BAKKAOUI's dissociated-set counterexample

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bakkaoui_2026_dissociated_interval_counterexample: Saved posts 8701 and 8709 on the 13-element dissociated-set example, retaining the author's correction and AI-assistance disclosure.

evidence/: Checks both dissociated witnesses and all 1287 five-element subsets of the fixed set A*, using exact integer arithmetic.

interval_not_extremal: A fixed 13-element positive integer set has largest dissociated subset size four, compared with five for the interval from 1 to 13.

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BAKKAOUI, posts 8701 and 8709, 3 September 2026, in the Erdős Problems 963 thread. The saved source excerpt retains both posts, the author's correction, and the disclosure of AI-assisted searches and literature checking. It was read from the pinned saved thread, not the live site.

The source gives a concrete set A* of 13 positive integers and reports that its largest dissociated subset has size four, compared with five for [13]. The finite reconstruction retains that claim with exact inputs, a full subset-sum check and the elementary deductions needed to interpret the computation. The five-element interval witness and the short interval upper-bound proof are supplied in this reconstruction; they are not quoted from the forum post.

The correction in post 8709 is essential. This example gives only f(13) <= 4; the source's bounded searches do not establish equality for the minimum over all real sets. The initial interval is therefore not always extremal, but this example does not refute the requested logarithmic lower bound in Problem 963. The numerical comparison is 4 versus 5, while floor(log_2 13) is 3.

The larger searches through n=16 and window 34, the proposed exceptionality of n=13, the OEIS identification and the negative literature claim remain unverified reports here. The separate zero obstruction and other thread arguments are not part of this finite reconstruction. No current literature or status search was performed.

Proof standing. The frozen finite reconstruction received the independent mathematical verdict refutation-failed and distinct completed-record grading. The review record retains the exact subject, corrected report, attribution and limits. The owner checker and shared-harness adaptation passed full normal and optimized runs; the execution account records their outcomes and failure controls. These author runs do not independently certify the adaptation or extend the frozen finite report's scope. No native L-tier, formal verification or catalog resolution is claimed.

Bears on. Problem 963, by ruling out the initial interval as a universal minimizer; it supplies no catalog solution.