Constraining Classical Field Models of Self-Generated Particle Structure: Successive No-Go Theorems

We assemble a two-theorem obstruction stack for the classical-field programme of self-generated particle structure. Theorem A (with two independent proofs) shows that a finite-energy, source-free Maxwell field on unbounded vacuum ℝ³ cannot sustain a stationary or time-periodic energy density: the only such field is the zero field. Its scalar / Klein–Gordon corollary (Theorem A′), stated for a homogeneous medium, transports the same persistence obstruction to a linear wave carrier, cleanly separating the massless case (both proofs transfer) from the massive/gapped case (where only the spectral, RAGE-based proof survives) and noting that a trapping medium lies outside the hypotheses. A companion zero-net-flux proposition records that every closed surface carries vanishing electric and magnetic flux. Together these results locate precisely where charge or nonlinearity must enter for localized structure to exist. As a sharpened illustration of that boundary, Theorem B shows that even the natural self-coupled route—a fixed linear Beltrami mode—supplies no amplitude-independent, charge-normalized magnetic response, forcing the closure identity μ²_Bel,z / U = e²ℏ² / (4 m³c²) whose left-hand side contains no elementary charge, and we verify the mode-shape functional numerically on the lowest Chandrasekhar–Kendall spheromak mode.