Catalan's Constant, Mock Theta, Euler Product, Jones polynomial: All one geometry?

AI-collaborative research note. Starting from the mod-30 coprime lattice Z30* = {1, 7, 11, 13, 17, 19, 23, 29} with projection weight P(r) = sin^2(r*pi/15), this paper derives exact shifted Ramanujan-sum overlap formulas, a mod-15 lane-difference autocorrelation satisfying DFT[A15] = |c15|^2, and a rational Gram matrix whose eigenvalues split into Q(sqrt(5)) and Q(sqrt(2389)), where 2389 = 25^2 + 42^2.The paper gives the corrected Catalan identity G = beta(2) = L(2, chi_-4) and the mod-30 Euler product G = (15/16) * product over p in Theta30 of p^2 / (p^2 - chi_-4(p)). It also separates exact arithmetic results from proposed physical interpretations involving Malus's law, the electric/magnetic field split, Jones polynomials, Hopf projection from toroidal field states to spherical readouts, Mertens cancellation, and quantum gravity. Exact results are labeled derived or verified; broader physical identifications are explicitly labeled as hypotheses.