell_h dq)_e|^2,
+    + 1/2sum_e m_e(q),|(ell_h dq)_e|^2,
 
 with r_e=|d_eq|, d_eq = q_y-q_x for e=xto y, and where
 
@@ -1378,7 +1378,7 @@ Fix once and for all:
 Fix the parameters above and give the scale-j lattice
 Λ_j=(Z/L^jZ)^2 on the torus of side L^j. A density
 rho on ambient configurations Q=(q_x)_{xinΛ_j} is in the
-scale-j class C_j= C_j(b_j,t_j,scrC_j(b_j,t_j,delta_j,h)
+scale-j class C_j= C_j(b_j,t_j,C_j(b_j,t_j,delta_j,h)
 if:
 [label=(roman*)]
  It has the form
@@ -1409,7 +1409,7 @@ if:
  Each covering label ell has a bounded measurable
       weight k_ell(q), a finite complete support P_ell, and a
       non-empty inventory D_ell of bonds contained in P_ell,
-      with k_ellequiv0 unless D_ellsubset D(q)={e:r_e>t_j}.
+      with k_elleuiv0 unless D_ellsubset D(q)={e:r_e>t_j}.
       The covering gas
       
         Xi(q)=sum_{substack{Gamma: disjoint support
@@ -1451,8 +1451,8 @@ The exact dependence of the kinetic energy on V is reorganised by
 writing the kinetic part as
 
   b_jE_{h,t_j}(q)
-  = b_jE_{h,t_j}(q)-tfrac12 a,b_j|V-Qu|^2
-  +tfrac12 a,b_j|V-Qu|^2,
+  = b_jE_{h,t_j}(q)-1/2 a,b_j|V-Qu|^2
+  +1/2 a,b_j|V-Qu|^2,
 
 i.e. we insert and remove the V-dependent Gaussian term. Combined
 with the observation, we get a positive Gaussian density on the fine
@@ -1502,7 +1502,7 @@ Fix the parameters L,P_0 large and H large enough so that the
 hypotheses of Appendix~app:finite-choices hold. There are
 H_j-dependent constants C_L,c_L>0, and there is a scale-dependent
 data (b_j,t_j,delta_j,h_j) such that, for every
-j-level density in C_j(b_j,t_j,scrC_j(b_j,t_j,delta_j,h_j) and every
+j-level density in C_j(b_j,t_j,C_j(b_j,t_j,delta_j,h_j) and every
 period m=L^j large enough, the exact integration to scale j-1 gives
 a density in
 [
@@ -1552,7 +1552,7 @@ constants are fixed.
 ## Lemma 1: The Schur observation split
 
 [Schur observation]
-Let rhoinC_j(b,t,scrC_j(b,t,delta,h) and let Q(Vmid u) be the
+Let rhoinC_j(b,t,C_j(b,t,delta,h) and let Q(Vmid u) be the
 positive observation kernel (eq:obs-kernel). Write q_x^* for
 the conditional mean of q_x given V, and let C_V(x,y) be the
 conditional covariance matrix at pair (x,y). Then the exact
@@ -1574,9 +1574,9 @@ where
 
 Proof. 
 The exact identity (eq:free:schur-energy) writes
-tfrac12inner u{K_hu}+tfrac a2|V-Qu|^2 as a free energy in V
+1/2inner u{K_hu}+tfrac a2|V-Qu|^2 as a free energy in V
 plus the positive fluctuation cost
-tfrac12inner{u-P_hV}{C_h^{-1}(u-P_hV)}. Writing u_x for the
+1/2inner{u-P_hV}{C_h^{-1}(u-P_hV)}. Writing u_x for the
 fine spin inside block b and P_hV for its block-mean proxy, the
 difference u_x-(P_hV)_x is directly controlled by the residual
 estimate (eq:rg:residual),
@@ -1619,7 +1619,7 @@ satisfy:
  F_3 has a bound O(C_LM^D t);
  F_4 is O(C_LM^Dt^0) (bounded independently of t and L);
  R_5 has the exponential-small bound
-      |R_5|calR_5|le C_Lg^5poly(M,p) in the
+      |R_5|R_5|le C_Lg^5poly(M,p) in the
       free-kernel exponential norm.
 
 
@@ -1739,8 +1739,8 @@ F of output spins on one side, let QF be its conditional
 expectation given the input spins on that side. Conditional independence
 and reflection covariance give
 [
-  mu_{rm out}((thetaoverlineoverline F)F)F)F)
-  =mu_{rm in}((thetaQFbar) QF)calQF)ge0.
+  mu_{out}((thetaoverlineoverline F)F)F)F)
+  =mu_{in}((thetaQFbar) QF)QF)ge0.
 ]
 Thus reflection positivity passes through each blocking step. Translation
 and reflection compatibility on the retained lattice pass through the
@@ -1751,7 +1751,7 @@ covariance; that is the whole content.
 ## Lemma 6: Assembly
 
 [Assembly]
-Let rhoinC_j(b,t,scrC_j(b,t,delta,h). Then under one exact RG step,
+Let rhoinC_j(b,t,C_j(b,t,delta,h). Then under one exact RG step,
 rho is mapped to rho'inC_{j-1}(b',t',delta',h') with
 
   b'-b = alpha+kappa/b+..., delta' le H_j^{-1/20}delta,
@@ -1808,7 +1808,7 @@ axis and write
 
 Under the nearest-neighbor measure ddmu_beta the sum
 sum_e q_y· q_x becomes
-sum_e[1-tfrac12|u_y-u_x|^2+O(|u|^3)], so the quadratic fluctuation
+sum_e[1-1/2|u_y-u_x|^2+O(|u|^3)], so the quadratic fluctuation
 is a massive Gaussian of mass sim nearest-neighbor Laplacian, and the
 cubic and higher terms are suppressed by 1/sqrtbeta per spin.
 
@@ -1895,10 +1895,10 @@ D_0 is really doing).
 The precise value A(n) requires the standard scaling computation for
 the free energy of D bosonic modes with the chart density
 (eq:cal:chart-density): the coefficient of beta^{-1} equals
--tfrac12Dlog n^2 + tfrac12 times a universal finite part plus o(1).
-We denote this by A(n)=-tfrac32log n+C_A+o(1).
+-1/2Dlog n^2 + 1/2 times a universal finite part plus o(1).
+We denote this by A(n)=-3/2log n+C_A+o(1).
 
-Similarly Z_2(n) has form -tfrac32log n + C_{Z_2}+o(1), generated
+Similarly Z_2(n) has form -3/2log n + C_{Z_2}+o(1), generated
 by the one-loop determinant correction. Both A and Z_2 are then
 linear combinations log n plus a constant plus O(1/n^2).