diff --git a/spacetime_penrose_area_charge_v2.txt b/spacetime_penrose_area_charge_v2.txt
index 569adfc..a576bab 100644
--- a/spacetime_penrose_area_charge_v2.txt
+++ b/spacetime_penrose_area_charge_v2.txt
@@ -245,9 +245,9 @@ Outermostness and area-minimality [sec:outermost]. Two conditions are imposed on
 Outer area-minimality. B is outer area-minimal when
 |Gamma|_gge|B|_g for every enclosing cut Gamma.
 
-Partial-coincidence outermostness. Let mathcal TcalT be the class
+Partial-coincidence outermostness. Let T be the class
 of enclosing cuts that attain a_g(B). B is partially outermost if
-every Gammainmathcal TcalT with Gammane B omits at least one entire
+every GammainT with Gammane B omits at least one entire
 component of B.
 
 
@@ -341,13 +341,13 @@ densities transform with nonnegative factors when u_s is a positive
 supersolution of the conformal Laplacian. The ADM energy formula
 
 
-  E_{g_s}=E+scdot(1)/(16pi)lim_{rto∞}int_{S_r}
+  E_{g_s}=E+s·(1)/(16pi)lim_{rto∞}int_{S_r}
             ∂_nu(4u_s∂_nu u_s),d A
   =E+sA_phi
 
 
 with A_phiproptoint w>0 and P_s=P follows from the standard
-second variation of the mass under gmapsto u_s^4g.
+second variation of the mass under g|-> u_s^4g.
 
 Because supp w is disjoint from a neighborhood of B, the normal
 derivative of u_s vanishes there, and conformal covariance of the
@@ -436,7 +436,7 @@ Let u>0 be smooth and set g_u=u^4g, K_u=u^2K. Then
 
 16pi u^4mu_u &=16pimu-8u^{-1}Delta_gu-(1-u^{-8})((tr_gK)^2-|K|_g^2),
 [eq:conformal-mu]
-8pi J_u &=u^{-2}(8pi J+4K(nablalog u,cdot)), [eq:conformal-J]
+8pi J_u &=u^{-2}(8pi J+4K(nablalog u,·)), [eq:conformal-J]
 
 the second being an identity of covectors. Consequently [eq:conformal-dec]
 
@@ -462,8 +462,8 @@ u^4((tr_{g_u}K_u)^2-|K_u|^2_{g_u})=u^{-8}((tr_gK)^2-|K|_g^2).
 Adding the two contributions gives [eq:conformal-mu]. With pi=K-(tr_gK)g, pi_u=u^2pi and
 tr_gpi=-2tr_gK; contracting the connection difference in dimension
 three gives
-u^2J_u=J+2pi(nablalog u,cdot)-(tr_gpi)dlog u
-        =J+4K(nablalog u,cdot),
+u^2J_u=J+2pi(nablalog u,·)-(tr_gpi)dlog u
+        =J+4K(nablalog u,·),
 which is [eq:conformal-J]. Norms in g_u add a factor u^{-2},
 and the triangle inequality yields [eq:conformal-dec]. Finally
 nu_u=u^{-2}nu, H_{u^4g}=u^{-2}(H+4∂_nulog u), and
@@ -522,7 +522,7 @@ principle force a trivial kernel, so the operator is an invertible
 Fredholm perturbation. Uniformity in t: if the solution supremum
 norms on a truncation are unbounded, divide by the supremum, and with
 W^{2,p} estimates, derivative interpolation, and p>3 Sobolev
-compactness the limit v solves Delta_gv+Acdot dv=0 with
+compactness the limit v solves Delta_gv+A· dv=0 with
 |A|le b measurable, contradicting the maximum principles. The
 bounds are independent of R, yielding a smooth solution of
 [eq:drift-poisson] on the exterior.
@@ -556,7 +556,7 @@ Remark (Decay needed by the drift-Poisson source).
 The source w=r^{-3-delta}>0 of [eq:drift-poisson] must, at the
 end of the strictification step, also dominate the conformal residual
 from the [lem:n3:conformal-formulas] lemma. That residual has size
-|1-u_s^{-8}|cdot|(tr_gK)^2-|K|^2_g|
+|1-u_s^{-8}|·|(tr_gK)^2-|K|^2_g|
 =O(sphi|(tr_gK)^2-|K|^2_g|)=O(s,r^{-1-2q}) because
 phi=O(r^{-1}) and |(tr_gK)^2-|K|^2_g|=O(r^{-2q}). A strict gap
 w-|1-u_s^{-8}|,|(tr_gK)^2-|K|^2_g|ge c' r^{-3-delta} therefore
@@ -636,7 +636,7 @@ The second step is the asymptotic Lorentz boost. Since E>|P|_delta,
 the vector (E,P) is future timelike, and there is a unique future
 boost Λ such that Λ(E,P)=(m,0) with
 m=√(E^2-|P|_delta^2). On the asymptotic chart of [eq:decay],
-the change of coordinates xmapstoΛ x acts linearly on the end
+the change of coordinates x|->Λ x acts linearly on the end
 and is trivial on a fixed neighborhood of B. With respect to
 (g_s,K_s) it produces isometric data with the same DEC, the same
 theta_+(B)le0, and ADM vector Λ(E_s,P)=(m_s,0), where
@@ -755,7 +755,7 @@ distance except at countably many thresholds.
 Proof. 
 Monotonicity is immediate from the definition. Fix a countable dense
 set {x_j} in X. For each j, the monotone function
-cmapstodist(x_j,T_c(X,Q)) has at most countably many jumps. Away
+c|->dist(x_j,T_c(X,Q)) has at most countably many jumps. Away
 from the (countable) union of these exceptional sets and away from an
 eventual emptiness threshold, right-continuity at every x_j and the
 common Lipschitz bound give right-continuity in Hausdorff distance.
@@ -866,13 +866,13 @@ boundary expansions of Ω_0.
 
 Lemma (The Hessian square is coercive). [lem:n3:hessian-square] 
 Put H^f=Hess_gf and let
-mathcal T=calT=tfrac12{(tr_{A_d}H^f)^2-|H^f|_{A_d ⊗ A_d}^2} be
+T={(tr_{A_d}H^f)^2-|H^f|_{A_d ⊗ A_d}^2} be
 the second-form square of the comparison graph. Then there is c_0(n)>0
 such that [eq:hessian-square-coercive]
 
-mathcal TcalTge c_0((tr_gH^f)^2+|nabla w|_{A_chi}^2)
+T c_0((tr_gH^f)^2+|nabla w|_{A_chi}^2)
 
-and mathcal TcalTle C(n)(|nabla f|^2+|Hess_gf|^2).
+and T C(n)(|nabla f|^2+|Hess_gf|^2).
 
 
 
@@ -883,7 +883,7 @@ Write A_d and A_chi in a frame where A_d=diag(1,1,1,d) and
 A_chi=diag(1,1,1,chi). Direct contraction gives
 
 
-mathcal T=calT=tfrac12|H^f_perp|^2+chi,(H_{ee}^f)^2,
+T=|H^f_perp|^2+chi,(H_{ee}^f)^2,
 
 
 where H^f_perp is the trace-free part on the two transverse directions
@@ -964,10 +964,10 @@ polynomial losses.
 Lemma (Uniform separation). [lem:n3:apriori:separation] 
 Let AsubsetbarbarΩΩ be compact. In the first two homotopy stages
 one has the following. If A avoids the full black total region
-mathcal B_calB_{c_b} at threshold c_b, then hge b_-+eta_A on A
+B_{c_b} at threshold c_b, then hge b_-+eta_A on A
 for some eta_A>0, uniformly in n large, every sufficiently large
 truncation, and every admissible solution. If A avoids
-mathcal W_calW_{c_w}, then hle b_+-eta_A.
+W_{c_w}, then hle b_+-eta_A.
 
 
 
@@ -1027,24 +1027,24 @@ b_-<k<k'. For integers j,
 
 The lower relaxed limit of v_j is the indicator of E_k: at points
 of E_k it vanishes, off E_k it converges uniformly to one. If
-varphi has nonzero gradient and is a lower test of the indicator at
+phi has nonzero gradient and is a lower test of the indicator at
 a zero value, the same strict contact localization gives lower tests
-varphi+c_j of v_j at y_jto x with v_j(y_j)to0. Since
+phi+c_j of v_j at y_jto x with v_j(y_j)to0. Since
 psi_j(k')to1, one has underline h(y_j)<k' eventually.
 Composition with the increasing psi_j preserves the oriented level
 expansion:
 
 
-  Hess(thetacircvarphi)=theta'Hessvarphi
-    +theta''dvarphi ⊗  dvarphi,
-  frac{tr_{(dvarphi)^perp}Hess(thetacircvarphi)}
-       {|d(thetacircvarphi)|}
-  =frac{tr_{(dvarphi)^perp}Hessvarphi}{|dvarphi|}.
+  Hess(thetacircphi)=theta'Hessphi
+    +theta''dphi ⊗  dphi,
+  frac{tr_{(dphi)^perp}Hess(thetacircphi)}
+       {|d(thetacircphi)|}
+  =frac{tr_{(dphi)^perp}Hessphi}{|dphi|}.
 
 
-So psi_j^{-1}(varphi+c_j) is a smooth lower test of
+So psi_j^{-1}(phi+c_j) is a smooth lower test of
 underline h with contact <k', and the oriented-level inequality
-passes to varphi. Hence every smooth exterior support of E_k has
+passes to phi. Hence every smooth exterior support of E_k has
 expansion at most k'+C_b.
 
 The argument also rules out a hidden layer at a white face: the
@@ -1060,10 +1060,10 @@ applies. The resulting compact closed set contains the original inner
 barrier with a collar and avoids the outer barriers. Lemma on weak
 support then places it in a smooth black trapped region at any
 threshold c''>k'+C_b close enough to c_b, hence in
-mathcal B_calB_{c''}.
+B_{c''}.
 
-Hausdorff right continuity of cmapstomathcal B_ccalB_c at c_b supplies
-delta>0 with mathcal B_calB_{c_b+delta}cap A=∅; shrink
+Hausdorff right continuity of c|->B_c at c_b supplies
+delta>0 with B_{c_b+delta}cap A=∅; shrink
 delta so the enlarged threshold is still negative. Choose
 k,k',c'' with b_-<k<k'<k'+C_b<c''<c_b+delta. Then
 E_kcap A=∅ for every sequence in the class. A failing
@@ -1163,21 +1163,21 @@ third-stage bound.
 
 
 Lemma (Weighted collar traces). [lem:n3:apriori:trace] 
-Let mathcal CcalC be a fixed disjoint union of short inner collars. In
-the first two stages, for nonnegative smooth varphi supported in
-mathcal C,calC,
+Let C be a fixed disjoint union of short inner collars. In
+the first two stages, for nonnegative smooth phi supported in
+C,
 
 
-  int_B Lvarphid A_gle
-  Pi_nint_{mathcal C}calCu[(1+√(mathcal T))calT))varphi
-       +|dvarphi|_{A_chi}]d V_g,
+  int_B Lphid A_gle
+  Pi_nint_{C}[(1+√(T))
+       +|dphi|_{A_chi}]d V_g,
   
-  int_Bvarphid A_gle
-  Pi_nint_{mathcal C}calCsqrt D[(1+√(mathcal T))calT))varphi
-       +|dvarphi|_{A_chi}]d V_g.
+  int_Bphid A_gle
+  Pi_nint_{C} D[(1+√(T))
+       +|dphi|_{A_chi}]d V_g.
 
 
-The estimates extend to arbitrary varphi by a fixed collar cutoff
+The estimates extend to arbitrary phi by a fixed collar cutoff
 equal to one on B.
 
 
@@ -1201,7 +1201,7 @@ Differentiation gives
   (Div(sqrt D,W))/(sqrt D)=sqrt d[tr_{A_d}H^f+dp,a∂_{e_f}t].
 
 
-Coercivity bounds both right sides by Pi_n(1+√(mathcal T)):calT)): the
+Coercivity bounds both right sides by Pi_n(1+√(T)):calT)): the
 axial Hessian term has coefficient d^{3/2}, the transverse terms are
 traces, and the gradient term is O(sqrt d |∂_{e_f}t|).
 
@@ -1209,8 +1209,8 @@ Let ε_B=1 on a black collar and -1 on a white one. The
 signed slopes give w_nu=ε_B a on B with age1/2 for
 large n, hence
 uW_nu=ε_B La and sqrt D,W_nu=ε_B a. Apply
-the divergence theorem to -ε_B uWvarphi with a fixed
-collar cutoff. Its boundary flux is Lavarphi and the outer
+the divergence theorem to -ε_B uWphi with a fixed
+collar cutoff. Its boundary flux is Laphi and the outer
 contribution vanishes by support. The gradient bound and divergence
 formula give the first inequality; replacing u by sqrt D gives the
 second.
@@ -1224,7 +1224,7 @@ In stage one there is M_0=M_0(n)>0 such that, for all n large and
 all R and lambdain[0,1],
 
 
-  Xigedelta_0mathcal T+calT+rho+tfrac12delta_0tau asigma
+  Xigedelta_0T+calT+rho+tfrac12delta_0tau asigma
   quadon {Z>M_0}.
 
 
@@ -1232,7 +1232,7 @@ With k_0 a smooth cutoff supported on [M_0,M_0+1] equal to one
 on [M_0+1,∞),
 
 
-  int_Ω u k_0(mathcal T+calT+rho+tau asigma)d V_g
+  int_Ω u k_0(T+calT+rho+tau asigma)d V_g
   +int_Ω u k_0',|dZ|_{A_chi}^2d V_gle C(n).
 
 
@@ -1242,7 +1242,7 @@ on [M_0+1,∞),
 
 
 Proof. 
-Outside supp m_0, Xi=delta_0mathcal T+calT+rho+delta_0tau asigma
+Outside supp m_0, Xi=delta_0T+calT+rho+delta_0tau asigma
 is clearly ge the right side. On supp m_0 one has
 -εle tle -ε+1/n and lasympell; the identity
 D=e^{8(Z-t)} makes asigmato∞ uniformly as Zto∞,
@@ -1252,14 +1252,14 @@ Test Div V_lambda=uXi by k_0(Z). The outer boundary term
 vanishes by Z=0. The drift cross term is bounded by
 
 
-  uk_0'|< dZ,K(w,cdot)>_{A_chi}|
+  uk_0'|< dZ,K(w,·)>_{A_chi}|
   le 2u k_0'|dZ|_{A_chi}^2+Cuk_0'|K|^2,
 
 
 and the last term has bounded integral on a fixed compact set in the
 transition strip of k_0. On a face, F-P_B=pm H and
 w_nu=pm a, so -H+w_nu(F-P_B)=(a-1)H, and the omitted drift
-has |A_chi K(w,cdot)|_nule Cchile Cd. Thus
+has |A_chi K(w,·)|_nule Cchile Cd. Thus
 
 
   |(V_lambda)_nu|le Cud=CLsqrt dle Cfractauell L
@@ -1269,7 +1269,7 @@ has |A_chi K(w,cdot)|_nule Cchile Cd. Thus
 Integrating,
 
 
-  int_Ω uk_0(delta_0mathcal T+calT+rho
+  int_Ω uk_0(delta_0T+calT+rho
      +tfrac12delta_0tau asigma)d V_g
   +2int_Ω u k_0'|dZ|_{A_chi}^2d V_g
   le C(n)+Cfractauellint_B Lk_0d A_g.
@@ -1277,9 +1277,9 @@ Integrating,
 
 Bound the last term using the weighted trace with the fixed collar
 cutoff: it costs
-Pi_nfractauellint_{mathcal C}calCu(k_0(1+√(mathcal T))calT))
+Pi_nfractauellint_{C}+√(T))
 +k_0'|dZ|_{A_chi})d V_g. On the fixed collars rhogerho_0>0,
-so (1+√(mathcal T))calT))le C(rho+delta_0mathcal T)calT) there, and
+so (1+√(T)) C(rho+delta_0T) there, and
 Pi_ntau/ellto0 absorbs the first summand into the positive source.
 Young's inequality reduces the derivative summand to
 uk_0'|dZ|^2_{A_chi}+C(Pi_ntau/ell)^2uk_0', whose second term has
@@ -1292,7 +1292,7 @@ constant depending only on n and the fixed data.
 
 Proof (Proof of the [prop:n3:apriori).  proposition]
 The lemma above gives a level-set bound for the quantity
-mathcal T+calT+rho+tau asigma. The ordinary product graph
+T+calT+rho+tau asigma. The ordinary product graph
 Gamma_f in g+db_0^2 has measure √(1+sigma^2)d V_g
 comparable to sqrt Dd V_g, and on every fixed compact base patch
 Q,
@@ -1334,12 +1334,12 @@ estimates.
 
 
 Lemma (Graph Sobolev). [lem:n3:graph-sobolev] 
-For every compactly supported smooth varphi on the graph
+For every compactly supported smooth phi on the graph
 Gamma_f, [eq:graph-sobolev]
 
-(int_{Gamma_f}|varphi|^4)^{1/2}
-le C(n)int_{Gamma_f}(|nabla_{Gamma_f}varphi|^2
-+(1+mathcal T)calT)varphi^2).
+(int_{Gamma_f}|phi|^4)^{1/2}
+le C(n)int_{Gamma_f}(|nabla_{Gamma_f}phi|^2
++(1+T)
 
 
 
@@ -1359,22 +1359,22 @@ submanifold Sobolev embedding for compactly supported functions then
 gives, in dimension 3,
 
 
- (int_{Gamma_f}|varphi|^{6})^{1/3}
- le Cint_{Gamma_f}|nabla_{Gamma_f}varphi|^{2},
+ (int_{Gamma_f}|phi|^{6})^{1/3}
+ le Cint_{Gamma_f}|nabla_{Gamma_f}phi|^{2},
 
 
 with C depending only on the immersion geometry, hence only on n.
 H"older's inequality together with V_Q<∞ yields
 
 
- int_{Gamma_f}|varphi|^{4}
- le V_Q^{1/3}(int_{Gamma_f}|varphi|^6)^{2/3},
+ int_{Gamma_f}|phi|^{4}
+ le V_Q^{1/3}(int_{Gamma_f}|phi|^6)^{2/3},
 
 
 and substituting the Sobolev bound closes the estimate on pi^{-1}Q.
-Patching finitely many such Q that exhaust the support of varphi
+Patching finitely many such Q that exhaust the support of phi
 gives [eq:graph-sobolev], using
-|varphi|^2le(1+mathcal T)|calT)|varphi|^2 to absorb the remaining
+|phi|^2le(1+T)|calT)|phi|^2 to absorb the remaining
 L^2 factor. The inner-face boundary is handled by the even reflection
 of H^1_0 across the plane where f is constant, as in
 [sec:strictification].
@@ -1450,11 +1450,11 @@ Suppose along a sequence with deltato0 one has
 inf_{B_{1/2}}sto0. The inhomogeneous weak Harnack inequality for
 [n3:axial:s-supersolution], after dropping the favorable
 -2c_0|H|^2 term, gives sto0 in measure on B_{1/2}. Testing
-by varphi^{2}(b-s)_{+} with a cutoff varphi and fixed b>0, and
+by phi^{2}(b-s)_{+} with a cutoff phi and fixed b>0, and
 absorbing the trace term by ellipticity and Young's inequality, yields
 
 
- int_{{s<b}}varphi^{2}|Ds|^{2}
+ int_{{s<b}}phi^{2}|Ds|^{2}
  le Cb^{2}+o(1).
 
 
@@ -1537,7 +1537,7 @@ coupled system [eq:coupled-system] has the schematic form
   on Ω_R,
  lDelta_g f+Psi_chi(f,Df)= eta_n,Xi_2(f,Z,Df,DZ), &
   on Ω_R,
- h=b, (V_lambda)_nu/u=mathcal B,calB, &
+ h=b, (V_lambda)_nu/u=B, &
   on B_0,
  f=Z=0, & on S_R.
 
@@ -1587,7 +1587,7 @@ linearized Z-equation has no zero-order term and its source
 vanishes there, so by ellipticity Div(anabla Z)le0 forces Z to
 be constant, hence to equal the Dirichlet value, so Z vanishes. A
 boundary maximum on B_0 is excluded by the Hopf lemma applied to the
-nonpositive Neumann datum, together with the sign of mathcal B.calB. The
+nonpositive Neumann datum, together with the sign of B. The
 adjoint problem has the same form, so the cokernel is trivial. On the
 f-side, the maximum principle for Psi_chi with strict monotone
 zero-order dependence rules out both signs. Hence the linearization
@@ -1651,12 +1651,12 @@ the traced Gauss identity for the graph Gamma_f over (Ω_R,g)
 combined with the conformal-change formula in dimension three gives
 
 tfrac12e^{2t}S_{barbar g}g
-={}&tfrac12S_g-vRic_g(e,e)+mathcal T(H^f)calT(H^f)
+={}&tfrac12S_g-vRic_g(e,e)+T(H^f)
 &-3dHess_g t(e,e) -(3+pv)tr_perpHess_g t +O(eta_n^2),
 
 where e is the unit vector of sigma=nabla f (the d-term drops
 by homogeneity at sigma=0), p=p(t) is determined by the warp
-function l, mathcal T(H^f)calT(H^f) is the coercive second-form square of
+function l, T(H^f) is the coercive second-form square of
 the [lem:n3:hessian-square] lemma, and the O(eta_n^2) collects the
 bounded contributions of the F-source terms to the trace difference.
 
@@ -1670,7 +1670,7 @@ propagated by Hopf. Consequently
 
 
 tfrac12e^{2t}S_{barbar g}g
-getfrac12S_g+mathcal T(H^f)-vRic_g(e,e)+O(calT(H^f)-vRic_g(e,e)+O(eta_n^2).
+getfrac12S_g+T(H^f)-vRic_g(e,e)+O(calT(H^f)-vRic_g(e,e)+O(eta_n^2).
 
 
 
@@ -1679,7 +1679,7 @@ tfrac12S_gge-tfrac12|K|^2+8pimu. Dominant energy supplies
 muge|J|_g, and the coercivity estimate of
 the [lem:n3:hessian-square] lemma (with constants absorbed into
 C(n)) bounds the negative pieces -tfrac12|K|^2-vRic_g(e,e) by a
-fixed multiple of mathcal T(H^f)calT(H^f) plus C(n)eta_n^2 on the bounded
+fixed multiple of T(H^f) plus C(n)eta_n^2 on the bounded
 range of the [prop:n3:apriori] proposition. Therefore
 tfrac12e^{2t}S_{barbar g}gge0 for all sufficiently large n,
 i.e. R_{barbar g}gge0 on Ω_R.
@@ -1966,8 +1966,8 @@ function uge0 and a smooth vector field X on M_{ext} with
 u>0 in IntM_{ext}, such that, on M_{ext} and up to S,
 
 symnabla X&=-uK,[eq:stationary-kid]
-Delta u&=-Div(K(X,cdot)^{sharp}),[eq:stationary-lapse]
-Hess u&=u(Ric_g+tau K-2K^2)-mathcal L_XK+8calL_XK+8pi X_{(i}J_{j)}. [eq:stationary-hess]
+Delta u&=-Div(K(X,·)^{sharp}),[eq:stationary-lapse]
+Hess u&=u(Ric_g+tau K-2K^2)-L_XK+8calL_XK+8pi X_{(i}J_{j)}. [eq:stationary-hess]
 
 Here tau=tr_gK, (K^2)_{ij}=K_i{}^kK_{kj}, and parentheses denote
 averaged symmetrisation. The data extend smoothly to S, and on S,
@@ -1993,7 +1993,7 @@ derivative at equilibrium is a positive measure omega whose metric
 perturbation equals the minimum of the individual cuts' area
 derivatives (the [lem:n3:area-envelope] lemma). Combining this one-sided
 derivative with the supporting-energy functional
-mathcal E=b^0E_calE=b^0E_{rm new}+b^iP_{{rm new},i} (with
+E=b^0E_{rm new}+b^iP_{{rm new},i} (with
 (b^0,b^i)=(E/m,-P_i/m) the future-unit-normal of (E,P)) and
 integrating by parts against constraint variations gives the variational
 identity of the [lem:n3:multiplier] lemma; a Riesz representation and a
@@ -2011,7 +2011,7 @@ the componentwise alternative).
 
 
 Lemma (Compactness of minimizing cuts). [lem:n3:area-envelope] 
-The collection mathcal TcalT of enclosing cuts attaining a_g(B) is
+The collection T of enclosing cuts attaining a_g(B) is
 compact in local L^1 and in unoriented tangent-plane area measures,
 and off S every two cuts coincide on the common tangent plane.
 
@@ -2040,10 +2040,10 @@ At equality, for every smooth compactly supported constraint variation
 constraint multiplier eta, [eq:multiplier]
 
 cint_S h_s,d A
-=mathcal E'(h,p)+2calE'(h,p)+2int_S u,d(eta),
+=E'(h,p)+2calE'(h,p)+2int_S u,d(eta),
 
-for every unit speed s on S, where c=(16pi)^{-1/2}cdotfrac12 A^{-1/2}
-cdot16pi/(16pi)=1/(4m) is the derivative of √(a/16pi) at
+for every unit speed s on S, where c=(16pi)^{-1/2}·frac12 A^{-1/2}
+·16pi/(16pi)=1/(4m) is the derivative of √(a/16pi) at
 a=A.
 
 
@@ -2051,7 +2051,7 @@ a=A.
 
 
 Proof. 
-The functional mathcal EcalE anchors the minimizer S: at the minimum
+The functional E anchors the minimizer S: at the minimum
 of the supporting energy over S, its derivative equals the derivative
 of √(a/16pi). Under the variation the constraint functional and
 the area functional have explicit derivatives; the multiplier identity
@@ -2249,7 +2249,7 @@ r>2m; the future horizon and the bifurcation sphere of the maximally
 extended Schwarzschild spacetime are both smooth cross-sections of
 that boundary. The given K is the restriction of the ∂_t
 Killing-field Lie derivative to the time slice t=0 in the spacetime
-frame, so K=frac12mathcal L_ngcalL_ng for the future unit normal n of
+frame, so K=frac12L_ng for the future unit normal n of
 xi=∂_t.
 
 
@@ -2488,13 +2488,13 @@ with the four modifications below.
 
 
 Strictification and end reduction in dimension four [sec:four-reduction]. The drift-Poisson strictification of the [lem:n3:strictification] lemma
-adapts: in dimension 4 the conformal family gmapsto u^2g,
-Kmapsto uK preserves the constraints and DEC, and the drift
+adapts: in dimension 4 the conformal family g|-> u^2g,
+K|-> uK preserves the constraints and DEC, and the drift
 source exponent is w=r^{-4-delta} with 0<delta<beta<min(q,1).
 The repaired data satisfy theta_+(B)<0, mu-|J|_gge c_0r^{-4-delta_1}
 with c_0>0, and K compactly supported. The end-reduction of
 the [prop:n3:rest-frame] proposition works in dimension 4: a distant
-Lorentz boost sends (E,P)mapsto(m,0), and a conformal support
+Lorentz boost sends (E,P)|->(m,0), and a conformal support
 truncation on the far end adjusts m and A_e by O(R^{-3}).
 
 
@@ -2510,7 +2510,7 @@ boundary three-volume at least (1-ε)A_e. The regularity estimates
 of the [sec:existence] section (graph Sobolev, axial gradient estimate,
 four-stage homotopy) pass through in dimension 4 with the same
 structure; the scalar identity uses a Schur-type trace calculation
-3Delta_{barbar g}gZ=mathcal T-typecalT-type decomposition that gives
+3Delta_{barbar g}gZ=T-typecalT-type decomposition that gives
 R_{barbar g}gge0 from the DEC.