The Emergent Speed of Light: Unifying Quantum Mechanics and General Relativity Through Phase–Amplitude Stiffness

The speed of light c appears as a fundamental constant in both quantum mechanics and general relativity, yet its origin is not explained within either framework in isolation. We present a stiffness-based substrate description with two response channels: a compact phase field (quantum coherence) and a real amplitude field (geometric response). After diagonalization of the coupled quadratic response functional, we define the dimensionless stiffness ratio c_new^2 ≡ K_Θ^eff / K_A and argue that macroscopic physics requires its renormalization-group flow to a stable fixed point. Introducing a substrate coherence length ℓ_★ (defined independently of c) as the scale where phase–amplitude mixing becomes order unity, we obtain the emergence relation c^2 = ℏG/ℓ_★^2. The usual Planck length ℓ_P = √(ℏG/c^3) is then recovered as the empirically inferred spacetime image of ℓ_★ once the measured c is inserted. We summarize observational handles on any residual scale-dependence of c_new using existing constraints on Lorentz invariance violation, gravitational-wave dispersion, and searches for varying constants. Key notation translations: c_new^2 = c-new squared K_Θ^eff = effective K-theta (phase stiffness) K_A = K-A (amplitude stiffness) ℓ_★ = ell-star (substrate coherence length) ℏ = h-bar (reduced Planck constant) G = Newton's gravitational constant ℓ_P = ell-P (Planck length) √(ℏG/c^3) = square root of (h-bar times G divided by c cubed)