Consecutive uniquely represented differences have subquadratic length

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_proof: Exact counts for overlapping and adjacent intervals give a small discrete energy. Averaging translates of the resulting sequences bounds their squared norm and gives the m^(5/3) estimate without a diameter bound.

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This author-recorded result is due to Plasma AI. The proof uses two exact sliding-interval counts and an elementary inequality for finitely supported sequences. It uses no external theorem or computational premise.

Conventions and scope. A representation of a positive difference has the ordered form $d=b'-b$; its two entries are necessarily distinct. The set $B$ need not be Sidon for differences larger than $L$. A strong Sidon set, whose unordered pair sums include doubles, has at most one representation of every positive difference and therefore satisfies the hypothesis whenever $[1,L]\subseteq B-B$. Negative elements of $B$ are allowed, and neither its location nor its diameter enters the bound.

Significance. Along any family with $m\to\infty$, the conclusion implies $L/m^2\to0$. Thus the [[research/small_maximal_sidon_attack/integer_transfer|proposed integer transfer using a quadratic-length Sidon difference basis]] cannot work. This does not refute Problem 156: maximality involves $A+A-A$ and midpoints, and different translates of a difference set or interactions among added points can cover one another's gaps. The [[research/small_maximal_sidon_attack/consecutive_differences|research extension allows missing differences]] and records that separate scope.

Standing. Tier 0, with no independent review or novelty determination recorded. Verification consists of checking the two interval identities, the averaged norm inequality, and the choice of the integer smoothing parameter in the proof. No mathematical obligation remains inside this statement; the interval construction sought in the editable proof sketch remains open.