Half-ball chords

• 2026-08-28 [Verified] The half-disk chord constant (Zerr) decomposes into an integer and an area: P(the chord through two uniform points of the upper unit half-disk crosses the diameter) = I_diam/(3 Area(H)^2) by Blaschke-Petkantschin, the flat-face chord-cube integral is the integer I_diam = 4 and Area(H)^2 = Pi^2/4, giving 16/(3 Pi^2); the whole computation collapses to Integral_(-1)^(1) (-a u + sqrt(1 - a^2 + a^2 u^2))^3 du = 2 for every a, the even part of the cube being the exact derivative d/du [u (1 - a^2 + a^2 u^2)^(3/2)] with R(+-1) = 1, checked by the exact derivative, by differentiation under the integral in a, at 50 digits on 50 values of a, and with a symbolic residual of exactly zero at every step. Witness: lab/py/half-ball-mismatch.
• 2026-08-28 [Conjecture] The mismatch theorem: a design's coprime density equals a half-ball flat-face probability at dim = d = 2 and nowhere else, since design densities are rational multiples of 1/zeta(dim) while Version level gives rational/Pi^2 at even d and a pure rational at odd d and Version H gives Q + Q/Pi^2 at even d and Q + Q Pi at odd d, so even dim >= 4 is blocked by Lindemann and dim = 3 against Version level by Apery, both unconditionally; dim = 3 against Version H is conditional on zeta(3) not being algebraic over Q(Pi), odd dim >= 5 on zeta(dim) irrational (Rivoal and Zudilin give it only for infinitely many odd dim), and the dim = 2 uniqueness half is numerical, 11 base-2 and 502 base-3 designs against every Version level value to d = 24, base-2 numerators 4, 16/3, 6, 8, exactly one match, 16/3 at d = 2 carried by 3 designs. Witness: lab/py/half-ball-mismatch; coprime-density-above-dimension-one; A395134.
• 2026-08-28 [Conjecture] The Version H half-ball family, d uniform points and the hyperplane through them, has exact values 4 - 19845 Pi/16384 at d = 3, 4 - 549978112/(14189175 Pi^2) at d = 4 and 16 - 178919214166875 Pi/35184372088832 at d = 5, so odd d carries Pi^1 where Version L carries a pure rational, by an unoriented-normal Blaschke-Petkantschin reduction integrated in closed form, quadrature at 60 against 80 digits agreeing to 2.3e-62, 7.2e-64 and 1.5e-63, and an independent 10^8-sample random-point estimate whose deviations 2.39e-6, -6.74e-6, -3.55e-6 sit inside one sigma of 3.96e-5, 2.6e-5, 1.54e-5; d = 6 and d = 7 are exact too, 16 - 10363195833496113250304/(65656392092180764875 Pi^2) = 0.0074784083 and 64 - 403492347953923610203877211975 Pi/19807040628566084398385987584 = 0.0021206659, so the parity law "even d gives Q + Q/Pi^2, odd d gives Q + Q Pi" rests on six terms and no proof. Witness: lab/py/half-ball-mismatch.
• 2026-08-28 [Conjecture] No Euclidean body reproduces the bracket: a sweep of composite bases finds B(F) taking 1/2, 5/8, 3/4, 7/8, 1 across twenty-five base-4 designs of identical dimension 1.5, so B(F) is where a design's geometry lives, but no body whose chord-power integral reproduces B(F) was found, and the mismatch theorem makes the search futile above dim = 2; a failed search, not a proof of nonexistence, and the base-4 sweep has no generator in lab/. Witness: lab/py/half-ball-mismatch for the futility only.