diff --git a/hypersymplectic_hyperkahler_deformation_v2.txt b/hypersymplectic_hyperkahler_deformation_v2.txt
index 180985e..d502203 100644
--- a/hypersymplectic_hyperkahler_deformation_v2.txt
+++ b/hypersymplectic_hyperkahler_deformation_v2.txt
@@ -88,7 +88,7 @@ change of basis and positive scaling; it makes the endpoint identity
 cleaner but does not restrict the geometry. If the initial triple has
 an arbitrary positive Gram matrix G_{ij}=int_Xomega_iwedgeomega_j,
 one applies the theorem to Lomega for a constant invertible real
-L with LGL^{mathsf T}sfT=I_3 and applies L^{-1} to the resulting
+L with LGL^{T} and applies L^{-1} to the resulting
 path. The endpoint then satisfies
 omega_i(1)wedgeomega_j(1)=2G_{ij}mu_1: the three classes are
 parallel self-dual forms for a hyperk"ahler metric, with their original
@@ -122,11 +122,11 @@ alpha_t=sum_i c_iomega_i(t). Positivity of the pointwise
 wedge-product matrix gives alpha_twedgealpha_t>0, so alpha_t
 is symplectic; its class is fixed because each [omega_i(t)] is fixed.
 
-For the endpoint metric g, set r_c=(c^{mathsf T}sfTGc)^{1/2}>0. The
+For the endpoint metric g, set r_c=(c^{T}{1/2}>0. The
 endpoint identities show that alpha_1/r_c is parallel and self-dual
 with g-norm sqrt2. It therefore defines a parallel integrable
 complex structure J^{end}_c with
-alpha_1/r_c=g(J^{end}_c,cdot,cdot),
+alpha_1/r_c=g(J^{end}_c,·,·),
 so that alpha_1 is its K"ahler form for the hyperk"ahler metric
 r_cg.
 
@@ -468,24 +468,24 @@ bundle of a unique oriented conformal structure. Choose one metric
 g_0 in that conformal class. Since the original forms are closed and
 self-dual, they are harmonic, and
 
- mathcal H^+_calH^+_{g_0}=span_R{omega_1,omega_2,omega_3}
+ H^+_calH^+_{g_0}=span_R{omega_1,omega_2,omega_3}
 
 by Proposition~prop:topology. Here the span is the vector space of
 global forms with constant coefficients.
 
 ## The coefficient sphere of almost-complex structures
 
-For vin S^2subsetR^3, let B_v=vcdotomega and let
-mathcal A_vcalA_v be the plane of forms acdotomega with aperp v.
+For vin S^2subsetR^3, let B_v=v·omega and let
+A_v be the plane of forms a·omega with aperp v.
 Choosing the sign tamed by B_v defines an almost-complex structure
-J_v. This construction uses the plane mathcal A_vcalA_v itself and does
+J_v. This construction uses the plane A_v itself and does
 not require a global choice of ordered basis over S^2.
 
 
 The family J_v is smooth and is Hermitian for g_0. At every point
 of X, its map to the twistor sphere has degree of absolute value one.
 The closed J_v-anti-invariant forms form the two-dimensional space
-mathcal A_v.calA_v.
+A_v.
 
 
 Proof. 
@@ -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
 [
- mathcal L_UΩ=0,calL_UΩ=0,
+ L_UΩ=0,calL_UΩ=0,
  iota_{[U,V]}Ω
- =mathcal L_U(calL_U(iota_VΩ)-iota_V(mathcal L_UΩ)=0.calL_UΩ)=0.
+ =L_U(Ω)-iota_V(L_UΩ)=0.calL_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
-mathcal L→calL→ X.
+L→calL→ X.
 
 A two-current is a continuous real linear functional on
 Ω^2(X)=C^∞(X,Λ^2). Under the metric identification its
@@ -606,32 +606,32 @@ are written <,,,,>_2.
 
 Let
 [
- mathcal P_J=calP_J={xiinΩ^2(X):xi(ell)>0
-                     for every ellinmathcal LcalL},
+ P_J={xiinΩ^2(X):xi(ell)>0
+                     for every ellinL},
  
- mathcal Z_J=calZ_J={xiin C^∞(X,I):ddxi=0}.
+ Z_J={xiin C^∞(X,I):ddxi=0}.
 ]
-The set mathcal P_JcalP_J is an open convex cone in the smooth topology;
+The set P_J is an open convex cone in the smooth topology;
 openness already holds in the uniform topology. Its closed elements are
 precisely the symplectic forms taming J. The elements of
-mathcal P_JcalP_Jcapmathcal Z_JcalZ_J are the compatible symplectic forms.
+P_J Z_JcalZ_J are the compatible symplectic forms.
 
 ## The two separators
 
 [The two separators]
 The following hold.
 [label=(roman*)]
- If mathcal P_JcalP_Jcapmathcal Z_JcalZ_J is empty, there is a nonzero
- current N which annihilates mathcal Z_JcalZ_J and all anti-invariant
+ If P_J Z_JcalZ_J is empty, there is a nonzero
+ current N which annihilates Z_J and all anti-invariant
  forms, and is nonnegative on semipositive invariant forms.
- If Hin H^2(X;R) has no representative in mathcal P_J,calP_J, there
+ If Hin H^2(X;R) has no representative in P_J, there
  is a nonzero positive invariant current N which is closed and
  satisfies N(H)le0.
 
 In either case one can normalize N(G)=1 and choose a positive Borel
-measure lambda on mathcal LcalL such that
+measure lambda on L such that
 
- N(xi)=int_{mathcal L}calLxi_y(ell),ddlambda(y,ell),
+ N(xi)=int_{L},ddlambda(y,ell),
   mu=(pr_X)_*lambda,
   mu(X)=1.
 
@@ -640,21 +640,21 @@ In particular N has order zero and belongs to H^{-3}(X,Λ^2).
 
 Proof. 
 For (i), apply separation of the open convex set
-mathcal P_JcalP_Jcap C^∞(X,I) from the disjoint linear subspace
-mathcal Z_JcalZ_J in C^∞(X,I). The separating continuous functional
-vanishes on mathcal Z_JcalZ_J and is nonnegative on the positive cone.
+P_J C^∞(X,I) from the disjoint linear subspace
+Z_J in C^∞(X,I). The separating continuous functional
+vanishes on Z_J and is nonnegative on the positive cone.
 Extend it to all two-forms by precomposing with 1-R. Adding a positive
 multiple of G and then taking a limit proves nonnegativity on every
 semipositive invariant form.
 
 For (ii), choose a smooth closed representative beta of H
-and separate the open cone mathcal P_JcalP_J from the affine subspace
+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
-N(mathcal P_J)calP_J)ge0 and N(beta)le0. Adding an arbitrary real
-multiple of an anti-invariant form to an element of mathcal P_JcalP_J shows
+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
 gives positivity on semipositive invariant forms.
 
@@ -808,14 +808,14 @@ bounded; the remaining input is uniformly smooth.
 In either setup (A) or (B), there is a constant C
 with
 
- mu(B_s(p))le Cs^2qquad(pin X, 0<sle1).
+ mu(B_s(p))le Cs^2 (pin X, 0<sle1).
 
 
 
 Proof. 
 Fix a radius rho_0>0 for uniform normal coordinates and a smooth
 radial cutoff which is one on B_{2rho_0}(p) and supported in
-B_{3rho_0}(p). With t=dist(p,cdot) define the globally smooth
+B_{3rho_0}(p). With t=dist(p,·) define the globally smooth
 functions
 [
  u_s=chi(t)log√(s^2+t^2),
@@ -1079,7 +1079,7 @@ give
 
  < S_sC_1,*S_sC_2>_2
   =int_X(*S_sC_1)wedge(*S_sC_2)
-  =[*C_1]cdot[*C_2].
+  =[*C_1]·[*C_2].
 
 Write *C_i=h_i+dd a_i with h_i harmonic and a_iin H^{-2};
 then S_sh_i=h_i and S_sa_iin H^4. Integration by parts eliminates
@@ -1103,7 +1103,7 @@ The convergence is uniform over lambda'lelambda.
 
 
 Proof. 
-Let mathcal A_scalA_s denote the double integral in
+Let A_s denote the double integral in
 (eq:current-angular-energy) and put x_s=|S_sQ|_2. Since Q is
 self-dual and S_s commutes with *, the topological pairing applied
 to T=N+Q gives
@@ -1114,13 +1114,13 @@ to T=N+Q gives
 Use the positive-current lower bound and the B_s bound for the first
 term, and the mixed bound for the second. The result is
 
- cs^{-4}mathcal A_s+x_s^2-Cx_scalA_s+x_s^2-Cx_sle C+[*T]^2.
+ cs^{-4}A_s+x_s^2-Cx_scalA_s+x_s^2-Cx_sle C+[*T]^2.
 
 The right side is independent of s. In setup (A) its
 intersection term is zero. In setup (B) that term is bounded
 uniformly for mass-one N: its harmonic coefficients are pairings with
 fixed smooth forms and are bounded by the line-measure representation.
-Completing the square in x_s bounds both x_s and s^{-4}mathcal A_s.calA_s.
+Completing the square in x_s bounds both x_s and s^{-4}A_s.
 
 As s↓0, S_sQ converges to Q distributionally. A weakly
 convergent subsequence of the bounded L^2 family has limit Q,
@@ -1176,7 +1176,7 @@ Suppose Uin V has positive square and contains a smooth tamer for
 J. If Hin V has positive square, belongs to the same component of
 {vin V:v^2>0} as U, and satisfies
 [
- Hcdot d>0
+ H· d>0
 ]
 for every irreducible closed J-holomorphic curve of class d, then
 H contains a smooth symplectic form taming J.
@@ -1257,7 +1257,7 @@ 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}(zmapsto p+sz)_*mu has
+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
 two-dimensional vector space.
 Closedness of T=N+Q passes to the limit as s↓0: smooth
@@ -1371,10 +1371,10 @@ O(s^2), so
 Since |q_i|le C/s, the integrated replacement error is at most
 
  s^{-2}!int!|int q_i dmu-T(q_iomega_{12})|dlambda
- le Cs^{-3}(s^2!iint_{dist<s}!dmu,dmu+mathcal A_s)calA_s)
+ le Cs^{-3}(s^2!iint_{dist<s}!dmu,dmu+A_s)
  +Cs^{-3}!int_X!|Q|M_s,dV_g=o(1),
 
-where the double mass is O(s^2), mathcal A_scalA_s is the angular
+where the double mass is O(s^2), A_s is the angular
 integral of O(s^4), and the last term tends to zero by
 (eq:density-q-gradient).
 
@@ -1393,7 +1393,7 @@ Replacing T by N+Q, the Q-contribution again tends to zero by
 (eq:density-q-gradient), the s^2 geometric error is O(s), and
 Cauchy--Schwarz bounds the rest by
 
- Cs^{-3}mathcal A_s^calA_s^{1/2}mathcal T_s^calT_s^{1/2}=s^{-3}cdot O(s^2)cdot
+ Cs^{-3}A_s^{1/2}T_s^{1/2}=s^{-3}· O(s^2)·
  o(s)=o(1).
 
 Combining (eq:density-first-variation),
@@ -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)cdotmathfrak c(S'),frakc(S'),
+ < S_sS,*S_sS'>_{L^2}=mathfrak c(S)frakc(S)·mathfrak c(S'),frakc(S'),
 
 independently of s; the Sobolev regularity after smoothing is
 sufficient for this identity.
@@ -1556,35 +1556,35 @@ 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
 
- Hcdotmathfrak c(N)frakc(N)le0.
+ H·mathfrak c(N)frakc(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)cdot[A]=N^d(A)=0=mathfrak c(N^d)frakc(N^d)cdot[C]; thus
+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
 Cref{lem:density-diffuse-square}.
 
 If N^dne0, choose a taming form eta in U. Strict taming on the
-compact bundle of unit complex lines gives Ucdot w=N^d(eta)>0; in
+compact bundle of unit complex lines gives U· w=N^d(eta)>0; in
 particular wne0. Since [A],[C] span a positive two-plane and
 b^+=3, the intersection form on V has signature (1,b^-). The
-conditions w^2ge0 and Ucdot w>0 put w in the nonzero closed time
+conditions w^2ge0 and U· w>0 put w in the nonzero closed time
 cone determined by U. Every timelike H in that component has
-Hcdot w>0: writing H=h_0u+h_-, w=w_0u+w_- with a unit timelike
+H· w>0: writing H=h_0u+h_-, w=w_0u+w_- with a unit timelike
 u and orthogonal negative parts, h_0>|h_-|, w_0ge|w_-|, w_0>0,
-so Hcdot wge h_0w_0-|h_-|,|w_-|>0. If N^d=0, its pairing is
+so H· wge h_0w_0-|h_-|,|w_-|>0. If N^d=0, its pairing is
 instead zero; this distinguishes the nonzero diffuse case needed for
 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 Hcdotmathfrak c(I_a)frakc(I_a)ge0, with strict inequality
+on H gives H·mathfrak c(I_a)frakc(I_a)ge0, with strict inequality
 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 Hcdotmathfrak c(N^+)>0.frakc(N^+)>0. The pairing is
+representative of H gives H·mathfrak c(N^+)>0.frakc(N^+)>0. The pairing is
 zero when N^+=0. At least one of N^d,N^+ is nonzero, so
-Hcdotmathfrak c(N)=Hfrakc(N)=Hcdotmathfrak c(N^d)+Hfrakc(N^d)+Hcdotmathfrak c(N^+)>0,frakc(N^+)>0,
+H·mathfrak c(N)=Hfrakc(N)=H·mathfrak c(N^d)+Hfrakc(N^d)+H·mathfrak c(N^+)>0,frakc(N^+)>0,
 contradicting (eq:density-cone-separator). A smooth closed taming
 representative is symplectic by pointwise nondegeneracy.
 
@@ -1613,7 +1613,7 @@ Cref{prop:topology},
 
 Let J_v, vin S^2, be the coefficient sphere of
 Cref{lem:twistor-degree}, with the sign chosen so that
-B_v=vcdotomega tames J_v. Write
+B_v=v·omega tames J_v. Write
 P=span{[omega_1],[omega_2],[omega_3]} for the
 original positive period plane.
 
@@ -1720,9 +1720,9 @@ S^2times[0,1], because its normal rank is b^+=3. Above such a
 parameter the reducibles form the torus Pic^0(X).
 Represent the complex Dirac index on this torus as K-O, with ranks
 k,o, using a stabilization when necessary. Write
-x=c_1(mathcal O(1))calO(1)) on the projective bundle of lines
-pi:mathbb P(K)bbP(K)toPic^0(X). The obstruction bundle
-contributes e(mathcal O(1)calO(1) ⊗  O)=sum_ix^{o-i}c_i(O).
+x=c_1(O(1)) on the projective bundle of lines
+pi:P(K) The obstruction bundle
+contributes e(O(1) ⊗  O)=sum_ix^{o-i}c_i(O).
 Projective pushforward therefore gives
 
  pi_*(x^qsum_ix^{o-i}c_i(O))
@@ -1752,7 +1752,7 @@ regular family moduli space to S^2.
 
 [Evaluation identity]
 The evaluation line has first Chern class U_p+epi^*h, where
-h=c_1(mathcal O(1))calO(1)) is the twistor parameter. Its spherical pairing
+h=c_1(O(1)) is the twistor parameter. Its spherical pairing
 satisfies
 
  FSW_E(U_p)=U_p([M_E^{reg}])
@@ -1764,13 +1764,13 @@ then the exact family has an evaluation zero at p.
 
 Proof. 
 The evaluation line over the coefficient sphere is a line bundle
-mathcal O(1)calO(1) ⊗ mathcal E_E|_p.calE_E|_p. The evaluation transforms with
+O(1) ⊗ E_E|_p.calE_E|_p. The evaluation transforms with
 the same gauge weight as the spinor, and its sign on a reducible link
-can be checked on elltomathcal E_E|_p:calE_E|_p: it lies in
-Hom(ell,mathcal E_E|_p)=calE_E|_p)=mathcal O(1)calO(1) ⊗ 
-mathcal E_E|_p.calE_E|_p. On the quotient, the evaluation section is a
+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
 section of the tensor product of the based-gauge evaluation line with
-the pullback of mathcal E_E|_calE_E|_{{p}times S^2}; its first Chern
+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
 gives SW_X(E); equivalently, restrict the family to a
 regular base point. Formula (eq:families-evaluation-identity)
@@ -1821,8 +1821,8 @@ 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
 
- |mathcal A-calA-mathcal A_calA_{can}|
- +|F_{mathcal A}calA-F_{mathcal A_calA_{can}}|
+ |A-calA-A_{can}|
+ +|F_{A}-F_{A_{can}}|
  le Cr^{-1}+Crexp(-sqrt r,
           dist(x,alpha^{-1}(0))/C)
 
@@ -1831,12 +1831,12 @@ cite[Proposition~6.1]{Taubes1999}.
 
 Fix the disk bound. Write
 f_n=(i)/(2pi)F_{a_n}
-      =(i)/(4pi)(F_{mathcal A_n}calA_n-F_{mathcal A_calA_{can,n}}).
+      =(i)/(4pi)(F_{A_n}-F_{A_{can,n}}).
 The sharp estimate
 cite[Proposition~4.2, Equation~(4.9)]{Taubes1999}, specialized to
 |eta_n|=sqrt2, is
 
- |F_{mathcal A_n}calA_n^-|le(r_n)/(2sqrt2)(1-|alpha_n|^2)+C.
+ |F_{A_n}-|le(r_n)/(2sqrt2)(1-|alpha_n|^2)+C.
 
 Its constants depend on the background family in a fixed smooth way.
 If ell is a unit, positively oriented J_n-complex bivector, then
@@ -1939,9 +1939,9 @@ There is a primitive class FinΛ with F^2=0 such that the
 following implication holds for every dinΛ and every real
 number c>0:
 
- d^2ge-2, d^+=cF^+, dcdot Fle0
+ d^2ge-2, d^+=cF^+, d· Fle0
  ⇒
- cinZ_{>0}, dcdot F=0.
+ cinZ_{>0}, d· F=0.
 
 Moreover F^+ne0.
 
@@ -1955,26 +1955,26 @@ so it contains an integral null vector. Take a rational null subspace
 L of Λ of maximal dimension, namely dimension three: the
 intersection form restricted to L_{R} has signature (1,2) and
 intersects Λ in a lattice. Let lin L be a primitive vector
-with lcdot l=0 and such that L_{R}=l^perpcap L_{R} ⊕ R l.
+with l· l=0 and such that L_{R}=l^perpcap L_{R} ⊕ R l.
 Let
 [
- mathcal F=calF={Fin LcapΛ: F^2=0, Fhbox{ primitive},
-          Fcdot l=1}.
+ F={Fin LcapΛ: F^2=0, Fhbox{ primitive},
+          F· l=1}.
 ]
-Each Finmathcal FcalF is a nonzero null vector and belongs to L. Since
-l is null and Fcdot l=1, we have F^+notin P^perp: otherwise
-Fcdot l=F^+cdot l+F^-cdot l=F^+cdot l=F^+cdot l and F^+in P
-would force F^+cdot l=0, a contradiction. In particular F^+ne0.
-The set mathcal FcalF is infinite by completing any rational null vector
+Each FinF is a nonzero null vector and belongs to L. Since
+l is null and F· l=1, we have F^+notin P^perp: otherwise
+F· l=F^+· l+F^-· l=F^+· l=F^+· l and F^+in P
+would force F^+· l=0, a contradiction. In particular F^+ne0.
+The set F is infinite by completing any rational null vector
 to a hyperbolic pair with l.
 
-noindentStep~2: projecting and pinning d^+. Fix Finmathcal FcalF
+noindentStep~2: projecting and pinning d^+. Fix FinF
 and suppose dinΛ, c>0 satisfy the three left-hand conditions
 of (eq:lattice-chamber-implication). Then
 [
- r:=d-cFin P^perp, r^2=d^2-2c(dcdot F)ge-2,
+ r:=d-cFin P^perp, r^2=d^2-2c(d· F)ge-2,
 ]
-because Fcdot F=0, so r lies in the compact set
+because F· F=0, so r lies in the compact set
 [
  K={rin P^perp:r^2ge-2}.
 ]
@@ -1997,23 +1997,23 @@ 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
 [
- mathbb T=P/bbT=P/pi(L),
+ T=P/bbT=P/pi(L),
 ]
 because adding an element of L changes the projection by an element of
-pi(L). For Finmathcal FcalF define the circle
+pi(L). For FinF define the circle
 [
- mathbb S_F=bbS_F={tpi(F)+pi(L):tinR}subsetmathbb T.bbT.
+ S_F={tpi(F)+pi(L):tinR}subsetT.
 ]
 Two such circles with different directions intersect only at torsion
 points: if tpi(F)-spi(F')inpi(L) with F,F' not collinear, choose
 an integral linear functional on L vanishing on F and nonzero on
 F'; applying it to tF-sF'in L forces sinQ, so the common point
-is torsion. In particular a non-torsion point of mathbb TbbT lies on at
-most one circle mathbb S_F.bbS_F.
+is torsion. In particular a non-torsion point of T lies on at
+most one circle S_F.
 
-The elements of mathcal FcalF have distinct directions: two collinear
-elements with l-pairing one are equal. Since mathcal FcalF is infinite,
-we can choose Finmathcal FcalF whose circle contains none of the
+The elements of F have distinct directions: two collinear
+elements with l-pairing one are equal. Since F is infinite,
+we can choose FinF whose circle contains none of the
 non-torsion points among the finitely many possible coset points.
 
 noindentStep~5: integrality and vanishing. For this F, every
@@ -2025,14 +2025,14 @@ So there is an integer k>0 with kcFin L. Pairing with l shows
 kcinZ, so cinQ. It follows that r=d-cFin P^perpcapΛ_Q,
 and the lattice property of P^perp now gives
 
- c=lcdot dinZ_{>0}.
+ c=l· dinZ_{>0}.
 
-If m=dcdot F<0, then m is a negative integer and
+If m=d· F<0, then m is a negative integer and
 [
  r^2=d^2-2cmge-2+2cge0.
 ]
 Negative definiteness of P^perp forces r=0, which would imply
-m=0. Contradiction; hence dcdot F=0, and the implication is proved.
+m=0. Contradiction; hence d· F=0, and the implication is proved.
 
 
 ## Curve signs
@@ -2042,8 +2042,8 @@ change of the coefficient basis of the original triple and denote the
 result again by (A,C,B), choosing
 
  [B]=(F^+)/(lVert F^+rVert),
- [A]cdot F=[C]cdot F=0,
- beta=[B]cdot F=lVert F^+rVert>0.
+ [A]· F=[C]· F=0,
+ beta=[B]· F=lVert F^+rVert>0.
 
 Here the norm is the positive intersection norm on P. The three
 classes remain orthonormal, and V=[A]^perpcap[C]^perp has
@@ -2056,8 +2056,8 @@ Let J be an almost-complex structure with definite pair (A,C)
 satisfying the adapted periods (eq:lattice-adapted-periods) and
 tamed by B. For every irreducible closed J-holomorphic curve with
 class dinΛ (identified with its Poincar'e dual), let
-m=Fcdot d and write [B]cdot d=beta c with the constant
-beta=[B]cdot F>0. Then:
+m=F· d and write [B]· d=beta c with the constant
+beta=[B]· F>0. Then:
 [label=(roman*)]
  mge0.
  m=0 if and only if d=cF for some real c (in which case
@@ -2076,24 +2076,24 @@ cite[Theorem~2.8]{Wendl2020} give
 so in particular d^2ge-2.
 
 Curve classes couple trivially with the anti-invariant forms, so
-dcdot[A]=dcdot[C]=0. Equivalently din V=[A]^perpcap[C]^perp.
+d·[A]=d·[C]=0. Equivalently din V=[A]^perpcap[C]^perp.
 Since Vcap P=R[B] (both spaces contain [B] and V is the
 codimension-two plane cut out by [A],[C]), the projection d^+
 is a multiple of [B]=F^+/lVert F^+rVert.
 
 (i). Write d^+=c'F^+ with c'inR. If c'>0, apply
-Cref{prop:chamber-class} to d with c=c': if dcdot Fle0 then
-c'inZ_{>0} and dcdot F=0; in particular, either dcdot F=0 or
-dcdot F>0. If c'<0, apply to -d and -c': same conclusion, and
-the sign of dcdot F is unchanged. If c'=0, then d^+=0; by
-adjunction d^2ge-2 and by negativity of P^perp, dcdot F=d^-cdot
-F^-le0, so the implication traps dcdot F=0 (indeed dcdot Fle0
+Cref{prop:chamber-class} to d with c=c': if d· Fle0 then
+c'inZ_{>0} and d· F=0; in particular, either d· F=0 or
+d· F>0. If c'<0, apply to -d and -c': same conclusion, and
+the sign of d· F is unchanged. If c'=0, then d^+=0; by
+adjunction d^2ge-2 and by negativity of P^perp, d· F=d^-·
+F^-le0, so the implication traps d· F=0 (indeed d· Fle0
 would give c'inZ_{>0}, which c'=0 violates; a direct inspection
-shows dcdot Fge0). In all cases mge0.
+shows d· Fge0). In all cases mge0.
 
 (ii). If d=cF with cinR then m=cF^2=0 and
 cinZ since both d,F are integral and F is primitive. Conversely
-assume m=0. From din V and m=dcdot F=0, the class d is
+assume m=0. From din V and m=d· F=0, the class d is
 orthogonal to the nondegenerate plane span([B],F)subset
 V; the orthogonal complement in V is negative definite. So
 d^2le0, and adjunction bounds d^2ge-2. If d^+ne0, then
@@ -2102,10 +2102,10 @@ and then d^2le0 combined with 0=m=d^+F^+ + d^-F^- yields
 d^2=0 (by completing the square in the Lorentzian space). In that case
 d^pm are both null, and d is a null vector in V. But in a
 Lorentzian space, a null vector in the same closure of the positive cone
-as F (which it is since dcdot[B]=beta c'>0) is a positive
+as F (which it is since d·[B]=beta c'>0) is a positive
 multiple of F; hence d=cF for some cinZ_{>0}. If d^+=0, then
 din P^perp and d^2le0; adjunction gives -2le d^2le0, and
-m=dcdot F=d^-F^-le0 forces d^-F^-=0; together with F^-ne0
+m=d· F=d^-F^-le0 forces d^-F^-=0; together with F^-ne0
 this forces d^-=0 (the orthogonal complement of F^- in P^perp
 being spacelike with intersection ge-1), so d=0=cF with c=0.
 
@@ -2132,7 +2132,7 @@ For (i), write a positive configuration as
 ]
 where d_i are the classes of its distinct irreducible images. Taming
 excludes zero real component classes. By Cref{cor:curve-signs},
-m_i=d_icdot Fge0, whereas
+m_i=d_i· Fge0, whereas
 [
  0=F^2=sum_{i=1}^rn_im_i.
 ]
@@ -2285,7 +2285,7 @@ Represent nearby immersions by normal graphs over u, removing
 infinitesimal reparametrizations. If u is simple, its stabilizer is
 trivial, so the normal-graph slice parametrizes the local
 unparametrized moduli. The symplectic form A identifies the real
-normal bundle with T^*L by wmapstoalpha=u^*iota_wA. Write,
+normal bundle with T^*L by w|->alpha=u^*iota_wA. Write,
 along the source,
 
  H=(C-xA)/(y), Ω=A+iH,
@@ -2296,7 +2296,7 @@ one-forms for the induced conformal structure and orientation. The
 complex volume identity gives u^*iota_wC=xalpha+y*_Lalpha; Cartan's
 formula then gives the normal derivative
 
- mathcal DcalDalpha=(ddalpha,dd(xalpha+y*_Lalpha)).
+ D,dd(xalpha+y*_Lalpha)).
 
 The nonlinear pullbacks have values in pairs of exact two-forms: their
 integrals are the two fixed zero periods.
@@ -2305,7 +2305,7 @@ Given exact two-forms zeta_1,zeta_2, choose alpha_0 with
 ddalpha_0=zeta_1. Adding dd f changes only the second equation,
 by
 
- mathcal PcalP f=dd(xdd f+y*_Ldd f)
+ P f=dd(xdd f+y*_Ldd f)
  =dd xwedgedd f+dd(y*_Ldd f).
 
 After division by a positive area form its principal part is the
@@ -2320,14 +2320,14 @@ is constant along this homotopy
 cite[Section~7, Propositions~7.3--7.4]{TaylorFunctionalNotes}. Its
 image has integral zero by Stokes' theorem, so its cokernel has
 dimension one and its range is exactly the integral-zero two-forms.
-We can therefore solve mathcal PcalP f=zeta_2-dd(xalpha_0+y*_Lalpha_0),
+We can therefore solve P f=zeta_2-dd(xalpha_0+y*_Lalpha_0),
 proving surjectivity onto the stated target with bounded elliptic
 right inverses in Sobolev or H"older spaces.
 
-The kernel of mathcal DcalD is thus {alpha:ddalpha=0} together
+The kernel of D is thus {alpha:ddalpha=0} together
 with the solutions of the scalar equation, which are the constant
 functions; the holonomy from L=L_0 to nearby L_t identifies the
-two kernels. The kernel of mathcal DcalD is therefore isomorphic to
+two kernels. The kernel of D is therefore isomorphic to
 H^1(L;R), of dimension b_1(L). The linearization is Fredholm of
 index b_1(L), and the graph map from the normal graph is a
 diffeomorphism onto an open set of common Lagrangian immersions.
@@ -2647,30 +2647,30 @@ model.
 
 The proof of part (iv) deserves a short additional comment
 because it uses Grauert's theorem in a way that is easy to lose track
-of. Let L=mathcal L_calL_rho be the relative polarization on
+of. Let L=L_ be the relative polarization on
 p:Y_rhotoDelta_R. The nodal fiber N=p^{-1}(0) is the image of
-a nodal plane cubic in mathbb P^2.bbP^2. Its normalization
-nu:mathbb P^1bbP^1to N is a rational curve, and the pullback
-nu^*L is mathcal O_calO_{mathbb P^1}bbP^1(3) up to a scalar. The gluing
+a nodal plane cubic in P^2. Its normalization
+nu:P^1 N is a rational curve, and the pullback
+nu^*L is O_{P^1} up to a scalar. The gluing
 condition gives the exact sequence
 [
  0→ L→
- nu_*mathcal O_calO_{mathbb P^1}bbP^1(3)
+ nu_*O_{P^1}
  xrightarrow{ev_0-lambdaev_∞}
  C_{node}→0,
 ]
 with lambda the appropriate modular parameter of the nodal cubic.
 The map on global sections is surjective, and the normalization is
-finite, so the cohomology of nu_*mathcal O_calO_{mathbb P^1}bbP^1(3) is
-that of mathcal O_calO_{mathbb P^1}bbP^1(3). In particular h^0(N,L)=3 and
+finite, so the cohomology of nu_*O_{P^1} is
+that of O_{P^1} In particular h^0(N,L)=3 and
 h^1(N,L)=0.
 
 Grauert's base-change theorem now applies: p is proper and
-surjective, the disk Delta_R is reduced, the sheaf mathcal L_calL_rho
+surjective, the disk Delta_R is reduced, the sheaf L_
 is flat, and the fiber dimensions h^0 and h^1 are constant. So
-E_rho=(p_rho)_*mathcal L_calL_rho is a holomorphic rank-three bundle,
+E_rho=(p_rho)_*L_ is a holomorphic rank-three bundle,
 and its fiber at s is H^0(p^{-1}(s),L_s). Evaluation gives a
-relative closed embedding in mathbb P(E_bbP(E_rho^*). At the nodal
+relative closed embedding in P(E_*). At the nodal
 point, the Zariski tangent space of the fiber is the total tangent
 space of the smooth surface; tangent separation there makes the whole
 map an immersion of the surface, and properness gives a closed
@@ -2705,7 +2705,7 @@ Locally, take alpha=dd f on the base; the fields
 Y_{dd f} are Hamiltonian, their brackets vanish, and their complete
 flows on each compact fiber give a transitive
 T_s^*S^circ-action with a rank-two stabilizer lattice
-mathcal L_s.calL_s. Locally the lattices vary smoothly, and the two
+L_s. Locally the lattices vary smoothly, and the two
 periods continue smoothly.
 
 
@@ -2720,9 +2720,9 @@ p^*dd f a base form, so Y_f is Hamiltonian for
 are pulled back from the base and their fields are vertical. Complete
 flows therefore give a local group action.
 
-To identify the lattice, use a local section smapstosigma(s) of
+To identify the lattice, use a local section s|->sigma(s) of
 p; each element of the stabilizer satisfies
-int_{F_s}alphainmathcal L_scalL_s for a small piecewise-smooth disk
+int_{F_s}alphainL_s for a small piecewise-smooth disk
 whose boundary is the corresponding translation. The two periods
 continue smoothly by the implicit function theorem. Equivalently,
 a local lattice basis {alpha_1,alpha_2} of closed one-forms is a
@@ -2756,7 +2756,7 @@ when the input is closed.
 The main statement of this section is the following.
 
 
-Let (A_0,C_0,B_0) and pcolon Xto S be the positive triple and
+Let (A_0,C_0,B_0) and p: Xto S be the positive triple and
 pencil of Cref{thm:pencil}. Retain the normalized original periods
 and the class F of Cref{cor:curve-signs}. On disjoint disks U_j
 about the critical values, use the model of Cref{thm:nodal-model}
@@ -2959,8 +2959,8 @@ eps shrinks.
 
 On the regular region X^circ, consider the form
 
- E_eps=eps,Psi+(1)/(eps)cdot A^2
- cdot(semi-flat horizontal)/(int_X A^2).
+ E_eps=eps,Psi+(1)/(eps)· A^2
+ ·(semi-flat horizontal)/(int_X A^2).
 
 Concretely, in local invariant action-angle coordinates
 (s;,a,b) adapted to the torus action, the semi-flat form is
@@ -3001,14 +3001,14 @@ fixed smooth data).
 
 The three wedge residuals
 [
- mathcal R_calR_eps=
+ R_
  (-Awedge E_eps,,
        -Cwedge E_eps,,
        (A^2-E_eps^2)/(2))
 ]
 satisfy
 
- |mathcal R_calR_eps|_{H^k}
+ |R_|_{H^k}
    le Ceps^{-s}e^{-c/eps}
 
 for every fixed k, with C,s,c>0 depending only on k and the
@@ -3053,7 +3053,7 @@ the explicit bounds
  b_eps:=int_XE_epswedge C=O(e^{-c/eps}),
 
 which hold because the exact gluing adjusted the two pairings. The
-remaining residual mathcal R_calR_eps is exponentially small in the
+remaining residual R_ is exponentially small in the
 fixed Sobolev norms after the standard Sobolev inequalities on the
 fixed chart cover.
 
@@ -3135,7 +3135,7 @@ polynomial derivative bounds.
 
 Euclidean Sobolev estimates on this cover give polynomial intrinsic
 Sobolev constants; since dim X=4 and kge6, derivatives of order
-at most lfloor k/2rfloor are L^∞-bounded by H^k. The
+at most floor( k/2) are L^∞-bounded by H^k. The
 product rule on each derivative of hwedgewidetilde h gives
 (eq:auxiliary-product).
 
@@ -3144,14 +3144,14 @@ product rule on each derivative of hwedgewidetilde h gives
 For every fixed kge6 and every sufficiently small eps, there is
 a bounded linear map
 [
- mathcal L_calL_eps: (t_1,t_2,t_3)in H^ktimes H^ktimes H^k
-    mapsto hin H^k
+ L_: (t_1,t_2,t_3)in H^ktimes H^ktimes H^k
+    |-> hin H^k
 ]
 such that
 [label=(roman*)]
 
-The form h is exact: int_Xhwedgevarphi=0 for every closed
-varphi; equivalently its harmonic projection is zero.
+The form h is exact: int_Xhwedgephi=0 for every closed
+phi; equivalently its harmonic projection is zero.
 
 h solves
 [
@@ -3159,7 +3159,7 @@ h solves
        < h,C>_{L^2},
        int_Xhwedge h)=(t_1,t_2,t_3),
 ]
-in the sense that h is the unique element of mathcal E^k_calE^k_eps
+in the sense that h is the unique element of E^k_
 (exact H^k two-forms) with those three moments, and its H^k norm
 satisfies
 [
@@ -3219,24 +3219,24 @@ bounded by Ceps^{-N}e^{-c/eps} in every fixed Sobolev norm.
 
 Proof. 
 Fix kge6. Define the closed subspace
-mathcal E^k_calE^k_epssubset H^k of exact two-forms. Writing
-D_eps=D^0_eps+h with hinmathcal E^k_calE^k_eps, the desired
+E^k_ H^k of exact two-forms. Writing
+D_eps=D^0_eps+h with hinE^k_, the desired
 equations become the fixed-point equation
 
- h=mathcal T_calT_eps(h):=mathcal L_calL_eps
+ h=T_:=L_
  (-Awedge D^0_eps,,
        -Cwedge D^0_eps,,
        (A^2-(D^0_eps)^2-hwedge h)/(2)).
 
-This map is defined on all of mathcal E^k_calE^k_eps: the first two data
+This map is defined on all of E^k_: the first two data
 have integral zero by (eq:auxiliary-exact-moments), as does the
 free term in the third slot; for each exact iterate,
 int_Xhwedge h=0 by approximation and Stokes' theorem. Every
 iteration satisfies the three mean conditions of
 Cref{lem:auxiliary-inverse}.
 
-The map mathcal T_calT_eps is a contraction on a small ball of
-mathcal E^k_calE^k_eps: by the exponential smallness of the free terms
+The map T_ is a contraction on a small ball of
+E^k_: by the exponential smallness of the free terms
 (Cref{lem:auxiliary-gluing}) and the polynomial bounds of
 Cref{lem:auxiliary-geometry,lem:auxiliary-inverse}, the quadratic part
 hwedge h is a contraction with a factor le1/2 on a ball of
@@ -3281,8 +3281,8 @@ of Cref{thm:auxiliary-solution} satisfies
 
 with C independent of eps, and the following uniform
 implication: for every positive real number eps and every
-dinΛ with d^2ge-2 and [D_eps]cdot d>0, we have
-[B]cdot d>0.
+dinΛ with d^2ge-2 and [D_eps]· d>0, we have
+[B]· d>0.
 
 
 Proof. 
@@ -3296,15 +3296,15 @@ identities. The exact construction data are independent of eps, so
 C and eps_0 are too.
 
 For the implication: by Cref{cor:curve-signs}, either
-d=cF for some cinR (in which case [B]cdot d =
-c[B]cdot F=cbeta>0 whenever c>0), or d has the sign of the
-coefficient of F^+ in d^+. In the first case, [B]cdot d=
-beta c and the sign of [D_eps]cdot d = (lambda_eps)(c)+O(eps)
+d=cF for some cinR (in which case [B]· d =
+c[B]· F=cbeta>0 whenever c>0), or d has the sign of the
+coefficient of F^+ in d^+. In the first case, [B]· d=
+beta c and the sign of [D_eps]· d = (lambda_eps)(c)+O(eps)
 agrees with c for small enough eps. In the second case, adjunction
 gives d^2ge-2, and Cref{cor:curve-signs} gives the sign of
-[B]cdot d; the condition [D_eps]cdot d>0 together with the
+[B]· d; the condition [D_eps]· d>0 together with the
 expansion (eq:auxiliary-class-expansion) then forces the same
-sign. In both cases [B]cdot d>0.
+sign. In both cases [B]· d>0.
 
 
 # The hyperk"ahler endpoint
@@ -3456,7 +3456,7 @@ of V and [D] pairs positively on every J-curve.
 The pair (A,B_1) is then also integrable of equal square, with
 complex structure J_1 (the same as (A,C_1): the pair (A,D)
 originates from the same J_1). Interpolate from (A,B) to
-(A,B_1) as before. The endpoint pair (A,B_1) has [A]cdot[B_1]
+(A,B_1) as before. The endpoint pair (A,B_1) has [A]·[B_1]
 =0 and B_1^2=A^2 (the first is [A][B]=0; the second is
 B_1^2=A^2). Concatenate with the interpolation to (A,C_1).