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Record, attribution and exact subject

Complete and correct at its stated scope. This audit is part of the full-proof review of 2026-09-06. It covers the statement and finite-block interface of the infinite-tail refinement, the nonzero-target schedule, the zero-target case and the exact transfer to Problem 967. Reviewer: a fresh review context distinct from the author of the reconstruction and from the compilation-supplied corrections; it did not build on the subject before reviewing it. No distinct grader is recorded, so no numerical claim tier is assigned.

The approved bytes of infinite_refinement.md are not retained; the current page contains the reviewed schedule and estimates. The exact reviewed copies are not retained in this repository. On 2026-09-16 the current pages were compared with the report's description of the reviewed statements, constants and proof steps and agree with it; the retained version history since the earliest corpus snapshot shows only attribution and standing wording changes on these pages. A match of description is not a byte match, and any substantive change to the mathematics requires a new assessment. The page the report names is identified as it stood on 2026-09-15, before this record's filing on 2026-09-16; the exact reviewed copy was a review-packet candidate and is not retained, and the comparison recorded in this section says how the committed page relates to it.

This record was filed on 2026-09-16 from a retained report, the audit text. The report text is retained below in full. The filing changed only the wrapper, participant identifiers, private paths and operating-history material; it records no new verdict, and the first-person readings and judgments below belong to the historical reviewer, not to the filing author.

Retained report

Verdict

The proof in infinite_refinement.md is complete and correct at its stated natural-language scope. It closes the four obligations that the arXiv-v1 proof leaves implicit and validates the exact E967 transfer. No substantive or literal candidate correction is required.

Statement and finite-block interface

The supplement quantifies over every real t != 0, complex lambda, positive integer N, and real delta>0. Its conclusion uses S subseteq Z_{>=max(2,N)}, which simultaneously preserves Theorem 1.3’s lower endpoint and implies the p.2 tail requirement S subseteq Z_{>=N}. It requires S infinite, reciprocal mass at most |lambda|+delta, and exact complex sum lambda.

The finite-block interface is precisely the independently checked Lemma 2.1 with K=1+|1+it|. Repeating the concrete block construction is useful because it proves that every invocation with a nonzero target creates a nonempty finite block, even when the real phase point x is not an integer. No stronger unproved lemma is imported.

Nonzero target

Let L=|lambda|>0, q=delta/[2(L+delta)], rho=q/K, and alpha=1-q. Then 0<q<1/2, rho>0, K rho=q<1, and L/alpha=L+L delta/(2L+delta)<L+delta. The algebra and strict endpoint are correct for every L,delta>0.

At stage k, a_k=min(rho,v_k/2)>0 and c_k=a_k w_k/v_k is radially aligned with the nonzero residual. Increasing integer cutoffs produce strictly ordered disjoint blocks. Lemma 2.1 gives an error e_k=c_k-z_k with |e_k|<=K a_k^2<=q a_k and block reciprocal mass at most a_k.

Because a_k<=v_k/2, the exact radial remainder has norm v_k-a_k. The reverse and ordinary triangle inequalities yield

v_{k+1} >= v_k-(1+q)a_k >= (1-q)v_k/2 = alpha v_k/2 >0

and

v_{k+1} <= v_k-(1-q)a_k = v_k-alpha a_k.

The first inequality inductively prevents exact termination, so every target and every block remain nonzero. While v_k>2rho, a_k=rho and the fixed positive decrement alpha rho forces entry into the small phase after finitely many stages. Once v_k<=2rho, a_k=v_k/2 and v_{k+1}<=((1+q)/2)v_k. Since this factor lies in (1/2,3/4), the small phase persists and v_k->0.

From alpha a_k<=v_k-v_{k+1}, every finite partial sum satisfies alpha sum_{k<=J}a_k<=L-v_{J+1}<=L. Monotone passage to the limit gives sum a_k<=L/alpha<L+delta. The union has at least one new element per stage and hence is infinite. Nonnegative block sums give the reciprocal-mass inequality, which also proves absolute complex convergence. The block-end identity is lambda-w_{J+1}, and the block tails have vanishing absolute mass, so the full sum is exactly lambda.

Zero target

Choose an integer m>=max(2,N) with 2/m<delta. The target beta=-m^{-(1+it)} is nonzero, has modulus 1/m, and the slack epsilon=delta-2/m is positive. Applying the nonzero case beyond cutoff m+1 gives an infinite correcting set disjoint from the singleton {m}, with mass at most 1/m+epsilon. Adjoining the singleton cancels the complex sum exactly and gives total mass at most 2/m+epsilon=delta. Infinitude, the tail endpoint, disjointness, and absolute convergence all remain valid.

Exact E967 transfer

The explicit choice t=1, lambda=-1, N=2, delta=1 produces an infinite subset of integers at least two. Successively choosing the least unused member gives a strictly increasing infinite sequence. Every set element appears because there are only finitely many positive integers below it. Absolute convergence identifies the enumerated series with the set sum, giving 1+sum_k a_k^{-(1+i)}=0 and sum_k 1/a_k<infinity. One such pair of a sequence and real t refutes the universal nonvanishing assertion.

Attribution and limitations

The source owns the p.2 assertion; the schedule and estimates above are explicitly labeled as a compilation-supplied completion and are not represented as printed in arXiv v1. The supplement uses no external research theorem beyond the included, independently checked Lemma 2.1. It gives no result about the finite-set conjecture or the fixed set {2,3,5}.

This independent review grants complete natural-language proof-chain credit for E967 to the composition of the frozen source chain and the frozen supplement. It grants no formal-verification, peer-review acceptance, publication, canonical-integration, or external Erdős–Ingham proof credit.