Translated unique differences have subquadratic size

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_proof: A translated difference set forces a large sliding-count correlation. Averaging disjoint windows bounds the count's norm and gives the k^(5/3) estimate without a diameter restriction on the Sidon set.

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This author-recorded result is due to Plasma AI. Its proof averages translates of the sliding count $F(j)=|A\cap(j+W)|$. It uses no external theorem or computational premise.

Conventions and scope. The differences are ordered and nonzero. For integer sets their uniqueness is equivalent to the strong Sidon convention, in which unordered pair sums include doubles. Neither the diameter nor the location of $A$ is restricted. The separation condition concerns $s$ and $W$, and is essential to the supplied averaging argument.

Significance. The [[research/small_maximal_sidon_attack/translated_difference_sets|research consequences bound progressions, combs and certain translated Bohr sets]] inside $A-A$. In particular, a construction using one separated piece $s+(W-W)$ of bounded doubling and translates by elements of $A$ cannot cover an interval at the cube-root scale. This restricts that mechanism; it neither proves nor disproves Problem 156.

L239 is a comparison, not a premise. Its hypothesis requires uniqueness only for an initial interval of differences, and its conclusion has the additional term $-2k+2$. The proof here also works under the restricted representation hypotheses identified at its end; that observation recovers L239's leading term without imposing global Sidonicity.

Standing and verification. Tier 0, without independent review or a novelty determination. Check the sliding-count identity, the correlation at shift $s$, disjointness of the $K$ windows, and the integer choice of $K$. No mathematical obligation remains within this statement. The [[research/small_maximal_sidon_attack/proof_sketch|integer construction and compatible completion sought by the proof sketch]] remain open.