Every block position on small separated prime grids

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From the repository root run:

uv run --no-sync python wiki/theory/number_theory/L199_separated_grid_third_layer/evidence/main.py

The grids are six separated grids of primes $q+a_i+b_j$, chosen by Plasma AI, with $q$ from $251$ to $13171$. Five have rows and columns in arithmetic progression, so their third layers have the largest possible number of coincidences; the sixth has irregular offsets. Each grid is tried with its full cell set and with smaller cell subsets, at $\alpha=3$ and at a second ratio for which $\lfloor\alpha P\rfloor<4q$.

For each case the checker verifies the hypotheses of L199 and solves the congruences. It then counts directly, each integer once, the multiples in every block of $\lfloor\alpha P\rfloor$ consecutive integers that contains all row and column locations, and in every block inside $[z-2q+1,z+4q-1]$. The checks are $|T_S|\ge\lceil u/2\rceil$; that the least count over the first family equals $r_S+s_S+|T_S|$ and is attained by $[z-q+1,z-q+K]$; and that the least count over the second family is at least $\lceil u/2\rceil$. For the five full grids with progression offsets it also checks the equality $|\{a_i+2b_j\}|=rs-(r-1)\lfloor s/2\rfloor$ stated on the research account.

All arithmetic is exact, primality is decided by trial division, and the run takes a few seconds at most. These instances test the statement's block families and conventions; they prove nothing beyond themselves.