Extending a fixed Sidon prefix at coefficient one over square root two

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_proof: A fixed number of deletions removes mixed collisions and a translate yields the claimed asymptotic extension coefficient.

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The proof is an author-recorded specialization by Plasma AI of Singer's prime-power difference-set theorem. That theorem is consumed as an external premise whose statement has been checked; its projective-geometric proof is not reconstructed here. No independent review or Lean proof of the specialization is recorded, and no novelty is claimed.

For Problem 44, the coefficient $1$ is still missing. The research account identifies the mixed equations exposed by trying to remove the extra interval length. There is no gap in the weaker coefficient stated here. Verification consists of checking the deleted differences and all three pair-sum ranges.