Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 16

Interface disposition

Examined on 2026-10-16 UTC. The elementary examination was graded first; the complete conditional-chain examination was graded second. Both grades are PASS as grading products at their stated scopes. They preserve two distinct adverse mathematical verdicts:

Unrestricted E1 is refuted-as-stated at $z=1/2$ , where the remainder level excludes the empty product. Its proved $z\ge1$ fragment survives.
The unrestricted auxiliary dyadic estimate is refuted-as-stated at $m=1$ , where its ratio to $X_AV(W)^2$ grows like $\sqrt{\log x}$ . Its proved balanced specialization survives.

The elementary grade's filing identifier is wiki/research/smooth_successor_estimates/evidence/verify/elementary_estimates/grade.md; the conditional grade's filing identifier is wiki/research/smooth_successor_primes/evidence/verify/conditional_chain/grade.md. These grades concern the retained frozen excerpts of the elementary pages under wiki/research/smooth_successor_estimates/ and the signed, extraction and counting pages under wiki/research/smooth_successor_primes/. They do not cover later native text.

The exact accepted interface comparison permits this composition:

Antecedent retained	Permitted consequence
The quoted classical statements at their recorded scope and reading depth	Operative E1 with $z\ge1$ , and the exact E2–E5 scopes. The weighted E3 consumer consequence additionally requires its stated Type II/P hypothesis.
Those elementary scopes and the complete explicitly assumed P endpoint statement	Signed Type II, Type I, the successor extraction using only balanced dyads, and A+ for every sufficiently large real argument.
A+	For every fixed $\epsilon>0$ and all sufficiently large real $x$ , $h(x)\ge h_{\rm sf}(x)>x^{2-\epsilon}$ , with strict $1\le a<b<x$ , coprimality, equal divisor sums and squarefreeness.

Thus, conditional on P and the exact quoted classical premises, the surviving E scopes and the graded chain support the terminal implication. Every consumed sieve cutoff is eventually greater than 1. Full-cell restrictions, coefficient bounds, compact density ranges, early mass constants and the fixed exceptional index match. The signed and extraction mathematical texts and consumed Section 2/7 interfaces agree across the frozen packages. No proof repair or new premise supplies a match.

The narrow standing information read on 2026-10-16 supplies no accepted independent P weight. A prospective accepted P theorem must match both shift signs, the full integer-slot endpoint class, actual restricted discrepancy, conjugations, harmonic normalization and all uniformities. A favorable grade of this chain cannot supply that antecedent acceptance.

The source method remains OpenAI's unrefereed manuscript Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026 release. Original classical proofs, PDFs and the source Section 6-equivalence assertion remain outside the examinations. The source-owned successor and counting method is credited to Li and Zhuang; project exposition and parameter checks are credited to Plasma AI.

No unconditional catalog acceptance, unrestricted E theorem, unrestricted dyadic estimate, P grade or Lean credit follows. Public filing, preservation, any separate ruling on the current grader's disclosed scratch-path deviation, and subsequent acceptance remain with the commissioning authority.

Linked from (5)

Problem 824 Elementary estimates for the smooth-successor extraction Smooth successor primes from the weighted-dilation argument Examination of the conditional Problem 824 implication Primes 2u-1 with smooth u

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