The Evolving Faber-Jackson Relation: A Unifying Framework for Galaxy Ages and the Baryonic Tully-Fisher Connection

Referee: [Abstract] Abstract (equation M_b ∝ e^{-4∫ H(t) dt} σ^4): the claim of a direct derivation from the evolving BTFR requires explicit intermediate steps showing how the rotation velocity v_rot is replaced by velocity dispersion σ while preserving the identical exponential kernel and numerical prefactor 4; absent these steps, the unification appears to import the kernel from the Nexus Paradigm rather than derive it from the shared acceleration scale alone.

Authors: We agree that the intermediate mapping steps were insufficiently detailed in the abstract and main text. In the revised manuscript we will insert a dedicated subsection deriving the FJR from the BTFR: starting from the Nexus Paradigm acceleration scale a0 that sets the BTFR normalization M_b ∝ v_rot^4, we apply the virial relation for pressure-supported systems (σ^2 ≈ GM/R) together with the same time-dependent kernel arising from the cosmic expansion term in the action. This yields the identical prefactor 4 and exponential e^{-4∫H(t)dt} without additional tuning. The shared kernel follows directly from the common a0 threshold at which the paradigm modifies Newtonian dynamics for both rotation and dispersion. revision: yes

Referee: [Abstract] Sample of 39 galaxies and age validation: no details are given on data selection criteria, error propagation for the reported ages (e.g., 12±0.8 Gyr), or how the integral ∫ H(t) dt is evaluated for each galaxy to map to formation epoch; without these, the claimed agreement with HST ages within ~1 Gyr and the Pearson r=0.961 cannot be assessed for robustness or selection bias.

Authors: We will add an expanded Methods section (new subsection 3.1) that lists the exact literature sources and selection cuts for the 39 galaxies (e.g., isolation criteria, minimum S/N for kinematics, exclusion of ongoing mergers). The integral ∫H(t)dt is evaluated using the standard flat ΛCDM parameters (H0=70 km/s/Mpc, Ωm=0.3) from each galaxy’s redshift to its inferred formation time; we will tabulate the numerical values and the resulting age uncertainties propagated from both kinematic measurement errors and the 10% uncertainty in a0. The Pearson r=0.961 will be recomputed with bootstrap errors and shown only for the undisturbed subsample, with a clear statement of the selection criteria used to define that subsample. revision: yes

Referee: [Abstract] The weakest assumption (shared acceleration scale a0 ~10^{-10} m/s^2 and identical evolution kernel for pressure-supported systems): the paper must demonstrate that this scale and the coefficient 4 arise independently from the Nexus Paradigm dynamics rather than being chosen to reproduce the observed BTFR/FJR offsets; otherwise the derivation reduces to a re-expression within the new framework.

Authors: The value a0 ≈ 1.2×10^{-10} m s^{-2} is fixed by the Nexus Paradigm’s quantum-gravity action as the scale at which the modified Poisson equation transitions; it is not fitted to galaxy data. The coefficient 4 emerges analytically from the fourth-order time derivative term in the paradigm’s Lagrangian when the virial theorem is imposed, and applies identically to both v_rot and σ because both satisfy the same equilibrium condition at the a0 threshold. We will add a new paragraph in Section 2 that derives a0 and the kernel from the first principles of the paradigm (citing the relevant action integral) before any comparison to observations, thereby demonstrating independence from the BTFR/FJR data. revision: yes