From Spherical Light to Nested Maps: How One Jacobian Governs Frequency, Mass, and the Periodic Table

The preceding paper (Singh 2026w) established that wavelength is a sphere size and that the proton's internal density profile ρ ∝ A/r² follows from a single transmission fraction p = e^ (−3/4). This paper asks the next question: if the proton is a sphere described by the map r → r³ (the "z³ map"), what else does that map determine? The answer is nearly everything. The Jacobian of the z³ map, J = 3r²/Rp², simultaneously fixes the density (ρ ∝ 1/r²), the local frequency (f ∝ 1/r²), and the local time scale (T ∝ r²) inside the proton. The mass-size relation m × R = (4/3) π × ℏ/c, applied to both proton and electron, yields the mass ratio mp/me = Re/Rp = 1836, algebraically exact. The ratio of orbital to internal z³ scale is Rₒrb/Rᵢnt = 3/ (4πα) = 32. 7, algebraically exact. Two spherical waves overlapping in shared space reproduce the double-slit pattern exactly: dark fringes correspond to W = 1 (bright fringes to maximum OE), and total energy is conserved to 99. 999%. The same density-overlap mechanism, applied to two hydrogen atoms, predicts the H₂ bond length as √2 × a₀ = 74. 8 pm (0. 9%). The iron mass number follows from A (Fe) = (mp/me) (4/3) πα = 56. 13 (0. 2%). The hadron radius spectrum follows R/λC = (l+1) /3 × (4/3) π, resolving the kaon radius to 0. 3%. The strange-quark mass ratio mₛ/md = 20. 0 ≈ e³ (0. 4%). No free parameters are introduced.