Comparative Analysis Between the 5D τ -Delay Field Model and Gupta's CCC+TL Cosmology: Demonstrating the 4D Limit of a Unified Delay-Geometry Framework

Comparative Analysis Between the 5D τ-Delay Field Model and Gupta’s CCC+TL Cosmology: Demonstrating the 4D Limit of a Unified Delay-Geometry Framework Bahman Masarrat Zenodo DOI: 10.5281/zenodo.16734846 Email: [email protected] November 2025 Abstract This short communication demonstrates that the “CCC + TL Cosmology” proposed by Rajendra P. Gupta (2025) is mathematically equivalent to the four-dimensional limit of the 5D τ-Delay Field (TDF) model previously introduced by Masarrat (2025). While Gupta attributes the observed galactic and cosmological anomalies to a slow weakening of the fundamental forces—encoded in a phenomenological parameter α—the TDF model derives the same behavior naturally from the curvature and time-delay structure of an additional compact dimension τ. Hence, the α-term of CCC+TL corresponds directly to the projected coupling variation α↔ −β τ in TDF. This comparison confirms that the mechanisms behind galaxy rotation, lensing, and cosmic acceleration can all emerge from the 5D delay geometry without invoking dark matter or dark energy, thus situating Gupta’s formulation as a special 4D approximation of the broader TDF framework. 1 1 Background In Integrating a 5D Scalar Field with General Relativity and Photonic Contributions: A Unified Cosmological Model Beyond ΛCDM (Masarrat, 2025, Zenodo DOI: 10.5281/zenodo.16734846), a unified geometrical model was introduced where an extra compactified dimension τintroduces phase delays that mimic the gravitational effects normally attributed to dark matter and dark energy. Gupta (2025) later proposed the “CCC + TL Cosmology,” suggesting that the Universe’s forces gradually weaken over cosmic time. His empirical parameter αacts as an additive term in the Friedmann equations, fitting galaxy rotation curves and cosmic expansion without exotic matter. 2 Mathematical Relation When the 5D TDF field equations are projected into 4D and the higher-order τ-derivatives are neglected, the effective gravitational coupling becomes: Geff(t) = G0(1 −β τ(t)).(1) This yields an apparent acceleration term: ˙ Geff Geff ≈ −β˙τ. (2) Identifying α=−β τ, one recovers the exact form of Gupta’s correction term. Therefore: CCC+TL Cosmology ≈TDF ∂2 τ=0.(3) 2 3 Conceptual Comparison Aspect Gupta (CCC+TL) Masarrat (5D TDF) Dimensionality 4D (time + space) 5D (with delay dimension τ) Origin of αterm Empirical fit Geometric derivative of τ Force variation Assumed weakening Emergent from ∂τcurvature Lensing & Rotation Curves Phenomenologically fitted Derived via Yukawa modes + time delay Dark energy Apparent effect Replaced by quintessence in τspace Predictive scope Cosmological only Cosmology + galactic + quantum scale Photonic term None Explicit ϕγcontribution 4 Implications This mapping confirms that the empirical success of Gupta’s model arises from mechanisms already embedded in the 5D τ-Delay geometry. The TDF framework offers a deeper theoretical foundation that naturally connects cosmological dynamics, galactic rotation, and photon–gravity coupling, while remaining consistent with General Relativity. In essence: Gupta’s CCC+TL model is a phenomenological shadow of the full TDF geometry—its 4D projection under the assumption of a slowly varying τ-dimension. References •Masarrat, B. (2025). Integrating a 5D Scalar Field with General Relativity and Photonic Contributions: A Unified Cosmological Model Beyond ΛCDM. Zenodo. DOI: 10.5281/zenodo.16734846. •Gupta, R. P. (2025). Testing CCC+TL Cosmology with Galaxy Rotation Curves. Galaxies, 13(5):108. DOI: 10.3390/galaxies13050108. 3 Keywords 5D scalar field, time-delay cosmology, variable coupling, dark matter alternative, ΛCDM replacement, Gupta CCC+TL, Masarrat τ-field, unified gravity. 4