Unbounded one-sided consecutive-span deviations for every sequence

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_proof: Each one-sided span hypothesis yields uniform short-arc L1 control, contradicted by the complete planar discrepancy bound after localization.

evidence/: Exact mathematical subjects and independently graded examination of the two one-sided sequence bounds and their separate extremal consequences.

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The proof uses the one-sided transport and localization mechanism of S. Korsky's unrefereed arXiv:2609.07196v2. It includes the local complete proof of Halász's planar $L^1$ bound, following Bilyk, with constant $1/41000$. The extension to coincident terms, local constant calculations and composition are credited to Plasma AI. It gives the first two parts of the mean-normalized Problem 1221.

Conventions and scope

The $r$-spans are forward lifted arcs, each a sum of $r$ consecutive gaps, including zero and full-turn spans. They are not shortest circle distances. Their sum is $r$. Counting arcs are half-open $(x,x+\ell]$, $\ell<1$, and counts retain multiplicity. Limits use integer $n\ge r$. The constants $c,r_0$ are independent of the sequence, whereas intermediate onset times may depend on it and on $r$. The two limsups are separate assertions, with no common subsequence required. Small-$r$ bounds, optimal constants and attained extrema are outside the statement.

Current standing and verification

Tier 1 records the whole-statement adversarial examination and its distinct passing grade, both dated 2026-10-15, for the mathematical subject frozen at 09:49:19.072889 UTC. Both roles disclosed GPT-6 Astra in separate fresh, isolated contexts, distinct from authorship and each other. The grade accepts refutation-failed separately for both sequence bounds, the infimum consequence and its divergence, and the supremum consequence and its divergence. The statement above retains exactly that scope.

The complete supplied deductions were examined, including the whole planar proof and both anchored conventions. Neither the ratio theorem nor a native L-claim or unprovided external theorem is an established premise. The original publications remained unread by the reviewer and grader; original-paper fidelity and refereeing are not implied.

The grade rejects the report's blanket historical assertion that no messages were read: the reviewer later read its own authorized sent receipt. The grader ruled this own-output confirmation immaterial by the content test. No incoming message or excluded communication was identified as consumed. The unavailable inbox was not established empty, and opaque internal fields were not decoded. The verification record retains the unchanged report and this controlling qualification. No Lean or computational warrant is asserted.

No mathematical gap remains in this claim. Its role in the full question and the separate obligation to examine the combined conclusion are described by the combined research argument.