The third layer of a separated prime grid

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_proof: Outer-layer locations coincide only between adjacent rows, at most twice, and each missing following location forces a distinct preceding one into the block.

evidence/: Brute-force the counts of all core-containing blocks and all blocks near the Chinese-remainder point for small separated prime grids.

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The two-position construction of [[theory/number_theory/L37_prime_multiple_block_extremum/_index|L37, the exact extremum for 2<alpha<3]], uses a grid of primes $q+a_i+b_j$ and Chinese remaindering, so that each prime's two multiples sit at a row location $z-a_i$ and a column location $z+q+b_j$. The repeated prime patterns found in its proof have separated rows: consecutive row offsets differ by more than the largest column offset. This claim shows what happens to that construction when the block is lengthened to $\lfloor\alpha P\rfloor\ge3P$. Each prime gains a third multiple in the block. In the canonical extension it is the layer $z+2q+a_i+2b_j$, and separation forces that layer to have at least $\lceil u/2\rceil$ distinct locations. No block retaining the row and column locations does better, and no block in the window around $z$ has fewer than $\lceil u/2\rceil$ multiples. The proof counts the layers.

For Problem 1143 this is an exact obstruction to one construction, not a bound on the problem. The cost is linear in $u$, while Ruzsa's construction already has $O((u\log u)^{2/3})$ multiples and [[theory/number_theory/L201_type_class_long_block_bound/_index|L201 has exponent log(27/4)/log 27]]. So the separated grids of L37, with their Chinese-remainder placement, give nothing better than a linear count past $3P$. The statement fixes the Chinese-remainder solution $z$, and blocks far from $z$ realize other residue patterns, so it says nothing about the least count for the same primes over all blocks. The [[research/prime_multiple_long_intervals/_index|research account of long blocks]] explains the cell sets that avoid the obstruction. The full-grid bound $rs-(r-1)\lfloor s/2\rfloor$ is sharp for every $r,s$ and every $3\le\alpha<4$, including prime realization. Its proof uses [[theory/number_theory/L202_prime_realization_three_projections/_index|L202 to realize the equality pattern by primes using the Green--Tao theorem]]. The lower bounds and the exact minimum for any given grid are elementary and do not use that theorem. Without separated rows, [[theory/number_theory/L204_cartesian_prime_grid_obstruction/_index|L204 still gives a two-thirds-power obstruction for full Cartesian grids]].

This is author-recorded work by Plasma AI at tier 0, with no independent review or Lean proof. Verify it by checking the multiplicity bounds, the cluster disjointness and the exchange of each missing following location for a distinct preceding one in the proof. The [[theory/number_theory/L199_separated_grid_third_layer/evidence/_index|executable evidence]] brute-forces every block position in both families for small prime grids; it illustrates the statement and proves nothing beyond its instances.