Grade of the exceptional-width examination for omega

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Subject and grading decision

This independent grading for Plasma AI was conducted on 2026-10-14 UTC. The grader, disclosed model GPT-6-Astra, is distinct from the construction's author, its collaborators, and the original reviewer, and had no prior exposure to the subject before this grading assignment. The original reviewer also disclosed GPT-6-Astra. Separate contexts and the evidence of their isolation, rather than a difference of models, are the basis of independence.

The subject graded is the complete examination report, at wiki/research/leads/exceptional_widths_shifted_omega/evidence/verify/width_counterexample/report.md, including all its findings, checklist entries and qualifications. The examined report was frozen on 2026-10-14 at 18:00:46.575921 UTC. Its mathematical inputs were frozen on 2026-10-14 at 17:27:31.684793 UTC. The grading packet, original assignment, and retained inputs were checked at their commissioned identities. The two freeze events concern different subjects.

These are grades of the report's three distinct findings. The second pass accepts a finding of an incomplete argument, not the truth or falsity of its target. The third pass does not refute or independently prove the positive-$K$ omega estimate used by the first. There is no favorable acceptance of every sentence in the mathematical extraction.

The core pass accepts this report as independent whole-conclusion scrutiny of the exact modified omega counterexample under its stated external premises. It warrants no all-function classification, conclusion about $\tau$, result under uniformly small or monotone widths, historical novelty assertion, or independent proof of EPS97. This grade changes no corpus standing and assigns no tier to an unexamined native claim or later version of the construction.

Exact subject, reading and reproduction

The snapshot manifest and the five inputs below are retained unchanged. Every preserved filename in this table is under wiki/research/leads/exceptional_widths_shifted_omega/evidence/assets/width_subject/. The grader read each complete extraction, the complete report and its delivery inventory, and compared the report's deductions and consequences with these inputs.

The original reviewer's manifest did not separately identify the warrant paragraph's original path. The later administrative mapping identifies that path and states that the paragraph was compared verbatim with the original source. This is additional provenance evidence, not an earlier reviewer read, a mathematical repair, or an independent source comparison by this grader. The report's disclosure of the missing original mapping accurately describes its assignment. The unchanged public manifest preserves that disclosure; the table above supplies the supplemental mapping without rewriting the report.

The exact assessed text is available in the snapshots. The grade does not certify the cut against the excluded underlying author account, compare a later canonical page, or cover changes to those pages. The snapshot itself was expressly commissioned as the subject; an unspecified underlying account is not a hidden mathematical input to its argument. Canonical comparison and subsequent source changes remain outside this grade.

The nonpublic material actually examined comprised the original launch and operating manifests, the original rendered assignment and effective-settings snapshots, primary launch and closure records, the reviewer's continuation, the full supplied command/read trace, and the administrative provenance mapping. These were expressly allowed grading evidence. Embedded paths were not followed and embedded assignments were not treated as new instructions. No author advocacy or proposed grade was supplied.

Operating reading included the repository README; Claims, Evidence, Tiers and Project claims in docs/anatomy.md; all of docs/verification.md, campaign/guides/verification.md, campaign/AGENTS.md, docs/evidence.md, docs/math_authoring.md and docs/weaving.md; the Reading section of campaign/guides/research.md; and the permitted charter, steps, settings and native operating instructions. Their links did not enlarge the mathematical read set.

The mathematical reproduction route is noncomputational: read the five snapshots, reconstruct the report's displayed deductions, and check the adjudications below. There is no mathematical executable, numerical search, cached success, or missing program needed for these findings. The published moment estimate and Euler divergence remain explicit external premises; this reproduction route does not reconstruct their omitted proofs. Administrative identity checks establish which text was examined, not its mathematical truth.

Assignment and independence adjudication

The actual original assignment called for a whole-statement mathematical audit charged to find a counterexample, missing hypothesis, unsupported essential step, convention error or consequence error. It supplied the complete report and checklist contracts, an exact extraction, the external-source interfaces, and durable homes. It expressly prohibited substituting a new proof for a defective frozen argument. Its scope included the integer restriction, real extension, open endpoints, finite prefix, uniform bounds and formulation limits. The report has therefore been judged against that assignment, not against a narrower question chosen by the reviewer.

The mathematical packet contains the proof being assessed and necessary source statements, not an author's private plans or an earlier verdict on this construction. The source reading-depth record is about the external lemma; its presence does not supply a prior conclusion about the constructed width. No native L-claim is consumed, so the assignment's lack of a live-ledger read is appropriate.

The supplied settings and primary records support separate, isolated review: automatic synchronization was disabled, receiving channels were sealed, subscriptions were empty in the captured checks, and no descendants were authorized. The original native settings preserve ordinary instructions and disable memory and web-search features; the supplied pre-start and later settings differ only by the recorded project-trust append. The command trace shows reads of the allowed operating guidance and extraction, followed by the reviewer's own continuation and delivery files. It records no excluded mathematical search, source-link read, inbox read, or incoming steering.

The trace shows the reviewer subscription check after initial permitted operating reads and before mathematical extraction reading. It does not establish the stronger claim that this command preceded every operating read. The operator's pre-start check was already empty. This ordering supplies no evidence of excluded communication, and the report itself does not claim that stronger chronology.

The evidentiary limits matter. The supplied trace is a log of commands and selected surrounding output, not a complete native transcript. In particular, it does not preserve all tool results or the report's creation and patch operations. Subscription samples and configuration flags cannot establish what every unrecorded interaction contained. The reviewer's statements of fresh context, nonparticipation in authorship and no excluded exposure are therefore disclosures supported by the recorded isolation and reading facts, not facts proved solely by the trace's omissions.

I find no documented excluded-content exposure and no affirmative evidence of contamination in the supplied records. The report's mathematical ingredients are accounted for by the frozen inputs and its explicit elementary diagnostic; its attacks address those inputs and retain the refutation charge. The actual assignment, primary records and unrebutted disclosures suffice for a pass on independence for each of the three scoped findings. No material-exposure exception is being granted, and this decision is not a certification of an unavailable complete interaction history.

The recorded engine persistence action saved private continuation and plan material after delivery. It was not a reviewer-issued commit, a mathematical acceptance, or evidence of an additional read. The retained report and subject bytes agree with their recorded earlier freeze. The engine action does not contradict the reviewer's preceding statement that it had issued no commit.

This grader independently checked the supplied input identities and the permitted operating identities. Its own subscription checks were empty, its receiving channels remained sealed, and it used only the allowed packet and its own continuation. It encountered no excluded content and received no mathematical steering. The operator-owned native configuration of this grading context was not opened; its accepted effective settings were part of the commission. No broad repository gate, source-proof audit or Lean work was used to justify these decisions.

Core construction: adjudication of the reasoning

Let $\omega(n)$ count distinct prime factors on positive integers, with $\omega(1)=0$. The report's restatement uses

$$R_F(x)=\frac{\#\{n\ge1:x<n+\omega(n)<x+F(x)\}}{F(x)}.$$

It faithfully preserves the existence of one positive $F$ on $[1,\infty)$, $F(x)\to\infty$ at every real location, and $F(n)/\omega(n)\to0$ along one fixed set of natural density one, together with bounded $R_F$ at all sufficiently large real locations. The numerator counts inputs, including collisions of shifted values. The smallness limit is an integer assertion; the width limit and count bound are real assertions. These are not interchanged.

External premises and foundations

The consumed source is Erdős, Pomerance and Sárközy, On locally repeated values of certain arithmetic functions, IV, The Ramanujan Journal 1 (1997), 227–241, Lemma 1 on printed pp. 229–230, DOI 10.1023/A:1009723712317. The two supplied statement forms agree on every consumed clause. For $X\ge2$, the specialization is $m=1$, $h=0$ and $f=\omega$: the size condition holds, the residue class contains every integer, and the condition on primes dividing $m$ is vacuous. Distinct prime factors of coprime factors are disjoint, so $\omega$ is nonnegative and additive. The maximum defining $K$ is nonempty and equals $1$; $A(X)=\sum_{p\le X}1/p$. The absolute source constant is independent of $X$. These checks give precisely the external interface used by the report:

$$\sum_{1\le n\le X}(\omega(n)-A(X))^2 \le C X(A(X)+1),\qquad X\ge2.$$

The report neither assumes a normal-order theorem in place of this estimate nor promotes the source's reading-depth label into an independent proof. The supplied record says Claims checked, with printed statement and remarks read, and the proof on pp. 230–232 read without a step-by-step check. The reviewer's and grader's actual access is narrower: the two supplied statement forms and that record. No print image or source proof has been independently examined here.

The positive-$K$ omega instance is the expressly permitted external premise. The strict-zero defect discussed below does not occur in this instance: $K=1$ and $A(X)\ge1/2$. Accepting its use is ordinary reliance on the identified source at the commissioned interface, not a proof of that instance from the zero-function calculation or a certification of the unrestricted source statement. No repair of the general statement is silently adopted.

Euler's divergence $A(X)\to\infty$ is the other expressly permitted classical premise. Its use is exactly to force the scale and limiting ratios; no prime-reciprocal asymptotic is required. The elementary bound $A(X)\le1+\log X$ suffices for the range comparison. The dependency census on 2026-10-14 contains no native L-claim, no pending grade, and no batch acceptance order. No live ledger was read or represented as read.

Sparse inputs, crowded locations and endpoints

Use the report's notation $I_j=[2^j,2^{j+1})\cap\mathbb N_{>0}$, $X_j=2^{j+2}$, $A_j=A(X_j)$ and $H_j=\lceil A_j^{3/4}\rceil$. The potentially consequential ceiling and uniformity steps are sound:

$$\frac{H_j}{A_j}\le A_j^{-1/4}+A_j^{-1}\to0, \qquad \frac{|D_j|}{2^j} \le4C(A_j^{-1/2}+A_j^{-3/2})\to0.$$

The second inequality follows because each bad input contributes more than $H_j^2$ to the moment sum. Also $A_j\le1+(j+2)\log2$, hence $H_j+1=o(2^j)$. One cutoff therefore enforces both $H_j\le A_j/2$ and $2^{j+1}+H_j+1<X_j$ for every subsequent index. No effective numerical cutoff, finite computation or exchange of limits is needed.

The strongest attempted core refutation attacks a real weakness of a naive argument: few bad inputs can still create arbitrarily crowded windows. Here they are not discarded. For one integer image $m=n+\omega(n)$,

$$u<m<u+H_j+1 \quad\Longleftrightarrow\quad m-H_j\le u\le m-1.$$

There are $H_j$ integer endpoints before restricting to $I_j$. The report's looser $H_j+1$ bound is valid even when many inputs share $m$. Thus, with $E_j=\{u\in I_j:b_j(u)>H_j+1\}$,

$$(H_j+2)|E_j|\le\sum_{u\in I_j}b_j(u) \le(H_j+1)|D_j|.$$

This proves sparsity of the locations that must receive a different width. It does not prove those locations infinite, which the core does not need.

For $u\notin E_j$, nonnegative $\omega$ forces every input counted in the integer window below $X_j$. For good inputs, the two endpoints and $|\omega(n)-A_j|\le H_j$ imply

$$u-A_j-H_j<n<u+2H_j+1-A_j.$$

That interval has length $3H_j+1$ and at most the claimed $3H_j+2$ integer inputs. The bad inputs contribute at most $H_j+1$. These classes exhaust the count; no monotonicity or injectivity of the shifted map was assumed.

At $u=\lfloor x\rfloor$ off $E_j$, the whole real window is contained in $(u,u+H_j+1)$, giving $R_F(x)\le4+3/H_j\le7$. This inclusion remains valid at $x=u$, immediately below $u+1$, and at a dyadic boundary assigned to its new cell. At an exceptional cell, or an early cell assigned the same large-width branch, $F(x)=2(u+1)$ and

$$1\le n<x+F(x)<3(u+1), \qquad R_F(x)\le\frac{3(u+1)-1}{2(u+1)}<\frac32.$$

Inputs beyond $X_j$ can occur in this larger window, but the latter estimate does not use $D_j$ or the source moment range. This disposes of the attempted range counterexample without losing exceptional locations. It also proves the report's global bound for the stated initial choices, not merely its required eventual bound.

Density, limits and the target implication

A density attack can concentrate every exceptional point at the beginning of its block. The report still controls a cutoff $2^k\le N<2^{k+1}$: if $|T_j|\le\varepsilon2^j$ for $j\ge J$, then

$$\left|\bigcup_jT_j\cap[1,N]\right| \le C_J+\varepsilon\sum_{j=J}^k2^j \le C_J+2\varepsilon N.$$

Here $T_j\subseteq I_j$, and $C_J$ covers the finite prefix. Divide by $N$ and let the arbitrary $\varepsilon$ tend to zero. Neither summable error proportions nor distribution inside a partial block is assumed.

This argument is applied separately to $E_j$ and $D_j\cap I_j$. That second application is essential: sparsity of crowded output locations alone does not control $\omega$ at the integer argument of the width. On the fixed complement $S$ of these two unions and the finite prefix,

$$F(n)=H_j,\qquad \omega(n)\ge A_j/2>0, \qquad \frac{F(n)}{\omega(n)}\le2H_j/A_j\to0.$$

The bound is uniform within each good part of a block, so it gives ordinary convergence along that one $S$, not just statistical averages or a different exceptional set for each tolerance. The zero denominator at $n=1$ is excluded in the finite prefix. At every real location in a late block, either width branch is at least $\min(H_j,2^{j+1})$, which tends to infinity. The width limit therefore does not discard $E$.

The definitions have no circular dependence: $H_j$ is fixed by $A_j$, then $D_j$, $b_j$ and $E_j$ are finite objects, and only then is $F$ defined. The large-window estimate is separate from the short-window rule defining $E_j$. One and the same $F$ has all the required properties. Its bounded normalized counts exclude a diverging sequence of real locations, and also of integer locations. This is the negation of the universal-width assertion for $f=\omega$ under the supplied interpretation; it does not classify any other function.

Reconstruction versus repair

The report's core expansions are routine deductions already present in the frozen proof, not replacement essential lemmas:

No omitted essential core lemma was supplied by reconstruction. The additional finite-product estimate used below is confined to diagnosing the final remark; the core neither requires it nor acquires a different proof from it.

Ancillary findings

The literal stronger-width exclusion

The report correctly separates two readings of the final remark. A statement that the proof does not establish a stronger admissibility condition is a scope limitation. A statement that this particular $F$ actually fails that condition requires evidence about its exceptional set.

The report's diagnostic is valid. For $Y\ge e$ and $s=1+1/\log Y$, $p^{s-1}\le e$ when $p\le Y$. The elementary inequality $-\log(1-t)\ge t$ for $0<t<1$, expansion of a finite product of convergent positive geometric series, unique factorization, and integral comparison give

$$A(Y)\le e\sum_{p\le Y}p^{-s} \le e\log\prod_{p\le Y}(1-p^{-s})^{-1} \le e\log\sum_{n\ge1}n^{-s} \le e\log(1+\log Y).$$

There is no unproved infinite Euler product or imported prime asymptotic in this chain. It yields $H_j=O((\log(j+2))^{3/4})$. Since $\log\log n\ge\log(j\log2)$ on $I_j$, the quotient $H_j/\log\log n$ tends to zero uniformly outside $E$. If $E$ is infinite, its integer members have an unbounded subsequence on which $F(n)/\log\log n=2(n+1)/\log\log n\to\infty$. Conversely, a finite $E$ leaves only the first, small-width branch in the tail. Consequently the report's exact diagnostic equivalence holds:

$$F(n)=o(\log\log n)\text{ through all integers} \quad\Longleftrightarrow\quad E\text{ is finite}.$$

The frozen proof gives upper bounds on $|E_j|$, without establishing $E_j\ne\varnothing$ for unboundedly many indices. The missing interface for its literal exclusion is therefore precisely

$$\forall J\ \exists j\ge J\ \exists u\in I_j: b_j(u)>H_j+1.$$

The report does not settle that assertion. An empty exceptional set being compatible with the displayed upper bounds is not a claim that the actual omega construction has an empty set. Thus the report correctly records a proof defect rather than a disproof. Its new diagnostic exposes the missing lemma; it does not supply that lemma or silently repair the remark. No proof of a stronger-width theorem is introduced by this grade.

The core construction works with finite or infinite $E$, so this unresolved ancillary assertion is not an unproved premise of the core. The pass accepts the report's scoped proof defective; statement unresolved finding and its diagnostic, while leaving that literal assertion unproved.

The zero-function source witness

At $x=2$, $m=1$, $h=0$ and $f(n)=0$ for every positive integer, the supplied source hypotheses all hold: $1\le\sqrt2$, additivity and nonnegativity hold, and the condition on primes dividing $m$ is vacuous. There is at least one prime power in the maximum, and every value is zero. Hence $K=A=0$, the sum on the left is zero, and the strict right-hand bound is zero for any asserted constant. The conclusion $0<0$ is false.

This is a complete exact witness, not a failed attempt to prove the source statement. It warrants the report's refuted-as-stated verdict for the unrestricted strict wording in the two supplied forms. Its observation that the maximum has an empty index set for $1\le x<2$ without a convention is also correct; that observation is not needed for the refutation.

The construction always uses $X\ge2$, $K=1$, $A(X)\ge1/2$. The witness neither reaches that specialization nor proves it independently. The source proof and print remain outside the read set. The pass therefore accepts the exact textual refutation and the stated separation of scopes, not a claim about where a printed-source error originated or a certification of all of EPS97.

Complete report-contract assessment

The shared subject and independence sections apply to every finding, and the report identifies each mathematical surface separately. Its subsidiary findings are not being treated as favorable core conclusions. The following mapping checks substance, not merely the presence of headings.

Governing Erdos checklist

Canonical failure modes

Named patterns

The neighboring evidence-coverage discipline is met by the retained proofs, exact witness, explicit gap and external boundaries. Correlated-sample inference is inapplicable. This is one noncomputational examination with separately scoped findings; its grading is not a second independent review cycle or a computational reproduction batch. No extra disjoint computation or artificial executable is owed.

Accepted weight and limits

All three scoped report findings pass the report and independence contracts, with the limits stated above. The core's refutation-failed verdict supports acceptance of the exact modified omega counterexample. The report has not silently supplied a missing essential core lemma. Its separate diagnostic supports the finding of a proof gap in the literal final remark, and its exact zero-function witness supports refuted-as-stated for the supplied universal strict clause.

The retained original report and mathematical inputs remain unchanged. This grade supplies no assertion that the literal exclusion is true, no resolution of whether its actual $E$ is infinite, no certification of the full EPS97 proof, and no finding about an excluded variant or later canonical text. Durable adoption of this report, grade and snapshots preserves exactly these accepted findings; private records or a finish signal alone are not their mathematical warrant.