diff --git a/strong_hyperkahler_syz_v2.txt b/strong_hyperkahler_syz_v2.txt
index 8525aa9..7934191 100644
--- a/strong_hyperkahler_syz_v2.txt
+++ b/strong_hyperkahler_syz_v2.txt
@@ -39,16 +39,16 @@ Exercise: Positive polarization orthogonal to e. Why does an h exist with q(h) >
 Lemma (Polarized Family Construction, part 1). Let q have signature (3, r − 3) with r = rank H^2(X, ℤ), and e ≠ 0 primitive with q(e) = 0. Then there is a primitive h ∈ H^2(X, ℤ) with q(h) > 0 and q(h, e) = 0. Proof: the radical of e^⊥ is ℝ e, and after modding it out, signature(e^⊥ / ℝ e) = (2, r − 4). In particular e^⊥ contains positive vectors. Because e is rational, e^⊥ is Q-defined, so rational approximation gives a rational positive vector; scale to a primitive integral one and call it h. Role: h will be the polarization we impose on the deformation family. Since q(h, e) = 0, moving in the h-polarized deformation component never mixes h with e, so e stays exactly where we put it. This is the first example of *bookkeeping by construction*: the perpendicularity q(h, e) = 0 is not a technical convenience, it is what allows us to keep track of e while we deform.
 
 Exercise: The algebraic family and the arithmetic quotient Q_h. Why does this setup give an honest algebraic family?
-Lemma (Polarized Family Construction, part 2). Take a manifold X_h in the marked deformation component of X on which h is ample (by period surjectivity and generic vanishing of extra NS classes: for almost every period with q(p, h) = 0, the NS lattice is ℤ h). Fix a positive integer k with k h very ample, so H^1(X_h, k h) = 0; bases of sections embed X_h into ℙ^M, giving a Hilbert point. Deformations of the polarized pair plus frames of the section space are smooth by local Torelli; the Hilbert scheme component has finite automorphism group acting freely on frames, so it has a smooth point with submersive period map. Restrict the universal family to a smooth connected Zariski-open base S and keep the polarization k h. Let Γ ⊂ G_h(ℤ) = {g ∈ O(Λ, q): gh = h, g D_h = D_h} be a torsion-free finite-index congruence subgroup acting on the full lattice Λ = H^2(X, ℤ); pass to a finite étale cover S → S with monodromy in Γ (this uses SGA 1, Exposé XII, Theorem 5.1). The arithmetic quotient Q_h = Γ \ D_h (where D_h is the h-polarized period domain component containing [X_h]) is smooth quasi-projective by Baily–Borel 1966 (https://annals.math.princeton.edu/1966/84-3/p11); the descended period map Φ: S → Q_h is algebraic by Borel 1972 (https://doi.org/10.4310/jdg/1214430642). The Γ-action on D_h is *free*: the stabilizer of a positive oriented 2-plane is compact in O(V_ℝ, q), hence finite, hence trivial in the torsion-free Γ.
+Lemma (Polarized Family Construction, part 2). Take a manifold X_h in the marked deformation component of X on which h is ample (by period surjectivity and generic vanishing of extra NS classes: for almost every period with q(p, h) = 0, the NS lattice is ℤ h). Fix a positive integer k with k h very ample, so H^1(X_h, k h) = 0; bases of sections embed X_h into ℙ^M, giving a Hilbert point. Deformations of the polarized pair plus frames of the section space are smooth by local Torelli; the Hilbert scheme component has finite automorphism group acting freely on frames, so it has a smooth point with submersive period map. Restrict the universal family to a smooth connected Zariski-open base S and keep the polarization k h. Let Γ ⊂ G_h(ℤ) = {g ∈ O(Λ, q): gh = h, g D_h = D_h} be a torsion-free finite-index congruence subgroup acting on the full lattice Λ = H^2(X, ℤ); pass to a finite étale cover S → S with monodromy in Γ (this uses SGA 1, Exposé XII, Theorem 5.1). The arithmetic quotient Q_h = Γ  D_h (where D_h is the h-polarized period domain component containing [X_h]) is smooth quasi-projective by Baily–Borel 1966 (https://annals.math.princeton.edu/1966/84-3/p11); the descended period map Φ: S → Q_h is algebraic by Borel 1972 (https://doi.org/10.4310/jdg/1214430642). The Γ-action on D_h is *free*: the stabilizer of a positive oriented 2-plane is compact in O(V_ℝ, q), hence finite, hence trivial in the torsion-free Γ.
 
 Exercise: Rational ternary subspace P and arithmetic curve C_P with cusp.
-Lemma (Polarized Family Construction, part 3). In V_ℝ = h^⊥_ℝ, decompose e = e_W + e_− for W a positive 2-plane period in the open set U ⊂ D_h that S actually realizes. Since W^⊥ is negative-definite and e is not zero, e_W ≠ 0 and q(e_−) = −q(e_W) < 0; hence W + ℝ e = W ⊕ ℝ e_− has signature (2,1). This shows that *some* positive 3-space containing e has the right signature. Now approximate W by rational 2-planes W_j = <x_j, y_j>_ℝ, keeping {e, x_j, y_j} rational and of signature (2,1); set P = <e, x_j, y_j>_ℚ for large j and let D_P ⊂ D_h be the positive-plane domain of P containing W_j. Because S is algebraic and open in Q_h, D_P ∩ U ≠ ∅ for large j. To make the arithmetic structure compatible with the full lattice Λ = H^2(X, ℤ), set Λ_0 = (P ∩ Λ) ⊕ (P^⊥ ∩ Λ) ⊂ Λ; Λ_0 has finite index, and an isometry of P ∩ Λ_P congruent to the identity modulo m · N (N large enough that N Λ ⊂ Λ_0) extends identically on P^⊥ ∩ Λ to an isometry of Λ. Take such a congruence subgroup Γ_P ⊂ SO(P, q) acting on D_P; then Γ_P ⊂ Γ. The quotient C_P := Γ_P \ D_P is a smooth arithmetic curve, and because P contains the rational null line ℚ e, its smooth compactification C̄_P has a cusp c_e represented by ℚ e. The inclusion j: C_P → Q_h is algebraic (Borel 1972).
+Lemma (Polarized Family Construction, part 3). In V_ℝ = h^⊥_ℝ, decompose e = e_W + e_− for W a positive 2-plane period in the open set U ⊂ D_h that S actually realizes. Since W^⊥ is negative-definite and e is not zero, e_W ≠ 0 and q(e_−) = −q(e_W) < 0; hence W + ℝ e = W ⊕ ℝ e_− has signature (2,1). This shows that *some* positive 3-space containing e has the right signature. Now approximate W by rational 2-planes W_j = <x_j, y_j>_ℝ, keeping {e, x_j, y_j} rational and of signature (2,1); set P = <e, x_j, y_j>_ℚ for large j and let D_P ⊂ D_h be the positive-plane domain of P containing W_j. Because S is algebraic and open in Q_h, D_P ∩ U ≠ ∅ for large j. To make the arithmetic structure compatible with the full lattice Λ = H^2(X, ℤ), set Λ_0 = (P ∩ Λ) ⊕ (P^⊥ ∩ Λ) ⊂ Λ; Λ_0 has finite index, and an isometry of P ∩ Λ_P congruent to the identity modulo m · N (N large enough that N Λ ⊂ Λ_0) extends identically on P^⊥ ∩ Λ to an isometry of Λ. Take such a congruence subgroup Γ_P ⊂ SO(P, q) acting on D_P; then Γ_P ⊂ Γ. The quotient C_P := Γ_P  D_P is a smooth arithmetic curve, and because P contains the rational null line ℚ e, its smooth compactification C̄_P has a cusp c_e represented by ℚ e. The inclusion j: C_P → Q_h is algebraic (Borel 1972).
 
 Exercise: The algebraic curve B reaching the cusp, plus the transport of the marking.
-Form the fiber product Z = S ×_{Q_h} C_P. Choose a point z_0 of Z at which the marked period lift from S equals the chosen point of U ∩ D_P; there the map Z → C_P is submersive (since the local period map of S is submersive). Intersect a quasi-projective embedding of Z with general hyperplanes through a smooth point near z_0 whose tangent intersection is transverse to the fiber of Z → C_P; take the component through that point and normalize: this is an irreducible smooth algebraic curve B with a nonconstant map b: B → C_P, dominating C_P and carrying the pulled-back family from S. Take smooth projective completions; b̄: B̄ → C̄_P is a finite surjective morphism of curves. Remove a finite set F ⊂ C̄_P containing every cusp, every branch value, and every image of a point of B̄ \ B. Then over C_P \ F the morphism b is a finite unramified covering and its source lies in B. (If necessary slide the starting point slightly within the same local branch of S so it maps outside F.) This is the family we will use; from here on, the base is a punctured disk around some b_e ∈ B̄ over c_e.
+Form the fiber product Z = S ×_{Q_h} C_P. Choose a point z_0 of Z at which the marked period lift from S equals the chosen point of U ∩ D_P; there the map Z → C_P is submersive (since the local period map of S is submersive). Intersect a quasi-projective embedding of Z with general hyperplanes through a smooth point near z_0 whose tangent intersection is transverse to the fiber of Z → C_P; take the component through that point and normalize: this is an irreducible smooth algebraic curve B with a nonconstant map b: B → C_P, dominating C_P and carrying the pulled-back family from S. Take smooth projective completions; b̄: B̄ → C̄_P is a finite surjective morphism of curves. Remove a finite set F ⊂ C̄_P containing every cusp, every branch value, and every image of a point of B̄  B. Then over C_P  F the morphism b is a finite unramified covering and its source lies in B. (If necessary slide the starting point slightly within the same local branch of S so it maps outside F.) This is the family we will use; from here on, the base is a punctured disk around some b_e ∈ B̄ over c_e.
 
 Exercise: The path lift and the equality of period lifts.
-In the D_P cover, the inverse image of F ∩ C_P is a locally finite discrete set; its complement is path connected. Take a path from the initial marked lift of c_e-part in D_P towards the boundary point represented by e, avoiding that discrete set; eventually it lies in a cusp neighborhood of e. Project this path to C_P \ F, lift it through the finite unramified covering B → C_P \ F, and note that the lift tends to a point b_e ∈ B̄ over c_e. So a small punctured disk around b_e carries our desired smooth algebraic polarized family approaching the cusp. *Transport of the marking:* as the family is transported over the path, its period lift to D_h is a continuous lift of the period map S → Q_h; by uniqueness of lifts for the covering D_h → Q_h, this D_h-valued lift begins at the same point as the constructed D_P-valued lift, and both descend to the same map on the base. Therefore they are equal on the universal cover of the punctured disk. That equality of lifts — not just equality of images in Q_h — is exactly what lets us identify the family monodromy with the arithmetic monodromy next.
+In the D_P cover, the inverse image of F ∩ C_P is a locally finite discrete set; its complement is path connected. Take a path from the initial marked lift of c_e-part in D_P towards the boundary point represented by e, avoiding that discrete set; eventually it lies in a cusp neighborhood of e. Project this path to C_P  F, lift it through the finite unramified covering B → C_P  F, and note that the lift tends to a point b_e ∈ B̄ over c_e. So a small punctured disk around b_e carries our desired smooth algebraic polarized family approaching the cusp. *Transport of the marking:* as the family is transported over the path, its period lift to D_h is a continuous lift of the period map S → Q_h; by uniqueness of lifts for the covering D_h → Q_h, this D_h-valued lift begins at the same point as the constructed D_P-valued lift, and both descend to the same map on the base. Therefore they are equal on the universal cover of the punctured disk. That equality of lifts — not just equality of images in Q_h — is exactly what lets us identify the family monodromy with the arithmetic monodromy next.
 
 Exercise: Identifying monodromy M on the full lattice.
 Lemma (Monodromy Identification). For a loop in B, let M_B ∈ Γ ⊂ O(Λ, q) be its second-cohomology monodromy, and γ_B ∈ Γ_P ⊂ Γ the deck transformation acting on the D_P-lift. Equivariance and the equality M_B Φ̃ = γ_B Φ̃ at every lift point imply γ_B^{−1} M_B fixes Φ̃(b̃) ∈ D_h. Since Γ acts *freely* on D_h (proved above by compactness of the stabilizer of a positive oriented 2-plane), γ_B^{−1} M_B = 1, and M_B = γ_B as isometries of the *full* lattice Λ. Two consequences. (i) The actual family monodromy fixes our labeled e, is the identity on P^⊥, and on P it acts by the explicit Γ_P element defined below. (ii) Because the equality of lifts is exact — not just up to a finite index in the arithmetic subgroup — the identification is not only up to base change; it pins down the labeling of e on the nose. That is the technical core of "retain the prescribed class in monodromy" from the four-step plan.
@@ -70,7 +70,7 @@ Exercise: Maximal growth forces a maximal crossing with nonzero residue.
 Fix a point of Y_0 where exactly k components meet, 1 ≤ k ≤ d + 1. Take coordinates with t = z_0 ... z_{k−1} and consider the local generator g = (dz_1 / z_1) ∧ ... ∧ (dz_{k−1} / z_{k−1}) ∧ dz_k ∧ ... ∧ dz_d of K_log. Its extended holomorphic coefficient (call it h(z)) is bounded on a smaller coordinate polydisk. Fix such a chart; in the variables x_i = −log |z_i| for 1 ≤ i ≤ k−1, the fiber part of the chart is an expanding simplex: each x_i is bounded below, and Σ x_i ≤ s + O(1), of dimension k−1. Integrating the angular variables and the remaining ordinary coordinates yields ∫_{Y_t ∩ chart} |η_t|^2 ≤ C (1 + s^{k−1}). If k = d + 1 (maximal crossing), the crossing is an isolated point o. See what happens if the coefficient h(o) = 0. Dilation trick: put A = −log δ for a polydisk radius δ, and rescale x_i = A + (s − (d+1)A) u_i over the standard simplex {u_i > 0, Σ u_i < 1}. For almost every u, all d+1 coordinates z_0, ..., z_d tend to zero as s → ∞, uniformly in their arguments; the rescaled integrand |h|^2 tends to |h(o)|^2 = 0 pointwise on the simplex and is uniformly bounded. Dominated convergence gives ∫_{Y_t ∩ chart} |η_t|^2 = o(s^d) if h(o) = 0. Now cover the compact central fiber by finitely many such charts (properness, then shrink the disk so nearby fibers are covered). If there were no maximal crossings (all k ≤ d), the total ∫ would be O(s^{d−1}). If there were maximal crossings but all coefficients vanished, the total would be o(s^d). Either contradicts the lower bound c s^d ≤ ∫ |η_t|^2 = I_t. So at least one maximal crossing has nonzero residue h(o) ≠ 0. At that point the functions z_1, ..., z_d may be chosen to be restrictions of rational functions on the algebraic total space (algebraic local defining functions), which the intersection estimates of the next section need.
 
 Exercise: Proposition (Family). Assemble the ingredients.
-The prescribed-cusp family π: Y → Δ constructed above has: (1) Y smooth, Y_0 reduced SNC, every Y_t IHS in the marked deformation component of X; a line bundle A on Y restricts on each Y_t to an ample bundle with c_1 = k h; the family is the analytic restriction of an algebraic family. (2) A rational subspace P = <e, f, v>_ℚ ⊂ h^⊥ of signature (2, 1) with q(e) = q(f) = 0, q(e, f) = 1, q(e, v) = q(f, v) = 0, q(v) = a > 0; periods p(τ) = f + τ v − (a/2) τ^2 e; q(p, p) = 2 a (Im τ)^2; monodromy M as in the previous exercise; M is identity on P^⊥; (M−1) P = <e, v>_ℚ, (M−1)^2 f = −a w^2 e ≠ 0, (M−1)^3 = 0. (3) A single-valued holomorphic relative σ on Y|Δ* with q([σ], e) = 1, η = σ^n extends to K_log, c s^d ≤ ∫ |η|^2 ≤ C s^d, and there is a maximal crossing o ∈ Y_0 with residue h(o) ≠ 0. (4) For any prescribed irrational λ ∈ ℝ \ ℚ, there is a path in Δ* along which τ = λ + i y and y → +∞. Properties (1)–(3) supply all hypotheses of Theorem (Torus) on any sufficiently small fiber along this path. Property (4) is the input to the rotation step: choosing λ irrational forces the rotated period plane to contain no rational isotropic class except the original Q e.
+The prescribed-cusp family π: Y → Δ constructed above has: (1) Y smooth, Y_0 reduced SNC, every Y_t IHS in the marked deformation component of X; a line bundle A on Y restricts on each Y_t to an ample bundle with c_1 = k h; the family is the analytic restriction of an algebraic family. (2) A rational subspace P = <e, f, v>_ℚ ⊂ h^⊥ of signature (2, 1) with q(e) = q(f) = 0, q(e, f) = 1, q(e, v) = q(f, v) = 0, q(v) = a > 0; periods p(τ) = f + τ v − (a/2) τ^2 e; q(p, p) = 2 a (Im τ)^2; monodromy M as in the previous exercise; M is identity on P^⊥; (M−1) P = <e, v>_ℚ, (M−1)^2 f = −a w^2 e ≠ 0, (M−1)^3 = 0. (3) A single-valued holomorphic relative σ on Y|Δ* with q([σ], e) = 1, η = σ^n extends to K_log, c s^d ≤ ∫ |η|^2 ≤ C s^d, and there is a maximal crossing o ∈ Y_0 with residue h(o) ≠ 0. (4) For any prescribed irrational λ ∈ ℝ  ℚ, there is a path in Δ* along which τ = λ + i y and y → +∞. Properties (1)–(3) supply all hypotheses of Theorem (Torus) on any sufficiently small fiber along this path. Property (4) is the input to the rotation step: choosing λ irrational forces the rotated period plane to contain no rational isotropic class except the original Q e.
 
 Exercise: Theorem (Torus). State the special Lagrangian torus theorem.
 Theorem (Torus). Let π: Y → Δ be the restriction to a disk of a projective algebraic family over a smooth curve. Assume Y is smooth, Y_0 is a reduced simple-normal-crossings divisor, and the fibers Y_t (t ≠ 0) are smooth connected projective manifolds of complex dimension d ≥ 2 with trivial canonical bundle. Let a_t be polarizing classes given by a line bundle on Y; the line bundle need not be relatively ample at 0. Let η_t be nowhere-vanishing holomorphic d-forms that extend as a section of the relative logarithmic canonical line. Put s = −log |t|, and suppose for constants c, C > 0 that c s^d ≤ I_t := ∫_{Y_t} |η_t|^2 ≤ C s^d for all 0 < |t| << 1. Suppose that at some point o ∈ Y_0 exactly d+1 components meet and that the logarithmic coefficient of η at o is nonzero: so in some coordinate neighborhood of o, z_0 z_1 ... z_d = t and η_t = f(z) dz_1/z_1 ∧ ... ∧ dz_d/z_d with f(o) ≠ 0. Equip Y_t with the Ricci-flat Kähler metric in the polarization class a_t. Then for every sufficiently small nonzero t, Y_t contains a smooth embedded special Lagrangian real d-torus isotopic in Y_t to an angular coordinate torus in the chart z_0 z_1 ... z_d = t. Here "special Lagrangian" means the Kähler form restricts to zero on T and the holomorphic volume form η_t has constant phase on the oriented torus T. In the application d = 2n. The theorem is independent of hyperkähler theory; it is a general Calabi–Yau degeneration result.
@@ -88,7 +88,7 @@ Exercise: Lemma (Many Centers) + Lemma (Choose One). Getting one good center wit
 Lemma (Many Centers). There are δ, v > 0 and measurable sets E_t ⊂ T_t(B_0) with μ_t(E_t) ≥ δ μ_t(Y_t) and μ_t(B_{g_t}(q, 1)) ≥ v for all q ∈ E_t. Proof sketch: the region sB_0 contains ≥ c_0 s^d disjoint translates of a fixed small cube; multiply by a fixed angle cube to get disjoint coordinate boxes Q in T_t(B_0). By the first trace bound at least half satisfy ∫_Q tr_γ g_t dV_γ ≤ C_0. By coordinate density, each such box has μ_t(Q) between fixed m_0, m_1. For a, b ∈ Q, a convexity/Jacobian change-of-variables estimate bounds the double integral of d_{g_t}(a, b) by a constant. Then Markov's inequality picks E_Q ⊂ Q of measure ≥ μ_t(Q)/2 on which the average distance is small; Bishop–Gromov (Ric(g_t) = 0) converts "μ_t(B(a, R_0)) large" into "μ_t(B(a, 1)) ≥ v". Union over good boxes gives a fixed positive fraction of total volume that is noncollapsed. Lemma (Choose One). For a nonnegative integrable h on Y_t, let M_t h(q) = sup_{r>0} (1/μ_t(B_{g_t}(q, r))) ∫_{B_{g_t}(q, r)} h dV_t be the centered maximal function. Doubling gives weak-(1,1): μ_t{M_t h > λ} ≤ (D_N/λ) ∫ h dV_t. Choose A so large that D_N C_H / A < δ/3; put ε_t = √m_t + s^{−1} → 0. The two exceptional sets {M_t H_t > A} and {M_t K_t > ε_t} each have measure < δ μ_t(Y_t)/3 for small t, so E_t is not covered; there is p_t ∈ E_t with μ_t(B(p_t, 1)) ≥ v, and for every R > 0, the average of H_t and of K_t over B(p_t, R) are bounded by A and ε_t respectively. Pass to a sub with b_t := x(p_t)/s → b_* ∈ closure(B_0) ⊂ B_1. In particular ρ(b_t) = 1, U_0(p_t) = s, U_j(p_t) = x_j(p_t).
 
 Exercise: Poincaré + Chebyshev locate balls in the chart, and the flat regular set of the pointed limit.
-The ball Poincaré inequality for Ric ≥ 0 (Cheeger–Colding segment inequality, https://annals.math.princeton.edu/articles/13331) at p_t for U_j gives an upper bound CA r^2 on the "oscillation vs mean" of U_j in B(p_t, r). Combine with dyadic radii and doubling to get the centered L^2 estimate: (1/μ_t(B(p_t, R))) ∫_{B(p_t, R)} |U_j − U_j(p_t)|^2 dV_t ≤ C R^2 for all 0 ≤ j ≤ d and all R > 0. Divide by s^2 and use b_t → b_*: on each fixed-radius ball, U_0/s → 1 and U_j/s → (b_*)_j in L^2. Where U_0/s > 1/2, the point is in the coordinate tube with x_j/s = (U_j/s)/(U_0/s). Chebyshev's inequality gives, for every R, η > 0, volume-tending-to-zero of B_{g_t}(p_t, R) minus the set {q in chart : |x(q)/s − b_*| < η}. Bishop–Gromov bounds μ_t(B(p_t, R)) ≤ ω_N R^N. Now pass to the pointed measured Gromov–Hausdorff limit (Y_t, g_t, p_t, μ_t) → (Z, d_Z, p, μ). The compactness, curvature, and singularity theory of Cheeger–Colding (Cheeger–Colding I, https://doi.org/10.4310/jdg/1214459974; Colding, https://annals.math.princeton.edu/articles/13099) and Cheeger–Naber (https://doi.org/10.4007/annals.2015.182.3.5) gives: Z complete proper; μ is the N-dimensional Riemannian volume on the dense regular set G; S = Z \ G is closed with dim_H S ≤ N − 4 and μ(B_Z(q, r)) ≤ ω_N r^N; compact subsets of G admit smooth convergence identities with regions of Y_t; complex structures converge smoothly on G to a parallel integrable limit I_∞. Because every compact smooth region lies in a fixed-radius ball about p_t with curvature energy ≤ ε_t ω_N R^N → 0, the limit metric has Rm_g = 0 on G. Flat regular set. Then the embedding maps Φ_t are holomorphic between Kähler manifolds, hence harmonic; with their energy on regular compact sets tending to zero by the K_t bound, Lemma (Vanishing Energy, next section) forces dΦ_t → 0; every local constant is Φ(o), so Φ_t → Φ(o) uniformly on compact subsets of G. From the embedding and the log equation Σ_{i=0}^d log |z_i| = −s, the corresponding regions of Y_t lie in the coordinate chart with each x_j pluriharmonic and bounded; interior harmonic estimates improve L^2 convergence to C^∞_loc: x_j/s → (b_*)_j. Since b_* ∈ closure(B_1) has ρ = 1 there, every compact subset of G has a neighborhood where U_j = x_j for all small t; the all-radius bounds then apply to the actual logarithmic coordinates, ready for the cylinder identification.
+The ball Poincaré inequality for Ric ≥ 0 (Cheeger–Colding segment inequality, https://annals.math.princeton.edu/articles/13331) at p_t for U_j gives an upper bound CA r^2 on the "oscillation vs mean" of U_j in B(p_t, r). Combine with dyadic radii and doubling to get the centered L^2 estimate: (1/μ_t(B(p_t, R))) ∫_{B(p_t, R)} |U_j − U_j(p_t)|^2 dV_t ≤ C R^2 for all 0 ≤ j ≤ d and all R > 0. Divide by s^2 and use b_t → b_*: on each fixed-radius ball, U_0/s → 1 and U_j/s → (b_*)_j in L^2. Where U_0/s > 1/2, the point is in the coordinate tube with x_j/s = (U_j/s)/(U_0/s). Chebyshev's inequality gives, for every R, η > 0, volume-tending-to-zero of B_{g_t}(p_t, R) minus the set {q in chart : |x(q)/s − b_*| < η}. Bishop–Gromov bounds μ_t(B(p_t, R)) ≤ ω_N R^N. Now pass to the pointed measured Gromov–Hausdorff limit (Y_t, g_t, p_t, μ_t) → (Z, d_Z, p, μ). The compactness, curvature, and singularity theory of Cheeger–Colding (Cheeger–Colding I, https://doi.org/10.4310/jdg/1214459974; Colding, https://annals.math.princeton.edu/articles/13099) and Cheeger–Naber (https://doi.org/10.4007/annals.2015.182.3.5) gives: Z complete proper; μ is the N-dimensional Riemannian volume on the dense regular set G; S = Z  G is closed with dim_H S ≤ N − 4 and μ(B_Z(q, r)) ≤ ω_N r^N; compact subsets of G admit smooth convergence identities with regions of Y_t; complex structures converge smoothly on G to a parallel integrable limit I_∞. Because every compact smooth region lies in a fixed-radius ball about p_t with curvature energy ≤ ε_t ω_N R^N → 0, the limit metric has Rm_g = 0 on G. Flat regular set. Then the embedding maps Φ_t are holomorphic between Kähler manifolds, hence harmonic; with their energy on regular compact sets tending to zero by the K_t bound, Lemma (Vanishing Energy, next section) forces dΦ_t → 0; every local constant is Φ(o), so Φ_t → Φ(o) uniformly on compact subsets of G. From the embedding and the log equation Σ_{i=0}^d log |z_i| = −s, the corresponding regions of Y_t lie in the coordinate chart with each x_j pluriharmonic and bounded; interior harmonic estimates improve L^2 convergence to C^∞_loc: x_j/s → (b_*)_j. Since b_* ∈ closure(B_1) has ρ = 1 there, every compact subset of G has a neighborhood where U_j = x_j for all small t; the all-radius bounds then apply to the actual logarithmic coordinates, ready for the cylinder identification.
 
 Exercise: Lemma (Vanishing Energy). Why do zero-energy harmonic maps on a smooth ball collapse to constants?
 Lemma (Vanishing Energy). Let h_ν be metrics converging smoothly on a fixed smooth domain D of real dimension N, and let ψ_ν: (D, h_ν) → P be smooth harmonic maps to a fixed compact Riemannian manifold P. If their energies tend to zero on every relatively compact subdomain, then sup_K |dψ_ν| → 0 for every K << D, and after extraction they converge smoothly to a constant on each connected relatively compact domain. Proof sketch. Smooth harmonic maps are stationary, so the stress tensor |dψ|^2 h − 2 ψ^* g_P has zero divergence. The energy functional Θ_ν(q, r) = r^{2−N} ∫_{B_{h_ν}(q, r)} |dψ_ν|^2 dV is nondecreasing in r up to a bounded factor (test the stress tensor against radial vector fields, use ∇^2(r^2/2) = h + O(r^2) in geodesic balls, discard a nonnegative term, integrate). If derivatives were unbounded, maximize dist_{h_ν}(q, ∂D') |dψ_ν(q)| over a fixed intermediate domain D'; at a maximizer q_ν, put L_ν = |dψ_ν(q_ν)| → ∞. On the ball B_{h_ν}(q_ν, L_ν^{−1}) we have |dψ_ν| ≤ 2 L_ν. Rescale h_ν by L_ν^2; the rescaled maps v_ν on unit balls have |dv_ν(0)| = 1 and |dv_ν| ≤ 2, and the rescaled metrics converge smoothly to a Euclidean metric. Harmonic-map Bochner inequality then gives Δ|dv_ν|^2 ≥ −C |dv_ν|^2; the mean-value inequality with uniform constants gives 1 ≤ C ∫_{B(0,1)} |dv_ν|^2 = C L_ν^{N−2} ∫_{B_{h_ν}(q_ν, L_ν^{−1})} |dψ_ν|^2 dV = C Θ_ν(q_ν, L_ν^{−1}) → 0, contradiction. Therefore the derivatives are uniformly bounded on interior domains, and the same Bochner + mean-value argument on fixed smaller balls gives sup_K |dψ_ν|^2 ≤ C E_ν → 0. Elliptic estimates give C^∞ convergence to a constant on connected domains. Reference for stationary harmonic maps: Lin 1999, https://annals.math.princeton.edu/articles/12464.
@@ -102,22 +102,22 @@ Lemma (Capacity Extension continued). We use the local harmonic gradient estimat
 Exercise: Path distance, F extends to Z, and the completion is a bi-Lipschitz homeomorphism to H.
 Lemma (Regular Paths). The intrinsic path distance on G equals the restriction of d_Z. In particular F extends to a Lipschitz map Z → H. Proof: fix a, b ∈ G and small balls A, B about them with closures in G. Minimizing segments between A, B stay in a compact ball K (segment inequality). Cover K's singular portion by small balls B(q_i, r_i), disjoint from A ∪ B after doubling, with Σ r_i^{N−1} small. Put f = Σ r_i^{−1} 1_{B(q_i, 2 r_i)}. The volume upper bound μ(B(q_i, 2 r_i)) ≤ ω_N (2 r_i)^N gives ∫ f dμ ≤ C Σ r_i^{N−1} small. The segment inequality for nonnegative integrable f on Ric ≥ 0 limits (Cheeger–Colding III, Theorem 2.15, https://doi.org/10.4310/jdg/1214342146) says the average over endpoints in A, B, of inf_γ ∫_γ f over minimizing segments γ, is bounded by C √(∫ f dμ) √(diam(A) + diam(B)) → can be made small. So some minimizing segment has ∫_γ f < 1. A segment that meets B(q_i, r_i) with endpoints outside B(q_i, 2 r_i) spends length ≥ r_i in the doubled ball, giving ∫_γ f ≥ 1; so the chosen segment misses S. Short paths inside A, B join it to a, b. Letting radii → 0 gives dist_G(a, b) ≤ d_Z(a, b); the reverse holds for every path. Lipschitz from the inverse derivative bound; density of G in Z and completeness of H extend F to F: Z → H.
 
-Lemma (Full Cylinder). The extension F: Z → H is a bi-Lipschitz homeomorphism. Proof: set E = F(S); dim_H E ≤ N − 4; we will prove closedness only after constructing the inverse on all of H. Local openness of F on G gives a starting point of F(G) \ E; lift any finite polygonal path in H \ E through the local inverse on G. A finite-parameter end cannot appear: the inverse-derivative bound makes the lift Cauchy, and Z is complete; its image is the prescribed path end, outside F(S), so in G where the local inverse continues. Injectivity on G gives uniqueness. The dimension-avoidance facts below guarantee H \ E ⊂ F(G) and give a Lipschitz bound d_Z(K(a), K(b)) ≤ C d_H(a, b) for a, b ∈ H \ E. Both the domain H \ E and the image of K are dense: domain by dim_H E < N, image by local openness at each regular neighborhood and density of G. Completing K gives Lipschitz K: H → Z. Compositions F K and K F equal the identity on dense subsets, hence everywhere. So F is a bi-Lipschitz homeomorphism. Also E is closed, since S is closed. So the metric completion of the regular set is the whole flat cylinder H.
+Lemma (Full Cylinder). The extension F: Z → H is a bi-Lipschitz homeomorphism. Proof: set E = F(S); dim_H E ≤ N − 4; we will prove closedness only after constructing the inverse on all of H. Local openness of F on G gives a starting point of F(G)  E; lift any finite polygonal path in H  E through the local inverse on G. A finite-parameter end cannot appear: the inverse-derivative bound makes the lift Cauchy, and Z is complete; its image is the prescribed path end, outside F(S), so in G where the local inverse continues. Injectivity on G gives uniqueness. The dimension-avoidance facts below guarantee H  E ⊂ F(G) and give a Lipschitz bound d_Z(K(a), K(b)) ≤ C d_H(a, b) for a, b ∈ H  E. Both the domain H  E and the image of K are dense: domain by dim_H E < N, image by local openness at each regular neighborhood and density of G. Completing K gives Lipschitz K: H → Z. Compositions F K and K F equal the identity on dense subsets, hence everywhere. So F is a bi-Lipschitz homeomorphism. Also E is closed, since S is closed. So the metric completion of the regular set is the whole flat cylinder H.
 
 Exercise: Avoidance facts in dimension codimension ≥ 1 and ≥ 2.
 Two elementary consequences of small dimension of a removed set E, used repeatedly. Fact 1. In a Euclidean ball or flat cylinder, if dim_H E < N − 1, any two points outside E can be joined outside E by polygonal paths of length arbitrarily close to their distance. Proof: put intermediate vertices arbitrarily near a shortest segment, on the correct side of the cones of rays through E from the endpoints. Each bad cone is a countable union of Lipschitz images of E × an interval, dimension ≤ dim_H E + 1 < N. Work in Euclidean lifts for the cylinder. Fact 2. If E is closed and dim_H E < N − 2, then the complement of E in a convex Euclidean ball is simply connected. Proof: a loop has positive distance from E and can be approximated, through a homotopy in the complement, by a polygonal loop. Cone the finitely many edges to a generic point of the ball. For each edge, the "bad" cone centers lie in countably many bounded Lipschitz images of E × edge × a real interval (writing intersection as (1 − t) x + t c solves for t, c; one restricts to t bounded away from 0). Dimension ≤ dim_H E + 2 < N; a center outside all bad sets gives the required filling. Applied to E = F(S) ⊂ H with dim_H E ≤ N − 4: both facts apply. They supply the two structural facts used in proving that F is bi-Lipschitz and that its metric extends across E.
 
 Exercise: Removing the apparent singularities E = F(S) by bounded holomorphic coframe extension.
-Transfer the metric on G to H \ E; it is a flat Kähler metric g_∞, uniformly bounded above and below by positive multiples of the standard γ on H (from the previous |dF| and |(dF)^{−1}| bounds). Consider a small convex complex coordinate ball D ⊂ H. By Fact 2 (with dim_H E ≤ N − 4), D \ E is simply connected. Hence a unitary parallel holomorphic coframe exists there: θ_a = Σ_j A_{a j}(w) dw_j, with g_∞ = Σ_a |θ_a|^2 on the complement of E; the coefficient matrix A and its inverse are uniformly bounded, by the two-sided metric bounds. Each coefficient extends holomorphically across E: direct removal. Cover the compact exceptional set E ∩ D' (for D' << D) by balls with Σ r_i^{N−1} → 0, take finite subcovers, use Lipschitz cutoffs χ_k vanishing near E. Then ||dχ_k||_{L^1} → 0. Testing the CR equation for the coefficient against χ_k times a smooth test form and passing to the limit gives the distributional CR equation on the smaller ball D', hence holomorphicity by Weyl's lemma. Nonsingularity of the extended matrix at E follows from continuity + the uniform lower bound on the inverse. The identities expressing that the coframe is closed, and that the resulting metric is flat and Kähler, extend from the dense complement by continuity. So g_∞ extends as a smooth flat Kähler metric across E, and extensions on overlapping balls agree by density. Consequence: the metric-distance completion already obtained has the smooth flat-Kähler-metric distance of the extension; the apparent singularities are removable.
+Transfer the metric on G to H  E; it is a flat Kähler metric g_∞, uniformly bounded above and below by positive multiples of the standard γ on H (from the previous |dF| and |(dF)^{−1}| bounds). Consider a small convex complex coordinate ball D ⊂ H. By Fact 2 (with dim_H E ≤ N − 4), D  E is simply connected. Hence a unitary parallel holomorphic coframe exists there: θ_a = Σ_j A_{a j}(w) dw_j, with g_∞ = Σ_a |θ_a|^2 on the complement of E; the coefficient matrix A and its inverse are uniformly bounded, by the two-sided metric bounds. Each coefficient extends holomorphically across E: direct removal. Cover the compact exceptional set E ∩ D' (for D' << D) by balls with Σ r_i^{N−1} → 0, take finite subcovers, use Lipschitz cutoffs χ_k vanishing near E. Then ||dχ_k||_{L^1} → 0. Testing the CR equation for the coefficient against χ_k times a smooth test form and passing to the limit gives the distributional CR equation on the smaller ball D', hence holomorphicity by Weyl's lemma. Nonsingularity of the extended matrix at E follows from continuity + the uniform lower bound on the inverse. The identities expressing that the coframe is closed, and that the resulting metric is flat and Kähler, extend from the dense complement by continuity. So g_∞ extends as a smooth flat Kähler metric across E, and extensions on overlapping balls agree by density. Consequence: the metric-distance completion already obtained has the smooth flat-Kähler-metric distance of the extension; the apparent singularities are removable.
 
 Exercise: Distance equality, S = ∅, and translation invariance.
-Distance equality. Any piecewise smooth path in a coordinate ball can be approximated in length by paths avoiding E: polygonal approximation + Fact 1 + uniform continuity of the smooth metric. Subdivision gives the assertion for arbitrary paths. Conversely any path off E has its original length. Lemma (Regular Paths) plus completion identifies the two distances. S = ∅. The extended metric is smooth flat at every point of Z, so every point has Euclidean tangent cones, so every point is regular in the original Einstein-limit sense. Hence the singular set S = Z \ G is empty. The smooth convergence theorem for Einstein metrics then applies at every point of the cylinder, including those not initially regular. Translation invariance. Lift the extended metric to ℂ^d, the universal cover of H; flatness + simple connectivity give a global parallel unitary holomorphic coframe. Its coefficients in the w_j coordinates are bounded entire functions, by the uniform metric bounds; Liouville makes them constant. So g_∞ is translation invariant. Together with the constant limits of ω_∞ and Ω_∞ (from the next exercise), the limit is exactly the standard flat Calabi–Yau structure on the cylinder H.
+Distance equality. Any piecewise smooth path in a coordinate ball can be approximated in length by paths avoiding E: polygonal approximation + Fact 1 + uniform continuity of the smooth metric. Subdivision gives the assertion for arbitrary paths. Conversely any path off E has its original length. Lemma (Regular Paths) plus completion identifies the two distances. S = ∅. The extended metric is smooth flat at every point of Z, so every point has Euclidean tangent cones, so every point is regular in the original Einstein-limit sense. Hence the singular set S = Z  G is empty. The smooth convergence theorem for Einstein metrics then applies at every point of the cylinder, including those not initially regular. Translation invariance. Lift the extended metric to ℂ^d, the universal cover of H; flatness + simple connectivity give a global parallel unitary holomorphic coframe. Its coefficients in the w_j coordinates are bounded entire functions, by the uniform metric bounds; Liouville makes them constant. So g_∞ is translation invariant. Together with the constant limits of ω_∞ and Ω_∞ (from the next exercise), the limit is exactly the standard flat Calabi–Yau structure on the cylinder H.
 
 Exercise: Lemma (Actual Logarithmic Convergence). Convergence in the *original* logarithmic coordinates.
 We have Z ≅ H with a smooth flat metric, but we need convergence in the original coordinates (the actual arg z_j), so we can perturb a torus in the labeled angular coordinate torus. Set L_t(q) = (x_j(q) − x_j(p_t) + i arg z_j(q))_{j=1}^d ∈ H, defined on the part of Y_t in the crossing chart; Q_r = {w ∈ H : |Re w| < r}. The real shift does not change the angular coordinates. Normalize η_t by a positive scalar to get a holomorphic volume form Ω_t with the Calabi–Yau normalization for g_t. Lemma (Actual Logarithmic Convergence). For each fixed r > 0, the chart L_t contains Q_r in its image for all small t along the selected sub; the inverse Ψ_t: Q_r → Y_t satisfies Ψ_t^* g_t → g_∞, Ψ_t^* ω_t → ω_∞, Ψ_t^* Ω_t → Ω_∞ in C^∞_loc(Q_r), with constant tensors and Ω_∞ ≠ 0; and the angular coordinates of Ψ_t are the original arg z_j. Proof sketch. The inclusion Q_r ⊂ L_t(chart) is immediate from x(p_t)/s ∈ B_0 << Σ: a bounded real shift leaves all z_0, ..., z_d exponentially small. To identify inverses with the abstract GH maps: take smooth local identifications compatible with pointed GH convergence, available at every point of Z since S = ∅. The embedding-map and harmonic-coordinate arguments of Section 4 apply on each neighborhood, giving smooth local subsequential limits of L_t. On the intersection with the old regular set G, these limits equal F, because old and new identifications send each point to g_t-distance-tending-to-zero points, and interior harmonic estimates bound derivatives of the shifted logarithms and angular maps. G is dense in Z, so every local limit equals F throughout its neighborhood. The derivative of F is nonsingular; the inverse function theorem gives smooth convergence of local inverses on smaller image neighborhoods. Those inverses agree with Ψ_t because the actual logarithmic chart is injective. A finite cover of each compact subset of Q_r proves the asserted convergence of metrics and Kähler forms. This argument uses the actual arg z_j at every stage; no change of angular marking. Finally, Ω_t = (positive scalar) × f(z) dw_1 ∧ ... ∧ dw_d in these coordinates; the scalar converges to a positive limit by the coordinate-density; z → o exponentially; f(z) → f(o) with all derivatives in the w coordinates. This gives nonzero constant Ω_∞. Same for ω_t; g_∞ is already translation invariant by the preceding exercise.
 
 Exercise: The angular torus T_0 in the flat cylinder is special Lagrangian for the limit.
-Fix a tube Q_{2r} ⊂ Q_r' (with r' > r), and use Lemma (Actual Logarithmic Convergence) to regard the CY data (g_t, ω_t, Ω_t) as tensors on one fixed domain Q_{2r}. The polarizing line bundle on Y_t is topologically trivial on a sufficiently small contractible coordinate neighborhood of the maximal crossing o (its restriction to that chart has a smooth trivialization of the underlying line bundle, regardless of amplification). Therefore ω_t is exact as a 2-form on Q_{2r}: ω_t = d β_t for a 1-form β_t. The constant-form limit ω_∞ has zero period on every angular coordinate 2-torus: in the coordinates w_j = x_j + i y_j, ω_∞ = Σ_j dx_j ∧ dy_j, so ω_∞(∂_{y_i}, ∂_{y_j}) = 0, and every coefficient of ω_∞ has to vanish when evaluated on angular tangent vectors. Constant coefficients then force ω_∞|_{\{Re w = 0\}} = 0 identically. Hence T_0 := {Re w = 0} ⊂ Q_{2r} is Lagrangian for ω_∞. Moreover the constant nonzero form Ω_∞ has constant phase on T_0, because T_0 = {Re w = 0} has tangent vectors all of the form Σ c_j ∂_{y_j}, on which Ω_∞ evaluates to a single fixed complex constant independent of the point. So T_0 is special Lagrangian for the limit structure. We now perturb T_0 to a torus that is special Lagrangian for the actual (t-dependent) structure.
+Fix a tube Q_{2r} ⊂ Q_r' (with r' > r), and use Lemma (Actual Logarithmic Convergence) to regard the CY data (g_t, ω_t, Ω_t) as tensors on one fixed domain Q_{2r}. The polarizing line bundle on Y_t is topologically trivial on a sufficiently small contractible coordinate neighborhood of the maximal crossing o (its restriction to that chart has a smooth trivialization of the underlying line bundle, regardless of amplification). Therefore ω_t is exact as a 2-form on Q_{2r}: ω_t = d β_t for a 1-form β_t. The constant-form limit ω_∞ has zero period on every angular coordinate 2-torus: in the coordinates w_j = x_j + i y_j, ω_∞ = Σ_j dx_j ∧ dy_j, so ω_∞(∂_{y_i}, ∂_{y_j}) = 0, and every coefficient of ω_∞ has to vanish when evaluated on angular tangent vectors. Constant coefficients then force ω_∞|_{{Re w = 0}} = 0 identically. Hence T_0 := {Re w = 0} ⊂ Q_{2r} is Lagrangian for ω_∞. Moreover the constant nonzero form Ω_∞ has constant phase on T_0, because T_0 = {Re w = 0} has tangent vectors all of the form Σ c_j ∂_{y_j}, on which Ω_∞ evaluates to a single fixed complex constant independent of the point. So T_0 is special Lagrangian for the limit structure. We now perturb T_0 to a torus that is special Lagrangian for the actual (t-dependent) structure.
 
 Exercise: Zhang–McLean persistence. From the flat torus T_0 to a smooth SL torus in Y_t.
 Persistence of Calabi–Yau torus tubes: Zhang 2017, Section 4, Lemmas 4.3–4.5 (https://arxiv.org/abs/0911.1028v2). Its linearization is McLean's special Lagrangian deformation operator (McLean 1998, https://doi.org/10.4310/CAG.1998.v6.n4.a4). Setup. Identify normal vector fields along T_0 with one-forms α via contraction with ω_∞; use a fixed normal graph chart ι_α: T_0 → Q_r. Pick a large integer k and 0 < β < 1. Source space:
@@ -130,7 +130,7 @@ Exercise: Full torus theorem for every sufficiently small nonzero t, not just al
 The hidden quantifier. The compactness construction chose an arbitrary sequence of nonzero parameters t_ν → 0, extracted a subsequence and produced centers p_{t_ν} with noncollapsed unit balls and small curvature energy on every ball. The subsequence ambiguity came from compactness (Bishop–Gromov, Cheeger–Colding, singularity theory). Lemma (Actual Logarithmic Convergence) then gave smooth convergence of Ψ_t^* (g_t, ω_t, Ω_t) to constant tensors along that sub. The 3-Jordan-block structure of M and the nonzero residue made the resulting limiting structure the standard flat Calabi–Yau cylinder. All the intermediate arguments were uniform in t. So the construction produced SL tori on a tail of the subsequence, starting from any t_ν → 0. Hence no failure is possible at any point of a sufficiently small punctured disk. Formal version: Suppose for contradiction that the conclusion failed at some sequence t_ν → 0: choose t_ν so that no Y_{t_ν} contains a torus with the stated properties. The compactness construction gives such tori on a tail of a subsequence, a contradiction. So the assertion holds for every sufficiently small nonzero t, closing Theorem (Torus). This is the pattern for "universal quantifier from compactness": the proof iteratively extracts so many subsequences that the limit is forced; therefore a counterexample cannot exist.
 
 Exercise: Rotation setup. What are I, J, K, and the class formulas on the chosen fiber?
-Fix λ ∈ ℝ \ ℚ. Use the family of Proposition (Family) and go to a fiber Y = Y_t along the vertical ray τ = λ + i y, y >> 0, which contains the SL torus T of Theorem (Torus) (isotopic to a coordinate angle torus in the crossing chart z_0 z_1 ... z_{2n} = t). Write the present complex structure as I. Choose a hyperkähler triple (I, J, K) for the polarized Ricci-flat metric g, with IJ = K, so that (rotation–normalization):
+Fix λ ∈ ℝ  ℚ. Use the family of Proposition (Family) and go to a fiber Y = Y_t along the vertical ray τ = λ + i y, y >> 0, which contains the SL torus T of Theorem (Torus) (isotopic to a coordinate angle torus in the crossing chart z_0 z_1 ... z_{2n} = t). Write the present complex structure as I. Choose a hyperkähler triple (I, J, K) for the polarized Ricci-flat metric g, with IJ = K, so that (rotation–normalization):
     ω_J + i ω_K = c σ_t,     c > 0,     [ω_I] ∈ ℝ_{>0} h.
 The positive scalar normalizes the length of σ_t; its phase fixes the ordered pair (J, K). Since [σ_t] = p(τ) = f + τ v − (a/2) τ^2 e:
     [ω_K] = c y (v − a λ e),      q(e, [ω_J]) = c > 0.