Grade

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PASS for E1, E2, E3, E4 and E5 at the separate scopes below. This accepts the examination as a grading product. It does not accept the unrestricted conjunction E as true: the report's literal E1 verdict is refuted-as-stated. Its complete proof for $z\ge1$, and E2–E5 relative to their stated classical premises, have verdict refutation-failed. No catalog solution, P acceptance or Lean credit follows.

The grading date is 2026-10-16 UTC. The subject is the complete elementary report, both original keepers weak steps and proof coverage, its subject map, and the ten unchanged mathematical excerpts that map identifies. The original examination is dated 2026-10-15; its mathematical extraction was frozen on that date at 20:36:13.943931 UTC. The grading packet was frozen on 2026-10-16. No later mathematical text is covered.

The native subjects are all six pages under wiki/research/smooth_successor_estimates/: _index.md, block_sieve.md, proxy_discrepancy.md, proxy_densities.md, candidate_mass.md and classical_inputs.md. Supporting inputs are _index.md, signed_type_ii.md and successor_prime_extraction.md under wiki/research/smooth_successor_primes/, and the supplied portions of library/arithmetic_functions/openai_2026_weighted_dilation_graphs_smooth_shifted_primes_totient_fibers/openai_2026_weighted_dilation_graphs_smooth_shifted_primes_totient_fibers.md. The retained excerpts, rather than the current native pages, are controlling.

The source method is OpenAI's Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026 release, an unrefereed manuscript. Project reconstruction and original project reasoning are credited to Plasma AI. The source's original PDF and the original classical proofs were not supplied or read.

Independence and actual coverage

The grader is a fresh adversarial grader, disclosed model gpt-6-astra, distinct from both mathematical authorship and the original reviewer, and has not collaborated on either. This is grading of an existing examination, not a replacement examination or an additional independent mathematical cycle. Both reports were available for inventory; E's disposition was completed before grading the conditional chain.

The original reviewer's complete mathematical read claim is supported by actual returned displays, not merely file-open commands or subscription flags. Matching calls and returns covers the ten inputs: the six E pages, three consumers, and the manuscript excerpt. The original report and both keepers were read in full for this grading, as were all ten inputs. The original commission, operating requirements, continuation, effective settings and supplied primary actual-read evidence were inspected for scope and eligibility. The manuscript excerpt contains Section 2, Section 6 definitions and statement, full Section 7 and bibliography. Reading those parts does not certify every source assertion: Lemmas 2.6–2.7, Corollary 2.10, P's proof and the source Section 6 proof are outside the examination's proof coverage.

The supplied primary record has matched returns for all recorded calls. The mathematical displays used to support the full-read claim cover the complete input ranges. Some later duplicate reads and an operating-skill display were truncated; they are not counted as complete additional reads. A line-match aid was checked against the requested ranges and returned text. It is administrative evidence, not a proof or a sufficient eligibility test. The two other supplied startup records contain no substantive tool reads. Opaque internal continuation data are not independently inspectable; the eligibility judgment rests on the readable instructions, continuations and matched actual calls and returns, not on those opaque data.

The recorded inputs and continuations contain no author advocacy, earlier grade, sibling ruling or mathematical steering. Recorded subscriptions are empty; no inbox, feed, live ledger, outside search, source checkout or additional mathematical subject was opened by the reviewer. The continuation uses the reviewer's own work. These content and route findings support PASS on reviewer eligibility within the supplied primary record. They do not assert visibility into unrecorded activity.

One timing qualification is necessary. The bootstrap manifest and accepted settings were displayed before the administrative identity comparison. Named static operating sources were checked before their substantive use. The report's pre-use comparison sentence is accepted in that latter sense; a broader assertion covering every bootstrap metadata read is not supported. The displayed settings contain operational controls and identities, not mathematics, an attack suggestion or a verdict. Under the verification standard's content test this supplies no material mathematical exposure and does not invalidate the examination. Empty subscriptions alone would not justify that ruling.

The current grader read only the commissioned frozen packet, named operating files and its own continuation. No excluded mathematical exposure was observed. A newly authored administrative parser was briefly written to the wrong scratch directory. Its first launch failed before execution; it was moved to the designated directory before the successful launch. The deviation was disclosed immediately. No excluded input was read by that action. This records the fact; any separate ruling on the grader's operating eligibility remains with the commissioning authority.

Assertion dispositions and exact surviving scope

All variables called fixed precede $x\to\infty$. Factor variables are positive integers; $P^-(1)=\infty$, $P^+(1)=1$, and distinct-prime counts ignore multiplicity. Put $L=\log x$, $T=\log L$, $W=e^{\sqrt L}$ and $V(y)=\prod_{p\le y}(1-1/p)$. A dyad is the stated half-open interval or fixed enlargement. No effective threshold is inferred from Siegel–Walfisz. Each grade covers the restatement, proof coverage, attack and conclusion together, not just the displayed answer.

Keeper assessment and attacks

The grading reproduction account records the indispensable calculations. It assesses the original keepers without adding a premise or mathematical repair.

The strongest successful attack is exact. Use no bad primes, $z=1/2$, one object of weight 1, $X=0,g(1)=1,r(1)=1$. All density hypotheses hold vacuously and the empty intersection has weight 1. For every positive even $h$, the remainder cutoff $z^{4h+2}<1$ excludes $d=1$, forcing $1\le0$. The support cutoff also forbids the only constant monomial. An imposed $z\ge2$ convention would exclude the witness, as the report says; the supplied conventions do not impose that restriction.

The principal failed attacks are substantive. Cutting proxy cells and maximizing the allowed character/twist ranges still leaves errors smaller than $L^{-B}$. The density proof keeps positive distance from the first-factor boundary and preserves $l/l!$ in minimum-coordinate slicing. Moving the exceptional slot with $x$ would destroy mass but violates the fixed-index quantifier; for a fixed late slot, exact harmonic sampling and the free first-band prime give the claimed local mass. E1's coefficient disjointness and factorial tail, and E5's conductor sums and choice order, were also reproduced.

The original proof-coverage keeper has a malformed delimiter in its principal-error display at line 336. Surrounding text and the retained consumer's correct display explicitly impose $(d,u)>1$. The grading reproduction uses that already supplied condition. This is a typesetting defect, not a missing inference supplied by the grader; the exact original remains unchanged.

The consumer's separate unrestricted “Dyadic bound” is not accepted here. The E report checks its balanced specialization using $M_i\ll x^{0.54}$. Grading the complete consumer and its wider wording belongs to the second report in this batch. The balanced interface finding is not blanket consumer approval.

Premises and standing

The narrow supplied premise-standing channel was read at grading on 2026-10-16 UTC. It records no accepted independent weight for P and says E is pending this batch's grade. The same metadata was visible at packet intake; no live standing source was substituted. No P grade or exposure assessor's decision was supplied.

The E proofs consume no established native L-claim. Their external premises, exactly as stated in the original report and frozen inputs, are:

• PNT with error $O_A(y(\log y)^{-A})$ for every fixed $A>0$, attributed to Tao's 254A, Notes 2, 9 December 2014, Corollary 39 and Exercise 40.
• Mertens' harmonic estimate with error $O(1/\log y)$ and product asymptotic, attributed to Tao's Notes 1, 23 November 2014, Theorems 15 and 26. The sharper harmonic error is explicit; the manuscript's weaker opening display alone would not supply it.
• Chebyshev's dyadic harmonic bound, also implied by the stated PNT.
• Siegel–Walfisz for every fixed accuracy and logarithmic modulus exponent, uniformly in reduced classes, with possibly ineffective constants; source Theorem 2.2, attributed to Tao's Notes 2, Exercise 64.
• The multiplicative large sieve for primitive characters, with $(H+1+R^2)\sum|a_n|^2$; source Theorem 2.3, attributed to Montgomery–Vaughan, Multiplicative Number Theory II, Theorem 19.16, the identified undated draft.

Statements and applications were checked in the supplied text. The original classical sources and their proofs remain unread; original-source fidelity is an explicit hypothesis/limitation, not acceptance created by this grade. Binomial algebra, unique factorization, CRT, character orthogonality, Cauchy–Schwarz, partial summation, Fourier inversion and the stated elementary integration/convergence facts supply ordinary background. The quantitative sieve, divisor, Fourier and density steps are rederived in the keepers. Ford–Halberstam and Buchstab supply attribution, not additional unexamined load-bearing asymptotics.

The order is elementary algebra and those classical premises; operative E1, harmonic sums and E4; E2; E3; remaining divisor/Fourier tools and Type I after their actual inputs; then the consumed sieve and mass interfaces. E5 is an inventory, not a cyclic antecedent of itself. The original report already gives this acyclic order. Any later use must select the proved operative sieve fragment, never the false unrestricted statement.

Whole-contract and checklist judgment

Subject identification, independence, restatement, weakest-step derivations, strongest attacks, premises, exact inputs and written reproduction are present for every E assertion. The original nineteen-mode table gives each of E1–E5 a separate column; its ten-item Erdős table retains those dispositions. The following grader judgments apply to each assertion at its scope above.

Each Erdős-specific item is met: quantifiers and scope by the separate assertion table; circularity by the dependency order; model and convention changes by the exact transfers; finite and statistical overreach by the absence of sampled inference; uniformity by the proved parameter dependencies; extremal conclusions by coefficient/density checks; consequences and composition by the restricted E/P interfaces; computation by explicit inapplicability; reproduction by the unchanged package and written keepers; and source and verdict fidelity by comparison with supplied text while retaining unread-original limits. Correlated-sample inference is inapplicable. Evidence coverage is assessed clause by clause, not by counting checks.

Preservation and permitted use

Reproduction requires the complete original package and grading reproduction account. Follow the exact source copies and substitute the witness and estimates by hand; no numerical simulation or repository runtime is needed. Original omission markers remain intact. Private input identities and actual-read provenance are retained separately for adoption.

This grade permits using operative E1 and E2–E5 under the explicit classical boundary when comparing the conditional-chain interfaces. It grants no unrestricted E theorem. The second grade must check its own full report and snapshot; matching names alone is insufficient. P remains assumed. Public filing, any separate eligibility ruling on the grader, and later settlement or standing obligations remain with the commissioning authority.