Testing the Density Profile of the Unified Informational Theory (TURI) on the Proton/Neutron: A Contribution to the Unification of Scales of Order 1034

Starting from a cyclic universe and the strict conservation of total entropy (ΔS _cycle = 0), we derived in a previous publication [1] the universal density profile ρ(r) = ρ₀ · sech⁸(r / r _e ), which structurally resolves the Core-Cusp problem in galactic dynamics. The present work tests its application at the nuclear scale on the proton and neutron. For the proton, the triple convolution of valence quarks exactly reproduces the electric RMS radius r _E = 0.84 fm, consistent with the CODATA 2022/CREMA consensus. The asymmetric approach (valence + chiral component δ _M = 0.074, r _{e_cloud} = 4.00 fm) reproduces r _Magn ≈ 0.811 fm and improves G _M /μ _p at intermediate Q², but strongly underestimates at high Q² (no sea quarks, no perturbative QCD). G _E (Q²) captures the initial slope but underestimates curvature at 0.05–0.10 GeV² relative to PRad 2019; this tension is less constraining than the PRad vs. Mainz 2010 discordance. The extension to the neutron reproduces ⟨r²⟩ _E ⁿ ≈ −0.115 fm² and r _M ⁿ ≈ 0.82 fm within uncertainties, via an adjusted chiral fraction (δ _n = 0.12, r _{e_val} ≈ 1.29–1.30 fm), naturally capturing isospin breaking and the non-vanishing of G _E ⁿ (Q²). Falsifiable predictions: positive curvature of G _E from Q² = 0.020 GeV² (+5.8% vs. Taylor, accentuated to +14.3% at Q² = 0.050), testable by PRad-II; perfect lepton universality (MUSE); stability of the ratio R = μ _p G _E /G _M ≈ 2.78–2.79 up to Q² ≈ 2 GeV² (EIC phase 1). If confirmed, these signatures would validate sech⁸ universality and a unification of scales of order 10 ³⁴ .