Causal Membrane Dynamics and the 3.3% Overflow: Cyclic Moment Accumulation, Trigonometric Derivation of the 0.2791 Force Metric, Monte Carlo Validation, and the 7% Margin as the Cosmological Zero-Point Fluctuation

This submission extends the UAT/UCP framework from static equilibrium to dynamic membrane behavior. The cyclic moment equation Mₚ = ∮ Ψ₈phase · κcrit / √ (1- (v/c) ²) dt is introduced and numerically evaluated, revealing that the causal membrane operates on a truncated cycle of 348. 24° rather than the classical 360°. Key findings: (1) Causal overflow: The 11. 76° deficit (ε = 3. 3% = 1 - kₑarly) is the energy fraction expended per cycle to sustain cosmic expansion against gravitational collapse. When doubled by the quadratic scaling of the vacuum potential, it yields the 6. 6% ≈ 7% thermal calibration margin. (2) Trigonometric derivation of 0. 2791: The force metric Rgeom = sin (8 × 43. 530°/2) / sin (43. 530°/2) = 0. 279182 is derived from the 8-phasor sum with non-Euclidean step Δθ = 45° × kₑarly = 43. 530°. The derivation requires no free parameters beyond kₑarly = 0. 967. (3) Distinction of three related values: The causal overflow (0. 0327), the per-step overflow (0. 270), and the force metric (0. 2791) are formally distinguished. The 3. 4% difference between 0. 270 and 0. 2791 is shown to arise from collective phase coherence in the 8-phasor sum, not from measurement error. (4) Monte Carlo validation: A 10, 000-iteration simulation rejects the null hypothesis that 0. 2791 arises from thermal noise (p < 0. 001, maximum null amplitude 0. 0588 vs. observed 0. 2791, a factor of 4. 7× above the noise ceiling). (5) Numerical integration: Mₚ (UAT) ≈ 5. 16 × 10⁻⁸¹, Mₚ (Classical) ≈ 2. 03 × 10⁻⁸¹, ratio ≈ 2. 54. The temporal difference of 176, 122 ns per cycle at fbase = 187. 37 Hz corresponds exactly to 1 - kₑarly. (6) Cosmological zero-point fluctuation: The 7% thermal calibration margin is interpreted as the cosmological analogue of the quantum zero-point energy (½ħω). Just as a quantum particle cannot be at rest at the minimum of its potential, the universe cannot be at rest at the Ivancho limit (η = 4. 978). The 7% displacement is the irreducible minimum of causal action that the cosmos must expend simply to exist. (7) Search for ħ–7% relationship: A systematic algebraic search for a direct equation linking the reduced Planck constant and the 7% margin yielded a negative result. The connection is shown to be a scale correspondence rather than a mathematical equality, mediated by the causal overflow rather than by κcrit. The search script is included with a correction note documenting an initial numerical error and the productive reinterpretation that followed. The causal overflow constant εfrac = 0. 0327 is proposed as a new fundamental parameter governing the rate at which the universe expends energy to maintain its expansion. The package includes the theoretical manuscript (LaTeX) and three Python scripts that reproduce all numerical results.