The prime-multiple block extremum between two and three

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_proof: Distinct translates give the lower bound, and repeated prime patterns with Chinese remaindering attain it in an exact integer block.

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The proof is a rectangular form of the Erdős–Selfridge construction recorded in Erdős's 1978 paper, Section 6, Theorem 1 and its lemma. The argument is given in full here, using the prime number theorem and the Chinese remainder theorem. This exposition and extension to arbitrary $u$ are author-recorded by Plasma AI, with no novelty, independent review or Lean proof claimed. The earlier square-case claim keeps its own credit and standing.

For Problem 1143, this determines the extremum over prime sets for $2<\alpha<3$. It does not compute the minimum for every prescribed prime set, and says nothing sharp for $\alpha\ge3$. The research account records these distinctions. Verification is the lower bound, repeated prime-pattern count, congruence construction and integer endpoints; there is no computational premise.