slice-coprimality

• Coprimality of the points of a 3D parity design on a diagonal slice x + y + z = s.
• Checks A_s = Sum_{d | s, d squarefree} mu(d) N_s^(d) at every height, the base-prime peel N_s^(base)(level) = [000 in P][base | s] N_{s/base}(level-1), and the prime-slice visibility bound.
• Measures the local price of a prime aggregated over the heights it divides, against 1/p^2, and at p = 2 against the parity-walk formula.
• Counts the central slice s* = 3(base^level - 1)/2: its visible density, its per-prime locals, the independence product, its peel, the (9, -12) recurrence and the peel ratio.
• Designs are the parity codes carpet {000,100,010,001} (fill 20 at base 3), net {110,101,011,111} (fill 7), tree {000,001} (fill 12).

RUN

• uv run python research/lab/py/slice-coprimality/slices.py
• One core, about three seconds.

DOMAIN

• Enumerated: carpet base 3 to level 4, base 4 to level 3, base 5 to level 3; net base 3 to level 7; tree base 3 to level 7.
• The level 6 aggregated locals come from a residue transfer on (Z/p)^3 and the central slice from a meet-in-the-middle over half levels; the transfer is checked against enumeration at level 3, the meet-in-the-middle against the height recursion at level 2..7.
• The central count and its peel run to level 14 on the one-dimensional height recursion.

WITNESSES

• coprime.md:266,268 - zero Mobius mismatches at every height over the four design-base pairs, worst hidden count on a prime slice 3, peel exact at base 3, 5, no net point with 3 | gcd on all 7^7 points.
• coprime.md:270 - 0.040902 against 1/25, 0.020446 against 1/49, p = 11, 13 still converging.
• coprime.md:271 - 0.2850378 = 9121792/32002048, walk formula and count equal on the integer.
• coprime.md:272 - tree dichotomy, zero visible on even heights and zero even gcds on odd, 1.49 million points each side.
• coprime.md:274,275 - 0.89216, 0.89776 against 0.57143, 0.61067, 0.65218; R_7 = 1093; base 5 reading 0.345, 0.492, 0.560; independence product 0.64780, 0.89764, 0.55741 against measured 0.65218, 0.89776, 0.56006.
• coprime.md:276,277 - 6, 42, 306, 2250, 16578, 122202, 900882, the peel 3, 27, 207, 1539, 11367, 83835, 618111, both on the (9, -12) recurrence to level 14, ratio 0.093070331 against (sqrt(33) - 5)/8 = 0.0930703308.
• the claims line (was DISCOVERIES.md:35,36) - the same numbers, plus 2^1092 = 1 mod 1093^2 and the tree dichotomy carried to level 7.
• slices.md:415 - the pointer: 1/p^2 on the slice against 1/p^3 in the solid.