Time‑Accumulated Vacuum Cosmology v3.6 Unified Proper‑Time Tracking Model for Evolving Dark Energy ρ₍vac₎(x) = ρ₀ [1 + k Φ(x)/c²]

FEU Cosmology v3.6 — Corrected & Updated Release Time-Accumulated Vacuum Cosmology with Early-Time Suppression and Density-Dependent Screening FEU v3.6 is a phenomenological extension of ΛCDM in which the vacuum sector contains a smooth proper-time-dependent contribution. The model incorporates time accumulation, early-time suppression, and density-dependent environmental screening. This release includes important mathematical and statistical corrections intended to ensure that the stated FEU vacuum law, equation-of-state calculation, numerical implementation, and interpretation of observational results are internally consistent. Core FEU formulation The FEU vacuum energy density is defined as ρvac(z)=ρΛ[1+αD(z)γS(z)], where D(z)=t(z)t0 is the normalized cosmic proper time, with dtdz=−1(1+z)H(z), and the early-time suppression function is S(z)=11+[(1+z)/(1+z∗)]n. The density-dependent environmental screening function is f(ρ)=11+(ρ/ρcrit)m. For homogeneous cosmological background calculations, f(ρ)=1. The screening function is a model ingredient intended to suppress the accumulated contribution in sufficiently dense environments; galaxy-scale screening is not claimed to be observationally established by the cosmological analysis. Corrected equation of state A major v3.6 correction is the use of the full derivative of the FEU vacuum law. The required derivative is ddz[DγS]=γDγ−1D′S+DγS′. Thus the numerical equation of state includes both the proper-time derivative and the suppression-function derivative. The previous claim that w0=−1 exactly for all α applied only to the simplified S(z)-only implementation. In the full FEU formulation, D′(0) contributes to the derivative, so w0 should be calculated numerically and is not imposed to equal −1. Early-time behaviour The FEU modification is strongly suppressed toward the early Universe through the combined behaviour of D(z)γS(z). For the tested baseline configuration, the resulting background modification is numerically negligible at approximately recombination redshift, z∼1100. This statement applies to the tested background implementation and is not intended to claim that all aspects of BBN, recombination, perturbation evolution, or primordial physics are automatically unchanged. Planck constraints FEU v3.6 has been tested using the stated Planck 2018 TTTEEE + lowE + lensing implementation. The reported FEU result is approximately α=0.099±0.232, with the model remaining consistent with the tested Planck likelihood. The appropriate interpretation is that Planck does not provide a statistically compelling rejection of the tested FEU model. No CMB detection of FEU is claimed. Pantheon+ and late-time observations The corrected supernova implementation includes treatment of the supernova absolute-magnitude nuisance parameter and avoids interpreting improvements arising from earlier unnormalised or simplified implementations as evidence for FEU. The corrected results indicate that FEU and ΛCDM can provide statistically similar descriptions of the tested supernova data. Any preference for non-zero α in late-time combined analyses should be regarded as dataset- and likelihood-dependent until reproduced using the complete Pantheon+ covariance, consistently treated nuisance parameters, and a fully self-consistent combined likelihood. DESI BAO DESI BAO provides an important test of the late-time and intermediate-redshift FEU modification. The tested DESI comparisons do not establish a statistically significant FEU detection. A representative Planck + DESI sensitivity forecast gives approximately σ(α)≈0.12. This value is a forecast rather than a measured FEU constraint. The DESI results therefore demonstrate that current and forthcoming BAO measurements can provide meaningful tests of the FEU parameter space. w0−wa interpretation Earlier versions included simplified mappings between the FEU parameter α and CPL parameters w0,wa. These are now treated explicitly as toy approximations only. The authoritative FEU equation of state is obtained from the full numerical derivative of ρvac(z)=ρΛ[1+αD(z)γS(z)]. Any reported w0 or wa values should therefore be calculated from the numerical FEU implementation rather than inferred from the simplified toy mapping. Statistical interpretation FEU v3.6 deliberately distinguishes between: observational consistency; parameter constraints; forecasts; localized or subset-level features; and statistically significant evidence for a non-zero FEU parameter. No global FEU detection is claimed in this release. Earlier large Δχ2 improvements obtained from simplified or inconsistent likelihood implementations are not used as evidence for FEU. Any Bayesian evidence or statistical significance quoted for future analyses should be tied explicitly to the likelihood, covariance treatment, nuisance parameters, priors, and FEU implementation from which it was obtained. Scientific status FEU is presented as a testable phenomenological extension of ΛCDM, not as an established replacement for ΛCDM and not as a model-independent theory. The central observational question is whether allowing a smooth proper-time-dependent vacuum contribution provides a statistically preferred description of cosmological observations when the major datasets are analysed simultaneously and consistently. A future statistically significant preference for non-zero α, if reproduced with complete likelihoods and independent datasets, would support the FEU phenomenology within the specified model class. It would not constitute a mathematical proof that ΛCDM is impossible. v3.6 release statement FEU Cosmology v3.6 incorporates corrected FEU derivatives, an updated treatment of w0, revised Planck and early-time interpretations, clarified DESI forecast language, removal of inconsistent w0,wa mappings, a more precise description of density-dependent screening, and a statistically conservative interpretation of current results. These corrections provide a mathematically consistent and testable basis for further combined Planck + Pantheon+ + DESI analysis without overstating the present observational evidence. Time‑Accumulated Vacuum Cosmology v3.6 (2026). UPDATES v3.8 dynamic cosmology model visualised through audio‑signal processing tools And [v4.0] — WORKING CODE IN ZIP feu_background.py FINAL is inside the zip. • NO scipy needed, works standalone + as Cobaya Theory FEUBackground • z_grid 0-1100 for CMB distance priors R/la • Tested: CMB Δχ² 3.73, DESI BAO Δ0.09, Pantheon+ Δ0.09, Lyα z=2.33 χ² 0.36 vs ΛCDM 13.02, fσ8 & H(z) pass, total 1616 pts Δ+7.7 <1σ • python -m cobaya run cobaya_feu_planck_REAL_FEU.yaml --test → "Test initialization successful" (w:null wa:null wiring) After hours got Python working on macpro— this version runs. New update 0.4.2 v0.4.2 Hemispherical Isotropy Test (Planck TTTEEE + lowE + lensing findings α = 1.058 ± 0.021) Declaration of Generative AI Use in the Scientific Process The FEU concept, its motivation, and the decision to test it against cosmological datasets were entirely initiated by the author. The author defined the behaviour the model should exhibit and directed the development of the FEU framework. Generative AI tools (including Meta AI, Google’s AI systems, and Microsoft Copilot) were used to help translate the author’s conceptual description into formal mathematical expressions, to assist in structuring Python and YAML files, and to provide guidance on appropriate cosmological tests. These tools also supported the organisation and explanation of the material. All scientific responsibility for the FEU concept, its interpretation, and the conclusions of this work remains solely with the author.