Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
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Whole-statement examination and grade
grade: Independent PASS grade of the unchanged whole-statement ratio report, including its complete finite-prefix proof, exact extremal consequences, report-contract coverage and primary independence warrant.
grading_derivations: Mathematical checks used to grade the frozen ratio report: label transfers, finite-prefix discrepancy with repeated entries, and the absolute ratio threshold and extremal consequences.
report: Adversarial review of the all-sequence circle-span ratio bound, including the multiset transfers, finite-list discrepancy bound, uniform constants, and extremal consequence.
The whole-statement report and its distinct grade, examined on 2026-10-15 UTC, cover the coefficient- ratio theorem, its exact infimum inequality and integer- divergence. The grade is PASS, and the mathematical verdict is refutation-failed. Both roles disclosed GPT-6 Astra and worked in separate fresh, isolated contexts, distinct from authorship and each other. The grade supplies the tier-1 warrant recorded on the native claim.
The subject manifest names the exact frozen extracts and their original source ranges. The mathematical freeze is 2026-10-15 at 09:49:19.072889 UTC. The submitted report and grade are retained with their mathematical inputs and supporting derivations. No mathematical software or missing private witness is needed to reconstruct this analytic assessment.
Scope and independence qualifications
The reviewer and grader checked the complete supplied local deductions. The original source publications were unread; supplied attribution is preserved without claiming independent original-paper fidelity. Korsky's cited preprint remains unrefereed. No native claim or unprovided external theorem is consumed as an established premise. No Lean or computational warrant is asserted.
The private inbox could not be inspected and is unread, not established empty. The grader evaluated the complete available received-input and action records and found no excluded content consumed. Opaque internal fields were retained but not decoded. The full grade states these access limits and its own eligibility ruling; neither empty subscriptions nor a preparer's waiver replaces that ruling.
The ratio grading derivation contains a presentation defect in its displayed
four-block recurrence: a literal tab precedes frac12 in the second term.
The readable copy restores \tfrac12. The surrounding original prose and
the original review's recurrence both supply half of each within-block range.
The submitted grading derivation remains unchanged under assets. This is
only a typesetting correction, not a mathematical repair or new grade.
The coefficient , the one-sided bounds, equality of the extrema over distinct and unrestricted sequences, the unprovided optimized Theorem 4.1, and the optional Schmidt route have no warrant from this record.
Canonical correspondence and editorial treatment
On 2026-10-15 the integrator compared every ranged mathematical payload with its canonical source selection. They agree exactly after the extraction's stated cuts and removal of the author-standing labels. The prepared provenance map is a source account, not a canonical-page extract. The native claim states the same examined assertion. Its proof gives a concise navigation and composition account of the unchanged supporting deductions; the exact examined proof remains in the snapshots. This correspondence work is not another independent examination.
The readable report, grade and derivations retain their mathematical text, verdicts, dates and qualifications. Wiki metadata, separators and maintained relative links are added. The original submitted report, grade and derivations are retained unchanged under assets, including their original relative layout. In the subject manifest only the private mathematical-source revision and per-file hashes are removed under convention 9; source paths, selection ranges, freeze time and scope remain. The ratio record additionally lists its single typesetting normalization above. No report is represented as having examined a later substantive repair.
The whole-conclusion and exact-formulation assessment for the conjunction of the ratio and one-sided results is a separate obligation. This record does not itself accept Problem 1221 as solved.