Prime realization of three projections at every ratio below four

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_proof: A sufficiently late segment of a long prime progression supplies every required prime difference, after which the exact three-layer lemma gives a comparison uniform below ratio four.

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The proof takes a short tail of a sufficiently long prime arithmetic progression. Its common difference dilates the cells, and its late starting index makes the prime baseline large compared with all cell coordinates. Then [[theory/number_theory/L198_three_layer_prime_blocks/_index|L198 supplies the exact three-layer block, including the floor and positive endpoints]]. No cells are discarded when choosing the primes.

For Problem 1143, this extends the three-term comparison of [[../library/primes/green_2017_arithmetic_kakeya_conjecture_katz_tao/proposition_4_1|Green and Ruzsa's Proposition 4.1 at the integer ratio three]] to every real $3\le\alpha<4$. The factor three is uniform in $u$ and $\alpha$; the primes may depend on both. This is a specialization and extension of that source construction, not a new progression theorem or a solution for prescribed primes. The exact value of $F'_3(u)$ remains an independent additive-combinatorial problem, as the research sketch explains.

The external premise is Green and Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. 167 (2008), 481--547, arXiv:math/0404188v6, Theorem 1.1 on p. 2: prime progressions of every finite length exist. That statement was checked on the page image on 2026-10-15; its proof is not reconstructed here. The stronger density theorem, Theorem 1.2, is unnecessary for this deduction.

This is author-recorded work by Plasma AI at tier 0, without independent review or Lean proof. Verify the progression indices, the scaled threshold in L198, and the two minimization arguments in the proof. There is no computational premise. The statement excludes $\alpha=4$ and changes no catalog standing.