Half Ball Mismatch

• The half-disk chord constant split into an integer and an area, and the two half-ball families whose parity blocks any repeat of that coincidence.
• zerr.py: the inner integral Integral_(-1)^(1) (-a u + sqrt(1 - a^2 + a^2 u^2))^3 du = 2 for every a, symbolically and at 50 digits on 50 values of a, then I_diam = 4 and P = I_diam/(3 Area(H)^2) = 16/(3 Pi^2).
• version_l.py: the two-point family from its Wallis recurrences in exact rationals to ambient dimension d = 11, with an independent 10^7-sample random-point check at d = 2..7.
• version_h.py: the d-point hyperplane family, derived here by an unoriented-normal Blaschke-Petkantschin reduction and integrated in closed form at d = 2..7, cross-checked by 60- and 80-digit quadrature and by a 10^8-sample random-point estimate at d = 3, 4, 5.
• mismatch.py: every base-2 and base-3 design at dim 2 against every Version L value to d = 24.
• Beyond the page: Version H is exact here at d = 6 and d = 7, and the dim 2 sweep is a record to d = 24 rather than to d = 11.

RUN

• uv run python research/lab/py/half-ball-mismatch/zerr.py
• uv run python research/lab/py/half-ball-mismatch/version_l.py
• uv run python research/lab/py/half-ball-mismatch/version_h.py
• uv run python research/lab/py/half-ball-mismatch/mismatch.py
• Under four minutes together, almost all of it the 10^8-sample check in version_h.py. Prints only, writes nothing.

WITNESSES

• pi.md:165 and pi.md:167 16/(3*Pi^2) = 0.5403796460924681 and 1 - 16/(3*Pi^2) = 0.4596203539075319.
• pi.md:169 and research/claims/:185 the inner integral is 2 for every a, I_diam = 4, Area(H)^2 = Pi^2/4, P = 16/(3*Pi^2), three symbolic residuals exactly zero, 50 digits at 50 values of a.
• pi.md:171 and research/claims/:350 the parity law and the uniqueness at dim 2 and d = 2: 11 base-2 and 502 base-3 designs, base-2 numerators 4, 16/3, 6, 8, exactly one match, 16/3 at d = 2, carried by 3 designs.
• pi.md:173-180 Version L 16/(3*Pi^2) = 0.5403796, 3/8, 128/(45*Pi^2) = 0.2882025, 15/64, 1024/(525*Pi^2) = 0.1976246, 175/1024.
• pi.md:182-187 Version H 1 - 16/(3*Pi^2), 4 - 19845*Pi/16384 = 0.1947689081, 4 - 549978112/(14189175*Pi^2) = 0.0727502984, 16 - 178919214166875*Pi/35184372088832 = 0.0244047160.
• pi.md:189 and research/claims/:351 exact rationals to d = 11, and the 60- against 80-digit agreements 2.3e-62, 7.2e-64, 1.5e-63 at d = 3, 4, 5.
• Not regenerated: the three 10^8-sample deviations 2.1e-05, 9.7e-06, 4.7e-06 at pi.md:189 and research/claims/:351 belong to one sample. This study draws its own and reports the deviation against its own one sigma.