A smooth complex projective variety X of dimension n has a canonical
bundle K_X, the bundle of holomorphic n-forms; on projective space it is
very negative. For an ample line bundle L and an integer m, the adjoint
bundle is K_X + mL, with + denoting tensor product. Fujita's freeness
conjecture asserts that K_X + (n+1)L is generated by its
global sections, meaning that for every point x of X some global section
s satisfies s(x) not equal to 0. The same word means spanned: the
sections determine a regular map to a projective space with no point left
undetermined.

Connectedness of X is bookkeeping, since the statement applies to one
component at a time; in dimension zero every line bundle is generated.
Once the case m = n+1 is known, every larger twist follows. Write
r = m-(n+1) and apply the theorem on the product X x P^r with the ample
bundle L tensor O(1) on the P^r factor. The adjoint bundle of that
product is (K_X+mL) tensor O(n) on the P^r factor, so its restriction to
a slice X x point is K_X+mL, and a restriction of a globally generated
bundle is globally generated. For r = 0 the slice is X itself.

The bound n+1 cannot be lowered. On P^n with L = O(1), the canonical
bundle is O(-(n+1)), so K_X+(n+1)L is the trivial bundle, whose constant
section never vanishes. But K_X+nL is O(-1), which has no nonzero
section at all. No uniform threshold below (n+1) can work for every pair
(X,L). This also shows that the theorem cannot be proved by producing a
section of L, or by any argument that first needs one.

The difficulty is that one section has to avoid one prescribed point. A
general ample bundle may have many sections vanishing there, or none; the
canonical bundle may be arbitrarily negative; the dimension is unbounded.
The strategy converts failure at x into a numerical contradiction, then
lifts a nonzero value of the fiber at x to a genuine section.

Some vocabulary, kept terse. A projective variety is the common zero
locus of finitely many homogeneous polynomials in a projective space,
equipped with its algebraic structure; it is compact, and smooth means
that near each point it looks like an open ball in C^n. A line bundle
attaches a complex line to every point, algebraically; the standard
model is a divisor D, a finite integral combination of codimension-one
subvarieties, written O(D), with O(D)+O(E) = O(D+E) and dual O(-D). A
divisor is effective when all its coefficients are nonnegative. Ample
means that some positive power embeds X into a projective space. A
section is a vector in the line over each point, chosen algebraically;
the zero set of a nonzero section is a divisor, and H^0(X,M) denotes the
vector space of sections of a bundle M. Nef means nonnegative on every
curve, big means that the top self-intersection grows like an ample
bundle. A log resolution of a divisor D is a birational map pi: Y -> X
from a smooth Y on which the preimage of D, together with all exceptional
divisors, is a simple normal crossing divisor: locally a union of
coordinate hyperplanes, which makes every local computation monomial.

The argument has five stages. Failure of generation at x is first
translated into a strict inequality for how fast sections of large mL
vanish at x: the normalized mean order of a Taylor basis has lower limit
strictly above n/(n+1), which is a statement about lattice points. Next,
a finite-degree exponential average over bases and weighted orders is
minimized for one fixed t below n+1; the minimum is attained, and the
discrepancy of every minimizing order lies in a fixed compact interval
away from zero, uniformly in the degree. Third, every minimizer is shown
to be centered at the single point x: a positive-dimensional center would
force a weighted tangential moment to be small by a first variation, and
large by a uniform restriction estimate combined with Brunn-Minkowski.
Fourth, a point-centered minimizer yields rational supporting data on
finitely many polytopes of normalized section orders, from which an
effective ideal and a normal-crossing boundary are built, with exact
equality of boundary coefficient and log discrepancy on a reduced
divisor over x and strict inequality on every neighbor. Fifth,
Kawamata-Viehweg vanishing lifts a nonzero element of the fiber at x from
that divisor, and descent turns the lift into a global section of
K_X+(n+1)L not vanishing at x.

Earlier work forms a ladder toward the sharp twist. Curves follow from
Riemann-Roch. Surfaces follow from Reider's theorem. The threefold,
fourfold, and fivefold cases are due to Ein-Lazarsfeld, Kawamata, and
Ye-Zhu; Angehrn-Siu reached a quadratic bound in all dimensions, Helmke
supplied multiplicity estimates for the centers in the vanishing
argument, Heier combined the two, and Ghidelli-Lacini and Han pushed the
bound toward linear. Kawamata's argument for threefolds and fourfolds is
the closest relative of the local mechanism reused at the end.
pdf/kawamata-1997-alg-geom-9510004-fujita-3-and-4-folds.pdf
Fujita's own reduction of the spannedness question to a discrepancy
computation, in the remarks on the Ein-Lazarsfeld criterion, is the model
for the isolation step.
pdf/fujita-1993-alg-geom-9311013-ein-lazarsfeld-criterion.pdf
The fivefold case, with Ye and Zhu controlling centers of large
dimension, shows how much a bound of n+1 costs beyond dimension four.
pdf/ye-zhu-2020-1511.09154-fujita-dimension-5.pdf
The local positivity survey of Ein-Kuchle-Lazarsfeld sets out the
adjoint-induction method whose bookkeeping the present proof replaces by
minimization.
pdf/ein-kuchle-lazarsfeld-1995-9408003-local-positivity.pdf
Semicontinuity of complex singularity exponents, due to Demailly-Kollar,
is what makes the family of thresholds in this proof closed, and so what
makes the minimum exist.
pdf/demailly-kollar-2001-math-9910118-semicontinuity.pdf
Okounkov's observation that multiplicities of a filtered algebra behave
convexly underlies the lattice counting that opens the argument.
pdf/okounkov-2003-math-0002085-multiplicities.pdf
The multiplier ideal of a monomial ideal has a volume formula, due to
Howald, which is quoted once, for the threshold of the leading forms on
an exceptional divisor.
pdf/howald-2001-math-0003232-monomial-multiplier.pdf

    Main Theorem. Let a be a monomial ideal in affine n-space and let P be
    its Newton polygon. Then J(r a) is a monomial ideal, and contains
    exactly the monomials x^lambda with lambda + 1 in the interior of rP.

The linear bound of Han is the strongest published approximation to the
sharp twist, and its examples calibrate how much room remains.
pdf/han-2026-2609.01574-linear-bound.pdf
A stronger intersection-number form was announced by Chan and withdrawn
after a local extension step failed, an outcome that fixes attention on
the exact coefficient comparisons used below.

Notation throughout the proof. X, L, n are as in the theorem. Put
    P = K_X + (n+1)L,   R_j = H^0(X, jL),   N_j = dim R_j.
Ampleness gives the Hilbert asymptotic
    N_j = (L^n / n!) j^n + O(j^(n-1)),   with L^n a positive integer.

All centers are CLOSED centers. So a divisor centered along a
subvariety W is an admissible test at every point of W, not just at its
generic point.

WEIGHTED MONOMIAL ORDERS. Fix a smooth projective birational model
pi: Y -> X and local regular coordinates adapted to the ENTIRE relative
canonical divisor: every component of that divisor meeting the
coordinate neighborhood is a coordinate hyperplane. Choose a subset
E_1,...,E_r of the coordinate divisors and assign positive real
weights u_1,...,u_r; all remaining coordinates get weight zero. The
order of a germ is the least weight of a nonzero monomial in the local
equations of the E_j, with coefficients in the remaining coordinates.
The log discrepancy of this test is
    A(v) = sum_j u_j A(E_j),
where for a prime divisor F on a model, A(F) = 1 + ord_F(K_{Y/X}) is
the log discrepancy. A coordinate divisor absent from the relative
canonical divisor has A = 1. The chosen test has a positive-support
stratum; the closed image on X serves as the center. This adaptation
condition is exactly what makes the displayed formula for A(v) correct.
Positive multiples of a prime order are allowed, and scaling discrepancy
by the same factor. Approximating real weights by positive rational
weights on a fixed support approximates both the discrepancy and the
orders of any fixed finite family of sections.

LEMMA (local computation of a monomial order). A monomial order with
fixed positive support can be computed in the Taylor ring at any smooth
point of its open stratum. Its value is unchanged by replacing each
positive-weight coordinate by a unit multiple, or by changing the
complementary zero-weight coordinates.
Proof. For a threshold c, the relevant ideal is generated by monomials
in the positive-weight parameters of weight at least c, and these are
preserved up to units. In the completed local ring the ideal is
extended from a finite-colength monomial ideal in the positive-weight
variables, hence primary to the intersection of those coordinate
hyperplanes. Localization at the
generic point of the stratum followed by contraction does not change
membership, and completion is faithfully flat. So a coefficient that
vanishes at the chosen point but not identically on the stratum is a
nonzero series in the zero-weight variables; it does not raise the
weighted order.

LEMMA (SNC retraction). Let D_1,...,D_N be effective divisors on X and
let Y -> X be a log resolution of the union, with SNC exceptional
components E_j. For a positive multiple v of a prime-divisor order over
Y set u_j = v(E_j). Then
    v(D_i) = sum_j u_j ord_{E_j}(D_i),   and   A(v) >= sum_j u_j A(E_j).
If some u_j is nonzero, the monomial order with these weights at the
stratum containing the generic center of v has the same orders on the
D_i, no larger discrepancy, and a closed center on X containing that
of v.
Proof. Each pulled-back divisor is locally a monomial times a unit,
giving the first equality. The reduced SNC divisor is log canonical: a
logarithmic volume form has at most a simple pole along each prime on a
higher model. So the log discrepancy relative to Y is at least sum u_j;
adding the order of K_{Y/X} gives the inequality and the claimed
discrepancy of the monomial order. The stratum closure contains the
generic center, so its image contains the original closed center.

Resolution with a prescribed normal crossing boundary is due to Hironaka
and, in the economical form, to Wlodarczyk; the divisors whose
discrepancies enter are the components of that boundary together with the
exceptional locus. Log canonical centers, and the way a single center is
isolated from the strata of a boundary, are catalogued by Kollar.
pdf/kollar-1107.2863-sources-log-canonical-centers.pdf

THRESHOLDS. For a coherent ideal a on a smooth variety, the local log
canonical threshold at x is computed on a log resolution by
    lct_x(a) = min over F with x in pi(F) and ord_F(a) > 0 of
               A(F) / ord_F(a).
The minimum is +infinity for the unit ideal, and 0 for the zero ideal.

LEMMA (closed threshold loci). Let V be a smooth complex projective
variety, S a complex algebraic variety, and a an ideal on V x S given
locally by algebraic families of generators. For a real c, the locus
    {(y,s) in V x S : lct_y(a_s) <= c}
is Zariski closed, where a_s is the ideal generated by the restricted
generators (no flatness required).
Proof. The analytic semi-continuity of complex singularity exponents of
Demailly-Kollar gives closedness in the classical topology, because
phi = (1/2) log sum |g_i|^2 has e^phi locally Holder continuous and
translation handles a moving marked point. Constructibility comes from
stratifying: over the generic point of each component resolve the ideal,
spread the resolution over a dense open, and the fiber maps are log
resolutions with fixed orders and coefficients, so the threshold
condition is a finite union of proper images (algebraic, proper, hence
constructible); induction on the complement finishes. A subset that is
constructible and classically closed is Zariski closed, since a Zariski
open subset of an irreducible variety is classically dense.

The local threshold is also the quantity that must be compared between
the fibers of a family, and the closedness obtained above is exactly that
semicontinuity; how such thresholds behave under birationally varying and
adjacent data is surveyed by Hacon-McKernan-Xu. Multiplier ideals, whose
local integrability exponents are these thresholds, are treated
systematically by Lazarsfeld.
pdf/hacon-mckernan-xu-2014-1208.4150-acc-lct.pdf
pdf/lazarsfeld-multiplier-ideals.pdf

THEOREM (Kawamata-Viehweg vanishing). Let T be smooth projective over C.
Let Delta be an effective rational SNC divisor with coefficients in
[0,1), and let D be an integral divisor. If D - (K_T + Delta) is nef and
big, then H^i(T, O_T(D)) = 0 for every i > 0.
Equivalently, using Q = D - K_T - Delta and its rounding: if M is
integral and B is an effective rational SNC divisor with M - B nef and
big, then H^i(T, O_T(K_T + M - floor(B))) = 0 for i > 0. This is the
form used below.

The result is due to Kawamata and, independently, to Viehweg; the
rounding formulation quoted, in which a fractional boundary is replaced
by its round-up, is the one recorded in Fujino's lecture note. Positivity
statements of the same family, including weak positivity for direct
images of adjoint bundles, are collected in Fujino's later notes.
pdf/fujino-2017-1506.08873-notes-weak-positivity.pdf

LEMMA (descent). Let pi: T -> V be a proper birational morphism of
smooth varieties, P_V a line bundle on V, and C an integral divisor on T
whose positive part is exceptional. A section of pi^*P_V + C defines a
regular section of P_V on V. If a prime F on T maps to a point x, the
coefficient of C on F is zero, and the section is nonzero generically on
F, then the descended section is nonzero at x.
Proof. View the section as a rational section of P_V on the common
function field. Its orders at all nonexceptional primes are
nonnegative, because the coefficients of C there are nonpositive. So it
is regular in codimension one on the normal variety V, hence
everywhere. Under the extra assumptions its pullback has order zero
along F. A local function vanishing at x has positive order along every
prime centered at x, so the descended section cannot vanish at x.

Fix a closed point x in X. Choose regular parameters at x and a local
frame of L; use powers of that frame as frames of jL. Order Taylor
monomials first by total degree, then lexicographically, always taking
the smallest monomial. Let
    H_j = { in(s) : 0 != s in R_j }   inside Z_{>=0}^n
be the set of initial exponents. Then |in(s)|_1 = mult_x(s), the
multiplicity (order of vanishing) of s at x.

The identification of the number of initial exponents with the dimension
of the section space, and the convexity of the resulting set of exponents
under addition, are the elementary parts of the theory of Newton-Okounkov
bodies as developed by Lazarsfeld-Mustata and by Kaveh-Khovanskii, and
whose asymptotic jumps for filtered linear series are studied by
Boucksom-Chen. The order in use is the plain Taylor order at a point,
which keeps the identification elementary; only the multiplicativity is
used.

LEMMA (initial exponents give a basis). The set H_j has cardinality N_j,
and representatives of its distinct exponents form a basis of R_j.
There is a constant C > 0, independent of j and s, with
    mult_x(s) <= C j   for every nonzero s in R_j.
Moreover H_j + H_q is contained in H_{j+q}.
Proof. Taylor expansion is injective on R_j: a section with zero germ
vanishes on a nonempty open set, hence on the integral variety X. Each
monomial coefficient ranges over a one-dimensional space. Successively
select a smallest initial monomial and eliminate its coefficient from
the remaining vectors; this produces a basis with distinct initials, and
any linear combination has the initial of its first nonzero summand.
That proves the cardinality claim. For the multiplicity bound, choose a
fixed sufficiently ample multiple kL, and general members of its linear
system through x cut out a curve smooth at x and not contained in the
divisor of a prescribed nonzero section of jL. Local intersection
multiplicity bounds mult_x(s) by the degree of that divisor on the
curve, at most j k^(n-1) L^n. In dimension one this is just the degree
bound for an effective divisor on X. Finally the monomial order is
multiplicative, so initial exponents add under products of sections.

PROPOSITION (ordinary-order alternative). If P = K_X+(n+1)L is not
generated at x, then
    liminf over j of (1/(j N_j)) * sum_{gamma in H_j} |gamma|_1
        > n/(n+1).

Proof. Failure of the strict inequality will be shown to force
generation at x. Three steps: lattice near-equality forces
L^n = 1 along a subsequence; the leading forms on the exceptional
divisor of the blowup at x then have a strict multiplier threshold;
vanishing lifts a nonzero section from that divisor.

Put V = L^n. The number of nonnegative integral points of total degree
at most r is binom(r+n, n). Filling points in order of increasing total
degree, and using N_j = V j^n / n! + O(j^(n-1)), gives
    liminf (1/(j N_j)) sum_{gamma in H_j} |gamma|_1
        >= V^(1/n) * n/(n+1).
Indeed the last occupied degree for the N_j smallest lattice points is
V^(1/n) j + O(1), and summing total degree over these points gives mean
degree V^(1/n) n j/(n+1) + O(1). Since V is a positive integer, V^(1/n)
is at least 1, so if the strict inequality of the proposition fails,
then necessarily V = 1 and equality holds along a subsequence.

Let Delta = {u in R^n : u >= 0, |u|_1 <= 1}. Along the equality
subsequence, every fixed open ball with closure in the interior of Delta
meets j^{-1} H_j for all large j. (Compare H_j with the N_j smallest
lattice points T_j. If such a ball were missed, at least c_1 j^n points
of T_j of degrees at most (1-c_2) j would be absent; each replacement
outside T_j has degree at least j-O(1); the remaining replacements
cannot lower the degree sum, so the normalized mean would exceed the
minimum by a fixed amount, contradicting equality.)

Choose a rational t with n < t < n+1 and put c = (1/t,...,1/t), a
point in the interior of Delta. Choose finitely many small balls with
closures in that interior such that any choice of one point per ball has
convex hull containing c in its interior. For one large degree j on the
subsequence, choose initial exponents gamma_nu in H_j, one per ball after
scaling by 1/j. The rational polytope Q = conv{gamma_nu / j} contains c
in its interior. Its slice {|u|_1 = n/t} has rational points p_1,...,p_b
whose convex hull contains c in its relative interior (in dimension one
that slice is the single point c).

Write each p_l as a rational convex combination of the gamma_nu / j and
choose sections s_nu with those initials. If t = a/b in lowest terms,
choose Q_0 divisible by a and by every denominator; then products of Q_0
sections give sections r_1,...,r_b of j Q_0 L. Put h = j Q_0 / t, a
positive integer by construction. The initial exponents beta_l satisfy
    r_l in H^0(X, h t L),   beta_l = j Q_0 p_l,   |beta_l|_1 = n h.
Thus conv{beta_l / h} contains the all-ones vector 1 in relative
interior inside the hyperplane {|u|_1 = n}.

Let p: B = Bl_x X -> X be the blowup, E the exceptional divisor
(P^{n-1} for n>1; for n=1 take p the identity and E = x). Removing
the common order n h from the pullbacks of the r_l gives a subsystem of
    H = p^*(h t L) - n h E
with base ideal b. Its restriction to E is generated, in local frames,
by the leading homogeneous forms g_l of the r_l, each of degree n h.
CLAIM. lct_z(E, b|_E) > 1/h for every z in E. (For n=1 the restricted
ideal is the unit ideal.)

For n > 1, replace the g_l by the initial monomials z^{beta_l}. On the
chart of E where z_k != 0 the exponents are obtained by omitting
coordinate k. Projection from the hyperplane |u|_1 = n is an affine
isomorphism, so the normalized exponent polytope contains the all-ones
vector in ordinary interior. Hence for every nonzero u in R_{>=0}^{n-1},
    (1/h) min_l <u, beta_l^(k)>  <  sum_i u_i.
For a prime order over this chart use its values on the coordinates as
u. The left side is the order of the monomial ideal over h; the right
side is at most the log discrepancy (SNC inequality of the retraction
lemma). So the monomial ideal is klt at coefficient 1/h; computing its
threshold on a log resolution gives a strict threshold > 1/h at every
point. (If u=0 the order is 0 and the discrepancy is positive.)

Transfer to the original forms by an algebraic degeneration. Only
finitely many monomials occur; choose an integral weight w making the
lex initial of each g_l its unique smallest-weight term. The forms
    g_{l,lambda}(z) = lambda^{-<w,beta_l>} g_l(lambda^{w_1} z_1, ...)
have polynomial coefficients in lambda with monomial special fiber.
Such forms define an algebraic family of ideals on E x A^1. By the closed-
threshold-loci lemma, the locus where the fiber threshold is <= 1/h is
closed; its image in A^1 is closed (E is proper) and misses 0. For some
nonzero parameter the threshold exceeds 1/h everywhere on E. Every
nonzero fiber is a diagonal automorph of the original forms, so the
claim holds for the g_l.

It remains to lift from E. The blowup formula gives
    p^*P - (K_B + E) - H/h = (n+1-t) p^*L.
Take a projective log resolution sigma: T -> B of E and b, with strict
transform Eprime and b O_T = O_T(-G). Eprime is not a component of G
(the restricted system is not identically zero). Set
    D_T = K_T + sigma^*(p^*P - K_B - E) - floor(G/h),
an integral divisor. Then
    D_T - K_T - {G/h} = (n+1-t)(p sigma)^*L + (sigma^*H - G)/h.
The first summand is nef and big; the second is nef, since sigma^*H - G
is globally generated. The fractional boundary has SNC support. By
Kawamata-Viehweg vanishing, H^1(T, O_T(D_T)) = 0.

Let sigma_E: Eprime -> E be the induced morphism. Adjunction identifies
the restriction of D_T + Eprime with
    sigma_E^* (p^*P |_E) + K_{Eprime/E} - floor(G|_{Eprime}/h).
Indeed Eprime + Supp(G) is SNC, each component restricts with
multiplicity one, distinct components cannot restrict to the same prime,
so taking floors commutes with restriction; and the generators on
Eprime generate O_{Eprime}(-G|_{Eprime}), the pullback of the
restricted ideal. The strict threshold implies
    K_{Eprime/E} - floor(G|_{Eprime}/h) >= 0,
an effective correction. The first bundle is trivial with fiber P_x. A
nonzero vector in that fiber, times the canonical section of the
effective correction, gives a section on Eprime nonzero at its generic
point. The exact sequence for Eprime and the vanishing lift it to a
section of D_T + Eprime.

Relative to sigma^* p^* P, the correction of this lifted bundle is
    C_T = K_{T/B} - sigma^* E + Eprime - floor(G/h).
Its coefficient on Eprime is zero, and on every other sigma-
nonexceptional divisor it is nonpositive; its positive part is thus
exceptional over X. Since Eprime maps to x and the lifted section is
nonzero at its generic point, the descent lemma applied to p sigma gives
a section of P nonzero at x. This proves the proposition, including
dimension one, when Eprime is the point itself.

Both the basis and the test now vary; the quantity to minimize is
    F_{m,t}(v, s) = (1/N_m) sum_i exp( A(v)/t - v(s_i)/m ),   t > 0,
over ordered bases s = (s_1,...,s_N_m) of R_m and monomial tests v. Let
    f_{m,t}(z) = inf over v with z in c(v), and over bases, of F_{m,t}.
The symbol c(v) denotes the closed center. Positive rescaling of v is allowed:
approximating its positive weights by rational weights preserves the
center and approximates the orders of finitely many sections and the
discrepancy. So the same infimum is obtained using positive multiples of
prime-divisor orders.

LEMMA (order bound). If the center of v contains x, then every nonzero
section s satisfies
    v(s) <= A(v) mult_x(s).
In particular, for s in R_m, 0 <= v(s)/A(v) <= C m, with C from the
initial-exponent lemma.
Proof. For an effective divisor with local equation g of multiplicity
r > 0 at a smooth point, lct_x(g) >= 1/r. (Weierstrass preparation: on
a small polydisc g is a unit times a monic degree-r polynomial in one
coordinate; for any c < 1/r Holder bounds the integral of |g|^{-2c} by
products of finite one-variable integrals, so Fubini gives local
integrability.) Computing the threshold on a log resolution gives the
bound v(s) <= A(v) mult_x(s) for prime orders whose center contains x,
and rational approximation gives it for monomial tests.

LEMMA (uniform sublevel bounds). Suppose P is not generated at x. Then
there are numbers 0 < t_0 < n+1, c > 0, 0 < delta < 1, 0 < a_star <=
a^* < infinity, and an integer m_0 such that for every m >= m_0 and
every t in [t_0, n+1],
    c <= f_{m,t}(x) <= 1 - delta.
Moreover any pair centered with image containing x and with
F_{m,t}(v,s) <= 1 - delta/2 satisfies a_star <= A(v) <= a^*. All these
constants are independent of m and t in the stated ranges.
Proof. By the ordinary-order alternative there is epsilon > 0 with
    (1/(m N_m)) sum_{gamma in H_m} |gamma|_1 >= n/(n+1) + 3 epsilon
for all large m. Choose t_0 < n+1 so close that n/t_0 <=
n/(n+1) + epsilon. Take a basis with distinct ordinary initials and use
v = lambda * ord_x, the ordinary point order, with A(ord_x) = n (for
n=1 this is the order of the point). The linear coefficient at
lambda = 0 is at most -2 epsilon uniformly in m and t; by the
multiplicity bound the quantities n/t - mult_x(s_i)/m are uniformly
bounded, so the quadratic remainder is O(lambda^2). A fixed small
lambda gives F_{m,t} <= 1 - delta for a fixed delta > 0.

Next a uniform supply of low-order jets. Choose k with kL very ample
and sections s_0,...,s_n of kL with s_0(x) != 0 and z_i = s_i/s_0
regular parameters. For each residue class mod k choose a fixed large
b_r in that class with b_r L globally generated and tau_r in
H^0(X, b_r L) nonzero at x. Write m = qk + b_r. For |alpha|_1 <= q the
sections tau_r s_0^{q-|alpha|} s_1^{alpha_1}...s_n^{alpha_n} have germs
equal to a common unit times z^alpha in a local frame of mL. Let
    r_m = min(q, floor(m/(2(n+1)))).
The jets of these sections for |alpha|_1 <= r_m are independent modulo order
> m/(2t). For large m, r_m is at least a fixed positive multiple of m,
so Hilbert asymptotics give a constant 0 < kappa < 1 with
    codim { s in R_m : mult_x(s) > m/(2t) } >= kappa N_m.
By the order bound the subspace { s : v(s) > m a/(2t) } sits inside
this one, where a = A(v). Thus at least kappa N_m vectors of any basis
(adapted or not) have v(s_i) <= m a/(2t), so
    F_{m,t}(v,s) >= kappa exp( a/(2t) ) >= kappa exp( a/(2(n+1)) ).
This gives the lower bound c = kappa and bounds a above on the sublevel
set. Finally F >= exp(-C a) by the order bound and multiplicity bound;
choosing a_star small enough that exp(-C a_star) > 1 - delta/2, no
sublevel pair has a < a_star. Enlarging the upper bound if needed gives
a^* >= a_star.

LEMMA (closed realizability loci). Fix m with N_m > 0 and a decreasing
list alpha_1 >= ... >= alpha_{N_m} >= 0 of rationals. The set of points
z in X for which some ordered basis and some prime-divisor order v with
z in c(v) satisfy v(s_i)/A(v) >= alpha_i (all i) is Zariski closed.
There is also a closed incidence locus in the product of X with the
complete flag variety of R_m, recording realizability by bases adapted
to a given flag.
Proof. Let F be the projective flag variety; a flag has subspaces
U_i of dimensions i and base ideals b_i, by evaluation after twisting by
-mL. If the list holds, the flag spanned by the first i basis vectors
satisfies v(b_i) >= alpha_i A(v) (because the list is decreasing);
conversely these inequalities imply the list for every adapted basis.
If some alpha_i > 0, choose M making M/alpha_i integral for each
positive entry and set a_alpha = sum_{alpha_i>0} b_i^{M/alpha_i}. The
simultaneous inequalities are equivalent to v(a_alpha) >= M A(v). A
divisorial witness with center containing z exists iff
    lct_z(X, a_alpha) <= 1/M,
since the local threshold is computed by a component on a log
resolution. The universal subbundles on F give algebraic families of
generators on X x F, hence for the powers and sum; the closed-
threshold-loci lemma applied to this condition gives a closed incidence
locus, whose projection to X is closed (F proper). If all entries
vanish, both loci are the whole parameter spaces.

Attainment of a minimum for a functional built from a valuation and a
graded linear series has a precedent in the valuative thresholds of ample
bundles, where Blum-Jonsson prove that the minimum is attained; the
functional is a finite-degree exponential average rather than a
threshold, and its attainment is proved directly.

PROPOSITION (existence and uniform bounds for minimizers). Suppose P is
not generated at x. For the constants and ranges of the uniform
sublevel bounds, f_{m,t}(x) satisfies c <= f <= 1-delta and is attained
by an ordered basis of R_m and a nonzero real-weight monomial test on a
projective log resolution of the basis divisors. Every minimizing test
has a_star <= A(v) <= a^*. The same bounds hold eventually along every
minimizing sequence, so a_star is uniformly positive.

Proof. Only attainment remains. Fix m,t in range and take divisorial
pairs whose objective decreases to f_{m,t}(x); eventually such pairs lie in
the sublevel set. Order each basis by decreasing normalized values and
pass to a subsequence with A(v_b) -> a in [a_star, a^*] and
q_{b,i} = v_b(s_{b,i})/A(v_b) -> q_i. This is possible since
0 <= q_{b,i} <= C m. The limit list is decreasing and
    f_{m,t}(x) = (1/N_m) sum_i exp( a/t - a q_i/m ).
At least one q_i is positive, else the right side is exp(a/t) > 1,
contradicting f <= 1 - delta.

Choose decreasing nonnegative rational lists alpha^(k) increasing
componentwise to (q_i), strictly below q_i whenever q_i > 0. Each list
is satisfied by all sufficiently late pairs. By the closed-realizability
lemma the flags admitting a witness at x for alpha^(k) form a nonempty
closed subset F_k of the flag variety; these are nested, so projectivity
gives a flag in the intersection. Fix a basis s_1,...,s_{N_m} adapted
to this flag. Resolve the divisors of these sections on one smooth
projective model Y -> X, including exceptional divisors in the SNC
support. For each large k, choose a witness for the fixed flag and
alpha^(k) on a common smooth model dominating Y. Retracting to the fixed
SNC model (retraction lemma) preserves all section orders, does not
increase discrepancy, and has center on X containing x. Normalize to
discrepancy one; it satisfies tilde_v_k(s_i) >= alpha_i^(k).

There are finitely many SNC strata and components on Y. Pass to a
subsequence using the same stratum component, coordinate divisors indexed
by a fixed set J. The normalized weights belong to the compact simplex
    { (u_j) : u_j >= 0, sum_j A(E_j) u_j = 1 }.
All A(E_j) are positive since X is smooth. Pass to a limit of the
weights; some u_j may become zero. Take the component of the remaining
positive-support intersection that contains the old stratum component;
its closure contains the old center, so its image still contains x. The
limit is a nonzero monomial test tilde_v of discrepancy one, and section
orders are linear in the weights, so tilde_v(s_i) >= q_i for all i.

Set v = a tilde_v. Comparing with the limit objective,
    F_{m,t}(v,s) <= (1/N_m) sum_i exp( a/t - a q_i/m ) = f_{m,t}(x).
The reverse inequality holds by definition of the infimum, so this pair
attains it. The discrepancy bounds for all minimizers and eventually for
minimizing sequences follow from the uniform sublevel bounds.

LEMMA (specialization comparison). Fix m with N_m > 0 and a closed
irreducible W in X containing x. At a very general closed point y of W,
for every t > 0,
    f_{m,t}(y) >= f_{m,t}(x).
Proof. There are countably many rational lists as in the closed-
realizability lemma. For each, intersect its closed locus with W and
discard it if the intersection is proper in W; the complement contains
very general closed points. For any remaining y, every such list
realized at y is realized at x. Take a test and basis at y, reorder so
q_i = v(s_i)/A(v) is decreasing, a = A(v) > 0. Approximate (q_i) from
below by rational lists; each is realized at x, and after scaling to
discrepancy a gives an objective at x at most
    (1/N_m) sum_i exp( a/t - a alpha_i^(b)/m ).
Taking the limit gives f_{m,t}(x) <= F_{m,t}(v,s). Infimizing over tests
at y proves the claim. The discarded loci are independent of t, giving
the simultaneous statement for all positive t.

PROPOSITION (point center). Suppose P is not generated at x. Then there
is a real 0 < t < n+1 such that, for arbitrarily large integers m, the
infimum f_{m,t}(x) is attained and EVERY minimizing monomial pair has
center exactly {x}. (A minimizing pair is a monomial test together with a
basis attaining the infimum, over all such pairs.) The chosen t and
this point x stay fixed from now on.

Before the argument, a uniform estimate: the image of the restriction
map from sections of hL to a subvariety W has dimension bounded below by
a function of h alone, uniform over all W of a fixed dimension.

Fix k with kL very ample and embed X by its complete system. For an
integral d-dimensional closed W in X put
    I_W(h) = dim image( R_h -> H^0(W, hL|_W) ).

LEMMA (uniform restriction estimate). For each 1 <= d <= n there are
constants C_d, H_d depending only on (X,L,k,d) with
    (d! I_W(h))^{1/d} >= h - C_d    for all h >= H_d,
for EVERY integral d-dimensional closed W in X.
Proof. Set D_0 = k^d. The degree of W in the fixed embedding is
D = (kL)^d . W = k^d (L^d . W) >= D_0, since L^d . W is a positive
integer. A general linear projection gives a finite morphism
pi: W -> P^d of degree D, with separable generic fiber (characteristic
zero). Write K = C(P^d), E = C(W). Choose a nonzero linear coordinate
s_0 of the projection; all points of the geometric generic fiber lie in
s_0 != 0. Select D_0 distinct such points. Over an algebraic closure of
K, affine linear forms separate any two of these points; for each chosen
point, multiplying at most D_0 - 1 such forms gives a polynomial
vanishing at the others and not at that one. Homogenizing with s_0 gives
ambient degree-D_0 forms whose normalized values span the functions on
these D_0 points.

Let V_{D_0} be the space of ambient degree-D_0 forms. Restriction and
division by s_0^{D_0} give a K-linear map from V_{D_0} tensor K to E of
rank at least D_0; the rank is checked after extending scalars, where the
evaluation argument applies. Choose ambient degree-D_0 forms
F_1,...,F_{D_0} whose normalized images F_i/s_0^{D_0} are K-independent.
The forms F_i are taken from a fixed monomial spanning set, so the
coefficients lie in the base field. For q >= D_0, multiply each F_i by a
monomial basis of forms of degree q - D_0 in the projection coordinates.
The products restrict to independent sections on W (a relation, divided
by s_0^q, would contradict K-independence; the base forms then vanish
on P^d).
The products are restrictions of sections in R_{qk}, so
    I_W(qk) >= D_0 binom(q - D_0 + d, d),
hence (d! I_W(qk))^{1/d} >= k(q - D_0) = qk - k^{d+1}. To cover all
degrees, choose once and for all an integer h_r in each residue class r
mod k with h_r L globally generated; there is a section of h_r L not
identically zero on any W, and multiplication by it injects the
degree-qk restriction space into the degree-(qk + h_r) one (W integral).
For h = qk + h_r, q >= D_0, this gives (d! I_W(h))^{1/d} >=
h - (h_r + k^{d+1}). Taking maxima over the finitely many residue
classes proves the lemma, with both C_d and H_d at most
max_r h_r + k^{d+1}, independent of W.

The proof has two halves. For an arbitrary minimizing pair with a
center of positive dimension, the tangential moment T_m is first bounded
above by d/t, with t still a free positive parameter. The moment is then
bounded below by a quantity that grows with the slice count, and the two
bounds are incompatible for large degrees once t is chosen close enough
to n+1. The choice of t and of the degree sequence is made in the second
half.

Suppose a minimizing pair in degree m has a test v whose center W has
dimension d > 0. Keep m, v and one minimizing basis fixed for the whole
calculation; the parameter t enters the estimates only through the
eventual choice made in the second half. Let Z be the closure of the
defining stratum of v on a smooth model Y -> X, let r be the number of
positive weights, and put
k' = n-d, so that r is at most k'. Since the center is closed and W is
its image, every point of W is an admissible place to test the same
order, and a very general smooth point y of W may be chosen outside the
countably many proper closed loci of the specialization comparison. At
such a y,
    f_{m,t}(y) >= f_{m,t}(x).
Above y choose a point z of the open positive-support stratum at which
Z -> W is smooth and which avoids the extra components of the relative
canonical divisor; generic smoothness gives such a point over C.

Take regular parameters at z in three groups: the r parameters cutting
out the positive-weight crossing components; k'-r parameters along the
fiber of Z -> W; and d parameters pulled back from affine linear
coordinates on W vanishing at y, chosen from the fixed very ample
multiple used in the restriction estimate and completed by the fiber
parameters. On the open stratum the positive equations change only by
units and all complementary parameters carry weight zero, so by the local
computation lemma the weighted order of a germ equals the least
weighted degree of its Taylor expansion in these parameters.

Write Taylor exponents as pairs (p, beta) in Z^{k'} x Z^d, where ell(p)
is the old weighted order of the transverse part and beta records the
tangential part. Order monomials by increasing ell(p), then by
increasing |beta|_1, then by one fixed additive monomial well-order such
as total degree followed by lexicographic order. The result is an
additive well-order: below any fixed weighted bound only finitely many
transverse exponents occur, so the least element always exists.

Let Gamma_j be the set of initial exponents of nonzero sections of R_j
in this order. Each leading coefficient has one-dimensional leaves, so
elimination gives
    #Gamma_j = N_j,   and   Gamma_j + Gamma_q is contained in Gamma_{j+q},
the second because the order is additive and the local frames in all
degrees are powers of one frame. A degree-m basis representing the
distinct elements of Gamma_m is adapted to the filtration by v: its
sorted values dominate those of any other basis, so replacing the
minimizing basis by it cannot increase the objective and the pair is
still minimizing. For this test put
    Q_j = sum over (p,beta) in Gamma_j of exp(-ell(p)/m),
so that Q_m = N_m f_m exp(-A(v)/t) in degree m.

Minimality bounds the weighted mean tangential degree from above.
Perturb v at z by giving weight eps to each of the d tangential
parameters and weight eps^2 to each of the k'-r fiber parameters while
leaving the positive weights alone. For eps > 0 this is again a
monomial test, with center z on the model and y on X, and since the added
coordinate divisors have discrepancy one near z,
    A(v_eps) = A(v) + d eps + (k'-r) eps^2.
For a section whose initial exponent is (p,beta), the perturbation obeys
    v_eps(s) = ell(p) + eps |beta|_1 + O(eps^2)   as eps decreases to 0.
The expansion is legitimate section by section: only finitely many
transverse exponents lie below a fixed old weighted bound, so there is a
positive gap to the next larger old weight in a bounded interval, and
nonnegativity of the exponents excludes larger old weights for small
eps; among the terms of least old weight the least tangential total
degree fixes the coefficient of eps, and the remaining differences touch
only the eps^2 coefficient. No uniformity in m is needed, because one
finite basis is differentiated at a time.

Let F(eps) be the objective of this basis and v_eps in degree m. The
specialization comparison at y gives
    F(eps) >= f_{m,t}(y) >= f_{m,t}(x) = F(0),
so the right derivative of F at zero is nonnegative. Dividing by the
positive common factor and using the two expansions,
    T_m := ( sum over (p,beta) in Gamma_m of |beta|_1 exp(-ell(p)/m) )
           / ( m Q_m )  <=  d/t.
This is the upper bound that the growth estimate will contradict.

The same moment is now bounded below, using only sections restricted to
W. For a fixed transverse exponent p put
    G_p(j) = { beta : (p,beta) in Gamma_j },   b_p(j) = (d! #G_p(j))^(1/d),
with b_p(j) = 0 when the slice is empty. A section whose restriction to
W is nonzero has old weight zero and no fiber variables, so eliminating
within the restriction space produces distinct pure tangential initials
and so gives
    #G_0(h) >= I_W(h).
The uniform restriction estimate therefore gives b_0(h) >= h - C_d for
all large h, with C_d and the threshold depending only on X, L, k and d,
never on the center, its singularities, or the chosen test.

One exact inclusion in bounded degree is also needed. Put h_0 = kd and
write the selected affine coordinates as s_i/s_0, with s_0(y) nonzero
and s_i(y) = 0. For each beta in {0,1}^d the section
    product over i of s_i^(beta_i) times s_0^(d-|beta|_1)
lies in R_{h_0} and has initial exponent (0,beta): the change from the
projective trivialization to the local frame is a unit, and a unit has
initial exponent zero. Hence
    {0,1}^d is contained in G_0(h_0).

The cube turns the continuous Brunn-Minkowski inequality into the
discrete statement required. For nonempty finite A and C in Z^d,
    (A+[0,1]^d) + (C+[0,1]^d) is contained in A+C+{0,1}^d+[0,1]^d,
and the volume of a finite lattice set thickened by the unit cube is
its cardinality, because distinct unit cubes have disjoint interiors.
Brunn-Minkowski for these finite unions of cubes, in the form given by
Gardner, therefore yields
    #(A+C+{0,1}^d)^(1/d) >= #A^(1/d) + #C^(1/d).

Apply this with A = G_p(j) and C = G_0(h-h_0). The semigroup property
and the cube inclusion place the sum inside G_p(j+h), so whenever
G_p(j) is nonempty and h is large,
    b_p(j+h) >= b_p(j) + h - C'_d,   with C'_d = C_d + h_0.
Neither the constants nor the degree threshold depend on j, on p, on
the center, or on the coordinates chosen at z. Multiplication by the
restricted sections has forced every transverse slice to grow, and the
growth rate is uniform because the restriction estimate was uniform.

Weighted slices are summed next. Put
    B_j = (1/d!) * sum over p of exp(-ell(p)/m) b_p(j)^(d+1),
with the same test and the same reference degree m as in Q_j; all these
sums are finite. For h so large that h - C'_d is nonnegative, convexity
of the (d+1)-st power and the slice growth give
    B_{j+h} >= B_j + (d+1)(h - C'_d) Q_j,
since only slices already present in degree j are needed and new slices
contribute nonnegatively, and since b_p(j)^d/d! equals #G_p(j).

The power d+1 converts slice size into tangential moment. For a finite
A in Z_{>=0}^d put b = (d! #A)^(1/d). Among measurable subsets of the
positive orthant with volume #A, the simplex
    { u with u_i >= 0 and u_1+...+u_d <= b }
minimizes the integral of u_1+...+u_d, by comparing the integrand inside
and outside that sublevel set; its integral is d b^(d+1)/((d+1) d!).
The integral over A+[0,1]^d equals the sum of |beta|_1 + d/2 over beta
in A. Enlarging d/2 to d, applying the outcome slice by slice, and
inserting the exponential weights gives
    sum over (p,beta) in Gamma_m of exp(-ell(p)/m)(|beta|_1 + d)
        >= (d/(d+1)) B_m.
Together with the growth of B_j, this is the lower bound for the moment.

The two bounds meet in a Riemann sum. For each fixed center dimension
d > 0 the estimates above hold for every minimizing pair in the stated
degree and t ranges, with C_d, H_d and C'_d depending only on X, L, k and
d, and with the interval [a_*, a^*] of discrepancies coming from the
uniform sublevel bounds at x. Within that interval the integral
    I_* = integral from 0 to 1 of s^n exp( (1-s) a_*/(n+1) ) ds
strictly exceeds 1/(n+1), because the integrand is at least s^n and the
factor exceeds 1 for s < 1. Fix once and for all a real t < n+1 in the
common interval, close enough to n+1 that I_* > 1/t, and keep it in
f_j = f_{j,t}(x). Put c = liminf over j of f_j, which is positive by
the uniform sublevel bounds, and take an increasing sequence of degrees
m on which f_m tends to c.

Assume, for contradiction, that infinitely many of these degrees admit a
minimizing pair whose center is positive-dimensional. Choose one such
pair in each of those degrees and pass to a subsequence on which the
center dimension has a fixed value d > 0; the centers, tests, models and
bases are free to vary along the subsequence. The construction above is
then applied to each chosen pair separately: its own test and its own
coordinates serve all auxiliary degrees j attached to that m.

Fix an integer l >= 2 and put j_i = floor(m i / l) for 1 <= i <= l, so
j_l = m. As m tends to infinity the increments j_{i+1}-j_i tend to
infinity, so the growth estimate applies successively to this ladder.
Starting from B_{j_1} >= 0 and then converting B into the moment,
    T_m + d/m  >=  d * sum over i from 1 to l-1 of
                  ((j_{i+1}-j_i-C'_d)/m) * (Q_{j_i}/Q_m).
The uniformity of C'_d is what makes this step legitimate, since the
center and hence the whole ladder of auxiliary degrees moves with m.

Ratios of the Q's are compared using the definition of the minimum in the
auxiliary degrees. The test (j/m)v is admissible at x, because the center
of v still contains x; its discrepancy is j A(v)/m and its section
orders divided by j are v(s)/m. Applied to the basis with initials
Gamma_j, the defining property of f_j therefore gives
    Q_j >= N_j f_j exp( -j A(v)/(m t) )
for all sufficiently large j. This comparison happens at x and does not
require the point y to satisfy the specialization bound in degree j; only
the degree-m variation used that. Combining the displayed inequality
with Q_m = N_m f_m exp(-A(v)/t),
    Q_{j_i}/Q_m >= (N_{j_i}/N_m)(f_{j_i}/f_m)
                   * exp( (1 - j_i/m) A(v)/t ).
For each fixed i the Hilbert asymptotic gives N_{j_i}/N_m tending to
(i/l)^n; also j_i tends to infinity, f_m tends to c, and c is the
liminf, so the liminf of f_{j_i}/f_m is at least one, with no need for
j_i to belong to the subsequence that realizes the liminf. Finally
A(v) >= a_* and t < n+1 give
    exp( (1-j_i/m) A(v)/t ) >= exp( (1-j_i/m) a_*/(n+1) ).
Every summand in the partition bound is nonnegative for large m and
the number of summands is fixed, so the liminf over m at fixed l is at least
    (d/l) * sum over i from 1 to l-1 of (i/l)^n
            * exp( (1-i/l) a_*/(n+1) ).
Only now is l sent to infinity, and the right side tends to d I_*, which
exceeds d/t by the choice of t. That contradicts T_m <= d/t.

The order of the limits is what makes the contradiction honest: the
constants C_d, H_d, C'_d are fixed before any t, the single t is fixed
using a_* before the sequence of degrees, the degrees are then sent to
infinity with l fixed, and l is sent to infinity last. Consequently only
finitely many degrees on the chosen subsequence admit a minimizing pair
with positive-dimensional center. A nonzero test has a nonempty closed
center and every center under consideration contains x, so in all
remaining degrees the center is exactly the point x. That proves the
point-center statement: for the chosen t there are arbitrarily large
degrees in which the infimum is attained, and every minimizing pair in
those degrees is centered at x alone.

Fix, from now on and for contradiction, a point x at which P is not
generated, together with the t below n+1 and a degree m for which the
infimum is attained and every minimizing pair has center x. Choose such
a pair, consisting of a basis s_1,...,s_{N_m} of R_m and a monomial test
v, exhibited on a log resolution pi: Y -> X of the union of the divisors
D_i = div(s_i). Include all exceptional divisors in the normal crossing
divisor on Y and write its components as E_j. Put
    a_j = A(E_j),     z_j = ( ord_{E_j}(D_i) / a_j ) over i = 1..N_m.
Each a_j is a positive integer because X is smooth, each z_j has rational
nonnegative entries, and only finitely many such vectors occur.

For a component Z of an SNC stratum whose closure maps to a set
containing x, let J(Z) be the set of components crossing at Z and put
    C_Z = conv{ z_j : j in J(Z) },
using only nonempty J(Z). A convex representation z = sum lambda_j z_j
is realized by the monomial weights lambda_j/a_j, whose discrepancy is
one and whose normalized order vector is z, and rescaling gives any
positive discrepancy. When some weights vanish the center enlarges to
the corresponding stratum closure, which still contains x.

Write a = A(v) > 0 and q = ( v(s_i)/a ) over i = 1..N_m. The vector q
lies in the union of the polytopes C_Z, and in none of those indexed by a
stratum with positive-dimensional image: a representation there,
rescaled to discrepancy a, would give the same objective value and a
center containing that positive-dimensional image, which the point-center
result forbids.

Differentiating the objective at its minimum with respect to the scale of
v produces a functional. With
    b_i = (t/m) * exp(-v(s_i)/m) / sum over h of exp(-v(s_h)/m),
the derivative conditions read
    b.q = 1,    every b_i > 0,    m * sum over i of b_i = t < n+1.
For any C_Z containing q, move the normalized profile along the segment
from q to a point q' of C_Z while holding the discrepancy equal to a. The
one-sided derivative at the minimum is nonnegative, and the objective
decreases whenever a section order decreases, so
    b.q' <= 1    for every q' in C_Z with q in C_Z.

Vanishing needs rational coefficients, so both q and b must be
approximated without changing which polytopes contain the profile and
without losing the supporting inequalities. The elementary fact is this.
Let a finite family of rational polytopes in R^N be given, let q lie in
the union of the family, and let b in R^N with positive entries satisfy b.q = 1
and b.z <= 1 for every z in C whenever q lies in C. Fix an open
neighborhood V of b. Then rational vectors q^0 arbitrarily close to q and
b^0 in V with positive entries exist such that q^0 lies in C exactly when
q does, for each member C of the family, and b^0.q^0 = 1 with b^0.z <= 1
on every member containing q^0.

The proof is a face argument. For each member C containing q let F_C be
its smallest face containing q; the point q is in the relative interior
of F_C, and since b attains its maximum there, b equals one throughout
F_C. The intersection F of the finitely many F_C is a nonempty rational
polytope with q in its relative interior, its rational points are dense,
and a rational point q^0 of F can be chosen close enough to q to avoid
every member of the family that does not contain q, since those members
are closed and finitely many. Then b.q^0 = 1 and the incidences match.
For this fixed rational q^0 the conditions on a functional c are the
rational linear equation c.q^0 = 1 and finitely many rational linear
inequalities c.z <= 1 at the vertices of the members containing q^0.
The feasible polyhedron is rational and contains b, rational points are
dense in its smallest face through b, and a sufficiently close one lies
in V with strictly positive entries.

Apply this with V small enough to keep the strict budget m sum b_i < n+1.
The outcome is a rational vector q^0 with nonnegative entries and a
rational b^0 with positive entries such that
    b^0.q^0 = 1,   m * sum over i of b_i^0 < n+1,
    b^0.q' <= 1    whenever q^0 lies in C_Z and q' lies in C_Z.
The profile q^0 lies in no polytope with positive-dimensional image, and
in particular q^0 is not zero.

A rational point of a rational convex hull has a rational convex
representation, so clearing denominators produces a primitive rational
ray in the crossing. Toroidal extraction of that ray gives a prime
divisor F_0 over X whose normalized order vector is q^0, with equality in
the retraction inequality for monomial discrepancy, and with center x by
the exclusion just noted. The divisor is retained on every later common
resolution, because its orders, discrepancy, and the equality under
retraction depend on the divisorial valuation and not on the model that
exhibits it.

Let I be the set of indices with q_i^0 > 0 and choose an integer M > 0
such that M/q_i^0 is integral for every i in I. The effective ideal
    c = sum over i in I of O_X( -(M/q_i^0) D_i )
has, at any prime divisor F, order divided by M equal to the minimum of
ord_F(D_i)/q_i^0 over i in I. At the extracted divisor all these ratios
equal its discrepancy, and the minimum form of the order is what keeps
that equality stable under the perturbation below.

Choose u large enough that c(uL) is globally generated. Take a smooth
projective resolution rho: T -> X dominating Y, retaining F_0, and
principalizing c, so that
    c O_T = O_T(-G).
Arrange that G, the pullbacks of the D_i, and all exceptional divisors
have normal crossing support. Then u rho^* L - G is globally generated,
being a quotient of the pulled back generating system of c(uL). For a
sufficiently small rational number 0 < eta < 1 put
    B = (1-eta) rho^*( sum over i of b_i^0 D_i ) + (eta/M) G,
an effective rational divisor with normal crossing support. Since
D_i is linearly equivalent to mL,
    (n+1) rho^* L - B  is equivalent to
    ( n+1 - (1-eta) m sum over i of b_i^0 - eta u/M ) rho^* L
    + (eta/M) ( u rho^* L - G ).
The first coefficient is positive once eta is small, by the strict
budget on the b_i^0; hence the whole difference is nef and big, and the
small eta is fixed once and for all.

For a prime divisor F on T write a_F = A(F) and
    r_F = sum over j of ord_F(E_j) a_j,
which is at most a_F; the number r_F is the discrepancy of the normal
crossing retraction to Y, and it is zero exactly when the retraction
loses the divisor. Let S be the reduced sum of the primes in the chosen
normal crossing support that satisfy all three conditions
    x lies in rho(F),   ( ord_F(D_i)/a_F ) over i equals q^0,
    and r_F = a_F.
The extracted divisor satisfies all three conditions, so S is not zero. For
a prime F not touching S nothing needs to be said; the next statement controls
those that do.

CLAIM. Every component of S maps to x and has coefficient exactly a_F in
B, while every other prime of the chosen support that meets S has
coefficient strictly smaller than its discrepancy.

For the first claim, a component F of S has retraction with normalized
profile q^0, and the image of the retraction center contains rho(F) and
therefore contains x. That image cannot be positive-dimensional, by the
choice of q^0, so rho(F) is the single point x. Orders of a sum of
ideals are minima of orders, so for every prime F on T
    ord_F(G)/M = min over i in I of ord_F(D_i)/q_i^0,
and at a component of S this minimum is a_F. Together with b^0.q^0 = 1
the displayed form of B gives coefficient a_F at F.

For the second claim let F meet a component F_S of S and choose a point
of intersection; its image y on Y lies over x. Take the stratum through
y defined by all normal crossing components passing through y and the
polytope C associated to it. Every component with positive order along
F or along F_S contains the relevant generic center on Y and hence
contains y, so both retraction profiles lie in that single polytope
whenever the profiles are nonzero; in particular q^0 lies in C, and the
supporting inequality applies on C.

If r_F = 0, the retraction lemma gives zero order on every D_i and the
minimum formula gives zero order on G, so the coefficient of B at F is
zero, which is less than a_F. If r_F > 0, put
    q' = ( ord_F(D_i)/r_F ) over i,   mu = min over i in I of q'_i/q_i^0.
The numbers b_i^0 q_i^0 for i in I are positive and sum to one, so
    mu <= sum over i in I of b_i^0 q'_i <= b^0.q' <= 1.
Using the minimum formula once more, the exact coefficient ratio is
    (coefficient of B at F) / a_F
      = (r_F/a_F) ( (1-eta) b^0.q' + eta mu ) <= 1,
because r_F <= a_F and the bracket is at most one. Equality would force
r_F = a_F and, both coefficients in the convex combination being
positive, b^0.q' = mu = 1. The equalities in the previous display then
force q'_i/q_i^0 = 1 for every i in I, while strict positivity of every
b_i^0 forces q'_i = 0 for i outside I; hence q' = q^0 including its zero
coordinates. Since F meets F_S over x its image contains x, so F itself
satisfies all three defining conditions and is a component of S. That
proves the strict inequality for every other prime meeting S.

Define the integral Cartier divisor
    D = K_T + (n+1) rho^* L - floor(B).
Its difference from the canonical divisor with the fractional boundary is
    D - (K_T + {B}) = (n+1) rho^* L - B,
and the right side is nef and big, while the fractional part of B is an
effective rational normal crossing boundary with coefficients in [0,1).
Kawamata-Viehweg vanishing therefore gives H^1(T, O_T(D)) = 0, and the
exact sequence of the reduced Cartier divisor S turns that into a
surjection
    H^0(T, O_T(D+S))  ->  H^0(S, O_S(D+S)).

Compare the divisor upstairs with the adjoint bundle itself:
    C := D + S - rho^* P = K_{T/X} - floor(B) + S.
On a component F of S the two coefficient statements and integrality of
a_F give coefficient (a_F-1) - a_F + 1 = 0. On every other prime meeting
S the preceding argument gives coefficient of B at F < a_F, hence
floor(coefficient of B at F) <= a_F - 1 and coefficient of C at F >= 0.
Primes outside the chosen support contribute zero. So C is effective on a
neighborhood of S and no component of S lies in its support. Negative
components of C may well occur elsewhere, but such supports miss S, and
deleting those finitely many supports leaves the asserted neighborhood.

Because S is reduced and each of its components maps to x, the
restriction of rho to S factors scheme-theoretically through the point x:
the pullback of a function vanishing at x vanishes on every component of
S and therefore on the reduced scheme. Hence
    rho^* P restricted to S  is  P|_x tensor O_S.
Choose a nonzero element of the one-dimensional fiber P|_x and multiply
the corresponding constant section on S by the canonical rational section
of O_T(C). That section is regular near S, because C is effective there,
and it is nonzero at the generic point of each component of S, so the
product is a global section of O_S(D+S) that is nonzero at those generic
points. Notice that no global effectiveness of C enters: only the
behavior near S matters.

Lift this section along the surjection. The positive part of C is
exceptional over X, since on a nonexceptional prime outside S the
coefficient of C is minus the floor of a nonnegative number and on a
component of S it is zero. Since S is nonempty, since every component of
S maps to x, and since the lifted section is nonzero generically on each
component, the descent lemma applies to the composite rho and produces a
section s in H^0(X, P) with s(x) not equal to 0.

Dimension one is included rather than excluded. In that case a component
of S can be the nonexceptional point x itself, its coefficient in C is
still zero, and the same restriction and descent argument applies word
for word. So the assumption that P fails to be generated at some closed
point x leads to a contradiction, and K_X+(n+1)L is globally generated
on X. The reasoning is uniform in the pair (X,L) and uses no bound on the
dimension.

The sharp constant enters twice and nowhere else. In the minimizing
stage the strict budget m sum over i of b_i < n+1, inherited from
t < n+1, is what makes the coefficient of rho^* L in the nef and big
difference positive. In the center stage the same inequality t < n+1
combined with a_* > 0 is what makes the limiting integral exceed 1/t.
Any argument that relaxes either inequality loses the constant n+1, and
any sharpening must keep both, which is why the two stages cannot be
separated.