Physical Implications of the Minimal Discrete Épure_[4/6]

We derive three falsifiable, parameter-free physical predictions from the hypothesis that Planck-scale spacetime carries the geometry of the Minimal Discrete Épure G■ = (Z², d■), π■ = 1. THREE PARAMETER-FREE PREDICTIONS (P1) PHOTON LIV COEFFICIENT ξ₂ = −0.028549... (subluminal, n=2, zero free parameters) → Predicts: δv_g/c = ξ₂(E/E_P)² → Testable by CTA ~2030 → Only model in the CTA detection window with zero free parameters Derivation chain: U(1) gauge action on G■ → photon propagator G^γ_ij(k) = P^T_ij(k)/D(k) → D(k) = 2(1−cos k_x) + ½(1−cos k_y) → scalar-photon dispersion identity (Theorem 1) → isotropic S² average (Theorem 2) → ξ₂ = −(2/3)·(1/2π)∫₀²π [cos⁴θ/12 + sin⁴θ/192] / [cos²θ + sin²θ/4] dθ = −0.028549 (P2) BLACK HOLE MINIMUM ENTROPY S_min = 5π/16 ≈ 0.9817 k_B → Incommensurable with LQG spectrum {m ln 2} (ratio 5π/(16 ln 2) = 1.416 ∉ Q) → Implies QEC threshold p_th = 1 − e^(−5π/16) ≈ 0.626 → Testable via gravitational wave echoes (P3) BARBERO-IMMIRZI PARAMETER γ_G■ = 5/(16√3) ≈ 0.1804 → 24% off LQG standard value (γ_LQG ≈ 0.2375) → Discriminates épure from all LQG variants → Testable via gravitational wave echoes (LIGO/Virgo/KAGRA) ADDITIONAL RESULT Co-emergence of space and time: The temporal dimension emerges from the unfolding Φ₄(G■) as a projection — space and time are not independently fundamental but co-emerge from the discrete geometry. All results verified numerically in Python 3 (scipy, numpy). Code available upon request.