Abstract
<jats:p>This submission presents a phenomenological extension of the UAT/UCP framework describing causal membrane dynamics. The cyclic moment equation M_p = ∮ Ψ_8phase · κ_crit / √(1-(v/c)²) dt is introduced and numerically evaluated, revealing that the membrane operates on a truncated cycle of 348.12° rather than the classical 360°. Key findings: Causal overflow: The 11.88° deficit (ε = 3.3% = 1 - k_early) is the energy fraction expended per cycle to sustain cosmic expansion. When doubled by the quadratic scaling of the vacuum potential, it yields 6.6% ≈ 7%, the thermal calibration margin. Trigonometric derivation of 0.2791: The force metric R_geom = sin(8 × 43.515°/2) / sin(43.515°/2) = 0.279182 is derived from the 8-phasor sum with non-Euclidean step Δθ = 45° × k_early = 43.515°. This version corrects an arithmetic error present in earlier manuscripts where Δθ was incorrectly stated as 43.530°. The product 45 × 0.967 = 43.515, not 43.530. The resulting overflow values have been updated accordingly. Distinction of three related values: The causal overflow (0.0330), the per-step overflow (0.273), and the force metric (0.2791) are formally distinguished. Monte Carlo analysis: A 10,000-iteration simulation demonstrates that the observed residual amplitude (0.2791) cannot be explained by random thermal phase noise (p < 0.001, maximum null amplitude 0.0588 vs. observed 0.2791). The simulation rules out noise as the origin of the signal but does not independently confirm the UAT phase geometry. Cosmological zero-point fluctuation: The 7% margin is interpreted as a scale analogue of the quantum zero-point energy. Just as a particle cannot rest at the minimum of its potential, the universe cannot rest at the Ivancho limit (η = 4.978). Limitations explicitly acknowledged: The UAT/UCP framework is a phenomenological model under development. Its constants do not appear in standard references (CODATA, NIST). The Berry phase interpretation is a formal analogy, not a rigorous derivation. The GW170814 supplementary analysis confirms that LIGO strain data cannot validate the model (predicted signal is 10⁷ times smaller than source localisation uncertainty). Falsifiable predictions are provided for independent empirical testing. The package includes the theoretical manuscript (LaTeX) and three Python scripts that reproduce all numerical results.</jats:p>