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

Updated Oct 15

Grade of the White exact-one-failure review

Grade: PASS for each of the three commissioned clauses. The accepted mathematical verdict of the reviewed report is refutation-failed, under the classical interfaces stated below. This accepts the report at its exact scope; this grade supplies no additional independent theorem proof. It asserts no native claim tier, optimal estimate, full solution of Problem 1063, public adoption, or external community acceptance.

Subject and grading independence

The grader is an independent report grader, disclosed model gpt-6-astra, reasoning effort max, in a fresh context distinct from the source author, mathematical reviewer, and commissioner. The grader did not construct, collaborate on, or previously build upon the subject or the report. The mathematical reviewer disclosed the same model and effort. Their separation is established by the distinct assignments and retained input/action records, not by a difference of model.

The grading examination date is 2026-10-15 UTC. The graded report is the unchanged submission for wiki/problems/integer_sequences/E1063/claims/evidence/verify/white_upper_bound/report.md, examined on 2026-10-15 and frozen at 07:10:45.101304 UTC that day. This identifies the submitted copy, not a claim that the destination has already been populated. Its mathematical subject was frozen on 2026-10-15 at 05:52:55.296610 UTC:

Source	Exact examined scope
`wiki/problems/integer_sequences/E1063/_index.md`	Statement, lines 23–26.
`wiki/problems/integer_sequences/E1063/claims/2026_07_25_white.md`	Mathematical claim, lines 25–34, and supplied dependency sentence on line 56.
Patrick White's immutable README	Theorem, lines 47–54, and Sections 1–5, lines 61–190, as preserved in the commissioned extraction.

The complete report and all three mathematical fragments were read. The grade also examined the original launched commission and settings, primary session records and tool returns, launch observations, retained identity and delivery records, and the automatic-commit path listing. The unabridged private input closure is preserved. Derived navigation tables were checked against raw records and were not substituted for them.

The governing rules were the commissioned versions of docs/verification.md in full, including its canonical modes, named patterns, and Erdos-specific contract, and campaign/guides/verification.md in full. The permitted repository and operating instructions were checked before use. The corresponding original rules were supplied in the frozen packet and agree with the grading rules on the relevant requirements. The original commission's actual permissions govern the original review's history.

The grader read only the permitted operating context, supplied frozen packet, and its own continuation. Empty subscription observations support this isolation. No author inspection, additional report, live inbox, external source, or later correction was consulted. No excluded exposure to the grader is known. A failed checkpoint edit in preparation selected a nonexistent path; it returned no file content and changed nothing. It provides no mathematical information and does not affect independence.

The source construction and bound remain credited to Patrick White and the source's credited collaborators. The grading analysis is a Plasma AI assessment.

Exact clauses and separate findings

For integer $k\ge2$ , define

\mathcal A_k=\left\{n\in\mathbb Z:n\ge2k, \#\{0\le j<k:n-j\nmid\tbinom nk\}=1\right\}, \qquad n_k=\min\mathcal A_k.

Logarithms are natural. The report checks nonemptiness, the original positive integer divisibility convention, and the minimum before using them. In particular, “exactly one” is preserved throughout.

Commissioned clause	Accepted mathematical verdict	Contract grade and precise weight
Absolute constants $C>0,K$ exist such that every integer $k\ge K$ satisfies $n_k\le\exp(Ck\log\log k/\log k)$ .	refutation-failed	PASS. Whole supplied construction and quantitative deduction, including existence, admissibility, all prime classes, a strict failed divisor, and a common multiplier; relative to the stated classical interfaces. No numerical $C,K$ or sharpness is accepted.
$n_k=\exp(o(k))$ , meaning $\log n_k/k\to0$ .	refutation-failed	PASS. A separate squeeze argument from the eventual bound and positivity, with every sufficiently large integer covered.
$n_k=o(k\operatorname{lcm}(1,\ldots,k-1))$ .	refutation-failed	PASS. The separate logarithmic comparison, with the explicitly stronger $\psi(t)\sim t$ interface. The report also justifies the source's logarithmic rate $-k+o(k)$ for this ratio.

The all- $k$ nonemptiness check is accepted as support for the definition of $n_k$ . It does not extend the quantitative bound to every $k$ . The obstruction in source Section 2 is accepted only against the specified level-by-level surplus requirement, not as a necessity theorem about every possible construction or proof.

Reviewer eligibility, actual reads, and exposure rulings

The primary records support a fresh mathematical review. The preparatory probe contains the repository instructions and a content-free request to reply “ok,” with no tool call or mathematical input. The actual review begins with the native operating instructions and commissioned assignment. Its subsequent user messages are the next two prescribed steps; their commission text agrees exactly with the launched charter. No additional mathematical steering or inherited author conversation appears in those records.

All recorded tool calls were examined. The primary returns, not only the reviewer's assertion, establish reception of the complete mathematical manifest and three fragments. The numbered full-source return, with its line prefixes removed, equals the supplied source fragment; a second return contains that same complete fragment. The canonical statement and claim return equals their supplied fragments. The report's source/proof reading coverage is therefore supported by actual input bytes.

The actual operating reads consist of the supplied root instructions, README.md; the stated introduction, Claims, Evidence, Tiers, and Project claims sections of docs/anatomy.md; all of docs/verification.md, docs/evidence.md, docs/approach_vetting.md, docs/math_authoring.md, and docs/weaving.md; campaign/AGENTS.md and campaign/guides/verification.md; Reading in campaign/guides/research.md; the reviewer's own charter, steps, configuration, template, applicable operating skills, and newly written continuation. Other manifest-listed operating files were read as bytes for identity checking. Full verification-rule returns agree with the retained rules. Truncation of two combined operating/report outputs is visible; it does not hide an unexamined mathematical source, and the necessary material is present in separate returns.

The following rulings distinguish boundary facts from contamination:

Observed fact	Ruling
The reviewer read an operating manifest in another checkout.	Permitted in the original commission. It named that exact read-only input. The manifest contains operating identities/settings, not additional mathematical advocacy. The current grading boundary does not retroactively prohibit that read.
Empty subscription tables before launch and at the reviewer's recorded checks; blind, sealed, local, no-sync settings and zero child limits.	Supporting evidence, not the sole warrant. The tool/action and incoming-message record supplies the actual-reading basis. No inbox/feed read, subscription addition, unsealing, broad search, source-network access, or descendant call is recorded.
Native configuration acquired a project-trust append.	Accounted for, immaterial. The supplied before/after bytes differ only by the expressly accepted literal project-trust entry. The reviewer did not open the excluded native configuration; the operator's inspection is distinct from reviewer exposure.
Environment warnings named a different inherited virtual environment.	Immaterial operating metadata. The warnings say it was ignored in favor of the project environment. They contain no excluded mathematics or suggested attack.
An internal compaction occurred after the complete mathematical report had been written.	No external exposure established. Its retained user inputs come from the same review. The internal encrypted compaction payload cannot be inspected semantically, so this grade does not certify its wording. The surrounding raw input/action record, unchanged report, and independently read report content support the accepted finding; the encrypted payload supplies no evidentiary premise.
The original automatic commit listed three created files.	Permitted continuation persistence. Its actual paths are the reviewer's own `memory/frontier.md`, `plans/delivery.md`, and `plans/examination.md`. No corpus, subject, report, or other path appears in that commit listing. The exact path record is retained privately.

No excluded mathematical exposure is established. In particular, no report finding requires material absent from the authorized mathematical subject, and no recorded external communication changes an attack direction or weakens the refutation charge. The content test therefore gives no reason to replace the reviewer.

This is a finding about the retained primary record. It does not claim an independent observation of every process or filesystem event outside that record. The launch log records environment setup before the review and an unnamed ignored-path warning at automatic persistence; neither is silently treated as an additional mathematical read or a list of corpus changes. Recorded reviewer writes stay within its private delivery, continuation, and check records. A failed report patch changed no subject and was followed by an ordinary correction to the reviewer's own draft. No reviewer commit, repository gate, mathematical program, or Lean execution is recorded.

Complete report contract

The findings below apply separately to each clause in the preceding table. The report uses one construction for the bound and then makes the two consequence deductions explicitly; it is not a computational batch.

Required part	Quantitative bound	First consequence	Lcm consequence	Basis
Subject, conventions, identities, independence, allowed/actual reads	PASS	PASS	PASS	Opening subject and independence section; frozen fragment provenance and primary records checked above.
Restatement, all quantifiers and scope	PASS	PASS	PASS	Restatement defines $\mathcal A_k$ , the minimum, absolute eventual constants, and each limit in its own variables.
Every canonical mode and named pattern	PASS	PASS	PASS	Seven canonical rows and twelve named rows are substantive and explicit; individual grading rulings follow below.
Erdos-specific checklist	PASS	PASS	PASS	All ten required items are addressed and graded below.
Weakest steps, independently rederived and composed	PASS	PASS	PASS	Three groups cover valuation exactness, common realization/admissibility, and quantitative costs with separate consequences.
Strongest attack and precise witness	PASS	PASS	PASS	Cancellation and lost-deficit attack, explicit $k=7$ input, separate finite deletion identity; the cost and consequence attacks address the asymptotic clauses directly.
Premises, actual standing/interface reading, assumptions	PASS	PASS	PASS	No native L-premise is consumed. Classical interfaces are explicit; the lcm consequence's stronger premise and unread external proofs are disclosed.
Evidence, reproducibility, and durable scope	PASS	PASS	PASS	Complete symbolic reasoning and the exact diagnostic input are retained in the report. No omitted program is needed. Public filing and source-access limits are stated below.
Verdict, limitations, separate grading	PASS	PASS	PASS	Full verdict words, explicit exclusions, and no self-grade or standing change. This independent grade accepts only the stated scope.

Weakest-step and strongest-attack assessment

Valuations and the single failure. The report rederives the window count and Legendre carry identity, then keeps the contribution above $A_p$ as a nonnegative count outside the selected coordinate. It never equates an untruncated valuation with that of the offset in the potentially cancelling case. Small primes with positive extra depth, exceptional large primes, regular primes, and primes above $k$ each receive their own coverage argument. The chosen failure prime cannot be regular: regularity gives $A_p=\sigma_p=1$ . At the chosen prime the retained deletion calculation gives the strict deficit

v_{p_*}(n-E)-v_{p_*}\binom nk =v_{p_*}\!\left(k\binom{k-1}{E}\right)>0.

This tests exactly the required failure, including arbitrarily increased valuation of the multiplier. The alternative carry-tail derivation has the same sign and does not assume the desired nondivisibility. The reported diagnostic is arithmetically sound: $44102-6=44096=64\cdot689$ , whereas $2-6=-4$ has 2-adic valuation two. Its residues through 64 and at 128 support the stated six carry levels. This is a genuine challenge to the tempting shortcut, not evidence extrapolated from one example.

Common realization and admissibility. The report checks finite half-open pigeonhole boxes, a strictly positive index difference, and the empty regular-prime case. Its strict residue margin keeps the actual residue between 1 and $2E-1$ , below $k\bmod p$ , so signed approximation cannot wrap through zero. The source's finite prime universe is explicit in its anchor setup, $p\le k$ bound, and separate treatment of larger primes; making that universe uniform across the notation is routine reconstruction. Taking the least integer satisfying the anchors is a legitimate choice for the construction's upper-bound calculation.

The supplementary admissibility check also uses only those anchors. Each prime power dividing $k$ is covered by a small anchor or a large exceptional square, hence $k\mid S$ ; the forced anchor gives $S>k$ . Thus $S\ge2k$ , and the same permitted construction with $E=1$ supplies all- $k$ nonemptiness. This does not depend on the unread small- $k$ attribution in the source. These short deductions introduce no missing essential external lemma or replacement construction.

Quantitative strength and consequences. This was the strongest grading challenge to the report's completeness. A conclusion merely $\log S=o(k)$ cannot support the claimed quantitative scale: for example, $k/\sqrt{\log k}$ is $o(k)$ but exceeds that scale. The report does not rely on that insufficient inference. With $L=\log k$ and $F=k\log L/L$ , it substitutes the source's actual anchor estimates to obtain

\log S=O\!\left(\frac{k\log L}{L^2}+\sqrt{k}L+L^4\right)=o(F).

The three ratios to $F$ are respectively $1/L$ , $L^2/(\sqrt{k}\log L)$ , and $L^5/(k\log L)$ , each tending to zero. The approximation cost is $O(F)$ with absolute constants, and $n\le2Sm$ . This is an explicit completion of estimates already in the source, not a repair using an unprovided theorem. Ceiling errors, the forced prime cost, and interval overcounting retain their required inequality directions.

For the first consequence the report separately bounds $0\le\log n_k/k\le C\log\log k/\log k$ . For the second it uses the exact lcm identity and the permitted $\psi$ asymptotic to obtain $\log(n_k/(k\operatorname{lcm}(1,\ldots,k-1)))=-k+o(k)$ . Neither deduction strengthens an eventual estimate to an all-order one. The cost objection therefore fails against the complete report, although it correctly exposes why the source's weaker summary alone would be insufficient.

Premises and reading depth

On 2026-10-15 UTC, the grader derived the consumed-premise list from the entire supplied statement, argument, report, and consequence sentences. There is no native L-claim premise. The negative dependency sentence was not used as a substitute for this reading. No live ledger was read or needed; no pending grade or batch promotion order is involved.

Interface	Accepted use and reading depth
Legendre: $v_p(u!)=\sum_{a\ge1}\lfloor u/p^a\rfloor$ , prime $p$ , integer $u\ge0$	Source Sections 1 and 3.1; the report supplies the elementary counting derivation and checks its use in carries and the accounting identity. No separate external edition was read.
Absolute Chebyshev bounds for $\theta(t)$ above and below, and $\pi(t)\ll t/\log t$ eventually	Source Section 5; explicit permitted interfaces for anchor cost, original eventual admissibility, and approximation dimension. Their applications are checked; external proofs/editions are unread.
$\psi(k)-\theta(k)=O(\sqrt{k}\log k)$	Source Section 5; the report also derives the needed estimate by counting at most $\sqrt{k}$ prime bases, each contributing at most $\log k$ .
$\psi(t)/t\to1$	Source Section 5's lcm comparison; the specialization to $t=k-1$ and its stronger strength relative to Chebyshev are explicit. External proof/edition unread.

These are the classical interfaces expressly admitted by the original commission and retained by the grading commission. The PASS concerns the argument under those interfaces; it does not turn their unread proofs into independently inspected literature. Kummer is not an extra premise here, and neither Erdős–Selfridge nor another construction is needed. Pigeonhole, finite valuation identities, the binomial estimate, and elementary asymptotic comparisons are supplied directly. No hidden consumed premise was found within the assessed bytes.

Canonical failure modes

Mode	Grading finding
Almost all quietly upgraded to all	PASS. One eventual threshold covers every integer above it; no density-one substitute occurs.
Induction that presupposes termination	PASS, inapplicable. The multiplier comes from a specified finite pigeonhole set, not an unbounded search or induction.
Probabilistic or averaging heuristics presented as proofs	PASS, inapplicable. Deterministic pigeonhole and upper counting supply the relevant steps.
Circular use of an equivalent statement	PASS. Anchors and a positive multiplier are chosen without assuming an exact-one-failure integer; nonemptiness is then established.
Exceptional sets dropped from density arguments	PASS. Every exceptional prime is counted and covered; primes above $k$ are separately harmless. No density argument is used.
Finite verification cited as more than base-case coverage	PASS. The one numerical diagnostic tests a shortcut only; the theorem and consequences have symbolic derivations.
Relaxed or averaged convergence substituted for actual objects	PASS. A common integer realizes the residues, and the valuation argument concerns the original divisibility conditions.

Named patterns

Pattern	Grading finding
Model-class transport instead of entailment	PASS. Axiom-system classification is inapplicable; the relevant residue-to-divisibility implication is actually proved.
Uniformity over an infinite family from finitely many instances	PASS. Constants are absolute, the eventual dependence on $k$ is explicit, and no finite sample supplies uniformity.
Extremal claims in their own units	PASS. Nonemptiness and well-ordering justify $n_k$ ; a constructed admissible $n$ bounds that minimum. No sharp value is claimed.
Consequence sentences are claim surfaces	PASS. Both limits and the logarithmic exponential rate are checked separately. The unread small- $k$ attribution receives no acceptance.
Carry hypotheses used by quantified arguments	PASS. Integer domains, $1\le E<k$ , finite prime universe, residue margin, positive multiplier, and eventual parameter choice are all retained.
Composition inherits unproved premises	PASS within the explicit interfaces. Every internal transfer is supplied; unread classical proofs remain outside the accepted proof-inspection scope.
Reproducibility notes are claims	PASS. Raw records support zero mathematical code runs and no keeper-program claim. Identity/format checks are explicitly mechanical and supply no theorem evidence.
Verifier quotations are claims	PASS. The submitted report's actual verdict and limitations agree with its delivery record. No absent earlier ruling is invoked or strengthened.
Verdict words spelled in full	PASS. The accepted verdict is refutation-failed. The report does not confuse it with refuted-as-stated or proof defective; statement unresolved.
Certified-bracket functions fail loudly	PASS, inapplicable. There is no numerical enclosure routine or exhausted iteration cap in the mathematical warrant.
A harness leg with no failing input is decoration	PASS, inapplicable to the mathematics. No executable leg is claimed; package checks are not counted as proof. The cancellation example can and does falsify the identified equality shortcut.
A gate reading caches instead of rerunning is defective	PASS, inapplicable to the mathematical warrant. No gate or cached success supports it; this grade checks frozen inputs and reasoning directly.

Erdos-specific checklist

Item	Grading finding
Quantifiers and scope	PASS. Exactly one failure, $n\ge2k$ , absolute eventual constants, and both limits retain their scope. This is partial E1063.
Circularity	PASS. No target conclusion or equivalent existence claim is assumed.
Model and convention changes	PASS. Signed and least nonnegative residues are distinguished; their transfer to integer divisibility is justified.
Finite and statistical overreach	PASS. There is no statistical sample or extrapolated finite census.
Uniformity	PASS. Prime classes, anchor overlaps, ceilings, forced-prime cost, and all error terms have their claimed absolute bounds.
Extremal conclusions	PASS. Existence and the minimum are addressed; no unsupported optimality or attained lower estimate is introduced.
Consequences and composition	PASS. Unique failure, common realization, admissibility, cost, and each consequence have the necessary interfaces.
Computation	PASS, executable mathematical computation inapplicable. The short integer diagnostic and symbolic inequalities have exposed exact arithmetic.
Reproduction	PASS at the noncomputational scope. The report retains the derivations and diagnostic input; no hidden executable or cached output is needed. Source provenance and public-access limitations remain explicit.
Source and verdict fidelity	PASS for the supplied immutable extraction and submitted report. Routine reconstruction is disclosed. No remote authentication, omitted source section, prior verifier, or later correction is certified.

The neighboring disciplines are also accounted for. There are no correlated samples to turn into statistical evidence. The Section 2 obstruction and the cancellation diagnostic apply only at their stated hypotheses; neither is promoted to an impossibility statement about other approaches. Each proof or check is credited only with its actual coverage. The computational batch reproduction rule and a second cycle's disjoint-attack requirement are inapplicable: this is one noncomputational review and its separate grade, not two theorem-review cycles.

Reproducibility, retained evidence, and accepted weight

The nine-file submitted delivery agrees with its final frozen inventory, including the complete report. The mathematical warrant is the report's written reasoning, not the inventory, row counts, or success messages. The primary commands support the assertion that no mathematical program was run. Mechanical scripts used for identity and report structure are not mathematical keepers; their bare assertions do not become an optimization-safe theorem checker, and this grade makes no such claim.

There is no mathematical rerun command to test verbatim. A reader can recheck the report's three derivation groups, its explicit cancellation input, and its separate consequence calculations. The complete private packet, original report, and source extraction remain unchanged. The grader's private record-navigation code executes no logged command and is not additional mathematical evidence.

For public filing, the eligible mathematical snapshots and provenance belong under wiki/problems/integer_sequences/E1063/claims/evidence/assets/white_review_subject/, and the unchanged report and this grade belong under wiki/problems/integer_sequences/E1063/claims/evidence/verify/white_upper_bound/. Public adoption remains pending. The owner must preserve the submitted report and permitted snapshots, document any allowed editorial changes, and keep the distinction between this frozen subject and later text. This grade does not assert that those homes already resolve in a clone.

The complete copied White source and raw operational records remain private. The public provenance identifies the source by its immutable URL and exact ranges; no full offline public copy or independently checked remote availability is claimed. The supplied attribution was not authenticated by a network fetch. These are explicit source-access limits, not permission to substitute a different version. The report's retained mathematical derivations do not require a private program or success log.

No missing essential mathematical clause or disqualifying exposure was found in the commissioned report. The accepted weight is an independently graded, whole supplied proof examination of the eventual upper bound and its two consequences, under the explicit interfaces and provenance limits above. It excludes source checker claims, omitted source material, unread external proofs, other constructions, optimal constants, all-order quantitative claims, Lean verification, and any broader solution or tier. The three contract grades remain PASS at exactly that weight.

Linked from (3)

Problem 1063 White's upper bound exp(C k log log k / log k) for n_k problems/integer_sequences/E1063/claims/evidence/verify/white_upper_bound

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