Direct counts of three-layer blocks

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From the repository root run:

uv run --no-sync python wiki/theory/number_theory/L198_three_layer_prime_blocks/evidence/main.py

The instances list nine triples $(m,\alpha,q)$ with $m=1,2,3$ and $\alpha=3,7/2,39/10$, chosen by Plasma AI with $q$ just above the threshold $(16\cdot27^m+1)/(4-\alpha)$ of L198. For each one the checker builds the type class of the cells $(0,1)$, $(2,1)$, $(2,2)$ used by L201, verifies its size, its distinct differences and its three projection sizes $\binom{3m}{m}$, and checks the hypotheses of L198. It then keeps the cells whose $q+y-x$ is prime, solves the congruences, places the block given in the proof and makes it positive. Finally it counts the integers of the block divisible by a kept prime directly from the primes, each once.

The checks are that every kept prime has exactly three multiples in the block and that the direct count equals $|X|+|Y|+|W|$ for the kept cells. The retained prime and multiple counts are reproduced, among them $302$ primes with $220$ multiples for $m=3$, $\alpha=3$. All arithmetic is exact, primality is decided by trial division, and the run takes a few seconds at most. These instances test the endpoint inequalities, the floor in the block length, positivity and once-only counting; they prove nothing beyond themselves.