Grade

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PASS for the conditional examination and its separate adverse dyadic finding, at the exact scopes below. The original report's terminal verdict is refutation-failed under its explicit P/E assumptions. Its unrestricted auxiliary dyadic assertion is refuted-as-stated; the balanced specialization actually consumed has verdict refutation-failed. This grade does not accept P, the unrestricted E conjunction, an unconditional Problem 824 solution or any Lean result.

The grading date is 2026-10-16 UTC. The complete subject is the original report, its integral premise account, all three original keepers—signed, extraction and counting—and their seven unchanged retained inputs. The original examination is dated 2026-10-15; its extraction was frozen at 20:29:14.961128 UTC that day. The grading packet was frozen on 2026-10-16. No subsequent text is covered.

The retained subjects identify the following repository-relative mathematics:

These are source identities, not claims about current native pages. OpenAI's Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026 release, is an unrefereed manuscript. The successor application and product reduction are attributed to Shisheng Li and Zhong Zhuang's Coprime pairs with equal sum of divisors, assuming smooth shifted primes, 11 October 2026 draft, Theorem 1.3 and Section 3, pages 1–3. The full project exposition and parameter checks are credited to Plasma AI. No original PDF or original classical proof was supplied or read.

Independence and actual-read findings

The grader is a fresh adversarial grader, disclosed model gpt-6-astra, distinct from both mathematical authorship and the original reviewer, with no collaboration on either. This is whole-contract grading of the existing report, not a replacement examination or a second mathematical cycle. The elementary examination was graded first, with separate dispositions for every E assertion; that completed grade is used below only at the interfaces actually consumed here.

The complete original report, premise account, three keepers and all seven inputs were read for grading. The original commission, continuation, operating requirements, supplied settings and primary actual-read records were inspected for authority, coverage and exposure. Every recorded call in the substantive primary record has its matched return; the other supplied startup record contains no substantive tool read.

Actual returned mathematical displays support the original full-read claim: the neutral metadata, catalog and extraction were displayed together; the signed proof and product proof were displayed in full; the source excerpt was displayed in five contiguous ranges covering all 752 lines. Requested paths alone were not treated as proof of reading. A substantive-line matching aid was checked against the actual routes, range boundaries and displayed returns. Duplicate passages were not counted as evidence of reading unrelated intervening text. The mathematical displays used for this coverage contain no actual output truncation. Oversized operating displays were partial; the original report correctly limits their claimed depth. Both verification contracts received full readings.

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 original record shows only the frozen mathematical inputs, authorized operating material, the reviewer's own work and permitted outward notices. Its successive instructions preserve the same mathematical commission. No author advocacy, earlier grade, sibling ruling, live ledger, outside source acquisition, inbox or feed, other checkout, additional mathematical subject or native subagent read appears in that record. Subscriptions were empty, but this is corroboration rather than the basis of the content finding. The report's no-exposure and independence account is supported within the supplied primary record; unrecorded activity is not asserted to be visible.

The reviewer disclosed that the accepted-settings metadata was displayed alongside the manifest before its identity comparison. The comparison subsequently matched. Inspection of the displayed content shows operational settings, not mathematics, attack advice or a verdict. The named static operating sources were checked before substantive use. PASS on reviewer eligibility: this read-order fact is not material mathematical exposure under the verification standard's content test. It was disclosed rather than silently waived. There is no basis here for voiding the mathematical examination on that fact.

The current grader has read only the commissioned packet, named operating files and its own work. 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. This deviation was disclosed immediately and read no excluded input. Any distinct ruling on the grader's operating eligibility remains with the commissioning authority; this grade does not waive that ruling.

Restatement and assertion grades

The exact premise account is integral to the graded subject. All its fixed group and band geometry, damping, normalizations, coefficient classes and constant dependencies are retained. Put $L=\log x$, $T=\log L$, $W=e^{\sqrt L}$. All factor variables are positive integers; $P^-(1)=\infty$ and $P^+(1)=1$. Group counts count distinct prime divisors, band lists permit repetitions, and the exceptional band slot is an integer rough above $W$. Fixed choices precede $x\to\infty$; thresholds may depend on those choices.

P is the entire supplied shift-cancellation hypothesis. Its discrepancy condition tests every interval in $[1,x^2]$, all characters of modulus at most $L^B$, including principal and imprimitive characters, and every $|t|\le L^B$, bounding the harmonic sum by $L^{-B}$. An endpoint has the supplied convolution over $mpr=n_*$ with the original rough-supported coefficient $\alpha_m$, an explicit character and Mellin power on $m$, the integer slot $p$ in an interval in $[x^\tau,x^\eta]$, $0<\tau<\eta<1/4$, and unrestricted residual coefficient $c_r$. Coefficients, character modulus and frequencies obey the fixed logarithmic bounds.

For every fixed endpoint/group/cutoff data and requested $D_0>0$, P supplies a fixed $B$ and the uniform bound

$$\left|\sum_{z>0,\ z+k>0}\frac{\Psi(z/Z)}{z} z^{ia_1}(z+k)^{ia_2}\overline{F_0(z)}G_0(z+k) W_\ell(z)W_\ell(z+k)\right|\ll L^{-D_0}.$$

This includes both signs of nonzero $|k|\le L^C$, $xL^{-C}\le Z\le xL^C$, all stated outer frequencies, fixed positive compact support and derivative bounds $\|\Psi^{(j)}\|_\infty\ll_j L^{C(j+1)}$, and the stipulated conjugated cases. Only the actual restricted $\alpha$ in the designated endpoint must satisfy discrepancy; no such requirement is imposed on the arbitrary second Type II coefficient. This restates an assumed interface, not an acceptance of P or a verification of its identification with an omitted source theorem.

E comprises the exact stated elementary/classical interfaces, candidate mass, full-cell discrepancy and density bounds. The original implication assumes E and does not prove that complete antecedent. The separate E grade accepts only its operative sieve fragment $z\ge1$ and the other explicit scopes relative to their classical premises. Their use in this report is compared below; the false unrestricted E1 statement is never promoted.

For the stated fixed bands and groups and every fixed $D_*,b_*,C>0$ with $\eta_*<b_*/4$, signed Type II asserts some fixed $B$ works for all large real $x$, $UV\asymp x$, $x^{b_*}\le U,V\le x^{1-b_*}$, and all permitted $\alpha,\beta$, giving the weighted sum $O(xL^{-D_*})$ over $mn=2u-1$. The actual interval cuts of $\alpha$ satisfy discrepancy; $\beta$ is arbitrary within its bound.

Extraction asserts that for every fixed $0<\delta<1/4$ there are $c,C_A,C_w>0$ and a threshold such that every sufficiently large real $x$ has at least $c xL^{-C_A-1}e^{-C_w\sqrt L}$ distinct primes in $[2x-1,4x-1]$ with $P^+(p+1)\le x^\delta$. Consequently A+ gives, for every fixed $0<\delta<1/4,\eta>0$, at least $y^{1-\eta}$ such primes $p\le5y$ for every sufficiently large real $y$.

Define $\sigma(n)$ as the sum of the positive divisors of $n$ and

$$h(x)=\#\{(a,b)\in\mathbb N^2:1\le a<b<x,\ (a,b)=1,\ \sigma(a)=\sigma(b)\}.$$

Let $h_{\rm sf}(x)$ count those pairs with both entries squarefree, including the convention that 1 is squarefree. Each unordered pair is represented once by $a<b$; the diagonal is excluded. The terminal conditional assertion is

$$(\mathrm P)\ \&\ (\mathrm E)\quad\Longrightarrow\quad \forall\epsilon>0\ \exists x_0\ \forall x\in\mathbb R,\ x\ge x_0: \quad h(x)\ge h_{\rm sf}(x)>x^{2-\epsilon}.$$

This is the catalog's eventual exponent reading. A subsequence or an almost-all set would not suffice.

The source Section 6-equivalence assertion, original-source fidelity and P's proof are excluded from this grading. They receive no favorable warrant by inclusion of the source excerpt or by the report's comparison language. The standalone signed deduction above suffices for the graded composition. No new claim or repair is introduced to avoid that boundary.

Keeper assessment, weakest steps and strongest attack

The complete grading reproduction account records the indispensable calculations in all three existing keepers. Three weak bridges were reproduced:

1. The pre-Cauchy allocation and Fourier costs are bounded before $K_0$; ordinary second and sixth divisor moments pay the diagonal. The exact signed determinant correspondence, its converse, character units and endpoint conjugations meet P. Multiplying the test by its argument restores the indispensable factor $Z$ in the unweighted sum.
2. Type I, every sieve level, the proxy restrictions and finite-simplex density subtraction support the extraction. The balanced mask majorant, nonsquarefree CRT count, lattice error and compensation products prove the consumed dyadic specialization with an early constant.
3. The product count, overlap deletion and gcd-fiber calculation convert collisions into actual distinct coprime squarefree pairs with enough exponent margin and strict real endpoints.

The strongest successful attack is the original exact asymptotic witness. At $x=2^j$, use $m\in[1,2)$ and $n\in[2x,4x)$. Then $m=1$ is $W$-rough and the product condition holds. The sum in the unrestricted display is exactly $S_W=\sum_{P^-(2u-1)>W}A(u)\Psi(u/x)$. Type I and the operative block sieve give

$$S_W\sim\mathfrak S X_AV(W),\qquad \frac{S_W}{X_AV(W)^2}\sim \mathfrak S e^{\gamma_E}\sqrt{\log x}\longrightarrow\infty.$$

The smooth endpoints vanish; the level is $x^{o(1)}$; the error is $o(X_AV(W))$. This is a conditional mathematical witness using the same elementary/classical inputs, not an experimental prime-count assertion. The reusable defect is that a product-size condition permits one factor to collapse to 1 and loses one roughness saving.

The original report already distinguishes this false unrestricted reading from the inherited balanced one. Its complete balanced proof has $M_i\ll x^{0.54}$ and controls the summed error by $x^{0.64}L^C H_Q^{\rm mic}=o(X_AV(W)^2)$. Preserving that explicit restriction is not a new repair. No surviving use in this frozen chain requires the false wider estimate.

Other attacks tested a missing determinant divisibility condition, a lost sign or factor $Z$, uncontrolled Fourier dimension, precision circularity, a squarefree-only CRT count, unsupported cancellation of $\sigma(g)$, a large gcd fiber and a subsequence substituted for eventual real-variable quantifiers. The retained calculations defeat those attacks on the supplied hypotheses. None is dismissed by agreement with the author's answer.

Premises, standing and batch interfaces

The narrow premise-standing channel was read for filing on 2026-10-16 UTC. It supplies no accepted independent P weight and requires E to be graded first in this batch. The same metadata was visible at packet intake. No live ledger, P grade or exposure-assessor decision was substituted. Census from the actual proof sites found no established native L-claim consumed by this conditional argument. No missing essential native premise was found; if such a premise were needed, the closed packet could not supply its standing.

The exact classical interfaces remain assumptions:

Ford–Halberstam's The Brun–Hooley sieve, 2000, Section 1 Lemma 1, and Buchstab's 1937 work are attribution for supplied local constructions; their original papers are unread and no unspecified stronger theorem is imported. Elementary arithmetic, CRT, character orthogonality, Cauchy–Schwarz, partial summation and the stated integration/Fourier facts are used at their displayed scopes. The quantitative local estimates and deductions are retained in the report's keepers.

The separate E grade precedes this grade. Comparison of the exact frozen subjects, rather than their titles, gives the following interface result:

The signed and extraction mathematical texts agree across the two frozen packages apart from introductory scoping prose. The consumed Section 2 and Section 7 source text agrees; the additional source Section 6 proof in the conditional packet does not supply an extra antecedent. The terminal counting argument belongs to this report and was checked separately.

The current warranted reading is the conditional implication with P and the quoted classical/source premises explicit, using exactly the accepted E fragments. The original literal conjunction E contains the false unrestricted E1 statement; treating that entire conjunction as accepted would be invalid. The report's actual deduction uses only the restricted clauses above, so its useful conditional reading does not rely on the false clause or on vacuous entailment.

Prospectively, if an independent accepted P result supplies exactly the endpoint statement used here, and the stated classical boundaries are accepted at their recorded depth, the surviving E scopes and this chain can be composed in that order. This grade supplies no P acceptance, no source Section 6 equivalence and no extra endpoint hypothesis. Catalog acceptance, any eligibility ruling on the current grader, preservation and standing remain with the commissioning authority.

Whole-contract judgment and every failure mode

Subject, independence, restatement, weakest derivations, strongest attack, dependencies, preservation and written reproduction are present for each graded assertion. The original report contains explicit findings for all nineteen canonical modes and all ten Erdős-specific items. The grading findings below assess their substance, rather than merely the presence of headings.

Correlated-sample inference is inapplicable under the approach-vetting rules. Under the evidence rules, administrative identities and line matches establish no mathematical conclusion; the written derivations supply the actual proof coverage. No test was manufactured in place of reasoning.

Preservation and permitted use

Keep the original package, this grade and its reproduction account together. The original relative links resolve within the retained package. The seven inputs retain every mathematical omission marker. Their filed metadata and paths receive only the maintenance declared on this record's index. The exact private provenance and actual-read records are preserved separately for the commissioning authority; this public account states their content findings without making private records a mathematical premise.

Reproduction consists of reading the original premise account and frozen subjects, then recalculating the three keeper routes and the two scope witnesses recorded in this batch. No numerical simulation, source acquisition, repository runtime or revision lookup is required. The original package has not been edited to match a later argument.

The grade is PASS for the complete conditional report at its stated scope, including PASS for its refuted-as-stated unrestricted dyadic finding. No essential deduction is left unfinished within that conditional scope. P acceptance, original-source fidelity, public adoption and any separate ruling on the current grader remain outside this completed grading product.