sponge-visible-census

• Exact count A(level) of visible (gcd 1) nonzero points of the Menger sponge at that level, without enumerating the 20^level points.
• Split at a cutoff G: points with 1 < gcd <= G cancel exactly inside Sum_{d <= G} mu(d) (T_d(n) - 1), where T_d(level) counts points divisible by d from one transfer pass on (Z/d)^3.
• Points with gcd > G are g*y with y primitive in a box of side (3^level - 1)/g, tested for membership digit by digit and weighted by Sum_{d | g, d <= G} mu(d).
• Cross-checks: direct enumeration at level 1..6 matching 12, 270, 5916, 123504, 2538447, 51497040, and A(9) recomputed at G = 100 and G = 150, which split head and tail differently.
• Also prints the second-order gap (delta*20^level - A(level)) / 12^level with 12 = fill*lambda.
• The bracket is derived, not quoted: 19 of the 20 digits are nonzero, times 27/26 for the prime 3, gives 513/520, so delta = (513/520)/zeta(3) = 0.8207086195.

RUN

uv run python research/lab/py/sponge-visible-census/census.py 9

• From the repo root. Argument is the top level; default 9.
• Whole run takes about 1 min 24 s on one core, A(9) alone 23 s.
• Domain run is the full source domain: level 1..9, coordinates 0..19682, cutoff ladder G = round(3^(level/2)) capped at 120.

WITNESSES

• the claims line (was DISCOVERIES.md:160) - A(7) = 1038074187, A(8) = 20860210527, A(9) = 418429711224, and G = 100 and G = 150 agreeing on A(9).
• the claims line (was DISCOVERIES.md:320) - the gaps 0.347, 0.349, 0.344 in units of 12^level.
• coprime.md:251 - the base 3 carpet bracket 513/520.
• coprime.md:282 - the same three counts under "Counting without enumerating".
• coprime.md:283 - the same three gaps.