Three-layer Chinese-remainder blocks of prime multiples

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_proof: The multiples of every cell prime fall in layers near multiples of q, and a block of floor(alpha P) integers fits strictly between the layers t=-1 and t=3.

evidence/: Build type-class cell sets, keep the cells with prime differences, place the block of the proof and count its prime multiples directly.

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Each cell $(x,y)$ names the prime $p=q+y-x$. A Chinese-remainder choice of $z$ puts the multiples of $p$ at $z+tq+(1-t)x+ty$ for all integers $t$, so the multiples at $t=0,1,2$ are $z+x$, $z+q+y$ and $z+2q+2y-x$. The proof places a block of $\lfloor\alpha P\rfloor$ integers that holds exactly these three layers. The layers are disjoint, and within a layer two primes share a location exactly when their cells share the value of $x$, of $y$ or of $2y-x$. So the count is the sum of the three projection sizes, while the number of primes is $|S|$.

This is the three-position form of the rectangular prime-grid construction of [[theory/number_theory/L37_prime_multiple_block_extremum/_index|L37, whose two-position block has only the row and column layers]]. With $x=-a_i$ and $y=b_j$ on a grid of primes $q+a_i+b_j$, layers $0$ and $1$ are its row and column locations, and layer $2$ is the new set of locations $z+2q+a_i+2b_j$. The lemma needs no rectangle, so any cell set can be used; the [[research/prime_multiple_long_intervals/_index|research account of long blocks]] explains why the general shape matters. The lemma is the construction step of [[theory/number_theory/L200_long_block_sums_differences_exponent/_index|L200, which identifies the growth exponent for 3<=alpha<4]], and of [[theory/number_theory/L201_type_class_long_block_bound/_index|L201, an explicit bound with exponent log(27/4)/log 27]]. [[theory/number_theory/L202_prime_realization_three_projections/_index|L202 supplies the missing simultaneous primes after dilating any cell set]]. It makes no claim about blocks or prime sets of any other form.

This is author-recorded work by Plasma AI at tier 0, with no independent review or Lean proof. It uses only the Chinese remainder theorem and integer bookkeeping. It assumes the primes: finding cells and $q$ for which every $q+y-x$ is prime is left to its consumers. Verify it by checking the layer formula, the four endpoint inequalities and the disjointness of the layers in the proof. The [[theory/number_theory/L198_three_layer_prime_blocks/evidence/_index|executable evidence]] counts actual blocks for explicit instances; it illustrates the conventions and proves nothing beyond its instances.