diff --git a/singular_free_tangent_sheaf_v2.txt b/singular_free_tangent_sheaf_v2.txt
index cf318b2..6e9a8ca 100644
--- a/singular_free_tangent_sheaf_v2.txt
+++ b/singular_free_tangent_sheaf_v2.txt
@@ -86,12 +86,12 @@ of the surface below can be made commuting near its singular point.
 Conventions. Line bundles on curves are written additively; aL + bH
 denotes the tensor product L^a tensor H^b over O_C. Two-dimensional
 row vectors X = (X_1, X_2) and Y = (Y_1, Y_2) have wedge product
-X /\ Y = X_1 Y_2 - X_2 Y_1. A quadratic form Q on a two-dimensional
+X / Y = X_1 Y_2 - X_2 Y_1. A quadratic form Q on a two-dimensional
 space has bilinear polar form x . y = (Q(x+y) - Q(x) - Q(y)) / 2, and
 x^2 abbreviates Q(x). A residue on an annular overlap means (1/(2 pi
 i)) times a positively oriented contour integral. Adjunction and
 residue formulas put the normal differential first: locally
-Res_{y=0}((dy/y) /\ dz) = dz. All varieties and germs are complex
+Res_{y=0}((dy/y) / dz) = dz. All varieties and germs are complex
 analytic or complex algebraic as dictated by context. Background
 references are: Hartshorne, Algebraic Geometry, GTM 52 (Springer
 1977) for curve cohomology, adjunction and Riemann-Roch; Serre (Ann.
@@ -143,7 +143,7 @@ Since K_Sigma + Gamma = 0, there is a nowhere-vanishing logarithmic
 two-form Omega on Sigma whose pole divisor is Gamma. Its residue
 omega = Res_Gamma(Omega) is a nowhere-vanishing differential on
 Gamma. The residue convention places the normal differential first:
-locally Res_{y=0}((dy/y) /\ dz) = dz. Choose a basis u = (u_1, u_2)
+locally Res_{y=0}((dy/y) / dz) = dz. Choose a basis u = (u_1, u_2)
 of H^0(Sigma, F), and set eta_Gamma = u_1 du_2 - u_2 du_1 in
 H^0(Gamma, K_Gamma + 2F). The expression is independent of the local
 frame for F: in that frame it is the indicated coefficient
@@ -570,7 +570,7 @@ For the metric, choose coordinates (y, z) near a marked point and
 frames gm of O_Sigma(Gamma), fF of F, and m of M on Gamma with
 
   gamma = y gm,
-  Omega = e(y, z) (dy/y) /\ dz,   e(0, z) = 1,
+  Omega = e(y, z) (dy/y) / dz,   e(0, z) = 1,
   kappa = k(y, z) gm^2 fF^4,
   beta = b(z) m,   m^4 = gm^2 fF^4|_Gamma,
 
@@ -584,9 +584,9 @@ the contact expansion
 
 The normal-first adjunction sends the frame gm fF^4 to
 
-  Res_C( (e(y, z) dy /\ dz) / k(y, z) ) = (e|_C / k_y|_C) dz.
+  Res_C( (e(y, z) dy / dz) / k(y, z) ) = (e|_C / k_y|_C) dz.
 
-The sign is positive because dy /\ dz = (dk / k_y) /\ dz.
+The sign is positive because dy / dz = (dk / k_y) / dz.
 
 Let a = xi/z be the local frame of O_C(Z) and let tau = fF^3 a^2 be
 the resulting frame of Lambda. The fixed fourth-power identification
@@ -655,16 +655,16 @@ pair P = (P_1, P_2) in H^0(C, theta + Z)^2 with:
 
 (2) On the j-th disk, with coefficient rows in the indicated frames,
 
-  m_j = c_j / z_j,   R_j = m_j r,   W_j = R_j(0) /\ R_j'(0),
+  m_j = c_j / z_j,   R_j = m_j r,   W_j = R_j(0) / R_j'(0),
   p_j = P - m_j r',
 
 the pair P has principal part R_j(0) / z_j in the frame (l_j^fr)^2
 of theta, both R_j and p_j are holomorphic near 0, and R_j(0) != 0.
 
-(3) W_j = (v_j^*)^2, p_j(0) /\ R_j(0) = 1, and
+(3) W_j = (v_j^*)^2, p_j(0) / R_j(0) = 1, and
 sum_j (W_j - 1) = -9 deg L.
 
-(4) J = u /\ P lies in H^0(C, Lambda) and, in W^*,
+(4) J = u / P lies in H^0(C, Lambda) and, in W^*,
 
   <[J], [T]> = -V([T])  for every T in H^0(C, Lambda).
 
@@ -698,7 +698,7 @@ The isometry iota transports the equations for w_j to
 (vector-identities) and transports their evaluation directions. Since
 each l_j is nonzero, there is a unique nonzero c_j with c_j l_j = v_j^*.
 Locally R_j = c_j u, so R_j is holomorphic and nonzero at the center,
-and the norm formula gives R_j(0) /\ R_j'(0) = c_j^2 l_j^2 = (v_j^*)^2.
+and the norm formula gives R_j(0) / R_j'(0) = c_j^2 l_j^2 = (v_j^*)^2.
 
 Every prescribed set of simple principal parts of theta at Z is
 realizable by a section of theta + Z. The obstruction in H^1(C, theta)
@@ -710,7 +710,7 @@ obtain P with the stated pole parts. Since r = z_j u, the row m_j r'
 has exactly the same simple pole part as P, so p_j is holomorphic.
 
 The simple pole of P at q_j is proportional to u(q_j). Its wedge with
-u has zero residue coefficient, so J = u /\ P is holomorphic there;
+u has zero residue coefficient, so J = u / P is holomorphic there;
 away from Z it is already holomorphic. Hence J is a global section of
 Lambda.
 
@@ -742,18 +742,18 @@ which is (4).
 
 On a fixed disk suppress the subscript j. Since r = z u,
 
-  P = p + (c/z) u + c u',   c J = R /\ p + R /\ R'.
+  P = p + (c/z) u + c u',   c J = R / p + R / R'.
 
-In particular c_j l_j(J) = R_j(0) /\ p_j(0) + W_j. On the other hand,
+In particular c_j l_j(J) = R_j(0) / p_j(0) + W_j. On the other hand,
 the metric identification in (4) gives c_j l_j(J) = -v_j^* . V =
-W_j - 1. Hence R_j(0) /\ p_j(0) = -1, equivalently
-p_j(0) /\ R_j(0) = 1. Summing W_j - 1 = -v_j^* . V proves the last
+W_j - 1. Hence R_j(0) / p_j(0) = -1, equivalently
+p_j(0) / R_j(0) = 1. Summing W_j - 1 = -v_j^* . V proves the last
 identity in (3), together with the local expansion
 
   P = R_j(0) / z_j + p_j(0) + 2 R_j'(0) + O(z_j).
 
 If P is changed by a holomorphic pair S, the equality
-H^0(C, theta) = xi^2 H^0(C, 2H) shows u /\ S lies in
+H^0(C, theta) = xi^2 H^0(C, 2H) shows u / S lies in
 xi^2 H^0(C, 3H), so [J] is unchanged.
 
 Compatibility with the fourth-root structure. Let zeta = psi(z) be
@@ -774,7 +774,7 @@ row R_new is a scalar multiple of R with scalar g_0^2 at the center,
 and differentiating with respect to zeta shows
 W_new = g_0^4 W / psi'(0) = W. Finally, subtracting
 m_new d r_new / d zeta from P_new shows p_new(0) - g_0^{-2} p(0) is
-proportional to R(0), so p_new(0) /\ R_new(0) = p(0) /\ R(0). []
+proportional to R(0), so p_new(0) / R_new(0) = p(0) / R(0). []
 
 The finite-jet construction. On the total space of L^{-1} choose a base
 coordinate z and a fiber-linear coordinate t, so that t^4 corresponds
@@ -785,7 +785,7 @@ form
 
 Put E_t = t d_t. The two-form
 
-  sigma = t^{-4} (dt/t) /\ dz = t^{-5} dt /\ dz
+  sigma = t^{-4} (dt/t) / dz = t^{-5} dt / dz
 
 is invariant under such split changes, and has a pole of order five
 along the zero section C = {t = 0}.
@@ -801,14 +801,14 @@ in t whose coefficients are holomorphic in z, and with initial data
 
 Then
 
-  sigma(v_1, v_2) = t^{-2} (A /\ B),
-  v_1 /\ v_2 = t^2 (A /\ B) E_t /\ d_z,
+  sigma(v_1, v_2) = t^{-2} (A / B),
+  v_1 / v_2 = t^2 (A / B) E_t / d_z,
 
 where A = (A_1, A_2) and B = (B_1, B_2) are two-entry rows.
 
 The required logarithmic determinant is sigma(v_1, v_2) = 1 + O(t),
-equivalently A /\ B = t^2 G with G a unit of the form 1 + O(t). Then
-v_1 /\ v_2 = t^5 G d_t /\ d_z, so the ordinary determinant is t^5
+equivalently A / B = t^2 G with G a unit of the form 1 + O(t). Then
+v_1 / v_2 = t^5 G d_t / d_z, so the ordinary determinant is t^5
 times a nonzero unit along C.
 
 Gluing. Around each of the six marked points choose a small disk D_j;
@@ -861,9 +861,9 @@ Delta B_1 = -m_j r. Since P = p + m_j r' by the residue-data
 proposition, the first transition changes the inside leading row to
 (P, 0), exactly matching the outside leading row.
 
-The degree-one coefficient of A /\ B is A_0 /\ B_1 + A_1 /\ B_0.
-On the outside it is r /\ 0 + 0 /\ P = 0. On the inside it is
-r /\ (m_j r) + p /\ 0 = 0, since r /\ r = 0. So no obstruction
+The degree-one coefficient of A / B is A_0 / B_1 + A_1 / B_0.
+On the outside it is r / 0 + 0 / P = 0. On the inside it is
+r / (m_j r) + p / 0 = 0, since r / r = 0. So no obstruction
 arises at this order.
 
 Order two. The order-one transition acts on the row of index 2 via
@@ -900,7 +900,7 @@ The order-two obstruction is the residue matrix
 Decompose T_ij into symmetric and antisymmetric parts:
 
   T^sym_ij = -(2/3)(R_i p_j + R_j p_i + 2 R_i R_j' + 2 R_i' R_j),
-  T^alt_ij = (1/12) epsilon_ij ((R /\ p)(0) + (R /\ R')(0)).
+  T^alt_ij = (1/12) epsilon_ij ((R / p)(0) + (R / R')(0)).
 
 For the symmetric part, P_i P_j is a meromorphic section of
 L^4 = K_C with local expression P_i P_j dz. Using the expansion
@@ -911,7 +911,7 @@ P = R(0)/z + p(0) + 2 R'(0) + O(z) from the residue-data proposition,
 Summing over j and applying the residue theorem on C shows that
 sum_j T^sym_ij = 0.
 
-For the antisymmetric part, note R(0) /\ p(0) = -1, and
+For the antisymmetric part, note R(0) / p(0) = -1, and
 sum_nu (W_nu - 1) = -9 deg L. So
 
   sum_j T^alt_ij = (1/12) epsilon_ij sum_nu (W_nu - 1)
@@ -934,7 +934,7 @@ Higher orders. From order three onward, each step extends fields
 together with the required determinant. Write the
 determinant equation as
 
-  A /\ B = t^2 G,   G = 1 + G_1 t + G_2 t^2 + ...
+  A / B = t^2 G,   G = 1 + G_1 t + G_2 t^2 + ...
 
 The field equation in degree k uses G_{k-2}. To compare the next
 horizontal rows, that target coefficient must already be compatible,
@@ -970,27 +970,27 @@ changes. So A_{k-1} can be adjusted at degree k without disturbing
 determinant equations imposed earlier. G_k is chosen after fields
 match through index k, since its field equation occurs in degree k+2.
 
-Invariant. After step n >= 2: rows through index n match; A/\B is
+Invariant. After step n >= 2: rows through index n match; A/B is
 imposed through degree n; G_0 = 1, ..., G_n are chosen and compatible
 through degree n. For n = 2 both transitions preserve sigma exactly,
 so the invariant holds with G = 1.
 
 Step k >= 3. Assume the invariant through step k-1. On each chart,
 
-  r /\ B_k = G_{k-2} - sum_{a=1}^{k-1} A_a /\ B_{k-a},   (in L^{k-2}).
+  r / B_k = G_{k-2} - sum_{a=1}^{k-1} A_a / B_{k-a},   (in L^{k-2}).
 
 Under split changes, the only extra terms are multiples of the
-antisymmetric sum sum_{a=1}^{k-1} B_a /\ B_{k-a} = 0.
+antisymmetric sum sum_{a=1}^{k-1} B_a / B_{k-a} = 0.
 
 Making the RHS divisible at Z. Adding any a in H^0(C, L^k)^2 to the
 previously chosen A_{k-1} preserves the matching through index k-1 and
 all previous determinants: every previous nonsplit transition has
-positive order, and its first determinant contribution is a /\ B_1 in
+positive order, and its first determinant contribution is a / B_1 in
 L^k. Item (4) of the curve-data proposition (with k >= 3) gives a
 surjection H^0(C, L^k) -> H^0(Z, L^k|_Z), so a(q_j) may take any
 prescribed values. Since B_1(q_j) = R_j(0) != 0, the scalar map
-a(q_j) |-> a(q_j) /\ B_1(q_j) is onto, so a can be chosen so that
-(RHS - a /\ B_1)(q_j) = 0 for every j. Then the RHS is divisible by
+a(q_j) |-> a(q_j) / B_1(q_j) is onto, so a can be chosen so that
+(RHS - a / B_1)(q_j) = 0 for every j. Then the RHS is divisible by
 r. Since r = z u has only a simple common zero, dividing by z on
 each disk and applying Steinness on U_0 gives a holomorphic B_k.
 
@@ -999,14 +999,14 @@ Previous transitions act on the new B_k by
   delta A_k = ((k-1)/3) m' a + m a',
   delta B_k = -m a.
 
-The horizontal correction satisfies r /\ (-m a) = a /\ (m r),
+The horizontal correction satisfies r / (-m a) = a / (m r),
 matching the determinant change on the disk. These contributions
 are absorbed into the comparisons below.
 
 Matching horizontal rows. Choose holomorphic vertical lifts of the
 new B_k. After transport by previous transitions, all lower rows
-agree. The difference of the degree-k A/\B is r /\ delta B_k
-(because delta A_k /\ B_0 = 0), and target compatibility through
+agree. The difference of the degree-k A/B is r / delta B_k
+(because delta A_k / B_0 = 0), and target compatibility through
 degree k-2 makes it zero. The horizontal discrepancy is thus a
 holomorphic multiple of r on each overlap, and (1-k)b r + h r'
 removes it by choice of h in an order-k transition. Take b = h' when
@@ -1044,7 +1044,7 @@ and cancels both entries. The resulting vertical class splits,
 completing order three. The modified order-three transition still
 has b = h', so it preserves sigma exactly.
 
-The correction can affect the degree-four coefficient of A/\B.
+The correction can affect the degree-four coefficient of A/B.
 No conflict arises: G_2 = 0 is already chosen, but its first
 determinant equation is at step four; the degree-three equation is
 unchanged.
@@ -1069,7 +1069,7 @@ For k > 4 the new leading discrepancy lies in L^k, and
 
 because 4-k < 0. Thus it always splits and defines G_k. Every target
 extension leaves all previously imposed equations unchanged, since
-G_k enters A/\B only in degree k+2. The induction invariant is
+G_k enters A/B only in degree k+2. The induction invariant is
 preserved.
 
 Stopping. A row of index k contributes to degree k+1 of the
@@ -1082,12 +1082,12 @@ composition of flows of generating fields D = t^k(b E_t + h d_z) for
 1 <= k <= 7, together with holomorphic local rows A_k, B_k for
 0 <= k <= 7, such that: the transitions fix C and preserve its
 conormal bundle L; the vector fields v_i = t(A_i E_t + B_i d_z)
-match through row index 7; and A /\ B = t^2(1 + O(t)) on every chart.
+match through row index 7; and A / B = t^2(1 + O(t)) on every chart.
 
 On any surface neighborhood Y realizing these germs, with
 I = O_Y(-C) and F = T_Y(-log C), the rows give two sections of
 F/I^9 F inside I F/I^9 F. The determinant has leading term
-t^4 E_t /\ d_z, equivalently t^5 d_t /\ d_z, with coefficient one.
+t^4 E_t / d_z, equivalently t^5 d_t / d_z, with coefficient one.
 
 Proof. Perform the first two orders and the induction through
 k = 7. Only finitely many flows occur, each a genuine holomorphic
@@ -1095,7 +1095,7 @@ germ on an annular neighborhood of the zero section. Terms with row
 indices 0, ..., 7 have ordinary-degree coefficients 1, ..., 8 in the
 frame (d_t, d_z), so matching them is matching modulo I^9 F. The
 determinant equation with G = 1 + O(t) gives
-v_1 /\ v_2 = t^4(1 + O(t)) E_t /\ d_z = t^5(1 + O(t)) d_t /\ d_z. []
+v_1 / v_2 = t^4(1 + O(t)) E_t / d_z = t^5(1 + O(t)) d_t / d_z. []
 
 Analytic realization and Grauert contraction. The finite jets live
 in overlapping charts, and are realized
@@ -1201,7 +1201,7 @@ these coefficients through degree 8 in t. The matching in the
 finite-jets proposition is therefore precisely matching modulo
 I^9 F on the actual overlap maps, giving the claimed quotient
 sections. Their logarithmic determinant is
-t^2 (A /\ B) = t^4(1 + O(t)). Replacing t d_t by d_t multiplies
+t^2 (A / B) = t^4(1 + O(t)). Replacing t d_t by d_t multiplies
 this determinant by t, giving the ordinary determinant. []
 
 Vanishing and holomorphic lifting. The finite jets now live on a
@@ -1310,7 +1310,7 @@ lifted pair is still t^4(1 + O(t)); its leading coefficient is 1
 even at the common zeros of the leading row r. Its ordinary
 determinant is
 
-  v_1 /\ v_2 = t^5 u(t, z) d_t /\ d_z,   u(0, z) = 1.
+  v_1 / v_2 = t^5 u(t, z) d_t / d_z,   u(0, z) = 1.
 
 The unit condition holds on a neighborhood of each point of C.
 Compactness gives one neighborhood of C where all local factors are
@@ -1385,7 +1385,7 @@ Serre, "Un theoreme de dualite", Comment. Math. Helv. 29 (1955),
 
 By the ordinary determinant formula, the pullback of omega^o is
 
-  pi^* omega^o = t^{-5} u(t, z)^{-1} d_t /\ d_z.
+  pi^* omega^o = t^{-5} u(t, z)^{-1} d_t / d_z.
 
 It is meromorphic on Y with divisor exactly -5C. Thus