diff --git a/singular_free_tangent_sheaf_v2.txt b/singular_free_tangent_sheaf_v2.txt
index edf1fb9..e19f99d 100644
--- a/singular_free_tangent_sheaf_v2.txt
+++ b/singular_free_tangent_sheaf_v2.txt
@@ -161,7 +161,7 @@ degree-two involutions. One has iota_Phi = t_Delta o iota_F, where
 t_Delta is translation by the nontrivial point of order two
 represented by Delta. The four ramification points of F|_Gamma,
 namely the fixed points of iota_F, are paired without fixed points by
-iota_Phi. They form two distinct reduced fibers of the Phi-map. Their
+iota_Phi. These four points form two distinct reduced fibers of the Phi-map; the
 divisor is therefore the pullback of the zero divisor of a binary
 quadratic with two distinct roots. Scaling that quadratic produces
 -eta_Gamma / omega. Uniqueness follows because pullback from the
@@ -410,3 +410,1428 @@ Finally, in compatible frames r = z u at each marked point, so
 r'(q_j) = u(q_j) up to the frame. The six rows r'(q_j) in C^2 span
 C^2: their projective directions are the marked values of f, and
 f(q_j) are not all equal.
+
+The trace form uses Grothendieck's splitting theorem for vector
+bundles on PP^1 (Amer. J. Math. 79, 1957, 121-138), and Grothendieck
+duality as in Hartshorne, Residues and Duality, Lecture Notes in
+Mathematics 20 (Springer 1966). It converts the quadratic-space data
+on Gamma into a residue problem on C. Put theta = 2L,
+Lambda = theta + H = 3H + 2Z, and let f: C -> PP^1 be the morphism
+defined by u. With u the fixed basis of H^0(C, H), let
+eta_C = u_1 du_2 - u_2 du_1 in H^0(C, K_C + 2H); the alternating
+expression is independent of a frame choice for H. The analogous
+eta_Gamma is already defined on Gamma.
+
+Lemma (trace pairing). The rank-four vector bundle E = f_* Lambda on
+PP^1 carries a perfect symmetric pairing
+
+  E tensor E -> O_{PP^1},   (T_1, T_2) |-> Tr_{C/PP^1}(T_1 T_2 / eta_C).
+
+Its splitting type is (3, 0, 0, -3). Multiplication by xi^2 identifies
+O_{PP^1}(3) with the positive line subbundle P of E. The form on global
+sections has radical xi^2 H^0(C, 3H) and induces a nondegenerate
+symmetric form <.,.> on
+
+  W = H^0(C, Lambda) / xi^2 H^0(C, 3H),   dim W = 2.
+
+For every x in PP^1, evaluation identifies this quadratic space with
+(P^perp / P)_x.
+
+Let q in Z be a marked point, with local frames fF of H, tau of Lambda,
+and parameter z. Write, using the fixed isomorphism 2 Lambda =
+K_C + 2H,
+
+  tau^2 = s(z) dz fF^2,   eta_C = E(z) dz fF^2.
+
+Coefficient evaluation ell_q : T |-> (T/tau)(q) descends to W, and in
+the dual form on W^*,
+
+  ell_q^2 = E(q) / s(q).
+
+In particular ell_q is nonzero and nonisotropic.
+
+Proof. The morphism f is finite and flat of degree four. The identity
+2 Lambda = K_C + 2H identifies Lambda^2 with the relative dualizing
+bundle K_{C/PP^1} = K_C + 2H (equivalently, f^* K_{PP^1} = -2H).
+Perfection is checked locally. At a ramification point choose a
+parameter w on one component of f^{-1}(disk), with target coordinate
+x = w^e. Choose a frame fF of H pulled back from the target disk so
+that eta_C = dx fF^2. If tau^2 = a(w) dw fF^2 with a a unit, the local
+pairing is (b, c) |-> Tr(a(w) b c / (e w^{e-1})). On the basis
+1, w, ..., w^{e-1} the pairing without the unit a has matrix entries
+
+  Tr(w^{i+j} / (e w^{e-1})) = 1 if i + j = e - 1, else 0  (0 <= i,j < e).
+
+The pairing is holomorphic and perfect; multiplication by the unit a
+is an automorphism; taking the direct sum over the points above a
+target disk gives perfection. On an unramified fiber the trace is the
+sum of the four contributions, with no division by the degree.
+
+The splitting type follows from Grothendieck's theorem. Self-duality
+makes the splitting degrees pair as (a_1, a_2, -a_2, -a_1) with
+a_1 >= a_2 >= 0. Proposition (curve data) gives
+
+  h^0(E(-1)) = h^0(C, Lambda - H) = h^0(C, theta) = 3,
+  h^0(E(-2)) = h^0(C, Lambda - 2H) = h^0(C, H + 2Z) = 2.
+
+An O(a)-summand with a positive contributes a and a - 1 to these two
+counts; all nonpositive summands contribute zero. Hence a_1 = 3 and
+a_2 = 0, and E = O(3) + O^2 + O(-3).
+
+The map O(3) -> E given by multiplication by xi^2 vanishes in the
+lower-degree summands, so its image P is the positive summand, which
+is a subbundle including over f(Z). The pullback
+H^0(PP^1, O(3)) -> H^0(C, 3H) is injective between 4-dimensional
+spaces and hence an isomorphism; thus H^0(P) = xi^2 H^0(C, 3H).
+
+The line P is isotropic because Hom(O(6), O) = 0, and pairs trivially
+with the two O-summads. Perfection gives P^perp = O(3) + O + O and
+P^perp / P = O + O with a nondegenerate constant form. All global
+sections of E lie in P^perp, and H^1(P) = 0. Taking sections proves
+the assertions about W and its fibers.
+
+At q in Z, coefficient evaluation in tau annihilates P_{f(q)} because
+P is the image of xi^2. The fiber algebra over f(q) is Artinian, and
+the trace pairing is the orthogonal sum of its factors. Since f is
+unramified at q, the factor at q is one-dimensional with form
+coefficient s(q) / E(q); evaluation at q vanishes on the other
+factors. The metric dual of coefficient evaluation thus has square
+E(q) / s(q) in the fiber E_{f(q)}; it lies in P^perp_{f(q)} because
+evaluation kills P_{f(q)}; passing to the orthogonal quotient preserves
+the square and gives the functional on W. Finally E(q) != 0 by the
+unramifiedness of f at q. []
+
+The dual form on W^* satisfies x^2 = x . x. The reciprocal in the
+dual-evaluation identity comes from using this dual form rather than
+the primal one.
+
+Lemma (principal parts). There is an isomorphism
+
+  iota: W -> H^0(Gamma, Phi)
+
+identifying the full principal parts of T/xi^2 on C with those of
+iota([T])/beta^2 on Gamma through residue duality, and the projective
+evaluations at corresponding points of Z. Its dual
+iota^*: H^0(Gamma, Phi)^* -> W^* is an isometry for the quadratic
+form Q and the dual trace form.
+
+Proof. Principal parts are defined after fixing the logarithmic volume
+Omega and its residue omega, so the residue convention is the one
+introduced earlier. Set N_Sigma = Gamma + F, N_Gamma = N_Sigma|_Gamma,
+and D = 2Z. Fourth-order contact identifies D on C and on Gamma as the
+same embedded length-twelve subscheme of Sigma. The isomorphism
+N_Sigma|_C = K_C - 3H is the one supplied by adjunction with the fixed
+volume, and omega trivializes K_Gamma.
+
+For a line bundle B on a smooth curve, the local pairing between
+principal parts of B(D) and jets of K_C - B is
+
+  (a, s) |-> sum_{q in Z} Res_q(as).
+
+It is perfect. In a local parameter with mutually dual frames, the two
+coefficients of a_{-2} z^{-2} + a_{-1} z^{-1} pair with a jet
+s_0 + s_1 z as a_{-2} s_1 + a_{-1} s_0.
+
+The two principal-part spaces
+
+  H^0(C, O_C(3H + 2Z) / O_C(3H)),
+  H^0(Gamma, O_Gamma(-N_Gamma + 2Z) / O_Gamma(-N_Gamma))
+
+are therefore both identified with H^0(D, N_Sigma|_D)^* by residues.
+This defines their comparison without any choice of coordinates or
+frames.
+
+Both complete adjoint systems are restrictions of the same ambient
+system H^0(Sigma, N_Sigma). The kernels in the restriction sequences
+to C and to Gamma are respectively O_Sigma(-Gamma - 3F) = (-5, -2)
+and O_Sigma(F) = (1, 0), with vanishing first cohomology for each.
+Their images on the common scheme D therefore coincide. Equivalently,
+an ambient section vanishing on Gamma vanishes to order at least four
+on C at Z and has no effect on these residue tests.
+
+The principal-part sequence together with Serre duality now says that
+the image of H^0(C, Lambda) under T |-> pp(T/xi^2) is exactly the
+annihilator of this common adjoint jet space, and its kernel is
+xi^2 H^0(C, 3H). On Gamma the same annihilator is the image of
+
+  H^0(Gamma, Phi) -> H^0(Gamma, O_Gamma(-N_Gamma + 2Z) / O_Gamma(-N_Gamma)),
+  tilde(T) |-> pp(tilde(T)/beta^2),
+
+because Phi = 2M - N_Gamma and div(beta) = Z. This map is injective:
+a nonzero kernel element would be a nonzero section of -N_Gamma, but
+deg(N_Gamma) = 10 > 0, so H^0(Gamma, -N_Gamma) = 0.
+
+The two annihilators have the same dimension. On Gamma, h^0(N_Gamma)
+= 10, and restriction to D is injective because
+deg(N_Gamma - 2Z) = 10 - 12 = -2. So the kernel has dimension 10 and
+the annihilator in the twelve-dimensional space has dimension 2. On
+C the same dimension count gives dim W = 2. This gives the isomorphism
+iota, and identifies it with principal parts.
+
+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,
+  kappa = k(y, z) gm^2 fF^4,
+  beta = b(z) m,   m^4 = gm^2 fF^4|_Gamma,
+
+so omega = dz on Gamma. Here k(0, z) = b(z)^4 and
+b'(0) k_y(0, 0) != 0. Put K(z) = k_y(0, z). Solving y = 0 on C gives
+the contact expansion
+
+  y|_C = -b(z)^4 / K(z) + O(z^8),
+  e|_C = 1 + O(z^4),
+  k_y|_C = K(z) + O(z^4).
+
+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.
+
+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
+sends a^4 to (y|_C / z^4) gm. Using the adjunction formula, its square
+has the form
+
+  tau^2 = s(z) dz fF^2,   s(z) = (e|_C y|_C) / (z^4 k_y|_C),
+  s(0) = -b'(0)^4 / k_y(0, 0)^2.
+
+The negative sign comes from solving the equation of C.
+
+Write T = hA(z) tau, and write tilde(T) = iota([T]) in the frame
+tilde(tau) = m^2 / (gm fF) of Phi as tilde(T) = hAt(z) tilde(tau). For
+an ambient test section S(y, z) gm fF, the two residue tests are
+
+  Res_0( S|_C hA(z) e|_C / (k_y|_C z^2) dz ),
+  Res_0( S(0, z) hAt(z) / b(z)^2 dz ).
+
+The restrictions of S to the two curves agree modulo z^4. Equality of
+their residue functionals on all local test jets is exactly
+
+  pp( hA / (K(z) z^2) ) = pp( hAt / b(z)^2 ),
+  hAt equals (b(z)/z)^2 / K(z) * hA  mod z^2.
+
+To exhibit both coefficients, put b = b_1 z + b_2 z^2 + O(z^3),
+K = K_0 + K_1 z + O(z^2), hA = hA_0 + hA_1 z + O(z^2). The respective
+principal parts are
+
+  hA_0 / K_0 z^{-2} + (hA_1 / K_0 - K_1 hA_0 / K_0^2) z^{-1},
+  hAt_0 / b_1^2 z^{-2} + (hAt_1 / b_1^2 - 2 b_2 hAt_0 / b_1^3) z^{-1}.
+
+Hence
+
+  hAt_0 = (b_1^2 / K_0) hA_0,
+  hAt_1 = (b_1^2 / K_0) hA_1 + (2 b_1 b_2 / K_0 - b_1^2 K_1 / K_0^2) hA_0.
+
+This invertible triangular transformation is the local expression of
+the intrinsic residue-dual identification, so its expressions in
+different coordinates and frames agree.
+
+The metric agrees as follows. In these frames write
+eta_C = E_C dz fF^2 and eta_Gamma = E_Gamma dz fF^2. Tangency gives
+E_C(0) = E_Gamma(0). Let ell and tilde(ell) be coefficient
+evaluations in tau and tilde(tau). The two-coefficient formula gives
+iota^* tilde(ell) = alpha ell with alpha = b_1^2 / K_0. By the
+dual-evaluation formula and the formula for s(0),
+
+  (iota^* tilde(ell))^2 = alpha^2 E_C(0) / s(0) = -E_Gamma(0).
+
+The compatible bundle identification gives tilde(tau)^2 = fF^2, so
+the last quantity is exactly Q(tilde(ell)), since the section defining
+Q multiplies to -eta_Gamma/omega. The evaluations at Z have three
+distinct projective directions. A binary quadratic that matches Q at
+all three vanishes identically. Hence iota^* is an isometry with sign
+and scalar fixed. []
+
+Proposition (residue data). The basis u can be chosen with unchanged
+determinant so that (u_1)_0 is an unramified fiber of f disjoint from
+Z. Set r = xi u and define v_j^* = iota^* w_j in W^*, and
+V = sum_{j=1}^6 v_j^*. Then, for compatible coordinate disks at
+q_1, ..., q_6, there exist nonzero constants c_j in C and a global
+pair P = (P_1, P_2) in H^0(C, theta + Z)^2 with:
+
+(1) c_j l_j = v_j^*, V^2 = 90 = 9 deg L, and
+(v_j^*)^2 + v_j^* . V = 1.
+
+(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),
+  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
+sum_j (W_j - 1) = -9 deg L.
+
+(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).
+
+(5) Changing P by a holomorphic pair of sections of theta leaves [J]
+unchanged. The prescription is invariant under changes of compatible
+coordinates.
+
+Proof. A determinant-one change of basis preserves eta_C and
+eta_Gamma and all forms already fixed. Avoiding the finitely many
+branch values and the image of Z, one arranges that the zero divisor
+of u_1 is the required fiber.
+
+Fix on a small disk about q_j a parameter z_j and a frame l_j^fr of
+L whose fourth power is dz_j (such a frame exists as a holomorphic
+fourth root of a unit). Set
+
+  a_j = xi / z_j,   fF_j = l_j^fr / a_j,   tau_j = (l_j^fr)^2 fF_j.
+
+These frame O_C(Z), H, and Lambda respectively. With r = xi u, the
+coefficient row of r in l_j^fr is z_j u, where u here denotes the
+coefficient row in fF_j. Let t_j be the fiber-linear function on
+L^{-1} corresponding to l_j^fr, giving compatible split coordinates
+(t_j, z_j). Let l_j in W^* be coefficient evaluation in tau_j. Since
+tau_j^2 = dz_j fF_j^2, the trace pairing gives
+
+  l_j^2 = (u_1 u_2' - u_2 u_1')(0).
+
+Primes in the rest of this argument refer to z_j-derivatives.
+
+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.
+
+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)
+pairs, by Serre duality, with H^0(C, K_C - theta) = H^0(C, theta),
+by the sum of residues. All these test sections vanish at Z (item (3)
+of the curve-data proposition), so their residue against any simple
+principal part is zero. Apply this separately to the two entries to
+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;
+away from Z it is already holomorphic. Hence J is a global section of
+Lambda.
+
+For the residue pairing, fix T in H^0(C, Lambda) and consider the
+global meromorphic canonical differential
+
+  nu = P_1 T / u_1,
+
+with theta + (theta + H) - H = K_C. Its only possible poles lie in Z
+and in the simple zero fiber of u_1. At q_j the compatible frames make
+the canonical product exactly coefficient multiplication followed by
+dz_j, so Res_{q_j}(nu) = c_j l_j(T). At a zero q of u_1, the section
+u_2 is nonzero, and
+
+  J(q) = -u_2(q) P_1(q),   eta_C(q) = -u_2(q) du_1(q),
+
+with the same minus sign in both expressions. In a local parameter,
+
+  Res_q(nu) = (JT/eta_C)(q).
+
+The zero fiber is unramified, so the sum of these four residues equals
+Tr(JT/eta_C) at that fiber. This trace is a global holomorphic
+function on PP^1 (Lemma trace-pairing), hence constant. The residue
+theorem gives
+
+  <[J], [T]> = -sum_j c_j l_j(T) = -V([T]),
+
+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'.
+
+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
+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
+xi^2 H^0(C, 3H), so [J] is unchanged.
+
+Compatibility with the fourth-root structure. Let zeta = psi(z) be
+another centered parameter and g(z) l^fr the corresponding compatible
+frame of L, with g^4 = psi'. The other frames change as
+
+  a_new = (z / psi(z)) a,
+  fF_new = g (psi(z) / z) fF,
+  tau_new = g^3 (psi(z) / z) tau.
+
+With g_0 = g(0), l_new = g_0^{-7} l, so the same vector v^*
+prescribes c_new = g_0^7 c. Also r_new = g^{-1} r and
+R_new(0) = g_0^2 R(0). The coefficient row of the global theta-section
+transforms as P_new = g^{-2} P. Its coefficient of zeta^{-1} is
+psi'(0) g_0^{-2} R(0) = g_0^2 R(0), exactly the newly prescribed
+R_new(0). Thus the same global P realizes the new prescription. The
+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). []
+
+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
+to dz under the fixed isomorphism L^4 = K_C. Split changes take the
+form
+
+  z -> zeta = psi(z),   t -> t_hat = g(z) t,   g^4 = psi'.
+
+Put E_t = t d_t. The two-form
+
+  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}.
+
+Seek vector fields of the form
+
+  v_i = t (A_i E_t + B_i d_z),   i = 1, 2,
+
+with Taylor series A_i = sum_{k >= 0} A_{ik} t^k, B_i = sum_{k >= 0} B_{ik} t^k
+in t whose coefficients are holomorphic in z, and with initial data
+
+  A_0 = r = (r_1, r_2),   B_0 = 0.
+
+Then
+
+  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
+times a nonzero unit along C.
+
+Gluing. Around each of the six marked points choose a small disk D_j;
+the six D_j are mutually disjoint. A neighborhood of C in the total
+space of L^{-1} is described by a base chart U_0 over C minus the
+union of the D_j and inside charts U_j over the D_j, glued along
+annular overlaps by transition maps. The construction proceeds by
+modifying the transition maps through finitely many holomorphic
+parameter families. Admissible modifying maps are time-one flows of
+holomorphic vector fields
+
+  D = t^k (b(z) E_t + h(z) d_z),   k >= 1,
+
+on the annular overlaps. Using the flow of -D, the induced action on
+other vector fields is exp(ad D). Every generator vanishes on the
+zero section and changes its local equation only by t |-> t + O(t^2),
+so the zero section and its conormal bundle are preserved.
+
+Action on coefficients. For D as above and V = t^{l+1} (A E_t + B d_z),
+
+  [D, V] = t^{k+l+1} ( ((l+1-k) b A + h A' - B b') E_t
+                       + ((l+1) b B + h B' - k h A - B h') d_z ).
+
+The logarithmic divergence is
+
+  L_D sigma = t^k ((k - 4) b + h') sigma.
+
+For k != 4, choosing b = h' / (4 - k) preserves sigma exactly. On the
+leading row of index 0, the effect reduces to
+
+  Delta A_k = (1 - k) b r + h r',
+  Delta B_k = -k h r.
+
+Concretely, a modifying field of leading index k acts on the row
+indexed k by this rule, and on row indices greater than k by
+corrections that appear at strictly higher t-orders.
+
+Existence of flows. On a compact sub-annulus every D in the above
+form is holomorphic near the zero section and vanishes there. So its
+time-one flow and inverse exist after shrinking the fiber disc once
+and for all six disks.
+
+Order one. On the j-th overlap, take h = m_j and b = m_j'/3. Choose
+
+  (A_1, B_1) = (p, m_j r) on U_j,
+  (A_1, B_1) = (P, 0) on U_0.
+
+Lemma (bracket) with k = 1, A = r, B = 0 gives Delta A_1 = m_j r' and
+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
+arises at this order.
+
+Order two. The order-one transition acts on the row of index 2 via
+exp(ad D_1), D_1 = t (m' E_t/3 + m d_z), with m = m_j. Expanding in
+the two components gives
+
+  Delta A_2 = (1/3) m' p + m p' + (2/3) m m' r' + (1/2) m^2 r''
+              - (1/6) m m'' r,
+  Delta U   = -m p + (1/3) m m' r.
+
+Substituting m = c/z, so R = m r, R' = mr' + m' r, and using
+r = z R / c, gives the Laurent expansions
+
+  Delta A_2 = -(c R)/z^3 + (c R')/(3 z^2) + (c R'')/(2 z)
+              - (c p)/(3 z^2) + (c p')/z,
+  Delta U   = -(c p)/z - (c R)/(3 z^2).
+
+Define, using z = z_j and c = c_j on the j-th disk,
+
+  F_ij = Res_0(r_i Delta A_2, j dz),
+  H_ij = Res_0(r_i' Delta U_j dz).
+
+Direct computation from the Laurent expansions gives
+
+  F_ij = -R_i p_j / 3 - (2/3) R_i R_j' - R_i' R_j,
+  H_ij = -R_i p_j - R_i R_j' / 3 - (2/3) R_i' R_j.
+
+The order-two obstruction is the residue matrix
+
+  T_ij = F_ij + H_ij/4 + 3 H_ji/4
+       = -7 R_i p_j / 12 - 9 R_j p_i / 12
+         - 5 R_i R_j' / 4 - 17 R_i' R_j / 12.
+
+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)).
+
+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
+P = R(0)/z + p(0) + 2 R'(0) + O(z) from the residue-data proposition,
+
+  Res_0(P_i P_j dz) = R_i p_j + R_j p_i + 2 R_i R_j' + 2 R_i' R_j.
+
+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
+sum_nu (W_nu - 1) = -9 deg L. So
+
+  sum_j T^alt_ij = (1/12) epsilon_ij sum_nu (W_nu - 1)
+                 = -(3/4) epsilon_ij deg L.
+
+Hence the four column sums of T_ij in the two indices, i.e., the four
+totals of the entries (11), (12), (21), (22), equal
+
+  0,   -(3/4) deg L,   (3/4) deg L,   0.
+
+By the residue-data proposition, both the inside and outside data
+evaluate the same principal-part pairing V, so these totals are also
+the residue totals computed from the outside chart. Consequently the
+vertical difference lies in H^1(C, L^3)^2 and pairs trivially, by
+Serre duality, with every element of H^0(C, K_C - 2L) = H^0(C, 2L) =
+H^0(C, theta). The class in H^1(C, L^3)^2 is therefore zero, and
+holomorphic A_2, B_2 on the two charts solve the matching.
+
+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 + ...
+
+The field equation in degree k uses G_{k-2}. To compare the next
+horizontal rows, that target coefficient must already be compatible,
+so each target coefficient is chosen before the field equation it
+enters is imposed.
+
+Local bivector target. On a neighborhood Y of C, put I = O_Y(-C) and
+F = T_Y(-log C). The relevant line bundle is
+
+  M = I^4 det F = I^5 det T_Y,   M|_C = L^4 tensor T_C = O_C,
+
+using 4L = K_C and T_C = K_C^{-1}. Its local frame is sigma^{-1}; the
+degree-k associated graded piece is L^k. Transport under an order-k
+transition D = t^k(b E_t + h d_z) with divergence
+q = div_sigma D = (k-4)b + h' is
+
+  [D, G sigma^{-1}] = (D(G) - qG) sigma^{-1},
+  G |-> exp(D - q) G,
+
+where D - q acts by f |-> D(f) - qf. Every order-k transition changes
+a target with constant term 1 first in degree k, by -((k-4)b + h'). A
+leading degree-k discrepancy is a cocycle in L^k.
+
+Choice ordering. For field changes at step k, the first determinant
+degrees affected at each step are
+
+  - global addition to A_{k-1}: degree k (target G_{k-2})
+  - B_k: degree k (target G_{k-2})
+  - A_k: degree k+1 (target G_{k-1}).
+
+The target coefficients in the last column stay fixed during these
+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
+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}).
+
+Under split changes, the only extra terms are multiples of the
+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
+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
+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.
+
+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),
+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
+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
+k = 3 and b = 0 when k >= 4. The new transition first affects G in
+degree k, so it cannot disturb the target comparison of degree k-2
+just used.
+
+Matching vertical rows. After horizontal matching, the remaining
+class lies in H^1(C, L^{k+1})^2. For k >= 4 it vanishes, since
+H^1(C, L^{k+1}) = H^0(C, L^{3-k})^* = 0. Holomorphic vertical
+corrections on the charts solve it without touching the current
+determinant equation, because B_0 = 0.
+
+At k = 3 the obstruction lies in H^1(C, K_C)^2 = C^2, tested by
+constants in the two entries. The added freedom is: on the j-th
+overlap replace h by h + lambda_j/z, and on its disk compensate by
+
+  B_3 |-> B_3 + 3 lambda_j r / z.
+
+The compensation is holomorphic and wedges to zero with r. It
+preserves the current determinant equation and horizontal matching:
+the transition adds the opposite horizontal row -3 lambda_j r / z.
+Keep the vertical lift A_3 unchanged in the chosen disk coordinate.
+Changing a degree-three input row affects its transport under
+previous positive-order transitions only at higher indices. The
+change in the transported vertical difference at index three is
+
+  lambda_j (r'/z + 2 r/z^2),
+
+whose residue at 0 is 3 lambda_j r'(0). The two entries are
+coefficients of canonical differentials, since L^4 = K_C. The six
+rows r'(q_j) span C^2 by the curve-data proposition, so the map
+(lambda_1, ..., lambda_6) |-> sum_j 3 lambda_j r'(q_j) has rank two
+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.
+No conflict arises: G_2 = 0 is already chosen, but its first
+determinant equation is at step four; the degree-three equation is
+unchanged.
+
+Extending the target. After the order-k transition, choose G_k. For
+k = 3 all transitions preserve sigma, so G = 1 continues to work. At
+k = 4 the earlier transitions still preserve sigma, so the first
+target discrepancy is -h' in degree four, by the target transport
+formula with b = 0. The horizontal coefficient of an order-four
+generator has type L^4 tensor T_C = O_C; its split transformation
+h -> g^{4-k} h equals h at k = 4. Thus h' dz = dh is an exact
+differential on the overlap, with zero contour residue. Its sole
+obstruction to splitting is the pairing with the constant section of
+O_C, so it vanishes. There are holomorphic chart coefficients G_4
+with G_1 = G_2 = G_3 = 0. Transport by a previous positive-order
+modification changes a degree-four term only in degrees at least
+five, so the residue argument applies at degree four.
+
+For k > 4 the new leading discrepancy lies in L^k, and
+
+  H^1(C, L^k) = H^0(C, L^{4-k})^* = 0
+
+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
+preserved.
+
+Stopping. A row of index k contributes to degree k+1 of the
+logarithmic frame (E_t, d_z), so matching through index 7 prescribes a
+section modulo I^9 F.
+
+Proposition (finite compatible jets). There exist finitely many
+holomorphic disk-to-outside transition germs, each a finite
+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.
+
+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.
+
+Proof. Perform the first two orders and the induction through
+k = 7. Only finitely many flows occur, each a genuine holomorphic
+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. []
+
+Analytic realization and Grauert contraction. The finite jets of
+the previous section live in overlapping charts, and are realized
+on a single Hausdorff complex surface Y containing C; the vector
+fields are then lifted from their jets by a cohomology vanishing
+argument; and the curve C is contracted to a singular point.
+Throughout this section, powers of an ideal denote ordinary ideal
+powers, and cohomology is analytic sheaf cohomology.
+
+Coherence of analytic structure and ideal sheaves, and the Hartogs
+extension property for normal surfaces, are taken from Grauert and
+Remmert, Coherent Analytic Sheaves, Grundlehren 265 (Springer 1984),
+and from Demailly, Complex Analytic and Differential Geometry (2012).
+
+./demailly-2012-complex-analytic-differential-geometry.pdf
+
+Lemma (Hausdorff neighborhood). The overlap maps of the finite-jets
+proposition define, after shrinking their domains, a smooth Hausdorff
+complex surface Y containing C. Its conormal bundle is L, and its
+self-intersection is C^2 = -10. Writing I = O_Y(-C) and
+F = T_Y(-log C), the prescribed rows define two global sections
+
+  v_1bar, v_2bar in H^0(Y, F / I^9 F).
+
+In divisor coordinates (t, z), their local representatives have
+logarithmic determinant t^4(1 + O(t)) and ordinary determinant
+t^5(1 + O(t)).
+
+Proof. Let N = L^{-1} be the split normal bundle. All changes to
+its disk-to-outside identifications are finite compositions of flows
+of holomorphic vector fields
+
+  D = t^k ( b(z) t d_t + h(z) d_z ),   k >= 1,
+
+on punctured disk neighborhoods. On a compact annulus in any such
+overlap, these fields vanish along the zero section. The local
+existence theorem for holomorphic differential equations, applied
+uniformly on that compact annulus, gives the required time-one maps
+and their inverses after shrinking the fiber radius. There are only
+finitely many flows. Shrinking successively gives their compositions
+on neighborhoods of the same compact annular bands. Denote the
+resulting inside-to-outside maps by Phi_j for 1 <= j <= 6.
+
+Here is a quotient construction that also verifies the separation
+property. Choose mutually disjoint coordinate disks D_j about the
+six points, and in each radii
+
+  0 < a_j < a'_j < b'_j < b_j
+
+so that Phi_j and its inverse are defined on neighborhoods of the
+closed annulus a_j <= |z_j| <= b_j in their respective split
+bundles. Their domains may be taken somewhat wider than the closed
+annuli. Retain a small open neighborhood V_j of the zero section
+over |z_j| < b'_j, and a small open neighborhood V_0 in N over
+C minus the union of {|z_j| <= a'_j} for 1 <= j <= 6. Choose the
+transverse thicknesses small enough that the maps are defined
+throughout the retained parts of the closed annuli and
+
+  |z_j(Phi_j(p)) - z_j(p)| < (a'_j - a_j) / 2
+
+there. This is possible because each map restricts to the identity
+on the zero section. Make the images of the retained annuli lie over
+their respective disjoint disks D_j.
+
+Set A_j = { p in V_j : a_j < |z_j(p)| < b_j, Phi_j(p) in V_0 }, and
+identify p in A_j with Phi_j(p) in V_0. These are biholomorphic
+identifications between open subsets. A point of V_0 lies in at most
+one of their images, since those images lie over disjoint disks.
+Each equivalence class in the disjoint union V_0 union V_1 union
+... union V_6 thus has at most two elements. In particular the
+equivalence relation consists of the diagonal, these six graphs,
+and their inverses; no further identifications or triple-overlap
+equations occur.
+
+Each graph is closed relative to V_j x V_0. Indeed, suppose
+p_nu in A_j with p_nu -> p in V_j and Phi_j(p_nu) -> q in V_0. The
+source base radii have limit at least a_j and less than
+b'_j < b_j. The flow is defined across the closed annulus, so
+continuity gives q = Phi_j(p). If the limiting source radius were
+a_j, the collar inequality would give |z_j(q)| < a'_j, contrary to
+q in V_0. Thus a_j < |z_j(p)| < b_j and p in A_j, as required.
+Possible fiber boundaries cause no difficulty because both limiting
+points are assumed to belong to the retained open pieces.
+
+The equivalence relation is therefore closed. The quotient map is
+open, since it glues open sets by biholomorphisms. An open quotient
+map with closed equivalence relation has Hausdorff target: two
+inequivalent representatives have product neighborhoods disjoint
+from the relation, and the images of these neighborhoods are
+disjoint open sets. Moreover, the quotient map restricts on every
+V_i to an injective open map. The complex structures on these
+pieces consequently give the quotient a smooth complex surface
+structure. The images of their zero sections give the original
+compact curve C with its original complex structure.
+
+Every modifying flow fixes C and induces the identity on its
+conormal bundle. In the split identification its coordinate
+expansion has t' = t + O(t^2), although its base coordinate may
+change by O(t). Composing with the original split changes therefore
+leaves I / I^2 = L. This proves C^2 = -deg L = -10.
+
+In the logarithmic basis (t d_t, d_z) the prescribed fields have
+coefficients t A_i, t B_i. Rows with indices 0 through 7 prescribe
+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
+this determinant by t, giving the ordinary determinant. []
+
+Vanishing and holomorphic lifting. The finite jets now live on a
+Hausdorff surface. To lift them to holomorphic fields, the first
+cohomology of I^9 F vanishes after shrinking about C. The
+logarithmic tangent sheaf F is locally free of rank two.
+Restriction to C gives the exact sequence
+
+  0 -> O_C -> F|_C -> T_C -> 0.
+
+Here the first map sends 1 to the class of the normal Euler field
+t d_t, and the second is restriction of a tangent vector field to C.
+The Euler class is unchanged when the local equation of C is
+multiplied by a unit. The sequence is intrinsic; a splitting is
+unnecessary.
+
+Lemma (graded vanishing). For every n >= 9,
+
+  H^1(C, F|_C tensor L^n) = 0.
+
+Consequently, with E = I^9 F and every r >= 1,
+
+  H^1(Y, E / I^r E) = 0.
+
+Proof. Since K_C = L^4, tensoring the log sequence by L^n gives
+
+  0 -> L^n -> F|_C tensor L^n -> L^{n-4} -> 0.
+
+Serre duality identifies the duals of the first cohomology groups of
+the outer terms with
+
+  H^0(C, L^{4-n})   and   H^0(C, L^{8-n}),
+
+respectively. Both vanish for n >= 9, since deg L > 0. The first
+assertion follows from the cohomology sequence.
+
+Because C is a Cartier divisor and F is locally free,
+
+  I^n F / I^{n+1} F = F|_C tensor L^n.
+
+The quotient E / I^r E = I^9 F / I^{9+r} F has a finite filtration
+with precisely these graded terms for 9 <= n < 9+r. Their first
+cohomology vanishes by the first assertion. Successive cohomology
+sequences give the second assertion; the quotient is supported on
+the compact curve C. []
+
+Lemma (actual lifting). After shrinking Y about C, the sections
+v_1bar, v_2bar lift to holomorphic sections v_1, v_2 of F, producing
+a frame for T_{Y minus C}.
+
+Proof. The self-intersection matrix is the negative definite
+(-10). Grauert's contraction criterion therefore contracts C to a
+point x of a normal complex surface:
+
+  pi: Y -> X,   pi^{-1}(x) = C,
+  Y minus C is isomorphic to X minus {x},
+
+with pi_* O_Y = O_X. The criterion is stated in Grauert, "Ueber
+Modifikationen und exzeptionelle analytische Mengen", Math. Ann. 146
+(1962), 331-368. The contraction is proper and is a biholomorphism
+off the exceptional curve; it is the holomorphic reduction in that
+theorem, whose target is normal when the source is normal. Choose a
+sufficiently small Stein representative X about x and replace Y by
+its inverse image; then X minus {x} is smooth.
+
+Vanishing of R^1 pi_* E. Put E = I^9 F, and let m_x be the ideal
+sheaf of x in X, and J = m_x O_Y. Both J and I vanish exactly on C,
+and I is radical. The analytic Nullstellensatz and Noetherianity
+give an inclusion I^a subset J locally at each point of C for some
+positive integer a. A finite cover of the compact curve gives a
+single exponent, and shrinking the representative makes
+
+  I^a subset J subset I,
+  I^{ar} subset J^r subset I^r  for r >= 1.
+
+The proper analytic direct-image theorem makes R^1 pi_* E coherent.
+The theorem on formal functions identifies its completed stalk with
+
+  (R^1 pi_* E)_x completed = inverse limit over r of H^1(Y, E / J^r E).
+
+The inclusions above give J^{ar} E subset I^{ar} E subset J^r E. So
+the transition from index ar to index r factors through
+H^1(Y, E / I^{ar} E) = 0 by graded vanishing. The inverse system is
+therefore pro-zero, and its limit is zero. The stalk (R^1 pi_* E)_x
+is a finite module over the Noetherian local ring O_{X, x}, so its
+zero completion implies that the stalk itself is zero, by Nakayama's
+lemma. Away from x the map pi is an isomorphism, so the same sheaf
+vanishes there. Thus R^1 pi_* E = 0.
+
+The sheaf pi_* E is also coherent. Cartan's theorem B on the Stein
+space X, plus the Leray spectral sequence for pi, gives
+H^1(Y, E) = 0. The cohomology sequence of
+
+  0 -> E -> F -> F / I^9 F -> 0
+
+lifts v_1bar, v_2bar to holomorphic sections v_i of F.
+
+Any such lift still frames. In each divisor chart, let v_i^{(8)} be
+the polynomial representative of the prescribed jet. It belongs to
+I F, and v_i - v_i^{(8)} belongs to I^9 F. So the change in their
+wedge lies in I^{10} det F. The logarithmic determinant of the
+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.
+
+The unit condition holds on a neighborhood of each point of C.
+Compactness gives one neighborhood of C where all local factors are
+nonvanishing, and properness allows a further shrinking of X whose
+inverse image lies in this neighborhood. So v_1, v_2 frame
+T_{Y minus C}. []
+
+Counterexample. Proposition. There is a normal complex surface germ
+(X, x) with an isolated singular point and free analytic derivation
+sheaf of rank two:
+
+  T_X = Hom_{O_X}(Omega^1_X, O_X) = O_X^2.
+
+It has a resolution pi: Y -> X whose exceptional set is the smooth
+curve C of genus 21 and self-intersection -10.
+
+Proof. Take the contraction constructed in the lifting lemma.
+Suppose x were a smooth point. A sufficiently small neighborhood of
+x admits a nowhere-vanishing holomorphic two-form omega. Its
+pullback is holomorphic on Y and nonvanishing off C. Thus
+
+  div(pi^* omega) = a C  with a >= 0,
+  K_Y . C = a C^2 <= 0.
+
+Adjunction, however, gives
+
+  K_Y . C = 2 g(C) - 2 - C^2 = 40 + 10 = 50.
+
+The contradiction shows x is singular, and it is the only singular
+point after the chosen shrinking.
+
+Write X^o = X minus {x}, and j: X^o -> X. The two fields of the
+lifting lemma descend to a holomorphic frame w_1, w_2 on X^o.
+Normality in dimension two gives the Hartogs extension
+
+  O_X = j_* O_{X^o}.
+
+This property extends the fields as derivations, as follows. Embed a
+representative of (X, x) in a smooth analytic germ with coordinate
+functions x_1, ..., x_e. For each i and alpha, the holomorphic
+function w_i(x_alpha) on X^o extends uniquely to a function a_{i
+alpha} on X. If f is any local defining equation, then
+sum_alpha a_{i alpha} df/dx_alpha = 0 on X^o and hence on X, by
+density and reducedness. The tuple (a_{i alpha}) defines a section of
+Hom_{O_X}(Omega^1_X, O_X) extending w_i, still denoted w_i.
+
+For any local analytic derivation w on X, the frame on X^o gives
+unique holomorphic coefficients h_1, h_2 there with w = h_1 w_1 +
+h_2 w_2. Both coefficients extend by Hartogs. The extended equality
+holds on X, since applying its difference to embedding coordinates
+gives holomorphic functions vanishing on the dense open set X^o.
+This proves surjectivity of
+
+  O_X^2 -> T_X,   (h_1, h_2) |-> h_1 w_1 + h_2 w_2.
+
+If a pair lies in the kernel, both coefficients vanish on X^o because
+w_1, w_2 are a frame there, and therefore vanish on X. The map is
+also injective, proving the assertion. []
+
+Invariants. The construction also determines the canonical
+discrepancy and a lower bound for the geometric genus. The two-form
+omega^o dual to the frame w_1, w_2 generates the canonical sheaf on
+X^o. Extending reflexively across the normal surface point gives
+omega_X = O_X; equivalently, K_X = 0. A normal surface is
+Cohen-Macaulay, so this reflexive rank-one sheaf is its dualizing
+sheaf, and X is Gorenstein; see Flenner and Zaidenberg, "Rational
+curves and rational singularities" (2001; arXiv:math/0109221), and
+Serre, "Un theoreme de dualite", Comment. Math. Helv. 29 (1955),
+9-26.
+
+By the ordinary determinant formula, the pullback of omega^o is
+
+  pi^* omega^o = t^{-5} u(t, z)^{-1} d_t /\ d_z.
+
+It is meromorphic on Y with divisor exactly -5C. Thus
+
+  K_Y = pi^* K_X - 5C,
+
+so the discrepancy of C is -5.
+
+For the geometric genus, put O_{nC} = O_Y / I^n. Riemann-Roch and
+the established equality h^0(C, L) = 2 give
+
+  h^1(C, L) = h^0(C, L^3) = 12.
+
+In the sequence
+
+  0 -> L -> O_{2C} -> O_C -> 0,
+
+the constant section of O_C lifts to O_{2C}. Since H^2(C, L) = 0,
+its cohomology sequence gives
+
+  h^1(O_{2C}) = h^1(C, L) + h^1(C, O_C) = 12 + 21 = 33.
+
+For every n >= 1, the sequence with kernel L^n shows that
+H^1(O_{(n+1)C}) -> H^1(O_{nC}) is surjective. The theorem on formal
+functions with the cofinal filtrations above identifies their inverse
+limit with the completion of (R^1 pi_* O_Y)_x. This stalk has finite
+length, since the coherent higher direct image is supported at x,
+and thus equals its completion. The inverse limit maps onto
+H^1(O_{2C}), proving
+
+  p_g(X, x) = dim (R^1 pi_* O_Y)_x >= 33.
+
+The exceptional curve has genus g = 21 and its dual graph has first
+Betti number b = 0. Consequently p_g - g - b >= 12.
+
+Algebraization. The passage from the analytic germ to an affine
+variety approximates the defining equations together with a finite
+relation matrix; their exact zero product and the closeness of the
+approximation preserve the complete local ring. Tangent freeness
+then descends from the continuous derivations of that ring. This
+follows the equation-and-relation determinacy method of Hironaka,
+"Formal line bundles along exceptional loci" (Tata Inst. Fund. Res.
+Stud. Math. 4, 1969, 201-218), used in Artin's isolated-singularity
+algebraization, "Algebraic approximation of structures over complete
+local rings", Publ. Math. IHES 36 (1969), 23-58. The finite-
+determinacy statement is proved below, with fixed estimates and no
+complete-intersection hypothesis.
+
+Put P = C[[x_1, ..., x_e]], n = (x_1, ..., x_e), I = (f_1, ...,
+f_s), R = P / I, where f = (f_1, ..., f_s) in n P^s is a row.
+Write m_R = n R, and let an overbar denote reduction modulo I. Choose
+an s x t matrix D whose columns generate all relations on f. Thus
+
+  P^t -> P^s -> I -> 0
+
+is exact, with the first map given by multiplication by D and the
+second by f. Define
+
+  psi: R^s -> R^t,   v |-> v Dbar,
+  K = ker psi,
+  J = sum_{j=1}^e R * (d_j f-bar).
+
+Differentiating f D = 0 and reducing modulo I gives J subset K.
+Right-exact reduction of the presentation gives a presentation of
+I/I^2; its dual identifies
+
+  K = Hom_R(I/I^2, R),   T^1 = K/J.
+
+In particular, no exactness on the left after reduction is assumed.
+
+Lemma (finite determinacy with a relation matrix). Suppose T^1 has
+finite length. There exist integers N, n_0 depending only on f, D
+such that the following holds. If a row g in P^s and an s x t
+matrix B satisfy
+
+  gB = 0,
+  g - f in n^{n_0} P^s,
+  B - D in n^N P^{s x t},
+
+then there exist a continuous C-algebra automorphism Phi of P and an
+invertible matrix U in GL_s(P) such that
+
+  f = Phi(g) U.
+
+One can take Phi(x_j) = x_j mod n^2 and U = 1 mod n. Consequently
+P/(g) = P/(f) as complete local C-algebras.
+
+Proof. The equations are improved one order at a time. The exact
+relation gB = 0 makes the equation error, reduced modulo I, lie
+nearly in K. Artin-Rees corrects it into K; the finite length of
+K/J then expresses the corrected class by coordinate derivatives. A
+change of generators removes the remaining I-valued error. The
+corrections are arranged so equation precision increases at every
+step while relation precision stays fixed. The coordinate and
+generator changes then converge in P.
+
+Bounds. For a map u: M -> M' of finite modules over a Noetherian
+local ring with maximal ideal a, Artin-Rees applied to im(u) in M'
+gives a constant alpha(u) >= 0 such that
+
+  im(u) intersect a^r M'  is contained in  u(a^{r - alpha(u)} M)
+
+for r >= alpha(u). Choose alpha, beta, d >= 0 so that
+
+  im(psi) intersect m_R^r R^t  is contained in  psi(m_R^{r - alpha} R^s),
+  K intersect m_R^r R^s  is contained in  m_R^{r - beta} K,
+  I^{oplus s} intersect n^r P^s  is contained in
+     { fC : C in n^{r - d} P^{s x t} },
+
+for all relevant r. In the last line, Artin-Rees applies to the map
+C |-> fC: P^{s x t} -> P^s with image I^{oplus s}; the relevant
+filtration is the one induced from P^s.
+
+Choose l >= 0 with m_R^l K contained in J, and set
+
+  c_0 = beta + l,    c = c_0 + d,
+  N >= alpha + 1,    n_0 >= max{c + N, c + 2, 2c + 1}.
+
+Improvement step. Suppose n >= n_0 and
+
+  h = g - f in n^n P^s,   B - D in n^N P^{s x t},   gB = 0.
+
+Reduced modulo I, the exact relation and the closeness give
+
+  hbar Dbar = gbar Dbar = -gbar (Bbar - Dbar) = -hbar (Bbar - Dbar)
+
+in m_R^{n+N} R^t. The factor hbar on the right provides order n + N.
+Artin-Rees on psi gives epsilon-bar in m_R^{n+N-alpha}, contained in
+m_R^{n+1}, with psi(epsilon-bar) = psi(hbar). Then
+
+  kappa = hbar - epsilon-bar  in K intersect m_R^n
+        is contained in  m_R^{n-beta} K
+        is contained in  m_R^{n-c_0} J,
+
+where the last inclusion uses m_R^l K contained in J with remaining
+factor n - beta - l = n - c_0. Lift coefficients of kappa in the
+generators d_j f-bar to elements a_j in n^{n-c_0}, and lift
+epsilon-bar to epsilon in n^{n+1} P^s. Then
+
+  w = h - sum_j a_j d_j f - epsilon
+
+lies in I^{oplus s} intersect n^{n-c_0} P^s. The order of this
+ideal-valued remainder need only be n - c_0, since a derivative of a
+defining equation can be a unit. Artin-Rees on the ideal gives
+w = fC with C in n^{n-c} P^{s x t}. So
+
+  h = sum_j a_j d_j f + fC + epsilon,
+  a_j, C in n^{n-c},   epsilon in n^{n+1} P^s.
+
+The loss d for lifting the ideal-valued remainder is included in the
+fixed constant c.
+
+Correction. Put q = n - c, and define
+
+  phi(x_j) = x_j - a_j,     M = (1+C)^{-1},
+  g_new = phi(g) M,        B_new = (1+C) phi(B).
+
+The bounds give q >= N, 2q >= n + 1, n + q - 1 >= n + 1. In
+particular phi is a formal automorphism with identity linear part,
+and M is invertible. Formal Taylor expansion gives
+
+  phi(f) - f + sum_j a_j d_j f  in n^{2q} P^s,
+  phi(h) - h  in n^{n+q-1} P^s.
+
+Combined with the linear correction,
+
+  phi(g) = f(1+C)   mod n^{n+1} P^s,
+  g_new - f  in n^{n+1} P^s.
+
+Also g_new B_new = phi(g)(1+C)^{-1}(1+C) phi(B) = 0 exactly. And
+
+  B_new - D = C phi(B) + (phi(B) - B) + (B - D)  in n^N P^{s x t},
+
+since C has order at least q >= N and substitution by phi changes
+a power series by an element of n^q. The relation precision N is
+preserved while the equation precision increases from n to n+1.
+
+Iteration. Starting from (g_{n_0}, B_{n_0}) = (g, B), repeat for
+n = n_0, n_0+1, ... with cumulative corrections
+
+  Phi_{n_0} = id,   U_{n_0} = 1,
+  Phi_{n+1} = phi_n o Phi_n,
+  U_{n+1} = phi_n(U_n) M_n,
+
+so that g_n = Phi_n(g) U_n at every stage. Since
+phi_n - id has coefficients in n^{n-c} and M_n - 1 has entries in
+n^{n-c}, the consecutive images Phi_n(x_j) and matrices U_n differ
+in orders tending to infinity, and converge in the complete ring P
+to Phi(x_j) and U. The resulting substitution Phi has identity
+linear part and hence is an automorphism, while U = 1 mod n is
+invertible. Taking limits in g_n = Phi_n(g) U_n gives
+f = Phi(g) U. []
+
+The original D must generate all relations, in order that
+K = Hom_R(I/I^2, R) holds. No such assertion is needed for the
+approximating matrix B: its exact zero product and its closeness to
+D are sufficient. The lemma therefore concerns solutions of the
+finite system gB = 0, not arbitrary truncations of a list of
+defining equations.
+
+Proposition (algebraization of an isolated analytic germ). Let
+(X, x) be a normal complex analytic germ of dimension d >= 1,
+smooth away from x, with free holomorphic tangent module of rank d.
+There exist a finitely generated normal integral C-algebra A and a
+maximal ideal m in A such that
+
+  dim A = d,   A/m = C,
+  A-hat_m = O-hat_{X, x},
+  Der_C(A) = A^d.
+
+The affine variety Spec A can be chosen smooth away from m. If x is
+singular, A_m is nonregular.
+
+Proof. Completed analytic presentation. Embed the germ in a smooth
+analytic germ and write
+
+  S = O_{X, x} = C{x_1, ..., x_e} / I_an.
+
+Choose a finite row f generating I_an and a finite matrix D that
+generates all its relations over the convergent power-series ring.
+Completion is exact on finite modules, so these give the
+presentation of the finite-determinacy lemma over
+P = C[[x_1, ..., x_e]], with R = S-hat = P/(f). Complex analytic
+local rings are excellent (Cutkosky, "Etoiles and valuations",
+J. Pure Appl. Algebra 221, 2017, 588-610). Hence the completion of
+the normal local ring S is a normal local domain, and
+
+  dim R = dim S = d,   R regular if and only if S regular.
+
+Convergent conormal dual. The convergent conormal dual of the
+analytic embedding coincides with Hom_R(I/I^2, R). On the smooth
+locus the tangent-to-normal map is surjective, since a smooth
+analytic subspace of a complex manifold is locally a regular
+embedding. Its cokernel is a coherent module supported at the
+isolated point x, and has finite length. Flat completion identifies
+this cokernel with T^1 = K/J above. Finite determinacy therefore
+applies to f, D.
+
+Continuous tangents. The holomorphic differentials of S have the
+finite Jacobian presentation with generators dx_j and relations
+df_i; their dual is the holomorphic tangent module. Flat completion
+of this kernel gives, where continuous means with respect to the
+maximal-ideal topology,
+
+  Der_C^cont(R, R)
+    = ker( R^e -> R^s, (v_j) |-> sum_j v_j d_j f-bar )
+    = R^d.
+
+A continuous derivation of a power-series quotient is determined by
+its values on the x_j, and the displayed relations are precisely
+the conditions for it to descend from P.
+
+Approximation of equations and relations. Let
+
+  H = ( C[x_1, ..., x_e]_{(x_1, ..., x_e)} )^h
+
+be the henselization, whose completion is P. This is the
+henselization of a finite-type C-algebra at a prime, as covered by
+Artin's theorem. Apply that theorem over H to the polynomial system
+in the s + s t entries of a row G and a matrix B,
+
+  sum_{i=1}^s G_i B_{i nu} = 0   (1 <= nu <= t),
+
+with formal solution (G, B) = (f, D). It gives g in H^s and B in
+H^{s x t} with gB = 0 exactly and
+
+  g - f in n^{n_0} P^s,   B - D in n^N P^{s x t}.
+
+By the finite-determinacy lemma, P/(g) = R.
+
+The henselization is the filtered colimit of pointed etale
+neighborhoods of the origin, with the same residue field. The
+finitely many entries of g, B are represented on a single such
+neighborhood. Passing to a later neighborhood, the identities
+gB = 0 hold there as well. Finite presentation and clearing
+denominators spread this to a neighborhood of the origin in affine
+space; shrinking preserves the marked point and makes the morphism
+etale. After an affine shrinking the identities hold in the
+coordinate ring. This gives an affine etale neighborhood
+Spec E -> A^e_C with a marked point q over the origin, residue
+field C, and a row g of E vanishing at q. Its completed ambient
+local ring is P. Put
+
+  A' = E/(g),   p' = q/(g),   T = A'_{p'}.
+
+Then A' is of finite type over C, and exactness of completion
+together with P/(g) = R gives
+
+  T-hat = P/(g) = R.
+
+So this construction preserves the complete local ring, including
+its dimension and its normality.
+
+Descent of tangent freeness. Because E is etale over A^e_C, its
+differentials have basis dx_j near q. Completing the finite
+Jacobian presentation of Omega^1_{T/C} therefore gives the
+continuous Jacobian presentation of P/(g). For a finitely presented
+module, flat base change commutes with its dual (express the dual
+as the kernel of a map between finite free modules). This gives
+
+  Der_C(T) tensor_T T-hat = Der_C^cont(T-hat, T-hat) = T-hat^d.
+
+Let M = Der_C(T). The residue vector space of M has dimension d.
+Lift a basis to d elements of M. Nakayama's lemma gives a
+surjection T^d -> M; after completion this is a surjection between
+free modules of rank d over T-hat, hence an isomorphism. Faithful
+flatness of completion annihilates its kernel, proving M = T^d.
+
+Local ring properties. The ring T injects into its completion,
+which is a domain, so T is a domain. Normality also descends: if
+a/b in Frac(T) is integral over T, it is integral over the normal
+T-hat, hence lies in T-hat; then a in b T-hat intersect T = b T,
+the last equality by faithful flatness, so a/b in T. Completion
+preserves Krull dimension and the dimension of m_T/m_T^2. So T has
+dimension d and is nonregular whenever S is.
+
+Isolated singularity. Let j_an be the d-th Fitting ideal of the
+holomorphic differential module of S. The analytic Jacobian
+criterion identifies its zero set with the nonsmooth locus, so some
+power of m_S lies in j_an. Fitting ideals commute with base change.
+Completed Jacobian presentations plus T-hat = R therefore imply,
+by contraction from the faithfully flat completion,
+
+  m_T^b  is contained in  Fitt_d(Omega^1_{T/C})
+
+for some b >= 0. On the reduced pure-d-dimensional neighborhood
+constructed next, the algebraic Jacobian criterion interprets this
+as nonsmoothness supported only at the marked point.
+
+Integral normal affine neighborhood. The finite module Der_C(A')
+localizes at p' to Der_C(T), because Omega^1_{A'/C} is finitely
+presented. Represent the chosen basis after inverting an element
+outside p'; this defines a map from a free module of rank d to
+Der_C(A') whose kernel and cokernel vanish at p'. Their supports
+are closed, so a further principal shrinking makes this map an
+isomorphism.
+
+Since T is a domain, exactly one irreducible component of
+Spec A' passes through p', and the support of the nilradical
+avoids p'. Remove the other components and that support. The
+remaining neighborhood is reduced and irreducible. Its normal locus
+is open (finite type over C is excellent) and contains p' by the
+normality of T; restrict there. This neighborhood has dimension d,
+and its nonsmooth locus is closed; by isolated descent it has no
+positive-dimensional component through p'; remove the components
+that do not contain p'.
+
+All restrictions are open neighborhoods of p'. Choose a principal
+affine neighborhood contained in their intersection and write it
+Spec A, with marked maximal ideal m. Then A is finitely generated,
+normal, and integral, dim A = d, and
+
+  A_m = T,   Der_C(A) = A^d.
+
+The residue field and completed local ring are unchanged, and the
+variety is smooth away from m. []
+
+Main Theorem. There exist a finitely generated normal integral
+C-algebra A of Krull dimension two and a maximal ideal m in A such
+that
+
+  Der_C(A) = A^2  and  A_m is not regular.
+
+Proof. The analytic counterexample supplies a singular normal
+complex analytic surface germ, smooth away from its marked point,
+whose holomorphic tangent module is free of rank two. Apply the
+algebraization proposition and fix any of the approximations and
+affine neighborhoods furnished by its proof. Its coordinate ring A
+is a finitely generated normal integral C-algebra of dimension two
+with Der_C(A) = A^2 and A_m nonregular. The derivation module is
+free, in particular projective, while A is not regular. []
+
+The smooth locus. Let X = Spec A be the affine surface just
+constructed, and U = X_reg its smooth locus. The logarithmic
+Kodaira dimension of a smooth quasiprojective surface U is
+
+  kappa-bar(U) = kappa(K_Ubar + D),
+
+where Ubar is a smooth projective compactification and its reduced
+boundary D = Ubar minus U is a simple normal-crossing divisor. The
+value does not depend on the compactification.
+
+Corollary. kappa-bar(U) = 2.
+
+Proof. The free tangent sheaf on X restricts to a trivial tangent
+bundle on U. If kappa-bar(U) <= 1, the theorem of Biswas, Gurjar,
+and Kolte (J. London Math. Soc. 90, 2014, 270-286) would make the
+normal affine surface X smooth, contradicting the Main Theorem.
+The only remaining value of logarithmic Kodaira dimension for a
+surface is two. []
+
+./bgk-1403.5613-zariski-lipman-normal-surfaces.pdf
+
+Derivation blowups. For a finite A-module M of generic rank r, its
+blowup is universal among proper birational morphisms from
+integral varieties to X for which the pullback of M modulo torsion
+is locally free of rank r. The universal property is described in
+Barajas, Chavez-Martinez, and Romano-Velazquez, "The blow-up of a
+singularity at the module of derivations" (2026;
+arXiv:2510.07607v2), section 1.6.
+
+./barajas-chavez-romano-2510.07607-blowup-derivations.pdf
+
+Corollary (derivation blowups). Starting with the singular
+surface X of the Main Theorem, every iteration of blowing up at
+the derivation module is isomorphic to X. The same holds if each
+blowup is followed by normalization. Neither procedure resolves
+the singular point.
+
+Proof. The derivation module is free, so the identity morphism of
+X already has the universal property stated above. Its blowup at
+the derivation module is therefore X itself, and the operation
+remains the identity at every iteration. Normalization also leaves
+X unchanged, because A is normal. []
+
+The surface X is Gorenstein, its canonical divisor is zero, and
+the canonical discrepancy at the exceptional curve is -5. In the
+resolution, the exceptional curve has genus g = 21, self-
+intersection -10, and first Betti number b = 0 in the dual graph.
+The geometric genus is at least 33. Hence p_g - g - b >= 12,
+which exceeds the bound p_g - g - b <= 2 of Bergner and Graf
+(Forum Math. Sigma 8, 2020, paper e21). The surface is not log
+canonical.
+
+./bergner-graf-2020-lipman-zariski-genus-one-higher.pdf
+
+The surface X stands outside every positive result on the
+Lipman-Zariski conjecture for normal surfaces. Its tangent sheaf
+is free, but no frame can be made commuting at the singular point.
+This follows from the theorem of Jorder (Math. Z. 278, 2014,
+893-899) stating that a normal complex space with a local frame of
+pairwise commuting vector fields is smooth at the point. The
+surface is not covered by the log canonical extension theorem of
+Greb, Kebekus, Kovacs, and Peternell, by Druel, or by Graf and
+Kovacs, because its canonical discrepancy is -5, and it is not
+covered by the geometric-genus criteria of Graf or of Bergner and
+Graf, because p_g - g - b >= 12.
+
+./jorder-1311.5141-weak-lipman-zariski.pdf
+./gkkp-1003.5340-differential-forms-log-canonical.pdf
+./druel-1304.6000-log-canonical-spaces.pdf
+./graf-kovacs-1301.7315-optimal-extension-1forms.pdf
+
+Derivation blowups at the tangent module, and derivation blowups
+followed by normalization, leave X unchanged and do not resolve
+its singular point, so the Lipman-Zariski conjecture fails in
+characteristic zero for normal surfaces, and the analogous question
+of Barajas, Chavez-Martinez, and Romano-Velazquez about derivation
+blowups has a negative answer in characteristic zero.
+
+./barajas-chavez-romano-2510.07607-blowup-derivations.pdf