The Charge-Ring Radius from the Mass Profile: spectral theorems, verified (computer-assisted) computations, and a blind prediction for the charge ring of a two-sector Dirac-medium soliton

We study a coupled system of a Dirac bispinor and a scalar medium field in which the mass is the local value of the medium field, m = 1 − b, and b obeys dissipative relaxation with nonlinearly gated diffusion (the minimal dynamics of a spatial-field model, defined here self-containedly). In its two-sector soliton configuration a heavy spherical component digs a deep mass profile m(r), while a light component with unit topological index (~0.2% of the charge) settles into a ring mode on the profile wall; in the frozen limit the light sector reduces, within the J_z = 3/2 sector, to the lowest channel κ = −2 - a radial Dirac pair. We prove strict monotonicity of the Prüfer matching function, simplicity and sign change of the Wronskian at every eigenvalue, and automatic nodelessness of the lowest mode, whose density vanishes identically on the axis. Verified (computer-assisted) interval computations give rigorous eigenvalue brackets, ω₀ ∈ [0.76251; 0.76254] and [0.80505; 0.80513] for two profiles, and close the entire sector: the only bound mode below 0.95 is the κ = −2 mode, with spectral gap ≥ 0.187 / ≥ 0.144 - a theorem, not a scan. We further prove exact R_y(π) covariance of the evolution operator and, under a numerically controlled premise, the existence of a radially averaged self-consistent guided profile with operator bounds on the frequency and the mode. The main applied result is a preregistered blind prediction: the ring radius computed from a single heavy-component profile (3.27; rigorous interval [3.10; 3.45]; ±10% band) is confirmed by direct measurement of the live configuration at a point outside the calibration: plateau 3.28-3.45, midpoint +2.5%. The charge-ring radius is derived from the mass profile. The Russian and English versions of the manuscript and the reproducibility package (profiles, solvers, the verified-computation protocol and the preregistration files) are included in this deposit.