This document presents a formal correspondence between the causal membrane of the Unified Applicable Time (UAT) framework and the Penrose superradiance process in rotating black hole geometries. The phenomenological relation for the effective matter density —Ωₘ (eff) = Ωb × κcrit × φ × (1 - φ*/η) — is shown to be structurally identical to the efficiency formula for energy extraction from a Kerr black hole ergosphere. Key findings: The 7% thermal calibration margin as a geometric deficit: The quantum braking factor kₑarly = 0. 967 is interpreted as the lapse function N in an Arnowitt-Deser-Misner (ADM) decomposition of the causal metric. The square of the lapse is N² ≈ 0. 935, yielding a geometric deficit 1 − N² ≈ 0. 065, which coincides with the empirically calibrated thermal margin φ*/η = 0. 07 (7%) and the doubled causal overflow 2ε = 0. 066 (6. 6%). The amplification factor as a shift vector coupling: The product κcrit × φ = 4. 978 × 1. 618 ≈ 8. 05 is interpreted as the norm of the shift vector βⁱ that governs frame-dragging in the causal geometry, analogous to the angular momentum coupling in the Kerr metric. The efficiency formula: The surviving fraction (1 − 0. 07) = 0. 93 corresponds to the escaping component in the Penrose process, while the absorbed 7% corresponds to the negative-energy fragment that falls through the horizon, powering the amplification. Falsifiable predictions: If the causal membrane operates as a superradiant amplifier, the absorbed energy should be detectable as the causal overflow frequency (6. 18 Hz). The amplification should saturate at the Ivancho limit (κcrit = 4. 978). The force metric (Rgeom = 0. 2791) should appear in any 8-element phase-coherent array regardless of the physical substrate. Limitations explicitly acknowledged: The UAT metric tensor has not been constructed from the UAT Lagrangian. The ADM decomposition presented is hypothetical. The identification of the causal membrane with an ergosphere is a formal analogy, not a rigorous geometric proof. The document's value lies in providing a plausible physical mechanism —cosmological superradiance— for the numerical success of the effective matter density relation. The package includes the LaTeX manuscript.
Miguel Angel Percudani (Tue,) studied this question.