Lemma 18 ([Saturation and section counts]). Let f₁,…,f_s be a finite list of genuine rational functions on V , whose length may vary with the sequence, and let J(T)=0,_j(-T(f_j)), 0≤J≤C L₀ D, 1≤L₀=O(L). Thus J is an integral …

[Proof of the field assertions] Because J is integral and L_0=O(L) , choose a fixed small c>0 so that a(T)=0, LD(T)≤c J(T)=0. There is, after a subsequence, a single field plateau containing two cutoffs 0<β_0≤β_1≤c/L, Lβ_0⟶0, Lβ_1≥c'>0, K_ β_0 =K_ β_1 =K_*. Here is a precise reason for this choice. Pass to limits of the finitely many change locations multiplied by L , adding dummy endpoints if the number of changes varies. The last change tending to zero lies below a cutoff β_0 with Lβ_0→0 ; choose it decreasing slowly enough to lie beyond that change. The next change either has a positive limiting scaled location or lies beyond c/L . Choose β_1 strictly before it, with Lβ_1 fixed positive and at most c . This gives. Passing again fixes p= K_* . All I_ β_1 belong to L , so Lemma 16 ([Uniform cutoff models and order functions]) makes them horizontal over K_* . It follows, for every rational m>0 , that S(mP_ β_0 )=S(mP_ β_1 ). Indeed a function in the left side belongs to K_* : enlarge m to an integer and use the section-field description. Its order at every removed prime in I_ β_1 is zero by horizontality. Removing those pole allowances therefore leaves it in the right side. The reverse inclusion follows from P_ β_1 ≤P_ β_0 . For fixed C_1 , a function in S(C_1LD) has no pole at I_ β_0 eventually, because its order is integral and C_1Lβ_0<1 . Hence S(C_1LD)⊂S(C_1LP_ β_0 ) =S(C_1LP_ β_1 )⊂K_*. This proves (i). The cutoff generation statement at β_1 , and β_1≥c'/L , place a birational field-generating space for K_* inside S(B_0LD) for fixed B_0 . Enlarging B will therefore make S(BLD) generate K_* ; a basis contains p algebraically independent elements, by greedily choosing a transcendence basis from any field-generating set. We prove that some fixed B forces the asserted exact triviality. Otherwise choose B_i→∞ slowly and T_i∈ L that are nonnegative on S(B_iLD) but nontrivial on K_* . Since K_1=k , there is a change at β≥β_1 where T is trivial on K_β but nontrivial on the preceding field K_ β_- . Take β_- sufficiently close to β from below that β_->β/2 . Use H,φ_j from Lemma 16 ([Uniform cutoff models and order functions]) at β_- . Their pole bounds are C D/β , and 1/β=O(L) ; thus T(φ_j)≥0 eventually. Its nontrivial restriction gives H(T)≤-1. If L_0β→∞ , then L_0→∞ because β≤1 . The same bounded-cost field-generating space for K_ β_- is contained in S(L_0D) eventually. The hypothesis of triviality on the field generated by the latter space contradicts the choice of T . It remains to treat a subsequence on which L_0β is bounded. Restrict to the geometric generic fibre of the cutoff contraction U^β→W_β . There T is horizontal and D'≤β∑_ I_β Q'. Apply the extra-data part of Lemma 3 ([A uniform linear chamber]), over a point, with a=a+κJ, d=c_* = D/β, h=H, where κ>0 is fixed rational and sufficiently small; c_* denotes the additional nonnegative direction of that lemma. We verify all its hypotheses. The orders a have bounded denominator because a and J are integral. The order H has bounded denominator by the cutoff lemma. Both J and D/β are b-nef, and H is pulled back from a b-nef divisor on K_ β_- ; after a birational determination it restricts to a b-nef divisor on this fibre. Their Calabi--Yau realizations hold the total class rationally trivial. At floor primes on this fibre, H≤C D/β, J≤C D/β, where the second bound uses boundedness of L_0β . At other primes on the model, a=1 , D=0 , J=0 , and H≤0 . These inequalities give effective traces on uniform small positive ranges in, including the family with λH+sc_* for 0<s≤λθ_0 . The family a+td is klt for t>0 because D is positive on a=0 . Finally, a=0 a=J=0, -Cη_i c_*≤H≤Cc_* this zero set. The lower bound is part (iii) of the cutoff lemma, using the assumed uniform. Thus the exact sign conclusion of Lemma 3 ([A uniform linear chamber]) over a point prohibits H<0 at the lift of T , contradicting. This proves (ii). The proof of (iii), including its common normalization v_i , is given below after establishing the adjoint lattice-point statement. For the extension to rational monomial subcones, take any finite list of tested functions, resolve their divisors, and subdivide the cone so their orders are linear on each piece. Divisorial rational rays are dense in each rational piece. The exact zero assertions extend by continuity; applying this separately to each function gives triviality on the whole field.