The PiG Correspondence: Apollonian Geometry as the Origin of π and Gaussian Statistics in Sphere Packing

The fifth paper in a five-paper series connecting FCC lattice cohomology, Apollonian circle packing, post-quantum cryptography, and holographic cosmology. This paper introduces the PiG (Pi + Gaussian; equivalently, Packing-Induced Gaussian) Correspondence: the transcendental constant pi and Gaussian statistics are not external imports into physics but intrinsic and derivable consequences of Apollonian sphere packing geometry.PiG-A (proved): the tetrahedral symmetry Td of the Apollonian seed forces the Patterson-Sullivan measure on the conformal boundary of H3 to be proportional to the round measure on S2, whose total mass is 4*pi. Pi is the normalization constant of the unique SO(3)-invariant measure on this boundary, forced by the seed geometry. The identity Gamma(1/2) = sqrt(pi) directly connects this geometric pi to the Gaussian normalization.PiG-B (established by CLT argument): the Apollonian void-filling process generates approximately independent log-curvature increments; by the Central Limit Theorem the radial distribution is log-normal. Combined with the Hausdorff measure on the gasket, the void radius distribution is p(r) proportional to r^(D_H - 1) * exp(-r^2 / 2sigma^2) with D_H = 1.3057. This peaks at r = sqrt(D_H - 1) * sigma = 0.5529 * sigma with no free parameters.The bell curve in sphere packing (oranges, lemons, ball bearings, hourglasses) is not randomness. It is the deterministic statistical projection of Apollonian void-filling geometry. The falsifiable prediction: the void radius distribution in random sphere packing follows a generalized chi distribution with non-integer parameter k = D_H = 1.3057, testable against existing tomographic and simulation datasets. Falsified if best-fit k deviates from 1.3057 by more than 0.05 in a well-controlled experiment.The PiG Correspondence arose as a corollary of the dimension selection theorem in Paper 3, not as a constructed explanation. Companion papers are at zenodo.18793518 (Paper 1), zenodo.18828073 (Paper 2), zenodo.18828771 (Paper 3), zenodo.18826779 (Paper 4).