diff --git a/cohomological_transfer_equivariant_v2.txt b/cohomological_transfer_equivariant_v2.txt
index b9e7a96..a82c1d3 100644
--- a/cohomological_transfer_equivariant_v2.txt
+++ b/cohomological_transfer_equivariant_v2.txt
@@ -269,10 +269,10 @@ Let \(X\) be smooth connected projective, let \(L=-K_X\) have a smooth semiposit
 \]
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 The polynomial in Proposition~\ref{prop:index} is not zero at zero, so a fixed cohomological degree has nonzero invariant cohomology along an unbounded sequence. Average a K\"ahler form and the metric weights over the maximal compact torus; write \(\theta\) for the resulting curvature of \(L\). The smooth semipositive hard Lefschetz theorem \cite{DPS2001}, applied to \(mL=K_X+(m+1)L\), sends a harmonic \((n,q)\)-form \(u\) with values in \((m+1)L\) to a holomorphic form of type \((n-q,0)\), via a complex-linear star in the form indices. The construction is equivariant. For \(q=0\) this already produces invariant sections of \(mL\). For \(q=n\), the conjugate of the \((0,n)\)-valuation of \(u\) is a section of \((m+1)L\). If \(q>0\) and \(r=\max\rank\theta=n\), Bochner--Kodaira makes the holomorphic form zero on a nonempty open set, hence everywhere.
 
-In the remaining case, one extracts from the unbounded family \(\{u_m\}\) a determinant subsheaf of a positive exterior power of \(\Omega_X^{n-q}\). Fix a rational frame of \(L\) and divide each form by the corresponding power of that frame. The span over \(\C(X)\) has some rank \(e\); the specific frame only rescales each vector by a nonzero rational function, so the span is intrinsic. Saturate its top exterior power and call the resulting reflexive rank-one sheaf \(M\). Choose finitely many of the original forms as a generic basis, and wedge each unbounded member with \(e-1\) fixed ones. The resulting nonzero invariant wedge has twist degree \(m+1\) plus a fixed summand; its class is a section of \(M+a_iL\) with \(a_i\to\infty\).
+In the remaining case, one extracts from the unbounded family \(\{u_m\}\) a determinant subsheaf of a positive exterior power of \(\Omega_X^{n-q}\). Fix a rational frame of \(L\) and divide each form by the corresponding power of that frame. The span over \(\C(X)\) has some rank \(e\); the specific frame only rescales each vector by a nonzero rational function, so the span is intrinsic. Saturate its top exterior power and call the resulting reflexive rank-one sheaf \(M\). Choose finitely many of the original forms as a generic basis, and wedge each unbounded member with \(e-1\) fixed ones. In any fixed basis of the section space, at least one coordinate of the unbounded member is nonzero for unbounded \(m\); otherwise the entire family would vanish on a dense open set, and each \(u_m\) would be zero. A wedge of that coordinate form with \(e-1\) fixed basis forms is therefore nonzero. It is invariant, because each factor is. Its twist degree is \(m+1\) plus a fixed summand; its class is a section of \(M+a_iL\) with \(a_i\to\infty\).
 \end{proof}
 
 \paragraph{Consequence: natural invariant nonvanishing.}
@@ -281,7 +281,7 @@ In the remaining case, one extracts from the unbounded family \(\{u_m\}\) a dete
 Let \(X\) be smooth connected projective over \(\C\) with smoothly semipositive \(L=-K_X\). Suppose an algebraic torus \(T\) acts on \(X\) and \(\chi(X,\OO_X)\ne0\). Then \(H^0(X,mL)^T\ne0\) for some integer \(m>0\), with the natural linearization.
 \end{corollary}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Every element of the connected torus is homotopic to the identity, so its action on singular cohomology is trivial. By Hodge decomposition the action on every \(H^q(X,\OO_X)\) is trivial, and the invariant index at zero equals \(\chi(X,\OO_X)\ne0\). Lemma~\ref{lem:forms} supplies either the required section directly, or invariant sections of \(M+a_iL\) with \(a_i\to\infty\) and a nonzero map \(M\to(\Omega_X^1)^{\otimes p}\) for some \(p>0\). The cotangent-tensor theorem \cite[Theorem 4.1]{LMPTX2023} applied to the smooth zero-boundary pair with nef \(-K_X\) says that \(-M\) is pseudoeffective. Apply Theorem~\ref{thm:transfer} with \(P=L\) naturally linearized.
 \end{proof}
 
@@ -299,7 +299,7 @@ Let \(X\) be smooth connected projective of dimension \(n\), let \(L=-K_X\) have
 for every K\"ahler form \(\omega\). Each component has zero intersection separately, and \(\theta^r\) vanishes on its smooth locus.
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 For every K\"ahler class \(G\), Yau's prescribed-Ricci theorem \cite{Yau1978} gives a K\"ahler metric in \(G\) with Ricci form \(\theta\). K\"ahler curvature symmetry identifies the mean curvature of \(T_X\) with the Ricci form; the induced mean curvature of \(\bigwedge^e\Omega_X^p\) is nonpositive. If \(\deg_G M>0\), a scalar Laplace equation gives \(M\) a metric of positive constant mean curvature; its inverse tensor the cotangent construction then has negative mean curvature, contradicting the Bochner formula for its nonzero section. So \(c_1(M)G^{n-1}\le0\). Put \(G=[\theta]+t[\omega]\), divide by \(t^{n-1-r}\), and let \(t\downarrow0\). Since \(\theta^{r+1}=0\), \(c_1(M)[\theta]^r[\omega]^{n-1-r}\le0\). The same intersection for \(M+aL\) is nonnegative by effectivity and smooth semipositivity, and the contribution from \(aL\) is zero; it is therefore zero. Each component has zero intersection separately, and the nonnegative form \(\theta^r\wedge\omega^{n-r-1}\) vanishes on its smooth locus.
 \end{proof}
 
@@ -373,7 +373,7 @@ Let \(P\) be a compact simply connected nonflat K\"ahler factor, irreducible in
 \]
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 For a holomorphic covariant tensor, the contracted curvature on its bundle is the sum, over tensor slots, of the negative Ricci operator, so it is nonpositive. The Bochner identity on compact \(P\) makes the tensor parallel. The compact holonomy group \(H\) is connected and acts irreducibly on the complex tangent space. Consider the determinant character. If it is trivial, parallel transport of a canonical vector gives a global parallel volume form. If it is nontrivial, its differential is a nonzero central element of the Lie algebra, and by complex irreducibility and Schur's lemma it acts on the tangent space as a scalar \(ia\,\mathrm{Id}\), so on the \(m\)-fold tensor power as a nonzero scalar \(-mia\). The tensor power then has no \(H\)-fixed vector, so no nonzero holomorphic tensor of positive degree.
 \end{proof}
 
@@ -398,7 +398,7 @@ Let a torus \(T\) act on a smooth connected projective \(X\), let \(U\) be a \(T
 The trivial torus is allowed, with \(S=X\) and \(\mu=0\).
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 The first bound follows from positivity of the normal weights at \(S\): minus one normal weight is subtracted from each section weight. For ample \(U\), generation of a high enough power controls the fixed-component weights; for nef, apply the result to \(bU+A\), with \(A\) ample, and let \(b\to\infty\).
 
 For the full character identity, evaluate the equivariant fixed-point formula \cite{AtiyahSegal1968,AtiyahSinger1968} at \(a\lambda(t)\) as \(t\to\infty\), divide by \(t^{m\langle\mu,\lambda\rangle}\), and keep \(a\in T\) generic. The inverse normal Euler factors tend to one at \(S\) and to zero at every other fixed component. Truncating by ordinary degree gives the limit \(a^{m\mu}\chi(S,mU|_S)\). Since the equivariant index is a finite Laurent polynomial, existence of this limit excludes higher \(\lambda\)-weights, and the whole coefficient at the weight hyperplane is the single character \(m\mu\).
@@ -410,7 +410,7 @@ For the full character identity, evaluate the equivariant fixed-point formula \c
 Let \(X\) be smooth connected projective with smoothly semipositive \(L=-K_X\) and with \(H^0(X,\Omega_X^p)=0\) for all \(p>0\). Let an algebraic torus \(T\) act with a linearized ample bundle. For every generic integral cocharacter \(\lambda\) with \(X^\lambda=X^T\), let \(S\) be its source and let \(\mu\) be the full \(T\)-character of \(L|_S\). Then for some \(k_\mu>0\) there is a nonzero section of \(k_\mu L\) of character \(k_\mu\mu\). In particular a positive power of \(L\) has a nonzero invariant section, for the natural linearization.
 \end{theorem}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Take a generic integral cocharacter \(\lambda\) with \(X^\lambda=X^T\), and let \(S\) be its source. The projection of the attracting stratum to \(S\) is dominant, so \(S\) inherits the vanishing \(H^0(S,\Omega_S^p)=0\), and consequently \(\chi(S,\OO_S)=1\).
 
 By Lemma~\ref{lem:source-character} the polynomial in \(m\) (a specialization of the full character identity) has value one at zero. Some fixed cohomological degree thus has nonzero weight-\(m\mu\) classes for unbounded \(m\). Equivariant hard Lefschetz produces nonzero eigenforms
@@ -439,7 +439,7 @@ This corollary is a version of Corollary~\ref{thm:main} that requires no hypothe
 Let \(X\dashrightarrow Y\) be a dominant rational map of integral projective varieties over \(\C\). There are a smooth projective birational model \(Y'\) of \(Y\) and an integral projective flat family \(X_0\to Y'\) with a birational morphism to the graph of the original map. Every irreducible component of every fibre has dimension \(\dim X-\dim Y\). The normalization \(V\) of \(X_0\) is equidimensional over \(Y'\); in particular every vertical prime divisor of \(V\) maps onto a prime divisor of \(Y'\). If \(\C(Y)\) is algebraically closed in \(\C(X)\), the morphism from \(V\), and from any resolution of \(V\), to \(Y'\) has connected fibres. If a torus acts and the rational map is invariant, all these constructions and a resolution can be made equivariant, with trivial action on the base.
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Embed the projective graph over \(Y\) in \(\PP^N\times Y\). Over the flat open set the fibres define a map to the projective Hilbert scheme \cite{Grothendieck1961Hilbert}. Resolve the closure of its graph, pull back the universal family, and normalize. Flatness over an integral base embeds each coordinate ring into its generic localization; the generic fibre is integral, so the coordinate rings have neither nilpotents nor zero divisors, and the pulled-back family is integral. Hilbert-polynomial constancy bounds fibre dimension above; a local height bound bounds below. Purity of fibre dimension and equidimensionality follow, and every vertical prime divisor maps onto a prime divisor of \(Y'\) because a divisor over a codimension-\(\ge2\) locus would have dimension too small.
 
 For connectedness, apply Stein factorization; the finite intermediate base has function field the algebraic closure of \(\C(Y')\) in \(\C(X)\), which is \(\C(Y')\) under the field hypothesis, so the finite birational map is an isomorphism. The same reasoning applies after resolution. Equivariance is immediate from the action on the graph.
@@ -465,7 +465,7 @@ In the invariant case, the same condition with \(H^0(-)^T\) and \(P\) in place o
 whose squared fibre-maximum norms define a positive continuous metric with plurisubharmonic weights on \(Q\). In the invariant case \(Q\) has trivial linearization and the injection is equivariant.
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 In the invariant case, average a smooth semipositive metric on \(L\) over the maximal compact torus; the chosen linearization differs from the natural one by a character, which is unitary on this compact torus, so the same averaged metric is invariant for both actions.
 
 The rank one at \(k=0\) forces generic geometric connectivity. The constant functions on the fibre, multiplied by \(s_E\), inject into that adjoint space; in the invariant case the connected torus acts trivially on the finite Stein base, so constants are invariant; hence the dimension is one, and smooth general fibres are connected.
@@ -501,7 +501,7 @@ Let \(g:V\to Y\) be a morphism of smooth projective complex varieties, let \(A\)
 \]
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Fujino--Matsumura injectivity \cite{FM2016} says that on a compact K\"ahler manifold, multiplication by a section of a smooth semipositive line is injective on canonical cohomology with a multiplier ideal coefficient, provided the singular curvature dominates a positive multiple of the multiplying line's curvature. Apply this with the multiplying line \(g^*(mA)\); the comparison constant is \(\epsilon/m\), and the ideal is \(\OO_V\).
 
 We reduce to base cohomology by induction on \(\dim Y\). Choose \(m\) large for Serre vanishing on all direct images \(R^jg_*\OO_V(K_V+N)\). A general smooth divisor \(C\in|mA|\) contains no associated point of these sheaves; its restricted coefficient metric has trivial multiplier ideal. The long direct-image sequence of
@@ -517,7 +517,7 @@ breaks into short exact sequences, since multiplication by the cutting equation
 Let \(X\) be smooth connected projective over \(\C\), with \(L=-K_X\) smoothly semipositive. Suppose \(F_0\) is pseudoeffective and, for an unbounded \(S\subset\Z_{>0}\), there are effective integral divisors \(N_s\sim sL-F_0\) for \(s\in S\). Then \(H^0(X,mL)\ne0\) for some \(m>0\).
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Fix \(a<b\) in \(S\). The affine relations
 \[
  (s-a)N_b\sim(s-b)N_a+(b-a)N_s
@@ -576,7 +576,7 @@ The first ingredient is a regularised maximum that turns a continuous plurisubha
 Let \(Q\) be a line bundle on a smooth projective \(Y\) with a positive continuous Hermitian metric whose local weights are plurisubharmonic. For a fixed K\"ahler form \(\eta\), there are smooth metrics whose weights converge uniformly to these weights and whose curvatures are at least \(-\varepsilon_j\eta\), with \(\varepsilon_j\to0\).
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Write a weight as a smooth reference plus a global continuous function \(u\). For fixed \(\varepsilon>0\), add \(\varepsilon\eta\) to the background; on finitely many enlarged coordinate patches, choose smooth local potentials that dominate the background, convolve, and subtract. Take inner patches covering \(Y\). Add smooth bumps zero on inner patches and a small negative constant near the outer boundaries, with second derivatives smaller than the local margin. A regularised maximum of the active candidates is smooth; convexity, monotonicity, and translation equivariance of regularised maximum preserve the background lower bound and the transition rule for weights. Let the uniform error and then \(\varepsilon\) tend to zero. Equivalently, smooth the strictly positive metric on \(NQ+H\) with \(H\) positive, subtract the weight of \(H\), and divide by \(N\).
 \end{proof}
 
@@ -593,7 +593,7 @@ Then for every \(k\ge0\), \(q\ge0\), and \(p>0\),
 \]
 \end{theorem}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Fix \(k\). Choose a K\"ahler form \(\omega\) on \(V\) and a Dolbeault class \(c\in H^\ell(V,K_V+A)\) where \(A=D_V+kf^*Q\) and \(n=\dim V\). Lemma~\ref{lem:smoothing} gives product metrics \(h_j\to h\) uniformly on \(A\), with curvature at least \(\theta-o(1)\omega\). Let \(u_j\) be the harmonic \((n,\ell)\)-representative of \(c\) for \(h_j\). A fixed \(L^2\) bound comes from comparing to one fixed smooth closed representative and using uniform metric equivalence. Bochner--Kodaira gives
 \[
  \|(D'_j)^*u_j\|^2+\langle[\theta,\Lambda]u_j,u_j\rangle\le o(1)\|u_j\|^2.
@@ -645,7 +645,7 @@ Injection bounds the left side by the \(p=0\) summand; all other summands vanish
 In Theorem~\ref{thm:kernel}, replace the kernel hypothesis by the assumption that \(bD_V-f^*H\) is pseudoeffective for some \(b>0\) and some ample \(H\) on \(Y\). Then the same vanishing holds.
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Keep the uniformly convergent metrics. For \(\ell>0\), put
 \[
  S=i^{(n-\ell)^2}h\,v\wedge\bar v,\qquad T=S\wedge\omega^{\ell-1},
@@ -663,7 +663,7 @@ Pointwise positivity forces \(f^*\eta\wedge T=0\), so diagonalisation gives the
 Let \(\pi:V\to X\) be birational between smooth connected projective varieties, let \(L=-K_X\) be smoothly semipositive, put \(D_V=\pi^*L\), \(E=K_{V/X}=K_V+D_V\). Let \(f:V\to Y\) be surjective with \(Y\) smooth connected projective. Suppose a continuously semipositive line \(Q\) on \(Y\) and an injection \(f^*Q\to bD_V\), \(b>0\), satisfy either vanishing criterion above. Then \(H^0(X,kbL)\ne0\) for some \(k>0\). If a torus acts, trivially on \(Y\) and \(Q\), and all maps and the injection are equivariant, the section can be taken invariant.
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Let \(\mathcal G=f_*\OO_V(E)\) or its invariant summand. Vanishing gives \(h^0(Y,\mathcal G\otimes Q^k)=\chi(Y,\mathcal G\otimes Q^k)\) for every \(k\ge0\). The right side is a polynomial in \(k\) by Riemann--Roch for the coherent sheaf \(\mathcal G\). At zero it equals one: \(\pi_*\OO_V(E)=\OO_X\), and the exceptional canonical section is invariant. The polynomial is nonzero at some positive integer \(k\). Multiply the resulting section of \(E+kf^*Q\) by the injection's \(k\)-th power; the effective exceptional divisor \(E\) changes neither sections nor characters, by normality of \(X\).
 \end{proof}
 
@@ -714,7 +714,7 @@ In the setting above, with \(D_V=\pi^*(-K_X)\), \(E=K_{V/X}\), and \(bD_V=G+f^*Q
 viewed as sheaves of rational functions on \(Y\), stabilize. The line \(Q\) is nef, and \(G\) contains no entire fibre. These conclusions give a second proof that the fibrewise maximum is positive.
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 On \(V\), form the sheaves \(\mathcal P_k=(f_*\OO_V(E+kG))^T\) and view them as sheaves of rational functions on \(Y\) by dividing by \(s_Es_G^k\). They have rank one, contain \(\OO_Y\), and increase with \(k\). For each prime divisor \(P\) of \(Y\), take a lift \(\Gamma\) omitted by \(G\); regularity of \(f^*a\,s_Es_G^k\) bounds the possible pole order independently of \(k\) by one divisor \(C\). Normality embeds all \(\mathcal P_k\) in \(\OO_Y(C)\), so Noetherianity gives a stable value \(\mathcal P_\infty\).
 
 Take a very ample \(H\) on \(Y\). Apply Theorem~\ref{thm:kernel} with trivial base twist to \((1+kb)D_V+lf^*H\), \(k,l\ge0\). Its curvature dominates the original, so its kernel is still vertical. Projection to invariants and the projection formula give
@@ -786,7 +786,7 @@ Let \(S\) be a smooth connected projective rationally connected variety with smo
 Let \(S\) be as above, allowing the trivial action. Let \(S_0\) be the attracting component of the action. A nonzero eigensection of a linearized line \(U\) has weight at most \(w_{S_0}(U)\). If \(U\) is nef, then \(w_C(U)\le w_{S_0}(U)\) for every fixed component, and the multiplicity of weight \(m w_{S_0}(U)\) in the virtual character \(\chi(S,mU)\) is \(\chi(S_0,mU|_{S_0})\), for \(m>0\). If \(U\) is nef, on a smooth equivariant resolution \(d:\widehat S\to S\) of the source retraction \(p:\widehat S\to S_0\) there is an equivariant nonzero bundle map \(p^*(U|_{S_0})\to d^*U\).
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 A linearized line on a projective variety with \(\C^*\)-action exists: the map to the Picard variety from \(\C^*\) is constant (since a torus has no nonconstant homomorphism to an abelian variety), and the extension of \(\C^*\) by the scalar \(\C^*\) is a split torus. The section-weight bound and the comparison of fixed weights are Lemma~\ref{lem:source-character} for \(T=\C^*\). The retraction is Bia\l ynicki-Birula \cite{BB1973}. The bundle map comes from transporting a vector along the orbit, dividing by \(t^{w_{S_0}(U)}\), and taking the limit; the result is a multiplicative rational identification. Ampleness of a generated power of \(U\) rules out poles; nefness follows by considering \(kU+H\) with \(H\) ample and letting \(k\to\infty\).
 \end{proof}
 
@@ -794,7 +794,7 @@ A linearized line on a projective variety with \(\C^*\)-action exists: the map t
 Let \(S\) be smooth connected projective rationally connected, with smoothly semipositive \(L=-K_S\), and fix a \(\C^*\)-action with the natural anticanonical linearization. If \(w=w_{S_0}(L)\), then some \(\ell>0\) has a nonzero section of \(\ell L\) of weight \(\ell w\).
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 The source \(S_0\) is rationally connected, being a smooth rational image of \(S\), so \(\chi(S_0,\OO_{S_0})=1\). By Lemma~\ref{lem:source} the highest-weight index is a nonzero polynomial in \(m\); for unbounded \(m\) a fixed \(H^q(S,mL)\) contains weight \(mw\). Equivariant hard Lefschetz gives twisted sections
 \[
  v_m\in H^0(S,\Omega_S^{\dim S-q}\otimes(m+1)L)=H^0(S,\textstyle\bigwedge^qT_S\otimes mL)
@@ -816,7 +816,7 @@ Euler interpolation on the base. The continuous semipositive metric makes \(Q\)
 Every algebraic torus acting on the rationally connected \(S\) of Proposition~\ref{prop:extremal} has a nonzero invariant section of some positive power of \(-K_S\).
 \end{corollary}
 
-\begin{proof}[Sketch]
+\begin{proof}
 For a generic integral one-parameter subgroup \(\lambda\), let \(C\) be its source, \(\gamma_C\) the fibre character of \(L\). Choose a full torus eigenvector in the extremal section space of Proposition~\ref{prop:extremal}. It cannot vanish on \(C\), since a nonzero leading conormal jet would strictly lower its \(\lambda\)-weight; its character is therefore \(\ell\gamma_C\). The finitely many characters of such sources have convex hull containing zero; otherwise a separating functional approximated by a generic rational one would be negative on all of them, whereas the source for that one-parameter subgroup has \(\gamma_C(\lambda)>0\), the sum of its positive normal weights. Use the effective torus quotient if the action has a kernel, and multiply rational convex combinations to produce a nonzero invariant section.
 \end{proof}
 
@@ -909,7 +909,7 @@ Let \(Y\) be a smooth connected projective variety of dimension \(d>0\). Suppose
 \]
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Construct a globally strictly positive metric with the same multiplier ideal. For sufficiently small \(\delta>0\) and large \(l\), a section of \(lR-A\) in \(\mathcal I(l(1+\delta)\psi)\) exists: on \(lR\) use weights \(l(1+\delta)\psi-l\delta\psi_0\), add \(j_lA-K_Y\) with \(j_l=\lceil C\delta l+C'\rceil\), and add \(\eta l\) times a logarithmic pole with \(\eta\) small, chosen independently of \(\delta,l\). Nadel vanishing in the strict-current form \cite[Theorem 5.11]{Demailly2011} then gives surjectivity to jets of order \(\lfloor\eta l\rfloor-d\) at the chosen point. Continuity of \(\psi\) there makes \(\mathcal I_l\) trivial near the point, and polar-coordinate integration gives the ideal \(\mathfrak m^{\lfloor\eta l\rfloor-d+1}\) for the pole. So \(h^0(Y,(lR+j_lA)\otimes\mathcal I_l)\ge cl^d\) for some \(c>0\).
 
 Remove \(j_l+1\) copies of a very ample \(A\). At each step choose a smooth member of \(|A|\) containing no associated centre of \(\OO_Y/\mathcal I_l\); division by its equation preserves membership in the ideal because the equation is a nonzerodivisor. The loss at a step is at most \(C''l^{d-1}\) uniformly for \(0<\delta\le1\), since \(pA-R\) can be assumed globally generated for some fixed \(p\), and Hilbert-polynomial bounds on the restriction control the drop. The total loss is \(O(\delta l^d)+O(l^{d-1})\). Taking \(\delta\) small and then \(l\) large proves the initial section existence.
@@ -927,7 +927,7 @@ H\"older with the integrability of \(\exp(V/\delta)\) gives \(\mathcal I(\rho)\s
 Let \(p:V\to Y\) be a proper holomorphic submersion with compact fibres and K\"ahler total space. Let \(D\) have a smooth semipositive Hermitian metric. On an open set where \(p_*(K_{V/Y}+D)\) is locally free and its fibres are section spaces, suppose the coefficient curvature dominates the pullback of a positive base form on an open patch meeting \(V_{y_0}\), and let \(u_0\in H^0(V_{y_0},K_{V_{y_0}}+D|_{V_{y_0}})\) be a section nonzero on that patch. The direct-image metric curvature, paired with \(u_0\) at \(y_0\), is strictly positive in every nonzero tangent direction. For a torus action over a trivial base with metrics preserved and the invariant summand of rank one, the quotient metric has the same strictness at \(y_0\).
 \end{lemma}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Restrict to a disc in a chosen base direction; let the fibre dimension be \(a\). If \(a=0\), the direct image is a sum of nonnegative terms, with a positive term on the patch. For \(a>0\), a holomorphic extending section with vanishing covariant derivative has a smooth absolute \((a,0)\)-form \(u\) satisfying \(D'u=dz\wedge\lambda\), \(\bar\partial u=dz\wedge\eta\). On the central fibre \(\lambda\) is orthogonal to holomorphic adjoint forms and \(\eta\wedge\omega\) is \(\bar\partial\)-exact. Replace \(u\) by \(u-dz\wedge\xi\) to make \(\lambda=0\) and \(\eta\) primitive; Hodge decomposition solves the two equations simultaneously, and Lefschetz linear algebra identifies the relevant \((a,1)\)-forms with \(\omega\wedge\xi\).
 
 With \(c_a=i^{a^2}\), the curvature pairing is
@@ -947,7 +947,7 @@ Let \(X,V,Y\) be smooth connected projective, \(\dim Y>0\), \(\pi:V\to X\) birat
 If on a nonempty patch in the smooth base-change locus, with \(t\) nonzero, the curvature of \(L'\) dominates the pullback of a positive base form, then \(\psi_R\) is smooth and strictly positive on a base coordinate ball.
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 The candidate for the divisor \(B_0\) is determined by regularity over the generic point of a base prime: \(b_P=\min_{Q\mapsto P}\lfloor\ord_Q(t)/\ord_Q(f^*P)\rfloor\). This is the regularity test for \((f^*a)t\), and on the smooth base-change locus the invariant summand is a line whose fibre-integral metric is semipositive by the compact-torus orthogonal-quotient argument.
 
 The metric extends across every base prime. At a general point of a component attaining \(b_P\), write the map as \((z_1^e,z_2,\ldots,z_k)\), where \(e>0\), and let \(a\) be \(\ord(t)\). The relative canonical coefficient is a nonvanishing smooth factor times \(z_1^{a-b_Pe-(e-1)}\), whose exponent is nonpositive because \(0\le a-b_Pe<e\). Integration on a small vertical polydisk bounds the squared norm below. So the weight is locally bounded above and extends by first crossing the divisorial neighbourhoods and then codimension at least two.
@@ -963,7 +963,7 @@ The strictness assertion follows by applying Lemma~\ref{lem:strict} to the invar
 Let \(X\) be smooth connected projective with \(H^q(X,\OO_X)=0\) for \(q>0\), \(L=-K_X\) smoothly semipositive, and \(T\) an algebraic torus acting on \(X\). Then \(H^0(X,mL)^T\ne0\) for some \(m>0\), with the natural linearization.
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 Average a metric on \(L\) and a K\"ahler form over the maximal compact torus, and let \(\alpha\ge0\) be the resulting curvature. Proposition~\ref{prop:index} and Lemma~\ref{lem:forms} give: if the asserted invariant sections do not exist, there is a fixed \(q>0\) and unbounded \(j\) with a nonzero invariant form in \(H^0(X,\Omega_X^{n-q}\otimes jL)\). The Bochner curvature term is a sum of \(q\) eigenvalues of \(j\alpha\); if \(\alpha\) has full rank on an open set, this kills the form. So \(r=\max\rank\alpha<n\).
 
 Define the degree \(\ell(D)=[D][\alpha]^r[\omega]^{n-r-1}\). It is nonnegative on effective divisors and vanishes on \(L\). Lemma~\ref{lem:forms} gives a linearized determinant line \(S\) and invariant sections \(s_p\in H^0(X,S+pL)^T\) for unbounded \(p\). Lemma~\ref{lem:null-divisors} gives \(\ell(N_p)=0\) for their zero divisors \(N_p\).
@@ -990,7 +990,7 @@ The main additional input is a birational toroidalization, so that the model is
 Let \(X\) be smooth connected projective and let an algebraic torus \(T\) act on \(X\). Give \(L=-K_X\) its natural linearization and suppose it admits a smooth semipositive metric. Let \(M\) be a linearized line with \(-M\) pseudoeffective. If there are nonzero sections \(\sigma_j\in H^0(X,M+jL)^T\) for an unbounded \(J\subset\Z_{>0}\), then \(H^0(X,mL)^T\ne0\) for some \(m>0\).
 \end{proposition}
 
-\begin{proof}[Sketch]
+\begin{proof}
 \emph{Generic adjoint rigidity.} Fix \(j_0<j_1\) in \(J\) and choose \(k>0\) so that the invariant system \(H^0(X,kD(j_1))^T\) has maximal image dimension; write \(D(j)=M+jL\). Equivariant principalisation of its base ideal gives a smooth projective \(\pi_0:W_0\to X\) isomorphic off the base locus, and a connected-fibre Stein map \(f_0:W_0\to Y_0\) such that \(\pi_0^*kD(j_1)=f_0^*H+G\), with \(H\) ample and \(G\) effective, both fixed by the connected torus. The Stein base is trivially acted on. Maximality of image dimension forces
 \[
  \dim_K H^0(F,h\pi_0^*D(j_1)|_F)^T=1\qquad(h>0)