The Architectural Cosmos: Leveraging Hierarchical Cosmogonies to Resolve the Dark Matter Anomaly and the Big Bang via Scale-Dependent Dynamics_v2

ABSTRACTWe present the formulation of an explicit scale-dependent phenomenological model and a program of falsifiable testsaimed at evaluating whether a dynamical transition operating on real baryonic matter in ultra-low accelerationregimes can describe galactic rotation curves without resorting to unobservable dark entities in said regime.Furthermore, in contrast to non-stationary cosmological metrics where local covariant divergence (∇µTµν = 0)does not guarantee total energy invariance, this framework adopts as a foundational physical and philosophicalpostulate the strict validity of the global conservation of matter and energy (Noether’s temporal symmetryin a large-scale stationary substrate), proposing an eternal cosmos, without beginning or end, structured andcontinuous. Starting from the critical acceleration parameter (a0 = 1.20 × 10−10 m/s2, converging in itsasymptotic limit with Milgrom’s dynamics) and fixing the unique rational transition function µ(x) = x1+x, wederive a closed quadratic algebraic solution that continuously connects the Newtonian regime (1/r2) with theasymptotic vortex or whirlpool regime (1/r). In the Milky Way, implementing an explicit and uniform baryonicmodel—composed of a Hernquist spheroidal bulge (Mb = 1.00 × 1010 M⊙, ab = 0.60 kpc) plus a Freemanexponential disk (Rd = 2.60 kpc, evaluated across three parametric profiles: Branch A with Md = 4.50×1010 M⊙,Central Profile C with Md = 5.00 ×1010 M⊙, and Branch B with Md = 5.50×1010 M⊙)—, we generate viaa single reproducible Python code (scipy.special, included in the Technical Appendix) both the curve on auniform grid from 5.0 to 25.0 kpc (Table 1A) and the direct evaluation at the nine exact radii Rbin of Eilers etal. (2019) with two decimal places of precision without intermediate rounding (Table 1B). Under this algebraicphenomenological ansatz (which is not equivalent to solving the non-linear Poisson equation nor demonstrates byitself universal baryonic sufficiency across all cosmological scales), Central Profile C yields in Table 1A velocitiesof 228.67 km/s (5.0 kpc), 231.42 km/s (6.0 kpc), 230.19 km/s (8.0 kpc), 224.93 km/s (10.0 kpc), 218.64 km/s(12.0 kpc), 207.44 km/s (16.0 kpc), 199.62 km/s (20.0 kpc), and 193.57 km/s (25.0 kpc); and at the exact binsof Eilers et al. (Table 1B), velocities of 229.70 km/s at 5.27 kpc (residual +2.87 km/s), 231.63 km/s at 6.27 kpc(+0.83 km/s), 230.60 km/s at 7.78 kpc (+0.97 km/s), 229.63 km/s at 8.27 kpc (+0.74 km/s), and 193.73 km/sat 24.82 kpc (−4.69 km/s), achieving across the nine observational bins an exact global root-mean-square errorof RMSE =4.44 km/s for Central Profile C (RMSE = 9.33 km/s in Branch A and global RMSE = 6.34 km/s inBranch B, which achieves RMSEext = 2.58 km/s across the four outer bins from 12.26 to 24.82 kpc).At the scale of spiral galaxies and spheroidal clusters modeled under isotropic hydrostatic Jeans equilibrium,we explicitly distinguish between the observed baryonic mass (Mbar,obs), the baryonic mass inferred by theTully-Fisher scaling relation (Mbar,TF), and the effective dynamical mass (Mdyn,ef), preserving the accountingidentity Mt = Mc +Mr with operational equivalence separated by population (Mc ≡ MBH in 2D galaxies andMc ≡ MBCG in 3D clusters), and illustrating as an empirical working conjecture the bound Mc/Mt < 0.01.Beyond the galactic empirical conclusions, we explore on a strictly heuristic plane the conditional parametricscenario N7 (Mt(N7) = 7.00 × 1021 M⊙ starting from MN1 = 1.00 × 1021 M⊙, with algebraic consequencesvf = 102,754 km/s and rt = 9,300 Mly), the complete harmonic radial breathing cycle (R(t) = R0−Acos(2πt/T)with T = 2,000 million years), and the vectorial Hubble decomposition (Hobs = Hγ + ⃗ Hcin · ˆn), presenting themas parametric exercises under construction aimed at designing future observational tests