A rational seven-point bound for long prime-multiple blocks

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_proof: A fixed rational measure has projection entropy below 0.56213 times its joint entropy; exact type classes turn this strict margin into an all-size prime-block bound.

evidence/: Recompute the seven-point measure's marginals and use rational logarithm enclosures to certify that its largest projection entropy is less than 0.56213 times its joint entropy.

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For Problem 1143, this is an upper bound on the minimum over prime sets. It follows from type classes of a seven-point configuration of Marius Lemm, with an explicit rational probability measure. The proof derives the asymptotic construction from exact multinomial counts, and the [[theory/number_theory/L203_rational_entropy_prime_block_bound/evidence/_index|rational logarithm certificate]] checks the one numerical entropy inequality used. The constant $C$ is independent of $\alpha$; the primes and the block are allowed to depend on $\alpha$ and $u$.

Dependency role. [[theory/number_theory/L202_prime_realization_three_projections/_index|L202 realizes each whole type class by primes without a loss in cardinality]]. Its external premise is the Green--Tao theorem. The construction improves the simpler [[theory/number_theory/L201_type_class_long_block_bound/_index|three-cell bound of L201]], but is not a new sums--differences example or a claim of improvement over the published sums--differences record.

The source is M. Lemm, New counterexamples for sums-differences, Proc. Amer. Math. Soc. 143 (2015), 3863--3868, read in the held arXiv:1404.3745v2 edition. Its Theorem 2.1 supplies the seven-point configuration. The seven points and rounded probabilities were checked on pp. 4--5 on 2026-10-15; the canonical library records were read on the same date. The library result page distinguishes the claimed optimized ratio above $1.77898$ from the ratio of the printed rounded weights, which falls slightly below it. The measure here is stated exactly and has its own rational certificate; it does not rely on either numerical claim.

The library's Proposition 1.1 record qualifies the entropy-to-set implication as a limiting statement. The proof here constructs integer cells directly and uses a strict entropy margin below $0.56213$ to pass to every $u$. It does not infer attainment at a limiting exponent. The source pages retain their claims-checked reading coverage; that coverage is not independent acceptance of this prime-block deduction.

This is an author-recorded deduction by Plasma AI at tier 0, with no independent review or Lean proof. Verify the seven distinct differences, the marginal masses and logarithm enclosures, the type-class encoding, and the passage from its sizes to every $u$. The research sketch records the remaining gap in the exponent. This result does not compute a prescribed-prime-set minimum and does not resolve the catalog problem.