Schrödinger as the Envelope Equation of the Electromagnetic Carrier

The slowly varying envelope (SVE) reduction of a gapped classical electromagnetic carrier predicts the single-particle Schrödinger equation as an exact identity plus a closed-form residual: the constrained vector MacCullagh carrier admits an exact per-branch balance whose retained part is the Schrödinger operator and whose sole marked approximation leaves a remainder quadratic in the detuning, bounded per mode by c⁴k⁴ / (8ω_C³) and closing linearly in time. Read as a kinetic energy, the envelope dispersion is the relativistic function E(p) = √(m²c⁴ + c²p²), recovered consistently in three mathematical registers—the Foldy–Wouthuysen operator, the dispersion Taylor series, and direct numerics—with leading correction the fine-structure term −p⁴/(8m³c²). Evolution of the exact carrier confirms the predicted error structure: the fitted residual exponent converges to the predicted value 2 as the detuning tends to zero, and a wavepacket (kΔx)² artifact that would mimic it is excluded by the derived reparameterization. One action-scale identification, ℏω_C = mc², enters exactly once as an input; by Buckingham-π the reduction fixes only the ratio ℏ/m = c²/ω_C = ω″(0) and leaves one free scale, so ℏ is supplied, not derived. The result is a dynamical correspondence at the envelope stratum, not a derivation of quantum mechanics: |ψ|² is normalized wave action, never a Born probability, and an explicit ledger fences what the correspondence forces, permits, and provably cannot produce—the Born rule, state reduction, quantized charge, and entanglement—each with its standing theorem.