A Constraint-First Derivation of Planck's Law: Constant-c Forcing, Action-Time Geometry, and the Classical Origin of h

Abstract

I derive Planck's law and the relation E = hν within the CARD framework from constant-c forcing, curvature-work scaling, and action-time geometry in electromagnetic standing waves. Standing-wave antinodes provide a geometric representation of stored modal energy, with modal amplitudes directly characterizing the energy distribution at maximum displacement. Under constant-c forcing, modal progression yields the amplitude partition relation: nAₙ = A₁ in which a fixed fundamental amplitude invariant is distributed across an increasing number of antinodes. Together with curvature reconfiguration and the progressively shrinking action-time interval available for energy delivery as frequency increases, these geometric constraints produce the linear relation between per-cycle energy and frequency, E ∝ ν, without invoking quantization postulates.