The three mean-normalized consecutive-gap divergences

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_proof: Checks the common sequence model, absolute quantifiers, separate limits and extremum directions, then restricts to the distinct-point subclass.

evidence/: Exact subjects, whole-conclusion examination and independent passing grade for the mean-normalized consecutive-gap conjunction.

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The proof composes L163's universal ratio bound and L164's two separate one-sided bounds. Both inputs have their own whole-statement examinations and distinct passing grades. Their conjunction, its subclass consequence and its exact match to the modified Statement of Problem 1221 are recorded here by Plasma AI at tier 1. The whole-conclusion examination has verdict refutation-failed, including the match to the supplied modified Statement. Its independent whole-contract grade is PASS, examined on 2026-10-16 UTC. This is the warrant for the conjunction and its stated consequences; catalog acceptance remains pending.

Conventions and scope

Spans are forward arcs through $r$ consecutive gaps on the circle of circumference one. Labels preserve multiplicity and cyclic starting positions; real lifts distinguish a full turn from a zero span. The $n$ spans sum to $r$, so $M_n^{(r)}>0$. The two one-sided deviations need not occur at the same times. The sequence may depend on $r$; $c$ and $r_0$ may not. The extrema are taken after the limits, separately for each nonempty family. No equality of extrema between the two families is assumed or established.

The all-sequence conclusion answers the mean-normalized modified Statement, whose quantities are $\Lambda_r-r$, $r-\lambda_r$ and $r(\mu_r-1)$. It does not replace the literal erdosproblems.com first two expressions by an unannounced convention. The problem page records the printed normalization issue and its source evidence. No optimal constant or attained extremum is claimed.

Attribution and standing

The span-to-count mechanism and closing comparisons come from S. Korsky's unrefereed arXiv:2609.07196v2. Larcher's range splitting supplies the method of the finite-list input; Halász's planar theorem, as presented by Bilyk, supplies the other method. The complete local proofs with constants, coincident-point extension and composition are credited to Plasma AI as a technical completion of Korsky's full problem claim. The input grades examine the supplied local proofs, not the original publications. Their source and independence qualifications remain visible on L163 and L164. The conjunction has its own whole-statement examination and distinct passing grade; its warrant is separate from the two premise grades.

The current statement, conventions and proof are the examined mathematics. The grade accepts the complete conjunction, both families' extremal and divergence consequences, and the exact supplied mean-normalized formulation. At its final dependency reading on 2026-10-16 UTC, L163 and L164 were proved at tier 1 without stale standing and matched the frozen interfaces; filing confirmed those conditions. The grade supplies no original-publication fidelity, stronger coefficient $1/100$, optimality, attainment, equality of the two families' extrema, Lean credit or catalog acceptance. The independent records and their operating disclosures are retained with the evidence. The research argument keeps the complete chain and the unexamined stronger-constant alternatives.