# 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.