On the Entropic Origin of the Speed of Light in the Theory of Entropicity (ToE): A Complete, Rigorous Derivation from the Obidi Action and the No-Rush Theorem, and a Definitive Resolution of the Tautology Objection to the Theory of Entropicity (ToE)

Abstract This letter — Letter C in the Letter IIA extract of the Theory of Entropicity (ToE) Living Review Letters Series — provides the complete, rigorous, fully formal derivation of the universal speed of light c from the Obidi Action and the Obidi Field Equations (OFE). The central result is the No-Rush Theorem (Theorem C.2), which establishes that c is the maximum rate of entropic rearrangement on the entropic manifold — a finite, universal, and dynamically determined quantity, not a postulate, and not a tautologically defined constant. The derivation proceeds in six logical steps: (i) the quadratic entropic Lagrangian is established uniquely from five symmetry and consistency constraints; (ii) the Euler-Lagrange equations yield the entropic wave equation; (iii) the wave speed cent = √(κ/ρS) is identified as a pure ratio of response coefficients; (iv) dimensional analysis and Planck-scale matching derive κ and ρS independently from first principles; (v) the self-consistency equation is shown to be non-trivial by virtue of the No-Rush Theorem; and (vi) cent is identified with the empirically measured universal speed limit. The Letter responds comprehensively to all known forms of the Tautology Objection, demonstrates precise structural analogy with Maxwell's 1865 derivation, and articulates the novel predictions that distinguish the ToE derivation from both Maxwell's approach and Einstein's postulate. Maxwell-Obidi Reframing — electromagnetic waves are entropic phase waves, the speed of light is the entropic speed limit — is established as a deep theorem rather than a verbal metaphor. Reveals ToE’s entropic stiffness κ, entropic inertia ρ_(S ).