Physical Implications of the Minimal Discrete Épure₄/6

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: δvg/c = ξ₂ (E/EP) 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^γᵢj (k) = PTᵢj (k) /D (k) → D (k) = 2 (1−cos kₓ) + ½ (1−cos kᵧ) → 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 Sₘin = 5π/16 ≈ 0. 9817 kB → Incommensurable with LQG spectrum m ln 2 (ratio 5π/ (16 ln 2) = 1. 416 ∉ Q) → Implies QEC threshold pₜh = 1 − e^ (−5π/16) ≈ 0. 626 → Testable via gravitational wave echoes (P3) BARBERO-IMMIRZI PARAMETER γG■ = 5/ (16√3) ≈ 0. 1804 → 24% off LQG standard value (γLQG ≈ 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.