Logarithmic growth of the consecutive-span ratio for every sequence

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_proof: Converts a putative small ratio into uniformly small short-arc count errors, extracts a finite list, and contradicts its explicit prefix discrepancy lower bound.

evidence/: Exact mathematical subjects and independently graded examination of the ratio theorem and its exact extremal consequences.

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The proof combines the reconstructed span-to-count argument of S. Korsky's unrefereed arXiv:2609.07196v2 with a complete finite-list lower bound following Larcher's range splitting. The extension to coincident terms, finite-list argument with its explicit constant, and local composition are credited to Plasma AI. This gives the ratio part of the mean-normalized Problem 1221. The coefficient $1/100$ in Korsky's statement is outside this claim.

Conventions and scope

Spans are forward lifted arcs, or sums of $r$ consecutive nonnegative gaps; they are not shortest unoriented distances. Their sum is $r$, so $nm_n^{(r)}\le r\le nM_n^{(r)}$ and $M_n^{(r)}>0$. Counts in the proof retain label multiplicity in half-open arcs $(x,x+\ell]$, $\ell<1$. All limits in $n$ use integers $n\ge r$. The constant $r_0$ is independent of $X$, although intermediate time thresholds can depend on $X$ and $r$. No optimal coefficient, attained infimum or conclusion for every small $r$ is asserted.

Current standing and verification

Tier 1 records the whole-statement adversarial examination and its distinct passing grade, both dated 2026-10-15. The mathematical subject was frozen on that date at 09:49:19.072889 UTC. Both roles disclosed GPT-6 Astra and worked in separate fresh, isolated contexts, distinct from authorship and from each other. The accepted verdict is refutation-failed for the full sequence theorem, its exact ratio infimum and the divergence. The statement above has exactly that scope. The complete supplied proofs, including the finite-prefix lower bound, were examined; no native claim or unprovided external theorem was consumed as an established premise.

The original publications were unread by the reviewer and grader, so their mathematical acceptance is of the supplied local deductions, not independent source fidelity. The unavailable inbox was not established empty; the grader's eligibility ruling rests on the complete available received-input and action records. Opaque internal fields were retained but not decoded. The verification record retains these limits, the exact subject and the presentation correction in the grading derivation. There is no Lean or computational warrant.

No mathematical gap remains in this claim. The combined research argument explains its composition with the separate one-sided bounds. Whole-conclusion scrutiny of that combination and its match to the modified Statement are separate from this claim's warrant.