Time‑Accumulated Vacuum Cosmology v3 Unified Proper‑Time Tracking Model for Evolving Dark Energy ρ_vac(x) = ρ₀ [1 + k Φ(x)/c²]

Time‑Accumulated Vacuum Cosmology — v3 Unified Model Description and Release Summary This v3 release unifies two related ideas:(1) a cosmological model where vacuum energy accumulates over cosmic proper time, and(2) a phenomenological expression where vacuum energy tracks the local proper‑time rate through the gravitational potential. Together, these provide a consistent framework for exploring evolving dark energy, late‑time deviations from Lambda‑CDM, and a concrete laboratory null test using Casimir cavities. --- 1. Core Phenomenological Equation (Local Proper‑Time Tracking) The local vacuum energy density is modified according to: rho_vac(x) = rho_0 * [ 1 + k * (Phi(x) / c^2) ] where: • Phi(x) is the local Newtonian gravitational potential• c is the speed of light• k is a dimensionless coupling constant• rho_0 is the baseline vacuum energy density Interpretation: • In voids: Phi ≈ 0 → proper time runs faster → vacuum energy slightly larger• In clusters: Phi < 0 → proper time runs slower → vacuum energy slightly smaller This produces an effective evolving dark energy when viewed from inside a gravitational well, without introducing a new scalar field. BBN constraints require: |epsilon| = |k * / c^2| < 0.04 --- 2. Seasonal Casimir Null Test Prediction Because Earth moves through the Sun’s gravitational potential annually, the model predicts a measurable seasonal modulation in Casimir pressure: • Perihelion (Jan 3): Earth deepest in solar potential → proper time slowest → Casimir pressure minimum• Aphelion (July 4): Earth highest in solar potential → proper time fastest → Casimir pressure maximum Fractional modulation: |ΔP| / P = |k| * (ΔPhi_sun / c^2) ≈ |k| * 3.3e‑10 This is testable with CANNEX sensitivity. --- 3. Cosmological v3 Model (Global Proper‑Time Accumulation) The v3 cosmology model introduces a small accumulated vacuum component that grows with cosmic proper time while preserving early‑universe physics. Vacuum energy density: rho_vac(z) = rho_Lambda * [ 1 + alpha * D(z)^gamma * S(z) ] where: D(z) = t(z) / t0t(z) = integral from z to infinity of dz’ / [ (1+z’) * H(z’) ] S(z) = 1 / [ 1 + ((1+z)/(1+z_star))^n ] alpha = coupling strengthgamma = non‑linearity exponentz_star, n = suppression parameters --- 4. Modified Hubble Expansion Define: S0 = S(0)D0 = 1N0 = 1 + alpha * S0 Expansion law: H^2(z) = H0^2 * [ Omega_m * (1+z)^3 + (1 - Omega_m) * (1 + alpha * D(z)^gamma * S(z)) / N0 ] This ensures: • H(0) = H0• Lambda‑CDM recovered at early times• Mild late‑time deviation when D(z)*S(z) → 1 --- 5. Distances chi(z) = c * integral from 0 to z of dz’ / H(z’)D_L(z) = (1+z) * chi(z)mu(z) = 5 * log10(D_L) + 25 with c = 299792.458 km/s. --- 6. What We Did for v3 • Corrected H(0) normalization using N0 = 1 + alpha * S0• Removed circular nested quadrature• Added stable ODE form dt/dz = -1 / [(1+z) H(z)]• Restored correct distance units (explicit c)• Ensured early‑time Lambda‑CDM consistency• Produced mild late‑time deviations testable with SNe and BAO• Added full documentation and reproducible code• Unified the local proper‑time tracking equation with the global cosmology model --- 7. Contents of This Release • v3_model.pdf — full model definition• residuals_v3_timefraction.py — reference implementation• D_of_z_derivation.pdf — physical derivation Notes:• OCR‑corrupted code snapshot (not used in analysis) --- 8. Citation Time‑Accumulated Vacuum Cosmology v3 (2026).Robert John Mills, Oxford UK. Please cite this Zenodo record and specify parameters:{ Omega_m, H0, alpha, gamma, z_star, n, M }