diff --git a/fano_cartier_sections_v2.txt b/fano_cartier_sections_v2.txt
index 1973f54..b301ae2 100644
--- a/fano_cartier_sections_v2.txt
+++ b/fano_cartier_sections_v2.txt
@@ -746,6 +746,1155 @@ Section H (Reduction to the pinning input).
        the point ideal and existence of a good Cartier divisor
        through the point.
 
+
+================================================================================
+PART IX. EXACT STATEMENTS OF EVERY RESULT (VERBATIM, FROM ORIGINAL)
+================================================================================
+
+This appendix reproduces every theorem, lemma, proposition, corollary, and
+claim statement from the original file verbatim, in source order, organized
+by the eight sections A-H and the introduction. Proofs are excluded; those
+are in the original. Backslash-refs such as \ref{lem:...} point to labels
+visible in the source (and most of them appear in this very appendix); grep
+the label without the "lem:" prefix to locate the block you want.
+
+For one-line plain-English summaries of each of these results, indexed by
+section and number, see PART VIII above. PART VIII is what a first-time
+reader should read; PART IX is what a referee should verify against.
+
+
+Section A
+------------------------------------------------------------
+
+\begin{lemma}[Comparison along long parameter intervals]\label{lem:A-comparison} Let \(X\to S\) be a sequence of Fano type contractions as above.
+Suppose \(a_t\) are effective generalized CYs over \(S\), continuous finite rational piecewise affine families of discrepancies (with a finite subdivision of parameter per example, so the pieces can be treated on a common resolution), on \(0\le t\le L_i\) or \(-L_i\le t\le L_i\) with \(L_i\to\infty\).
+Pieces and endpoints can be taken rational.
+Values at each valuation are concave in \(t\), and \(a_0\) is klt.
+Assume the corresponding one of the following (one-sided for \(0\le t\le L_i\); two-sided for \(-L_i\le t\le L_i\)):
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item (one-sided) the discrepancy functions are nonincreasing, and whenever \(a_0(T)\to0\), \(a_t(T)/a_0(T)\to1\) for every fixed \(t\);
+\item (two-sided) for some fixed \(\delta>0\), whenever \(a_0(T)<\delta\), \(a_t(T)/a_0(T)\to1\) for every fixed signed \(t\).
+\end{enumerate}
+
+As throughout these lemmas, tests on sequences of valuations include along subsequences.
+Then on every fixed bounded parameter range the data are eventually klt
+over the generic point of \(S\).  Write \(b_t\) for their discriminant
+discrepancies on \(S\).  In case \textup{(i)}, they are nonincreasing
+and satisfy \(b_t(P)/b_0(P)\to1\) for every fixed \(t\) whenever
+\(b_0(P)\to0\).  In case \textup{(ii)}, some fixed \(\delta'>0\)
+gives the same convergence for every fixed signed \(t\) whenever
+\(b_0(P)<\delta'\).  These conclusions include every subsequence and
+primitive prime orders on every birational model of \(S\).
+
+\end{lemma}
+
+\begin{lemma}[Phases under pure pullback]\label{lem:A-pullback-phases} Let \(a\) be a sequence of klt effective generalized CYs over \(S\) in the same Fano type setting, and let \(H\) be a sequence of \(\mathbb Q\)-Cartier \(\mathbb Q\)-divisors on the respective bases \(S\).
+Suppose for some fixed positive integer \(m\), for every \(a(T)\to0\) the phase \(v_T(m f^*H)\bmod 1\) has lifts to \(\mathbb R\) of size \(O(a(T))\) bounded below by \(-o(a(T))\).
+Then with discriminant \(b\), for some fixed positive multiple \(m'\), \(v_P(m'H)\) satisfies the same phase condition wherever \(b(P)\to0\).
+
+\end{lemma}
+
+\begin{lemma}[A uniform linear chamber]\label{lem:A-linear-chamber} In the same setting suppose \(a\) is an lc effective generalized CY over \(S\), with values in \((1/p)\mathbb Z\) for fixed \(p\), and \(a+t d\) for \(0\le t\le t_0\) are effective generalized CYs, klt for positive \(t\), \(d\ge0\) on all divisorials; take \(L_t\) affine, and \(t_0>0\) fixed rational.
+Then eventually there is a uniform smaller \(t_1>0\) on which discriminants equal \(a'+t d'\) with \(a'\) of bounded denominator (i.e., the same \(p'\) works on all models).
+There is also the following version with extra data:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item \(a+\lambda h+s c\) are effective generalized CYs over \(S\) for \(0<\lambda\le\lambda_0,\ 0<s\le\lambda\theta_0\), \(\lambda_0,\theta_0>0\) fixed, \(h\) of bounded denominator, \(c\ge0\);
+\item on \(a=0\), \(d\le c\le C d\), \(-\eta_i c\le h\le Cc\) with fixed \(C\) and \(\eta_i\to0^+\).
+\end{enumerate}
+
+Then there are uniform \(\lambda_1,\theta_1>0\), with
+\(\lambda_1\le\lambda_0\) and \(\theta_1\le\theta_0\), such that
+eventually, on the sector
+\(0<\lambda\le\lambda_1\), \(0<s\le\lambda\theta_1\), the data
+are klt generically over \(S\), their discriminants are exactly
+\(a'+\lambda h'+s c'\), and \(h'\) has bounded denominator.
+Moreover, eventually there is \emph{no} divisorial \(T\) horizontal
+over \(S\) with \(a(T)=0,\ h(T)<0\).
+In particular for \(S\) a point exact nonnegativity on \(a=0\) follows even though only the approximate bound \(-\eta_i c\) was assumed.
+
+In this statement the minima are lexicographic: first minimize \(a/e\),
+then \(d/e\) on its minimum locus; for the extra data first minimize
+\(a/e\), then \(h/e\), then \(c/e\).
+Here \(e\ord_P=T|_{k(S)}\).
+Thus \(a'\) itself is the discriminant in discrepancy convention even
+if the endpoint is not klt generically.
+The family \(a'+t d'\), \emph{including its endpoint at zero}, has the generalized effective log data (with nef b-part and total logarithmic divisor \(L_t\)).
+
+\end{lemma}
+
+\begin{lemma}[Tools at an lc place]\label{lem:A-lc-place-tools}
+Let \(g:X_1\to W\) be a Mori step as above, and let \(A_*\) be an
+effective generalized CY discrepancy on \(X_1\) over \(S\).
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Suppose that \(A_*\) is lc over the generic point of \(W\),
+and that an effective generalized klt CY over \(S\) is available for
+perturbation.  Any prescribed horizontal divisorial lc place remains
+an lc place of an ordinary boundary \(\Omega\), effective on the
+generic fibre \(F\), with \(n(K_F+\Omega_F)\sim0\), for a bounded
+positive integer \(n\).
+\item For \(P\) on a birational model of \(W\), suppose \(A_*\) is klt generically and \(T\) computes the discriminant \(b_*(P)\).
+There is an ordinary \(n\)-CY over the DVR, lc in its b-discrepancies
+and effective horizontally on \(F\), with \(n\) bounded, keeping
+\(T\) as an lc place.  Here \(n\)-CY means that the ordinary total
+log canonical divisor, multiplied by \(n\), is linearly trivial
+over the trait.
+Moreover for the adjusted discrepancy \(A_*-b_*(P)e\), any two minimizing divisorials can be joined by a chain/interpolation of lc places in the following sense: on a log resolution over the generic DVR their strict divisors (or extractions of them) belong to a connected snc floor consisting of vertical coefficient-one divisors with monomial cones at the intersections.
+One can include any additional finite discrepancy data in the resolution.
+In particular normalized evaluations \(a/e\) of such other data interpolate continuously along the chain, using rational weights to approximate intermediate values by divisorial evaluations.
+\end{enumerate}
+
+\end{lemma}
+
+\begin{proposition}[Uniform charts]\label{prop:A-uniform-charts}
+At a chosen vertical place in Lemma~\ref{lem:A-lc-place-tools}\textup{(ii)},
+after a bounded degree extension of \(k(W)\) and a chosen extension of
+the DVR and of the place, there is a toroidal cone representing the
+place, with nonnegative coordinates
+\begin{equation}\label{eq:A-contact-cone}
+       (x,e_*),\qquad M x=r e_* .
+\end{equation}
+Here \(M\) ranges over a finite list of integral matrices of bounded sizes, \(r\) is integral (possibly huge), \(e_*\) is value of the new uniformizer, and original uniformizer value is \(\rho e_*\) with positive bounded \(\rho\).
+Nonzero integral solutions on the cone give divisorial valuations (not necessarily primitive after restriction); evaluations of discrepancy data always refer to homogeneous values by restriction.
+For horizontal places as in the first bullet one just has an orthant on a bounded generic resolution, without \(e_*\).
+For \emph{any} effective generalized CY data \(A\) on \(X_1\) over \(S\), values on these cones are finite continuous homogeneous piecewise linear, and are Lipschitz in \(x\) with uniform constant \emph{at fixed} \(e_*>0\) (uniform homogeneous Lipschitz in the horizontal case).
+Constants depend on the boundedness in this Mori step and the complement indices, \emph{not} on \(r\), vertical coefficients, or the nef part index or the chosen data \(A\).
+In particular finitely many differences at fixed parameter separations have uniform Lipschitz bounds as well.
+
+\end{proposition}
+
+\begin{lemma}[Bounded toroidal charts]\label{lem:A-toroidal-charts}
+For the prescribed place and tailored complement in
+Proposition~\ref{prop:A-uniform-charts}, there are charts with its stated
+contact equations and integral-valuation property.  The prescribed
+place is monomial on one of these charts.  They have rational functions
+\(z_1,\ldots,z_f\), \(f=\dim F\), with uniformly bounded pole degrees on
+a bounded embedding of \(F\), such that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item their logarithmic differentials form a relative log basis in the
+vertical case and an absolute log basis over the generic field in the
+horizontal case;
+\item their orders are linear functions of the chart coordinates with
+uniformly bounded slopes;
+\item every rational monomial valuation in the vertical chart with
+\(e_*>0\) restricts to a Gauss valuation on
+\(K_1(z_1,\ldots,z_f)\), where \(K_1/k(W)\) is the bounded extension
+used in constructing the chart.  Horizontally the same assertion holds
+with the trivial valuation on the generic field.
+\end{enumerate}
+Here Gauss means that evaluating a polynomial in the \(z_j\) gives the
+minimum of its coefficient-adjusted monomial weights, without
+cancellation among terms of equal weight.
+\end{lemma}
+
+\begin{lemma}[Degrees control angular variation]\label{lem:A-function-slopes}
+On a chart of Lemma~\ref{lem:A-toroidal-charts}, the order of a nonzero rational
+function is continuous, homogeneous and finite piecewise linear,
+including on the faces.  If its pole divisor on the embedded generic
+fibre has degree at most \(m\), where \(m\ge1\), its Lipschitz constant
+in \(x\) at fixed \(e_*>0\) is at most \(Cm\).  Horizontally this bound
+holds on the entire cone.  The constant \(C\) depends only on the fixed
+families and chart choices.  A function with pole degree less than one
+is constant on the generic fibre and has zero angular variation.
+\end{lemma}
+
+\begin{corollary}[Horizontal variants]\label{cor:A-horizontal-variants}
+The chart and metric conclusions have the following two variants.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item
+The same tailorable bounded complement and homogeneous chart argument works if we are working over an algebraically closed generic field, with the relevant effective CYs on a \(\mathbb Q\)-factorial model of Fano type, and have boundedness only up to isomorphism in codimension one of these models.
+\item
+A Cartier rational b-order function \(N\) with effective trace of degree
+\(O(\xi)\) on the bounded model, b-linearly \(\mathbb Q\)-equivalent
+to a difference of b-nef parts both of trace degrees \(O(\xi)\), has
+horizontal Lipschitz constant \(O(\xi)\), including when \(\xi\to0^+\).
+\end{enumerate}
+\end{corollary}
+
+\begin{lemma}[Integral approximation]\label{lem:A-lattice-approximation}
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item
+Let \(u_i\) lie in a fixed nonnegative orthant and
+\(\lvert u_i\rvert\to\infty\).  After passing to a subsequence there
+are \(s_i\to0^+\), nonzero nonnegative integral vectors \(n_i\), and
+a fixed, possibly subsequence-dependent, \(C_*\) such that
+\begin{equation}\label{eq:A-horizontal-approximation}
+ \lvert n_i-s_i u_i\rvert\le C_*s_i,
+ \qquad s_i\lvert u_i\rvert\longrightarrow\infty .
+\end{equation}
+\item
+Let \(M_i x_i=r_i\), where \(x_i\ge0\), \(r_i\) is integral and
+\(M_i\) belongs to a fixed finite list of integral matrices.
+If \(\alpha_i>0\) tends to zero, then after passing to a subsequence
+there are positive integers \(E_i\), nonnegative integral \(n_i\), and
+a fixed \(C_*\) such that
+\begin{equation}\label{eq:A-vertical-approximation}
+ M_i n_i=r_i E_i,\qquad
+ s_i=E_i\alpha_i\longrightarrow0^+,\qquad
+ \lvert n_i-E_i x_i\rvert\le C_*s_i .
+\end{equation}
+\item
+For every sufficiently small fixed \(\gamma>0\), the same matching
+equations admit \(n_i\ge0\) integral and \(1\le E_i\le J(\gamma)\)
+with
+\begin{equation}\label{eq:A-bounded-dirichlet}
+ \lvert n_i-E_i x_i\rvert\le\gamma ,
+\end{equation}
+where \(J(\gamma)\) is uniform in \(i\) and in the contact vectors
+\(r_i\).
+\end{enumerate}
+\end{lemma}
+
+\begin{claim}[Quantized chart cells]\label{claim:A-quantized-cells}
+The bounded-denominator functions \(a,h\), when present, have only a
+bounded finite set of possible slopes in a horizontal chart, or in
+integral kernel coordinates on a vertical slice \(e_*=1\).
+Their vertical affine constant terms belong to a rational grid with
+bounded denominator, although their magnitudes need not be bounded.
+Their common cells are cut out by a fixed finite set of possible
+normal vectors, with offsets of bounded denominator.  They are
+pointed, their vertices have bounded denominator, and they satisfy a
+uniform polyhedral distance bound.
+\end{claim}
+
+
+Section B
+------------------------------------------------------------
+
+\begin{lemma}[Phase nonvanishing]\label{lem:B-phase-nonvanishing}
+Let $(V_i,H_i)$ be a sequence in bounded dimension, where $V_i$ is
+projective of Fano type and $H_i$ is a nef and big $\Q$-Cartier rational
+Weil divisor.  Suppose that, for rational $t\in[0,L_i]$, where
+$L_i\to\infty$, there are generalized data with discrepancies $a_i(t)$
+such that
+\begin{equation}\label{eq:B-family}
+ B_i(t)\ge R(H_i),\qquad
+ K_{V_i}+B_i(t)+\mathbf M_i(t)_{V_i}\sim_{\Q}tH_i,
+ \qquad \mathbf M_i(t)\text{ is b-nef and $\Q$-Cartier}.
+\end{equation}
+Assume that $a_i(0)$ is klt and that $a_i(t)$ is nondecreasing in $t$
+at every divisorial valuation.
+
+Assume the following condition, also after restriction to any
+subsequence.  For every choice of rational $t_i\to\infty$ in the
+parameter range, one can pass to a further subsequence and choose a
+fixed positive integer $q$ such that, for every sequence of primitive
+divisorial valuations $T_i$ with $a_i(t_i,T_i)\to0$, the signed phases
+$\theta_i(T_i)$ of $qH_i(T_i)$ satisfy
+\begin{equation}\label{eq:B-phase-hypothesis}
+ -o\bigl(a_i(t_i,T_i)\bigr)
+ \le \theta_i(T_i)
+ \le O\bigl(a_i(t_i,T_i)\bigr).
+\end{equation}
+Then there is an integer $M>0$ such that for every $i$ some
+$m_i\in\{1,\ldots,M\}$ satisfies
+$|\lfloor m_iH_i\rfloor|\ne\varnothing$.
+Equivalently, after replacing $M$ by a common multiple, the same
+positive degree works for all $i$.
+\end{lemma}
+
+\begin{lemma}[Bundle construction and discrepancies]\label{lem:B-seifert}
+Let $V$ be projective normal, and let $H$ be a $\Q$-Cartier rational
+divisor.  There is a projective normal compactified Seifert bundle
+$p:Y\to V$ with disjoint $\Q$-Cartier zero and infinity sections
+$E_0,D$ and a rational fibre coordinate $z$ such that
+\begin{equation}\label{eq:B-bundle-identities}
+ \divisor(z)=p^*H+E_0-D,
+ \qquad
+ K_Y+E_0+D=p^*\bigl(K_V+R(H)\bigr).
+\end{equation}
+The second equality uses compatible canonical divisors.  The divisor
+$D$ is relatively ample; if $H$ is nef, then $D$ is nef, and if $H$
+is nef and big, then $D+p^*H$ is nef and big.
+
+In these identities, pullback of a rational Weil divisor along the
+equidimensional map $p$ is defined at the generic points of its prime
+components, using the multiplicities of their inverse images.  This
+agrees with Cartier pullback whenever the divisor is $\Q$-Cartier.
+Orders of $H$, $E_0$, and $D$ on higher models use their Cartier
+b-pullbacks.
+
+Suppose $B\ge R(H)$ and $\mathbf M$ is a nef $\Q$-Cartier
+b-divisor on $V$.  Give $Y$ boundary $p^*(B-R(H))$ and nef part
+$p^*\mathbf M$.  Write $a$ for the generalized discrepancy on $V$
+and $A$ for that on $Y$.  If a primitive divisorial valuation $v$ of
+$k(Y)$ restricts nontrivially to $w$ on $k(V)$, then
+\begin{equation}\label{eq:B-bundle-discrepancy}
+ A(v)=a(w)+|h(v)|+\operatorname{def}(v),
+ \qquad
+ h(v)=v(E_0)-v(D)\equiv-w(H)\pmod\Z,
+\end{equation}
+where $\operatorname{def}(v)$ is a nonnegative integer.  It vanishes
+precisely for the Gauss, or fibre-torus-invariant, valuations.
+If $w=0$, the primitive valuations are the ordinary places of the
+generic $\mathbb P^1$, and have discrepancy $1$.
+Consequently klt base data give klt data on $Y$ without the two
+sections; adding both sections gives lc data whose non-klt locus is
+contained in the sections.
+\end{lemma}
+
+\begin{lemma}[Clearing the small denominators]\label{lem:B-denominator-clearing}
+In bounded dimension, suppose $V$ is projective of Fano type and
+carries effective generalized klt Calabi--Yau data with
+$B\ge R(H)$.  After multiplying $H$ by a bounded positive integer,
+there is an ordinary klt log Fano boundary $\Theta\ge R(H)$.
+For this scaled divisor the Seifert bundle is of Fano type.
+The assertion does not require a bound on the Cartier index of the
+nef b-divisor.
+\end{lemma}
+
+\begin{lemma}[An unbounded-index obstruction]\label{lem:B-unbounded-index}
+Let $F_i$ be projective of positive bounded dimension and let $G_i$
+be ample $\Q$-Cartier rational divisors such that $(F_i,R(G_i))$
+is klt and
+\[
+ -(K_{F_i}+R(G_i))\sim_{\Q}I_iG_i,
+ \qquad I_i>0,\quad I_i\longrightarrow\infty.
+\]
+Let $C_i$ be the compactified Seifert cone obtained by contracting
+the zero section, and denote its infinity divisor by $D_i$.
+Then $C_i$ is klt of Fano type, $D_i$ is ample, and
+$-K_{C_i}\sim_{\Q}(I_i+1)D_i$.  It is impossible that
+\begin{equation}\label{eq:B-cone-small-height}
+ v_i(D_i)=o\bigl(A_{C_i}(v_i)\bigr)
+\end{equation}
+for every sequence of primitive divisorial valuations with
+$A_{C_i}(v_i)\to0$.
+\end{lemma}
+
+\begin{lemma}[Bounding a principalization order]\label{lem:B-principal-order}
+Let $F_i$ be projective of Fano type in bounded dimension, and let
+$P_i\sim_{\Q}0$ be $\Q$-Cartier rational divisors.  Suppose that
+$F_i$ carries effective generalized klt Calabi--Yau data with
+discrepancies $a_i$ and boundary at least $R(P_i)$.  Suppose,
+after a fixed integral scaling if necessary, that the signed phases
+of $P_i$ at every sequence with $a_i(T_i)\to0$ lie in
+$[-o(a_i(T_i)),O(a_i(T_i))]$.  Then the least positive integer $q_i$
+for which $q_iP_i$ is principal is bounded.
+
+More generally, the same cyclic-cover argument applies whenever
+the induced isotrivial fibration below has
+$u=o(A)$ at its small-discrepancy divisors over zero.  This
+formulation permits the argument to be used after an independent
+estimate of the height $u$.
+\end{lemma}
+
+
+Section C
+------------------------------------------------------------
+
+\begin{lemma}[Uniform cutoff models and order functions]
+\label{lem:C-cutoff-claims}
+Suppose $\dim W_\beta>0$.  After passage to a subsequence there is a
+fixed $\delta>0$ and a projective $\Q$-factorial Fano type extraction
+$\widehat W\to W_\beta$ such that $\widehat W$ has ordinary
+$\delta$-lc singularities and every positive coefficient of the pullback
+of $D_W/\beta$ is at least $\delta$.  A bounded constant $C$ can be
+chosen so that $S(CD_W/\beta)$ generates $k(W_\beta)$.
+
+There are algebraically independent functions
+$\phi_1,\ldots,\phi_{\dim W_\beta}$ in this space and a b-nef
+$\Q$-Cartier order function $H$ on $k(W_\beta)$, pulled back to
+$k(V)$, with uniformly bounded denominator of its values on primitive
+orders, such that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item If $a(T)=0$, then $H(T)\leq CD(T)/\beta$.  If $T$ is nontrivial
+on $K_\beta$ and all $T(\phi_j)\geq0$, then $H(T)\leq-1$.
+\item If $D(Q)=0$ and $a(Q)\leq1$, then $H(Q)\leq0$.
+\item If $a(T)=0$ and \eqref{eq:C-alpha} holds at $T$, then
+$H(T)\geq-C\eta D(T)/\beta$.
+\end{enumerate}
+In particular, if \eqref{eq:C-alpha} holds on all of $I_\beta$ with
+$\eta_i\to0$, those primes are eventually horizontal over $K_\beta$;
+that is, their restrictions to $K_\beta$ are trivial.
+\end{lemma}
+
+\begin{lemma}[Scaled phase bound]\label{lem:C-scaled-phase}
+Assume \eqref{eq:C-alpha} with $\eta_i\to0$ whenever
+$a(T)=0$ and $LD(T)\to0$, along arbitrary subsequences.  Let $\ell$
+be a rational principal order.  Suppose that for every nonfloor prime
+$Q$ on $V$ there is an integer $p_Q$ with
+\begin{equation}\label{eq:C-nonfloor-phase}
+ |p_Q-\ell(Q)|\leq G(Q),\qquad
+ G\geq0,\qquad G\sim_\Q c_gLD,
+\end{equation}
+where $c_g\geq0$ is rational and uniformly bounded.  There is a fixed
+positive integer $q$ such that
+\begin{equation}\label{eq:C-scaled-phase}
+ \operatorname{dist}(q\ell(T),\Z)=O(LD(T))
+ \quad\text{if }a(T)=0\text{ and }LD(T)\longrightarrow0.
+\end{equation}
+\end{lemma}
+
+\begin{lemma}[Saturation and section counts]\label{lem:C-saturation-count}
+Let $f_1,\ldots,f_s$ be a finite list of genuine rational functions on
+$V$, whose length may vary with the sequence, and let
+\begin{equation}\label{eq:C-neutral-polarization}
+ J(T)=\max\{0,\max_j(-T(f_j))\},\qquad
+ 0\leq J\leq C L_0\mathbf D,\qquad1\leq L_0=O(L).
+\end{equation}
+Thus $J$ is an integral basepoint-free Cartier b-divisor on a high
+model.  Suppose \eqref{eq:C-alpha} holds uniformly with $\eta_i\to0$
+on
+\[
+                       \mathcal L=\{T:a(T)=0,\ J(T)=0\}.
+\]
+In addition, on every subsequence on which $L_0\to\infty$, assume
+that all valuations in $\mathcal L$ eventually restrict trivially to
+the \emph{field generated by} $S(L_0D)$.
+
+After passage to a subsequence, there are cutoff fields $K_*$ of fixed
+transcendence degree $p$, a fixed constant $B$, and positive real
+numbers $v_i$ with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item For every fixed $C_1>0$, eventually $S(C_1LD)\subset K_*$.
+\item Eventually $S(BLD)$ generates $K_*$.  If $T\in\mathcal L$ is
+nonnegative on this space, its restriction to $K_*$ is trivial.
+\item There is $C'>0$, independent of the positive integer $n$, such
+that for every fixed $n$, eventually
+\begin{equation}\label{eq:C-uniform-counts}
+ \dim S(nLD)\leq C'v_i n^p,\qquad
+ \dim\operatorname{Span}\{g_1\cdots g_n:g_j\in S(BLD)\}
+                                      \geq v_i n^p.
+\end{equation}
+\end{enumerate}
+The eventual index in \textup{(i)} may depend on $C_1$, and that in
+\textup{(iii)} may depend on $n$; $B$ and $C'$ are fixed before $n$.
+All assertions concern divisorial evaluations.  Exact triviality also
+holds on rational monomial subcones cut out by the indicated
+conditions.
+\end{lemma}
+
+\begin{lemma}[Vanishing on a smooth orbifold]\label{lem:C-orbifold-vanishing}
+Let $\mathcal X$ be a smooth proper Deligne--Mumford stack over an
+algebraically closed field of characteristic zero, with trivial generic
+stabilizer and projective coarse space.  Let $\Theta$ be an effective
+rational divisor with simple normal crossing support and coefficients
+less than one.  If $D$ is an integral Cartier divisor and
+$D-(K_{\mathcal X}+\Theta)$ is nef and big, then
+$H^q(\mathcal X,\mathcal O_{\mathcal X}(D))=0$ for $q>0$.
+\end{lemma}
+
+\begin{lemma}[Adjoint Taylor exponents]\label{lem:C-adjoint-lattice}
+Let $X$ be a projective $\Q$-factorial variety of Fano type of
+dimension $p>0$, and let $N$ be a nef and big integral Weil divisor.
+Suppose that $H^0(X,\mathcal O_X(N))$ defines a birational map.
+Choose an effective rational boundary $\Delta$ with $(X,\Delta)$ klt
+and $K_X+\Delta\sim_{\Q}0$.  At a general smooth point outside
+the supports of $N$ and $\Delta$, fix regular parameters and take
+total-degree-then-lexicographic Taylor initial exponents, using the
+compatible local trivializations of the powers of $N$.  Let
+\[
+ K=\overline{\operatorname{conv}}
+ \left\{\frac{\nu(s)}j:
+ j\in\Z_{>0},\quad
+ 0\ne s\in H^0(X,\mathcal O_X(jN))\right\}.
+\]
+If $t$ is a positive integer and $\alpha\in\Z_{\ge0}^p$ satisfies
+$\alpha+\mathbf1\in\operatorname{int}(tK)$, then some section of
+$tN$ has initial exponent $\alpha$.
+\end{lemma}
+
+\begin{lemma}[Uniform count from the jet body]\label{lem:C-jet-count}
+In the setting of Lemma~\ref{lem:C-adjoint-lattice}, $K$ is bounded and contains the standard
+simplex $\Sigma=\operatorname{conv}(0,e_1,\ldots,e_p)$.  For every
+integer $n\ge1$,
+\[
+ h^0(X,nN)\le(n+p)^p\operatorname{vol}(K).
+\]
+Fix any integer $t>1+2p$.  The space spanned by products of $n$ sections
+of $tN$ has dimension at least $n^p\operatorname{vol}(K)$.
+\end{lemma}
+
+\begin{corollary}[Fixed generators and neutral algebraic extensions]
+\label{cor:C-neutral-field}
+In Lemma~\ref{lem:C-saturation-count}, let $g_{i,1},\ldots,g_{i,r_i}$
+be any finite list contained in $S(C_1L_iD_i)$ for one fixed $C_1$.
+For all sufficiently large $i$, the entire field
+$F_i=k(g_{i,1},\ldots,g_{i,r_i})$, and its relative algebraic closure
+inside $k(V_i)$, lie in $K_{*,i}$.  Thus every test from part
+\textup{(ii)} is exactly zero on every nonzero element of these fields
+at that same index, without a degree-dependent eventual qualification.
+More generally, a valuation trivial on a subfield remains trivial on
+any algebraic extension of it inside its valued overfield.
+\end{corollary}
+
+
+Section D
+------------------------------------------------------------
+
+\begin{proposition}[Cone dictionary]\label{prop:D-cone-dictionary}
+In the above setting the following statements hold.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item The discrepancy of $(X,H)$ at $(P,y)$ is $a(P)$, and its discrepancy
+at a pure weight is zero.  Thus $a(P)$ is a nonnegative integer for every
+primitive $P$, and $\mathbf u\in\operatorname{int}C$.
+\item For rational $\mu\in\operatorname{int}C$, the ray quotient $V_\mu$ is
+normal and projective, with function field $\mathcal K$.  The order function
+\begin{equation}\label{eq:D-polarization}
+ \mathbf L(\mu)(P)=\min_{y\in\Delta_P}\langle y,\mu\rangle
+\end{equation}
+is the b-pullback of an ample rational divisor on $V_\mu$.  Consequently
+\begin{equation}\label{eq:D-height-divisor}
+ \mathbf D(P)=\min_{y\in\Delta_P}(P,y)(h)
+             =\gamma(P)+\mathbf L(\mathbf u)(P)
+\end{equation}
+is the b-pullback of an effective ample rational divisor $D$ on
+$V_{\mathbf u}$.
+\item At every actual prime $P$ of $V_\mu$, a minimizer $y$ of
+\eqref{eq:D-polarization} can be chosen so that $e(P,y)$ is a primitive
+prime order on $X$, for some positive integer $e$.  For this choice,
+\begin{equation}\label{eq:D-actual-prime}
+ a(P)+(P,y)(h)=\frac1e,
+ \qquad e y\in\operatorname{Hom}(M,\Z).
+\end{equation}
+On each $V_\mu$, $a$ is the discrepancy function of an effective ordinary
+lc Calabi--Yau pair, and $a+\mathbf D$ is that of an effective generalized
+klt Calabi--Yau pair with nef part $\mathbf D$.  Each $V_\mu$ is of Fano
+type.  At an actual prime outside the floor $\{a=0\}$, the same minimizer
+satisfies
+\begin{equation}\label{eq:D-off-floor}
+ a(P)=1,\qquad e=1,\qquad (P,y)(h)=0.
+\end{equation}
+\end{enumerate}
+\end{proposition}
+
+\begin{lemma}[All divisorial lc tests are Gauss]\label{lem:D-lc-tests}
+Every nontrivial divisorial valuation nonnegative on $R$ and having
+zero discrepancy for $(X,H)$ is Gauss.  Its restriction to $\mathcal K$
+is either $eP$ with $a(P)=0$, or trivial, in which case it is a nonzero
+pure weight in $\sigma$.  Conversely every rational Gauss extension
+with $a(P)=0$, and every nonzero rational pure weight, has zero
+discrepancy and a center whose closure contains the contracting vertex.
+
+If $\Spec\breve R\to X$ is quasi-\'{e}tale and the pairs are related by
+log pullback, every divisorial $h$-lc test upstairs restricts to one of
+these tests, up to scaling.  Conversely all the rational tests just
+listed lift to divisorial $h$-lc tests upstairs whose center closures
+contain the upstairs vertex.  These statements suffice in particular
+for testing invariant rational functions.
+\end{lemma}
+
+\begin{lemma}[Sections as regular symbols]\label{lem:D-section-symbols}
+For every positive integer $q$ and every $g\in\mathcal K$,
+\begin{equation}\label{eq:D-symbol-equivalence}
+ g\in S(q\mathbf J_m)
+ \quad\Longleftrightarrow\quad
+ g\xi^q h^{qm}\in\breve R.
+\end{equation}
+In particular the neutral functions of cost $b\in\N$ are precisely
+\begin{equation}\label{eq:D-neutral-functions}
+ \{g\in\mathcal K:g h^b\in\breve R\}=S(b\mathbf D).
+\end{equation}
+\end{lemma}
+
+\begin{proposition}[Sandwich regularization]\label{prop:D-sandwich}
+Let $L\ge1$ vary with the sequence.  Choose integers $m>m_0$ with
+$m-m_0\ge cL$ for a fixed $c>0$, and suppose both
+$k+m_0\mathbf u$ and $k+m\mathbf u$ are interior.  Assume that whenever
+primitive divisorial valuations $P$ satisfy $a(P)=0$ and
+$L\mathbf D(P)\to0$, there are integers $j_P$ such that
+\begin{equation}\label{eq:D-sandwich-hypothesis}
+ j_P\le\mathbf J_{m_0}(P)\le\mathbf J_m(P)
+       \le j_P+O(L\mathbf D(P)).
+\end{equation}
+Then there is a bounded positive integer $q$ and a nonzero
+$g\in S(q\mathbf J_m)$.  Equivalently, $g\xi^qh^{qm}$ is regular on
+$\Spec\breve R$.
+\end{proposition}
+
+\begin{corollary}[Bounded-multiple character regularization]
+\label{cor:D-regularization}
+Suppose that the order condition \eqref{eq:C-alpha} on $V_{\mathbf u}$,
+with error $\eta_i\to0$, holds whenever $a(P)=0$ and
+$L\mathbf D(P)\to0$.  Fix $C_0>0$.  If $k\in M$ satisfies
+\begin{equation}\label{eq:D-width}
+ |\langle y,k\rangle|\le C_0L\langle y,\mathbf u\rangle
+ \qquad(y\in\sigma),
+\end{equation}
+there are a bounded positive integer $q$, a nonzero invariant rational
+eigenfunction $Y=fz^{qk}$, and a positive integer $b\le C_2L$ such that
+$h^bY\in\breve R$.  The constants are uniform in $k$.  Both signs of
+$k$ may be treated with the same bounded multiplier, and their costs may
+be increased to a common cost $O(L)$.
+
+The same bounds hold uniformly if $L$ is replaced by any $L'\ge L$ and
+\eqref{eq:D-width} is imposed at scale $L'$.
+\end{corollary}
+
+\begin{corollary}[Count for a single character and weight]
+\label{cor:D-character-count}
+Assume the hypotheses of Lemma~\ref{lem:C-saturation-count} on
+$V_{\mathbf u}$, and use its notation $p$ and $v_i$.  For a fixed
+$C_0>0$, let $W_{t,k}$ be the space of regular symbols in $\breve R$
+of finite-group character $\chi^t$ and torus weight $t\mathbf u+k$.
+For every fixed positive integer $n$, eventually
+\begin{equation}\label{eq:D-character-bound}
+ \dim_k W_{t,k}\le C_3 v_i n^p
+\end{equation}
+simultaneously for all integers $0\le t\le C_0nL$ and all weights
+satisfying
+\[
+ |\langle y,k\rangle|\le C_0nL\langle y,\mathbf u\rangle
+ \qquad(y\in\sigma).
+\]
+The constant $C_3$ is independent of $n,t,k$.
+\end{corollary}
+
+\begin{corollary}[Characters for a tilted height]
+\label{cor:D-tilted-characters}
+Suppose there are boundedly many invariant rational eigenfunctions
+$Y_j^\pm$ of weights $\pm k_j$, with
+\[
+ h^{m_j}Y_j^\pm\in\breve R,\qquad
+ 0<m_j\le A L_j,\qquad 1\le L_j\le L,
+\]
+where $A$ is fixed.  Fix bounded real numbers $u_j$ and $\delta>0$.
+Assume that, at every nonzero pure weight and at all Gauss extensions
+over primitive $P$ with $a(P)=0$, $L\mathbf D(P)\to0$, the evaluations
+satisfy
+\begin{equation}\label{eq:D-tilt-hypothesis}
+ y_j=v(Y_j^+)=-v(Y_j^-),\qquad
+ \sum_j\frac{u_jy_j}{L_j}\le\delta v(h).
+\end{equation}
+It suffices to check these inequalities on rational vectors.  For
+fixed $M_0$ sufficiently large compared with $1/\delta$, choose integers
+\[
+ r_j=M_0u_jL/L_j+O(1),
+\]
+with uniformly bounded rounding errors.  Then there is a regular
+nonzero symbol of finite-group character $\chi^{qm}$ and torus weight
+\begin{equation}\label{eq:D-tilt-weight}
+ q\left(m\mathbf u-\sum_jr_jk_j\right),
+ \qquad 0<m\le C_6M_0\delta L,
+\end{equation}
+for a bounded positive integer $q$.  The constant $C_6$ depends only on
+the fixed bounds for the number of functions, their costs, the $u_j$,
+and rounding errors; it is independent of $\delta$ and $M_0$.
+The bound for $q$ may depend on the fixed choices $\delta,M_0$.
+No order condition \eqref{eq:C-alpha} is assumed here.
+\end{corollary}
+
+
+Section E
+------------------------------------------------------------
+
+\begin{lemma}[Logarithmic jet estimate]\label{lem:E-log-jet}
+Fix finitely many regular affine functions on $S$. On a fixed resolution
+$P$, with $F$ reduced snc, there is a constant $C$ such that, for every
+polynomial $g\ne0$ of degree at most $D$ in those functions and every
+quasi-monomial $v$ over $x$,
+\begin{equation}\label{eq:E-jet}
+ r_F(v)(g)\le v(g)\le r_F(v)(g)+C(D+1)A_{P,F}(v).
+\end{equation}
+The same constant works when the centre is not closed and when some
+weights disappear on passing to a face of the monomial cone.
+\end{lemma}
+
+\begin{lemma}[Compactness and uniform centring]\label{lem:E-compactness}
+The slice $\mathcal Q$ is a compact finite polyhedral complex of
+monomial valuations on any log resolution adapted to $h$ and the
+$F_j$. Evaluations of fixed functions are continuous on it.
+For centred valuations with $A(v)=1$, there are constants $c,C>0$
+depending on the germ such that
+\begin{equation}\label{eq:E-centering}
+ \vol(v)\ge c,\qquad
+ \vol(v)\ge \frac{c}{v(\mathfrak m)}.
+\end{equation}
+Volume is continuous on the centred part of $\mathcal Q$.
+\end{lemma}
+
+\begin{corollary}[Simultaneous approximation]\label{cor:E-all-functions}
+In the situation of \eqref{eq:E-retraction-errors}, suppose
+$A_S(v)$ and $A_S(r)$ are bounded above and
+$v(\mathfrak m),r(\mathfrak m)\ge c_0>0$. Then a single constant $C$
+works for every such $v$ and every $f\in R$:
+\begin{equation}\label{eq:E-all-functions}
+ r(f)\le v(f)\le(1+Ce)r(f).
+\end{equation}
+In particular,
+$\vol(v)^{1/n}\ge(1+Ce)^{-1}\vol(r)^{1/n}$.
+\end{corollary}
+
+\begin{lemma}[Convex domination]\label{lem:E-convexity}
+For $v_1,\ldots,v_k\in\mathcal Q$ and $p_s\ge0$ with
+$\sum_s p_s=1$, there is $u\in\mathcal Q$ such that
+\begin{equation}\label{eq:E-convexity}
+ u(f)\ge\sum_s p_s v_s(f)\qquad(f\in R).
+\end{equation}
+If a positive-weight term is centred, then $u$ is centred. On the cone
+of common lc rays, the same statement holds with
+$A(u)=\sum_s p_sA(v_s)$, without requiring $A(v_s)=1$.
+For an allowed height,
+$b(u)\le\sum_s p_sb(v_s)$.
+\end{lemma}
+
+\begin{corollary}[Opposite ratios]\label{cor:E-opposite-ratios}
+Suppose rational functions $q,q'$ have regular numerators and
+power-of-$h$ denominators, and $v(q')=-v(q)$ on $\mathcal Q$.
+The interpolant in Lemma~\ref{lem:E-convexity} has
+$u(q)=\sum_s p_sv_s(q)$ and $u(q')=\sum_s p_sv_s(q')$.
+The assertion holds simultaneously for any number of such pairs.
+\end{corollary}
+
+\begin{lemma}[Volume interpolation and a strict gap]
+\label{lem:E-volume-interpolation}
+Let $w$ be centred, let $v$ be over $x$, and suppose
+$u\ge tw+(1-t)v$ on $R$ for $0<t<1$. For $f\ne0$ with $w(f)>0$,
+\begin{equation}\label{eq:E-volume-interpolation}
+ \vol(u)\le t^{-n}
+   \frac{tw(f)}{tw(f)+(1-t)v(f)}\vol(w).
+\end{equation}
+If $v$ is also centred, $t=1/2$, and
+$v(f)\ge(1+\gamma)w(f)>0$ for $0<\gamma\le1$, then
+\begin{equation}\label{eq:E-volume-gap}
+ \vol(w)+\vol(v)-2\vol(u)\ge c(n,\gamma)\vol(w),
+\end{equation}
+where $c(n,\gamma)>0$ depends only on $n,\gamma$.
+Consequently valuations whose volumes are within relative $o(1)$ of
+the minimum on the same $A=1$ common slice are multiplicatively
+$1+o(1)$ equal on \emph{all} regular functions. The conclusion is
+uniform when the dimension is bounded. A sufficiently small fixed
+relative error similarly gives any prescribed fixed multiplicative
+closeness to one.
+\end{lemma}
+
+\begin{lemma}[A boundary encoding the common slice]\label{lem:E-slice-boundary}
+Choose positive rational numbers $\theta_j$ with
+$\sum_j\theta_j<1$. For sufficiently general scalars $a_j$, put
+$E_j=\divisor(h^{m_j}+a_jF_j)$ and
+\begin{equation}\label{eq:E-slice-boundary}
+ \Psi=(1-\sum_j\theta_j)H+
+                     \sum_j\frac{\theta_j}{m_j}E_j.
+\end{equation}
+Then $\Psi$ is effective and lc, and its nonzero zero-discrepancy
+rays over the germ are exactly the common lc rays. On every such ray,
+\begin{equation}\label{eq:E-pencil-order}
+ v(h^{m_j}+a_jF_j)=m_jv(h).
+\end{equation}
+In particular, the coefficients $\theta_j$ may be chosen arbitrarily
+small while retaining these conclusions.
+\end{lemma}
+
+\begin{theorem}[Exact common-slice minimizer]\label{thm:E-exact-minimizer}
+For every allowed height $b$, the homogeneous functional
+$b(v)\vol(v)^{1/n}$ on centred common lc rays has a unique minimizing
+ray. Its $A=1$ representative depends continuously on the coefficients
+of $b$ while the height stays strictly positive on the fixed slice.
+It is fixed by every automorphism preserving the slice and the height.
+
+If the coefficients of $b$ are rational, its minimizer $w$ is the
+normalized-volume minimizer of an effective rational klt pair on
+$x\in S$. More precisely, one can choose an effective rational lc
+Cartier boundary $\Psi$ with exactly the common lc rays as its
+zero-discrepancy locus, and one positive rational number $\lambda$,
+fixed independently of the competing valuation, such that
+\begin{equation}\label{eq:E-klt-perturbation}
+ \Psi_\lambda=\Psi-\lambda b_0H
+                       +\lambda\sum_\ell b_\ell\divisor(G_\ell)
+\end{equation}
+is effective and klt, and $w$ minimizes
+$\widehat{\vol}_{S,\Psi_\lambda}$ among all centred valuations.
+On the common lc rays its discrepancy is $\lambda b$.
+
+For the same rational height, the algebra $\operatorname{gr}_wR$
+is finitely generated and normal.
+Its spectrum $S^0$ is a klt $\Q$-Gorenstein cone.
+Write $h^0,F_j^0,G_\ell^0$ for initial forms, $w^0$ for the grading
+valuation, and $\mathcal Q^0,b^0$ for the analogous central slice
+and height. Then $(S^0,\divisor(h^0))$ is lc,
+$A_{S^0}(w^0)=w^0(h^0)=1$, $w^0\in\mathcal Q^0$, and
+$b^0$ is strictly positive on $\mathcal Q^0$.
+The valuation $w^0$ minimizes $b^0\vol^{1/n}$ on its centred part.
+\end{theorem}
+
+\begin{corollary}[Near-minimizers for a height]\label{cor:E-height-near-minimizers}
+Fix an allowed height $b$ and normalize its common rays by $b=1$.
+If $w_i,v_i$ have functional values within relative $o(1)$ of the
+minimum, then $w_i=(1+o(1))v_i$ on all regular functions, with an error
+independent of the function. The same conclusion, after $A=1$
+normalization, holds for minimizers of heights whose coefficients
+converge to those of $b$.
+\end{corollary}
+
+\begin{corollary}[Domination by the minimizer]\label{cor:E-domination}
+For the minimizing ray $w$ and every $v\in\mathcal Q$,
+\begin{equation}\label{eq:E-domination}
+ \frac{v(f)}{b(v)}\le 2^n\frac{w(f)}{b(w)}
+                 \qquad(f\in R).
+\end{equation}
+For rational heights the same statement holds on the central slice
+with all symbols carrying a superscript $0$.
+\end{corollary}
+
+\begin{lemma}[One-sided height tilt]\label{lem:E-one-sided-tilt}
+Let $G\in R\setminus\{0\}$, $m\in\Z_{>0}$, and
+$g(v)=v(G)/m-v(h)$. Suppose $g(v)/v(h)\ge2c$ on some common lc ray,
+where $c>0$ is rational. Set
+\[
+ M=\max_{\mathcal Q}g\ge2c,\qquad
+ \mu_*=(M-c)^{-1},\qquad
+ b_\mu(v)=v(h)+\mu(cv(h)-g(v)).
+\]
+For $0\le\mu<\mu_*$ the height is positive. There is a rational
+$\mu$ in this interval whose $A=1$ minimizing valuation $w$ satisfies
+\[
+ g(w)>0,\qquad b_\mu(w)\ge1.
+\]
+Moreover $b_\mu\le2+1/c$ on $\mathcal Q$, and the corresponding
+central height obeys the same bound on $\mathcal Q^0$.
+\end{lemma}
+
+
+Section F
+------------------------------------------------------------
+
+\begin{lemma}[Weighted presentations]\label{lem:F-presentation}
+Choose $x_1,\ldots,x_N\in\mathfrak m_x$ whose symbols generate
+$\breve R$, and put $\alpha_i=w(x_i)>0$.  Any prescribed finite
+list of functions of positive $w$-value can be included in this list.
+Then
+\begin{equation}\label{eq:F-presentation}
+ \widehat{\mathcal O}_{S,x}=\C[[X_1,\ldots,X_N]]/I,
+ \qquad
+ \breve R=\C[X_1,\ldots,X_N]/\operatorname{in}_{\alpha}I,
+\end{equation}
+where initial forms have least weight.  The valuation ideals are
+exactly the images of the ideals generated by monomials of weight
+at least the prescribed threshold.  The same description holds
+before completion by intersection.
+
+For every sufficiently nearby linear functional
+$L:\Lambda_w\otimes\R\longrightarrow\R$, positive on the listed
+weights, the vector $\alpha_L=(L(\alpha_i))$ gives the same initial
+ideal, with its grading composed with $L$.  Its filtration defines
+a centred quasi-monomial valuation $w_L$.  The functionals can be
+chosen rational, and the resulting valuations can be computed
+monomially on a fixed smooth chart for $w$.
+\end{lemma}
+
+\begin{proposition}[Canonical forms under degeneration]
+\label{prop:F-canonical-degeneration}
+Under the hypotheses preceding Lemma~\ref{lem:F-presentation},
+$S^0$ has a divisor-free reflexive canonical generator $\omega^0$.
+It has the $G$-character of $\omega$.  It is homogeneous for
+$\mathbb T_w$, and its weight $\kappa$ satisfies
+\begin{equation}\label{eq:F-canonical-weight}
+ \langle w,\kappa\rangle=A_S(w).
+\end{equation}
+If an additional torus acts on the original germ, preserves $w$,
+and makes $\omega$ homogeneous, its character is preserved
+separately.  None of these assertions assumes $A_S(w)=w(h)$.
+Moreover the action of $G$ on $S^0$ is quasi-\'{e}tale.
+\end{proposition}
+
+\begin{corollary}[Lc equality and principal boundaries]
+\label{cor:F-lc-degeneration}
+If $w$ is $h$-lc, then $(S^0,\divisor(\bar h))$ is lc and
+$\omega^0$ has the same fine torus weight and $G$-character as
+$\bar h$.  Consequently $\omega^0/\bar h$ is invariant, and the
+quotient cone has the log-form data of Section~\ref{sec:D}.
+If $\Delta$ is any effective rational sum of principal divisors
+near $x$, with $(S,\Delta)$ lc and
+$A_S(w)=w(\Delta)$, its initial boundary $\Delta^0$ is lc on
+$S^0$.  All these assertions also hold with specified additional
+equivariant gradings.
+\end{corollary}
+
+\begin{proposition}[Sign transfer]\label{prop:F-bridge}
+Suppose $w$ is common $h$-lc and $w(f_j)=0$ for every $j$.
+Let $g=U/h^b$, where $0\ne U\in\mathcal O_{S,x}$,
+$b\in\N_{>0}$, and $w(g)=0$.  Let $\mathcal Q$ be the actual
+common slice and $\mathcal Q^0$ the common slice defined by
+$\bar h$ and the $\bar f_j$ on $S^0$.  Then
+\begin{equation}\label{eq:F-bridge}
+ q(g)\le0\quad(q\in\mathcal Q)
+ \quad\Longleftrightarrow\quad
+ q^0(\bar g)\le0\quad(q^0\in\mathcal Q^0).
+\end{equation}
+One may add any finite list of tests of value zero at $w$.
+There is no bound on the degree or the cost of $g$ in this statement.
+\end{proposition}
+
+\begin{corollary}[Exact tests and pure weights]
+\label{cor:F-exact-tests}
+Suppose $w$ is common $h$-lc. If every common actual valuation
+satisfies $q(f_j)=0$, then every common cone valuation satisfies
+$q^0(\bar f_j)=0$.
+The symbols $\bar f_j$ are invariant under $G$ and $\mathbb T_w$.
+Every nonzero pure weight valuation of the cone is $h$-lc and,
+after normalization by $\bar h=1$, is common.  Pure weights on
+the boundary of the dual weight cone are allowed under the
+closed-centre convention.
+\end{corollary}
+
+\begin{lemma}[Compression]\label{lem:F-compression}
+Suppose $w$ is as above, is $h$-lc, and satisfies $w(f_j)=0$ for
+every $j$. Let $v$ be a centred
+$G\times\mathbb T_w$-invariant, $\bar h$-lc quasi-monomial
+valuation on $S^0$, with normal finitely generated klt graded
+algebra $R^{00}$.  Suppose also $v(\bar f_j)=0$.
+The algebra $R^{00}$ has its joint grading by the old torus and
+the $v$-grading.  For arbitrarily small positive $\delta$ outside
+a countable exceptional set, it is the associated graded of an
+actual centred quasi-monomial valuation $w_\delta$ on $S$, whose
+grading evaluates a joint degree $(a,b)$ as $a+\delta b$.
+The valuation $w_\delta$ is $h$-lc, is exact on all $f_j$, and,
+after $h=1$ normalization, tends multiplicatively to $w$
+uniformly on all nonzero regular functions.  If $v$ is nonzero
+on an old degree-zero rational function, the rational rank of
+$w_\delta$ is strictly larger than that of $w$.
+\end{lemma}
+
+\begin{proposition}[Rank refinement]\label{prop:F-refinement}
+Consider a sequence of the preceding data in dimensions at most
+$n_0$.  Suppose that the common actual slices satisfy
+$q(f_j)=0$ identically, that $w$ is a centred point of the slice
+with the stipulated normal finitely generated klt graded algebra,
+and that the cones satisfy the following balance condition with
+one constant $C$ along the sequence:
+\begin{equation}\label{eq:F-balance}
+ q^0(s)\le Cw^0(s)
+ \quad\begin{gathered}
+ 0\ne s\in\breve R\text{ fine-homogeneous},\\[-2pt]
+ q^0\text{ common},\quad q^0(\bar h)=w(h)=1.
+ \end{gathered}
+\end{equation}
+After passage to a subsequence, the $w$ can be replaced by $w'$
+with all these properties, with a new fixed balance constant,
+and with
+\begin{equation}\label{eq:F-refinement-comparison}
+ w'(f)=(1+o(1))w(f)
+ \qquad(0\ne f\in\mathcal O_{S,x}),
+\end{equation}
+uniformly in $f$ in the normalization $w'(h)=w(h)=1$.
+On the angular model $V_{\mathbf u}$ of the new cone, the
+notation of Section~\ref{sec:D} then satisfies
+\begin{equation}\label{eq:F-angular-order}
+ T(\Gamma)\le(1+\eta_i)D(T),\qquad \eta_i\longrightarrow0,
+\end{equation}
+for every effective $\Q$-divisor $\Gamma\sim_\Q D$ and every
+primitive divisorial $T$ such that
+$a(T)=0$ and $T(\bar f_j)\ge0$ for all $j$.
+This is condition \textup{(C-$\alpha$)} of Section~\ref{sec:C}.
+The initial symbols here refer to $w'$.
+
+The balance hypothesis holds when $w$ is the exact minimizer
+on the common slice for the height $q(h)$.
+\end{proposition}
+
+
+Section G
+------------------------------------------------------------
+
+\begin{theorem}[Pinning obstruction]\label{thm:G-pinning}
+Fix an integer $n>0$ and $\epsilon>0$.  There is no sequence of the following
+data satisfying all the conditions below.  For each index $i$, let
+$x_i\in S_i$ be an $n$-dimensional complex klt germ with an action of a
+finite cyclic group $\Gamma_i$ fixing $x_i$, such that the quotient map is
+quasi-\'etale and $S_i/\Gamma_i$ is $\epsilon$-lc.  Suppose $K_{S_i}$ has a
+nowhere-vanishing local eigen-generator $\omega_i$, and let $h_i$ be a
+nonzero regular eigenfunction with the same character $\chi_i$ as
+$\omega_i$.  The character need not be faithful.  Assume that
+$(S_i,\divisor(h_i))$ is lc and admits a centred quasi-monomial valuation
+$T_i$ with
+\[
+ A_{S_i}(T_i)=T_i(h_i)=1.
+\]
+The excluded additional condition is: for every fixed integer $m>0$,
+there is an index $i(m)$ such that, for all $i\ge i(m)$ and every regular
+semi-invariant $s$ of character $\chi_i^m$,
+\begin{equation}\label{eq:G-input}
+                         T_i(s)\ge m.
+\end{equation}
+\end{theorem}
+
+\begin{lemma}[Non-cancellation in every degree]\label{lem:G-noncancellation}
+At the preceding stage, consider any finite linear combination of
+monomials in the accumulated and newly added $Y_j^+$ and the current
+$G_a$, with nonnegative exponents.  Suppose its least-$w$ group has a
+nonzero sum of initial symbols.  If all its terms have the same value at
+an actual point of the pre-pinning slice, their sum has that value.
+This assertion holds eventually simultaneously in all polynomial degrees.
+Moreover, at a later working valuation on this slice where these terms
+are tied, the same non-cancellation assertion transfers to its cone at
+every common test where the terms remain tied and the pair values are
+opposite.
+\end{lemma}
+
+\begin{lemma}[Uniform product growth]\label{lem:G-product-growth}
+Let $V$ be a finite-dimensional subspace of a function field over an
+algebraically closed field, containing $1$.  Put $g=\dim V$ and
+$a=\trdeg k(V)\le n$.  If $V^{(N)}$ is the span of $N$-fold products of
+members of $V$, there is $c_n>0$, depending only on $n$, such that
+\[
+                       \dim V^{(N)}\ge c_n gN^a
+                       \qquad(N\ge1).
+\]
+\end{lemma}
+
+\begin{proposition}[High-scale pin]\label{prop:G-high-pin}
+Consider a high stage with pre-pinning slice, working valuation $w$,
+baseline height $\mathfrak B$, scale $L$, and $s$ selected values as above.
+After passage to a subsequence, finitely many invariant tests of cost
+$O(L)$ can be added with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item The resulting slice contains a centred valuation.  The added tests
+are exact on that slice, all old tests and proportionalities remain in
+force, and the accumulated positive symbols remain algebraically
+independent at the exact minimizer and working valuation in
+\textup{(iii)} below.
+\item The selected values either vanish or lie on a single rational
+line.  In the line case their proportions have the bounds
+\eqref{eq:G-proportions}, with $L_-=L$.  An old anchor remains the anchor;
+if a new anchor is chosen, its scale is $L_1=L$.
+\item The baseline height is retained unless an anchor is first chosen,
+in which case it may change to the rational height
+\eqref{eq:G-baseline}.  It satisfies \eqref{eq:G-baseline-bounds} on the
+new actual slice.  At its exact minimizer $v'$, and at an arbitrarily close
+rank refinement $w'$, the anchor, when present, satisfies
+$|v'(Y_0^+)|,|w'(Y_0^+)|\ge cL_1$.  Both valuations dominate a fixed positive
+multiple of $w$ on every regular function.  In particular,
+\[
+                         w'\ge c w.
+\]
+\end{enumerate}
+Consequently \ref{it:G-tests}--\ref{it:G-working} hold with $L_-=L$.
+The next-scale assertions \ref{it:G-lattice}--\ref{it:G-field} are not
+assumed in constructing the pin or its rank refinement.
+\end{proposition}
+
+\begin{claim}\label{clm:G-new-row}
+After a subsequence there is an additional row of the form
+\eqref{eq:G-row}, exact at $v$, whose transverse part is independent of
+the previously imposed transverse rows.
+\end{claim}
+
+\begin{lemma}\label{lem:G-pin-consequences}
+The working valuation $w'$ furnished by Proposition~\ref{prop:G-high-pin}
+satisfies \ref{it:G-lattice}--\ref{it:G-field} at the next scale $L_-=L$.
+\end{lemma}
+
+
+Section H
+------------------------------------------------------------
+
+\begin{proposition}[Factorial saturation]\label{prop:H-saturation}
+For each example there is a primitive divisorial order $w$ over $X$ whose
+image centre on $Z$ contains the specified point and for which
+\begin{equation}\label{eq:H-saturation}
+ w(\Delta_s)\ge k A_X(w)\quad
+       (k>0,\ k\mid M_i),\qquad
+ w(\Delta_{s_b})=bA_X(w).
+\end{equation}
+After replacing the example by a geometric generic slice, one may assume
+that the image centre is a closed point $z$, that
+$1\le\dim X=\dim Z\le d$, and that all the inequalities hold for the
+complete local systems at $z$.
+\end{proposition}
+
+\begin{lemma}[The isomorphic case]\label{lem:H-canonical-cover}
+Suppose $f$ is an isomorphism near $z$.  There is a normal integral
+cyclic cover $S_0\to X$, with a unique point $o$ over $z$, a
+quasi-\'{e}tale cyclic action fixing $o$, and a divisor-free canonical
+generator of character one.  It has a regular eigenfunction
+$f_b=s_bt^b$ of character $b$ and a divisorial valuation $w_0$ centred
+at $o$ such that, with $A_0=A_{S_0}(w_0)>0$,
+\begin{equation}\label{eq:H-upstairs-saturation}
+ (S_0,\divisor(f_b)/b)\text{ is lc},\qquad
+ w_0(f_b)=bA_0,\qquad
+ w_0(a)\ge kA_0
+\end{equation}
+for every regular eigenfunction $a$ of character $k$ when
+$k>0$ and $k\mid M_i$.  The variety $S_0$ is klt.
+\end{lemma}
+
+\begin{lemma}[The signed anti-canonical torsor]\label{lem:H-signed-torsor}
+Suppose $f$ is not an isomorphism near $z$.  The signed algebra
+\begin{equation}\label{eq:H-signed-algebra}
+ R=\bigoplus_{k\in\Z}
+       H^0(X,\mathcal O_X(-kK_X))t^k,\qquad S_0=\Spec R,
+\end{equation}
+over the affine base is a finitely generated normal domain of dimension
+$\dim X+1$. Write $R_k=H^0(X,\mathcal O_X(-kK_X))t^k$ for its
+homogeneous part of degree $k$. Its \emph{positive chamber} is the open
+\[
+ S_0^+=\bigcup_{k>0}\ \bigcup_{0\ne a\in R_k}D_{S_0}(a),
+\]
+where $D_{S_0}(a)$ is the principal open on which $a$ does not vanish.
+This chamber has complement of codimension at least two. The form
+\begin{equation}\label{eq:H-torsor-form}
+ \Omega_0=t\omega_X\wedge d\log t
+\end{equation}
+is a divisor-free canonical generator of grading weight one, and every
+finite scalar grading subgroup acts quasi-\'{e}talely.
+
+The variety $S_0$ is klt, and
+$(S_0,\divisor(s_bt^b)/b)$ is lc.  There is a grading-fixed closed point
+$o$ over $z$ and a rational Gauss valuation $w_0$ centred there which
+satisfies \eqref{eq:H-upstairs-saturation} for every actual positive
+degree $k\mid M_i$.  Moreover there are numbers $\alpha>0$ and
+$F_w>\alpha$, depending on the example, for which every nonzero
+homogeneous section satisfies
+\begin{equation}\label{eq:H-two-slopes}
+ w_0(s t^k)\ge
+ \begin{cases}
+ k\alpha,&k>0,\\
+ |k|(F_w-\alpha),&k<0.
+ \end{cases}
+\end{equation}
+\end{lemma}
+
+\begin{lemma}[Padding and the actual quotient discrepancy]\label{lem:H-padding}
+In either construction, set
+\[
+ S=S_0\times\mathbf A^{b-1},
+ \qquad (o,0)\in S,
+\]
+and give every added coordinate $u_j$ character one.  Then $S$ is klt,
+the finite quotient $Y=S/\mu_r$ is $\epsilon$-lc, and the canonical
+generator on $S$ has character $b$.  There is a regular function $h$ of
+character $b$ such that $(S,\divisor(h))$ is lc and a centred
+quasi-monomial valuation $T_0$ such that
+\begin{equation}\label{eq:H-padding-equality}
+ A_S(T_0)=T_0(h)=bA_0.
+\end{equation}
+\end{lemma}
+
+\begin{proposition}[The pinning input]\label{prop:H-g-input}
+The failure sequence produces, after a subsequence, a fixed-dimensional
+sequence satisfying every hypothesis of Theorem~\ref{thm:G-pinning}.
+\end{proposition}
+
+\begin{lemma}[Point ideals and change of field]\label{lem:H-field-reduction}
+It suffices to prove Theorem~\ref{thm:main} over $\C$.  More precisely, a
+uniform positive lower bound for the log canonical threshold of
+$\mathfrak m_z\mathcal O_X$ implies the Cartier-section assertion after
+replacing the bound by a smaller number less than one.  Both the hypotheses
+and this ideal threshold are preserved by algebraically closed field
+extensions of characteristic zero.
+\end{lemma}
+
+
+Section Introduction
+------------------------------------------------------------------------
+
+\begin{maintheorem}[Uniform Cartier sections]\label{thm:main}
+For every integer $d>0$ and every real number $\epsilon>0$, there is a real number $\tau=\tau(d,\epsilon)>0$ with the following property. Let $(X,B)$ be an $\epsilon$-lc rational pair of dimension $d$, and let $f:X\to Z$ be a Fano type contraction of normal integral quasi-projective varieties with $\dim Z>0$. Suppose that $-(K_X+B)$ is nef over $Z$. For every closed point $z\in Z$, there are an open neighbourhood $U$ of $z$ and a nonzero effective Cartier divisor $D$ on $U$, with $z\in\Supp D$, such that
+\[
+\bigl(f^{-1}(U),B|_{f^{-1}(U)}+\tau f^*D\bigr)
+\quad\text{is log canonical.}
+\]
+\end{maintheorem}
+
+\begin{maintheorem}[Birational klt complements]\label{thm:birational}
+For each $d>0$ and $\epsilon>0$, there is a positive integer $n=n(d,\epsilon)$ such that the following holds. Let $X$ be an $\epsilon$-lc rationally Gorenstein variety of dimension $d$, and let $f:X\to Z$ be a birational Fano type contraction with $-K_X$ nef over $Z$. For every closed $z\in Z$, there is an effective rational divisor $\Delta$ over a neighbourhood of $z$ such that $(X,\Delta)$ is klt there and $n(K_X+\Delta)$ is Cartier and linearly trivial there, after shrinking the neighbourhood.
+\end{maintheorem}
+
+\begin{maincorollary}[Real boundaries]\label{cor:real-boundaries}
+Over $\C$, the conclusion of Theorem~\ref{thm:main} holds for effective real boundaries $B$ with $K_X+B$ real Cartier, with the same uniform dependence on $d$ and $\epsilon$.
+\end{maincorollary}
+
+
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