diff --git a/hypersymplectic_hyperkahler_deformation_v2.txt b/hypersymplectic_hyperkahler_deformation_v2.txt
index d502203..eb2169d 100644
--- a/hypersymplectic_hyperkahler_deformation_v2.txt
+++ b/hypersymplectic_hyperkahler_deformation_v2.txt
@@ -151,7 +151,7 @@ surface with a nowhere-vanishing holomorphic two-form. The compact
 hyperk"ahler surface classification cite[Section~2, p.~162]{Boyer1988}
 therefore identifies X itself as a complex two-torus or a K3 surface.
 A complex two-torus is smoothly T^4, while every complex K3 surface is
-diffeomorphic to a nonsingular quartic surface in mathbf P^3,bfP^3, the
+diffeomorphic to a nonsingular quartic surface in P^3,P^3, the
 standard K3 manifold cite[p.~58]{Kodaira1970}.
 
 
@@ -443,7 +443,7 @@ therefore vanishes: the space of its tamed almost-complex structures is
 contractible and contains compatible structures. Bauer's bound for a
 closed symplectic four-manifold with torsion first Chern class gives
 b^+le3 cite[Corollary~1.2]{Bauer2008}. The three independent
-positive cohomology classes give the reverse inequality.
+positive cohomology classes give the reverse ineuality.
 
 The characteristic-number identity c_1^2=2chi+3sigma, obtained
 from p_1=c_1^2-2c_2, the Euler-number identity and the signature
@@ -534,9 +534,9 @@ Set Ω=A+iD and use its kernel as T^{0,1} as in
 Lemma~lem:definite-pair. For vector fields U,V in that kernel,
 Cartan's formula and ddΩ=0 give
 [
- L_UΩ=0,calL_UΩ=0,
+ L_UΩ=0,L_UΩ=0,
  iota_{[U,V]}Ω
- =L_U(Ω)-iota_V(L_UΩ)=0.calL_UΩ)=0.
+ =L_U(Ω)-iota_V(L_UΩ)=0.L_UΩ)=0.
 ]
 The kernel is involutive. The smooth Newlander--Nirenberg theorem
 therefore gives integrability [cite], and
@@ -587,7 +587,7 @@ to identify bivectors and two-forms, so
   |G|_g^2=2.
 
 The bundle of these bivectors is denoted by
-L→calL→ X.
+L→L→ X.
 
 A two-current is a continuous real linear functional on
 Ω^2(X)=C^∞(X,Λ^2). Under the metric identification its
@@ -652,7 +652,7 @@ and separate the open cone P_J from the affine subspace
 beta+ddΩ^1(X) in Ω^2(X). If N is the separating
 functional, the unbounded translation directions in this affine
 subspace give N(ddalpha)=0. Multiplying a positive form by any
-positive scalar, the separation inequalities imply both
+positive scalar, the separation ineualities imply both
 N(P_J) and N(beta)le0. Adding an arbitrary real
 multiple of an anti-invariant form to an element of P_J shows
 that N annihilates anti-invariant forms. The same limiting argument
@@ -660,7 +660,7 @@ gives positivity on semipositive invariant forms.
 
 For either functional, comparison of an invariant form with sufficiently
 large positive multiples of G gives |N(xi)|le C N(G)|xi|_{C^0}.
-If N(G) were zero, this inequality and anti-invariance would make the
+If N(G) were zero, this ineuality and anti-invariance would make the
 functional zero; normalize N(G)=1. Locally the functional is
 represented by a positive Hermitian matrix of measures. Its trace is a
 positive measure mu; the Radon--Nikodym matrix relative to mu is
@@ -700,7 +700,7 @@ On sections of E consider P=Rdddd^*. If a is self-dual, the
 self-dual projection of xiwedgeiota_xi a is |xi|^2a/2: in a
 frame with first covector xi/|xi|, the two self-dual halves of a
 have equal length. Hence
-sigma_2(P)(x,xi)=tfrac12|xi|_g^2Id_E and
+sigma_2(P)(x,xi)=1/2|xi|_g^2Id_E and
 < Pa,a>_2=|dd^*a|_2^2. Thus P is elliptic,
 self-adjoint, and nonnegative. Its kernel consists of smooth sections
 with dd^*a=0; since a=*a, they also satisfy dd a=0, and hence
@@ -870,20 +870,20 @@ B_{rho_0}(p),
  beta(h_s)(ell)&ge c_2delta^{-1}-C_2(1+|logdelta|).
                   
 
-The inequalities also apply in setup (B), where the correction
+The ineualities also apply in setup (B), where the correction
 terms are absent.
 
 Choose 0<rho<rho_0 so small that
 C_2(1+|logdelta|)le c_2/(2delta) for 0<deltale2rho, and
 then choose a fixed multiplier Mge2C_1/c_2. If slerho and
-tlerho, the two displayed inequalities give
+tlerho, the two displayed ineualities give
 beta(u_s+Mh_s)(ell)ge c_1s^2delta^{-4}. On X∖ B_rho(p)
 all terms are bounded independently of s, including the nonlocal
 corrections, by the last assertion of
 Cref{lem:current-radial-operator}. Since deltale2s on B_s(p),
 globally
 [
- beta(u_s+Mh_s)(ell)ge c s^{-2}mathbf 1_bf1_{B_s(p)}-C.
+ beta(u_s+Mh_s)(ell)ge c s^{-2}1_bf1_{B_s(p)}-C.
 ]
 Apply N, use N(beta(u_s+Mh_s))=0 and mu(X)=1, and obtain
 (eq:current-quadratic-mass) for slerho. For larger s the
@@ -958,7 +958,7 @@ Take chige0 and use its leading coefficient to define
        int_0^∞ h^3e^{-h-d^2/(4s^2h)},dd h.
 ]
 For d<s, integration over 1le hle2 gives the lower bound. For
-every fixed k, the elementary inequality
+every fixed k, the elementary ineuality
 e^{-r^2/(ch)}le C_k(1+r)^{-k}(1+h^{k/2}) shows that the last integral
 is bounded by C_k(1+d/s)^{-k}, giving the upper bound.
 
@@ -1006,14 +1006,14 @@ Parallel transport preserves the Hodge star. From
 (eq:current-line-normalization), direct expansion gives
 
  <ell,*tau_{zto y}m>
- =tfrac12|ell-tau_{zto y}m|^2
-       -tfrac14|G_y-tau_{zto y}G_z|^2.
+ =1/2|ell-tau_{zto y}m|^2
+       -1/4|G_y-tau_{zto y}G_z|^2.
 
 The last squared difference is bounded by Cd^2, where d=dist(y,z).
 
 Express the pairing using the continuous kernel of S_s^2 and
 (eq:current-line-measure). Its leading scalar is nonnegative, so
-the lower-bound kernel inequality and the angular identity supply the
+the lower-bound kernel ineuality and the angular identity supply the
 positive term. The Cd^2 term, using the upper bound with exponent at
 least eight, and the error kernel, using exponent six, are bounded in
 absolute value by
@@ -1209,7 +1209,7 @@ so nu(B_s(p))le Cs^2, uniformly in p.
 Fix p, identify a normal-coordinate ball with a Euclidean ball, and
 freeze J_p and G_p as constant tensors J_0,G_0. Writing
 t=|zeta|=dist(p,exp_pzeta), define
-b_p(s)=s^{-2}T(mathbf 1_bf1_{B_s(p)}G_0). These evaluations are
+b_p(s)=s^{-2}T(1_bf1_{B_s(p)}G_0). These evaluations are
 well-defined because T has measure coefficients. Since G(ell)=1,
 G-G_0=O(t) and Q annihilates the invariant form G,
 
@@ -1218,7 +1218,7 @@ G-G_0=O(t) and Q annihilates the invariant form G,
 because both error integrals are bounded by Cs^{-2}s,nu(B_s(p)).
 
 In frozen coordinates
-b_p(s)=s^{-2}N(mathbf 1_bf1_{B_p(s)}G_0)+o(1). Approximating the
+b_p(s)=s^{-2}N(1_bf1_{B_p(s)}G_0)+o(1). Approximating the
 Euclidean ball from inside and outside by continuous test functions and
 dominated convergence give
 [
@@ -1258,7 +1258,7 @@ lambda-almost everywhere.
 Proof. 
 Fix p with a finite density θ(p) and such that the measure
 gamma_p obtained by the homothety s^{-2}(z|-> p+sz)_*mu has
-a tangent measure gamma=M(p)mathbf 1_bf1_{P_p} in a complex
+a tangent measure gamma=M(p)1_bf1_{P_p} in a complex
 two-dimensional vector space.
 Closedness of T=N+Q passes to the limit as s↓0: smooth
 test forms converge in C^1 after rescaling, and the Q term is
@@ -1292,11 +1292,11 @@ For either separator construction, let θ be the density of
 Cref{lem:density-existence}. If b:RtoR is smooth and bounded with
 b(0)=0, then the current b(θ)N is closed. The currents
 
- N^+=mathbf 1_bf1_{{θ>0}}N,
- I_a=mathbf 1_bf1_{{θ>a}}(pi)/(θ)Nquad(a>0)
+ N^+=1_bf1_{{θ>0}}N,
+ I_a=1_bf1_{{θ>a}}(pi)/(θ)Nquad(a>0)
 
 are also closed. In particular, if
-N^d=mathbf 1_bf1_{{θ=0}}N, then N^d+Q is closed.
+N^d=1_bf1_{{θ=0}}N, then N^d+Q is closed.
 
 
 Proof. 
@@ -1409,7 +1409,7 @@ closed, as claimed.
 
 For the remaining weights, take a smooth nondecreasing
 chi:Rto[0,1] vanishing on (-∞,0] and equal to one on
-[1,∞); the bounded functions chi(nt)tomathbf 1_bf1_{{t>0}}
+[1,∞); the bounded functions chi(nt)to1_bf1_{{t>0}}
 pointwise on [0,∞) and vanish at zero. Fix a>0 and set
 [
  b_{a,n}(t)=
@@ -1419,7 +1419,7 @@ pointwise on [0,∞) and vanish at zero. Fix a>0 and set
  
 ]
 These are smooth, vanish near zero, are uniformly bounded by pi/a,
-and converge at every tge0 to mathbf 1_bf1_{{t>a}}pi/t. Dominated
+and converge at every tge0 to 1_bf1_{{t>a}}pi/t. Dominated
 convergence and closedness of each smooth-weight current prove
 (eq:density-weighted-restrictions). No derivative bound uniform
 in n is used: the smoothing limit was taken first for each n.
@@ -1439,7 +1439,7 @@ are uniformly bounded, and B_s(z)to0 whenever θ(z)=0. In
 particular,
 
  int_XB_s(z),dmu^d(z)→0,
- mu^d=mathbf 1_bf1_{{θ=0}}mu.
+ mu^d=1_bf1_{{θ=0}}mu.
 
 
 
@@ -1458,7 +1458,7 @@ For closed currents S,S', the smoothing operator
 S_s=(1+s^2Delta_g)^{-3} preserves cohomology classes and commutes
 with the Hodge star. Hodge decomposition therefore gives
 
- < S_sS,*S_sS'>_{L^2}=mathfrak c(S)frakc(S)·mathfrak c(S'),frakc(S'),
+ < S_sS,*S_sS'>_{L^2}=c(S)c(S)·c(S'),c(S'),
 
 independently of s; the Sobolev regularity after smoothing is
 sufficient for this identity.
@@ -1468,7 +1468,7 @@ Suppose that no compatible symplectic form exists. The separator of
 Cref{lem:current-separators} gives a nonzero positive current N,
 normalized to trace mass one, and anti-invariant Q such that
 T=N+Q annihilates all smooth closed two-forms; in particular T is
-closed with mathfrak c(T)=0.frakc(T)=0. By Cref{prop:density-weights}, T^d=N^d+Q
+closed with c(T)=0.c(T)=0. By Cref{prop:density-weights}, T^d=N^d+Q
 is closed. (eq:density-smoothed-intersection) gives
 
  0&=< S_sT,*S_sT^d>notag
@@ -1479,7 +1479,7 @@ is closed. (eq:density-smoothed-intersection) gives
 The self-duality of Q accounts for the signs. Both mixed terms tend to
 zero by Cref{prop:angular-energy}. The positive-current lower bound of
 Cref{lem:current-positive-pairing}, applied with second submeasure
-lambda^d=mathbf 1_bf1_{{θ=0}}lambda, gives
+lambda^d=1_bf1_{{θ=0}}lambda, gives
 < S_sN,*S_sN^d>ge-Cint_XB_s,dmu^d=o(1). Since
 S_sQto Q in L^2, taking a limit inferior in
 (eq:density-compatibility-pairing) forces |Q|_2^2=0. Thus
@@ -1493,7 +1493,7 @@ a theorem representing integral cycles by curves.
 
 
 [Nonnegative diffuse square]
-If Q=0, then N^d is closed and mathfrak c(N^d)^2frakc(N^d)^2ge0.
+If Q=0, then N^d is closed and c(N^d)^2frakc(N^d)^2ge0.
 
 
 Proof. 
@@ -1501,7 +1501,7 @@ Closedness is Cref{prop:density-weights}. By
 (eq:density-smoothed-intersection) and
 Cref{lem:current-positive-pairing},
 [
- mathfrak c(N^d)^2=<frakc(N^d)^2=< S_sN^d,*S_sN^d>
+ c(N^d)^2=<c(N^d)^2=< S_sN^d,*S_sN^d>
  ge-Cint_XB_s,dmu^d,
 ]
 and the right side tends to zero by Cref{lem:density-diffuse-error}.
@@ -1532,7 +1532,7 @@ By Cref{prop:density-tangent-measure}, lambda-almost everywhere the
 current has a single complex tangent line with weight θ. The
 normalization of lambda makes (pr_X)_*lambda=mu;
 hence the trace measure of I_a equals
-mathbf 1_bf1_{{θ>a}}pi,dV_g and is rectifiable. Preiss's
+1_bf1_{{θ>a}}pi,dV_g and is rectifiable. Preiss's
 rectifiability theorem [cite] upgrades the a.e.~planar
 tangents to rectifiability; closedness of I_a and positivity then
 express it as an integral sum of simple curves. Positivity of the
@@ -1543,7 +1543,7 @@ gives a finite sum of irreducible closed curves with positive integer
 multiplicities.
 
 Finally, pointwise on {θ>0},
-int_0^∞mathbf 1_bf1_{{θ>a}}pi/θ,da=pi; for a
+int_0^∞1_bf1_{{θ>a}}pi/θ,da=pi; for a
 smooth two-form alpha, Fubini is justified by
 [
  int_0^∞|I_a(alpha)|,dalepi|alpha|_{C^0}mu^+(X).
@@ -1556,12 +1556,12 @@ Assume that H has no taming representative. The class separator of
 Cref{lem:current-separators}(ii) gives a nonzero closed
 positive current N, with Q=0, satisfying
 
- H·mathfrak c(N)frakc(N)le0.
+ H·c(N)c(N)le0.
 
 Its positive-density and diffuse parts are separately closed by
 Cref{prop:density-weights}. Anti-invariance of A,C gives
-mathfrak c(N^d)frakc(N^d)·[A]=N^d(A)=0=mathfrak c(N^d)frakc(N^d)·[C]; thus
-w=mathfrak c(N^d)frakc(N^d) lies in V and w^2ge0 by
+c(N^d)c(N^d)·[A]=N^d(A)=0=c(N^d)c(N^d)·[C]; thus
+w=c(N^d)c(N^d) lies in V and w^2ge0 by
 Cref{lem:density-diffuse-square}.
 
 If N^dne0, choose a taming form eta in U. Strict taming on the
@@ -1578,13 +1578,13 @@ strict positivity.
 
 For every a>0, Cref{lem:density-integral-thresholds} writes I_a as
 a positive integral sum of irreducible closed J-curves; the hypothesis
-on H gives H·mathfrak c(I_a)frakc(I_a)ge0, with strict inequality
+on H gives H·c(I_a)c(I_a)ge0, with strict ineuality
 whenever I_ane0. If N^+ne0, the layer identity has positive total
 mass, so the set of a for which I_ane0 has positive Lebesgue
 measure; pairing (eq:density-layer-cake) with a smooth
-representative of H gives H·mathfrak c(N^+)>0.frakc(N^+)>0. The pairing is
+representative of H gives H·c(N^+)>0.c(N^+)>0. The pairing is
 zero when N^+=0. At least one of N^d,N^+ is nonzero, so
-H·mathfrak c(N)=Hfrakc(N)=H·mathfrak c(N^d)+Hfrakc(N^d)+H·mathfrak c(N^+)>0,frakc(N^+)>0,
+H·c(N)=Hfrakc(N)=H·c(N^d)+Hfrakc(N^d)+H·c(N^+)>0,c(N^+)>0,
 contradicting (eq:density-cone-separator). A smooth closed taming
 representative is symplectic by pointwise nondegeneracy.
 
@@ -1682,15 +1682,15 @@ unit sphere of P.
 ## The fixed spin-c structure and the equations
 
 Fix the canonical Spin^c structure of one J_v, with
-isomorphism type mathfrak s_0.fraks_0. The structures J_v form a
+isomorphism type s_0.s_0. The structures J_v form a
 connected family, so all their canonical structures have the same
 fiberwise isomorphism type. For Ein H^2(X;Z), let
-mathfrak s_E=fraks_E=mathfrak s_0fraks_0 ⊗  L_E, where
+s_E=s_E=s_0 ⊗  L_E, where
 c_1(L_E)=E; the family extends over the contractible space of
 metrics with no twist from the parameter base. Its determinant class
 and dimensions are
 
- c_1(detmathfrak s_E)=2E,fraks_E)=2E, d(mathfrak s_E)=E^2,fraks_E)=E^2,
+ c_1(dets_E)=2E,s_E)=2E, d(s_E)=E^2,s_E)=E^2,
  ind_{C}not D_E=(4E^2-sigma)/(8).
 
 Fix a homology orientation once and for all.
@@ -1698,12 +1698,12 @@ Fix a homology orientation once and for all.
 ## Wall-crossing at the harmonic wall
 
 [Wall-crossing]
-Let mathfrak sfraks be the fixed spin-c structure of the family J_v
-over S^2, and let q=d(mathfrak s)/2fraks)/2 be its expected complex
+Let s be the fixed spin-c structure of the family J_v
+over S^2, and let q=d(s)/2fraks)/2 be its expected complex
 dimension. Then the family invariant in the chamber after crossing the
 harmonic wall gives
 
- FSW_{mathfrak s}fraks(S^2)
+ FSW_{s}s(S^2)
   =s_{b_1/2}(K-O),
 
 where K,O are the ranks of the complex Dirac index, and
@@ -1747,7 +1747,7 @@ spinor with weight one, as used in
 For E^2=0, write FSW_E(U_p) for the spherical
 pairing of the based-gauge evaluation line U_p in the large-parameter
 chamber, and SW_X(E) for the ordinary dimension-zero
-invariant of mathfrak s_E.fraks_E. Denote by pi the projection of the
+invariant of s_E.s_E. Denote by pi the projection of the
 regular family moduli space to S^2.
 
 [Evaluation identity]
@@ -1756,7 +1756,7 @@ h=c_1(O(1)) is the twistor parameter. Its spherical pairing
 satisfies
 
  FSW_E(U_p)=U_p([M_E^{reg}])
- =SW_X(E)+eFSW_{mathfrak s_E}fraks_E(S^2).
+ =SW_X(E)+eFSW_{s_E}s_E(S^2).
 
 For every prescribed point p, if FSW_E(U_p)ne0
 then the exact family has an evaluation zero at p.
@@ -1764,11 +1764,11 @@ then the exact family has an evaluation zero at p.
 
 Proof. 
 The evaluation line over the coefficient sphere is a line bundle
-O(1) ⊗ E_E|_p.calE_E|_p. The evaluation transforms with
+O(1) ⊗ E_E|_p.E_E|_p. The evaluation transforms with
 the same gauge weight as the spinor, and its sign on a reducible link
-can be checked on elltoE_E|_p:calE_E|_p: it lies in
-Hom(ell,E_E|_p)=calE_E|_p)=O(1) ⊗ 
-E_E|_p.calE_E|_p. On the quotient, the evaluation section is a
+can be checked on elltoE_E|_p:E_E|_p: it lies in
+Hom(ell,E_E|_p)=E_E|_p)=O(1) ⊗ 
+E_E|_p.E_E|_p. On the quotient, the evaluation section is a
 section of the tensor product of the based-gauge evaluation line with
 the pullback of E_E|_calE_E|_{{p}times S^2}; its first Chern
 class is U_p+epi^*h. Pairing pi^*h with the family moduli cycle
@@ -1821,7 +1821,7 @@ follows from the topological identity for the determinant line. Second,
 off the first-component zero set, the L^1-to-C^1 distance from the
 canonical connection is
 
- |A-calA-A_{can}|
+ |A-A-A_{can}|
  +|F_{A}-F_{A_{can}}|
  le Cr^{-1}+Crexp(-sqrt r,
           dist(x,alpha^{-1}(0))/C)
@@ -1852,7 +1852,7 @@ Proof. [Proof of Cref{thm:family-curves}]
 The projection F^+ is nonzero: otherwise F would be a nonzero
 square-zero vector in the negative-definite space P^perp. Lift F
 to an integral class and use the fixed spin-c structure
-mathfrak s_F.fraks_F.
+s_F.s_F.
 
 The ordinary invariant vanishes. Choose the opposite coefficient
 direction w=-F^+/|F^+|, so that F[B_w]<0. If
@@ -1997,7 +1997,7 @@ its kernel is null and lies in the negative-definite space P^perp,
 and both spaces have dimension three. Hence pi(L) is a lattice in
 P. Each of the finitely many cosets just found determines one point of
 [
- T=P/bbT=P/pi(L),
+ T=P/T=P/pi(L),
 ]
 because adding an element of L changes the projection by an element of
 pi(L). For FinF define the circle
@@ -2526,8 +2526,8 @@ with torus fibers of area rho in the dd t direction.
 
 The complex surface Y obtained by adding the point at X=0 is
 smooth because the Hopf chart of the charge-one instanton completes
-W_m,dmathbf X^2+W_m^bfX^2+W_m^{-1}theta_m^2 to a flat smooth ball. In that
-chart mathbf XbfX and r are smooth quadratic expressions, and
+W_m,dX^2+W_m^X^2+W_m^{-1}theta_m^2 to a flat smooth ball. In that
+chart X and r are smooth quadratic expressions, and
 rtheta_m is smooth. The coefficient 1/(4pi), with connection
 period one, is the smooth Hopf normalization. For instance, near the
 pole,
@@ -2536,7 +2536,7 @@ pole,
  frac{W^{-1}-W_m^{-1}}{r^2}
     =-((4pi)^2H_rho)/(1+4pi rH_rho),
 
-so g_rho/rho=W,dmathbf X^2+W^bfX^2+W^{-1}(theta_m+beta)^2 is smooth
+so g_rho/rho=W,dX^2+W^X^2+W^{-1}(theta_m+beta)^2 is smooth
 at the added point; the coefficient of the apparently singular
 squared-connection difference is smooth by the second identity.
 Subtracting the flat Hopf triple in
@@ -3054,7 +3054,7 @@ the explicit bounds
 
 which hold because the exact gluing adjusted the two pairings. The
 remaining residual R_ is exponentially small in the
-fixed Sobolev norms after the standard Sobolev inequalities on the
+fixed Sobolev norms after the standard Sobolev ineualities on the
 fixed chart cover.
 
 
@@ -3062,10 +3062,10 @@ fixed chart cover.
 
 Set
 
- widehatwidehat E_E_eps&=E_eps-a_eps A-b_eps C,&
- q_eps&=int_Xwidehatwidehat E_E_eps^2,notag
+ Ehat_E_eps&=E_eps-a_eps A-b_eps C,&
+ q_eps&=int_XEhat_E_eps^2,notag
  r_eps&=q_eps^{-1/2},&
- D^0_eps&=r_epswidehatwidehat E_E_eps.
+ D^0_eps&=r_epsEhat_E_eps.
 
 
 The period normalization gives int A^2=int C^2=1 and