carpet-stack-address

• Brightness of the odd parity-carpet stack at a rational point, in closed form, with no loop over layers anywhere.
• What layer n does at x = (a_1/q, a_2/q) depends only on r = n mod 2q, so B_N(x) = ceil(N/2) - Sum_{r in S(x), r <= N} (floor((N-r)/2q) + 1) with S(x) the odd residues whose two coordinates both land dark.
• Per-point cost is one pass over the 2q residues and never depends on N; the form is set against literal stacking on 48 points at q = 9973, and against a second route that counts the bright residues directly instead of subtracting the dark ones.
• Farey line-stack at N = 55: the node value floor(N/b) against literal membership in each of the 55 scales, hashed to the same digest.
• The 512 x 512 render at N = 55 by three routes sharing no inner loop: column bit-masks, layer accumulation, per-pixel triple loop.

RUN

• uv run python research/lab/py/carpet-stack-address/stack.py
• From the repo root. One core, about a second.
• Domain is the full source domain: 48 points at N = 55, 5555 and 10^18, the sweep N in {1, 2, 55, 5555, 19945, 19946, 19947, 40001}, the whole 512 x 512 render at N = 55.
• Nothing is written to disk; the render is hashed in memory.

WITNESSES

• farey.md:214 - the closed form itself, the only expression the point evaluator computes.
• farey.md:220-221 - 940 Farey nodes at N = 55, brightness sum 1540 = N(N+1)/2 landed by count, both routes on digest f42186ea5b3670ac677a5a4d42dc1acaed16cf34d10884de000277165c810834.
• farey.md:221-222 - the 512 x 512 render sha256-identical by three routes, 6cddd85b6298cffdeb374df76365be324ab96f37c212110f683e07c7d4e60239.
• farey.md:222 - 48 of 48 points equal, closed form against literal stacking, at N = 55 and N = 5555.
• farey.md:223 - 384 comparisons, 0 mismatches, over the 8 sweep values, both sides of the 2q = 19946 period boundary.
• farey.md:224-225 - N = 10^18, a stack of 5*10^17 layers, both routes agreeing on all 48 values in well under a tenth of a second; first four 375564022861726680, 374862127744911263, 374862127744911259, 375012533841371703.
• the claims line (was DISCOVERIES.md:52) - the same numbers on the ledger line for this section.