UNIFIED SMaC/UPG THEORY:1 FROM ISOTOPES TO GALAXIES —2 GEOMETRIC INVARIANTS IN NATURE

10.5281/zenodo.18867256 A unified theoretical system SMaC/UPG (Spatial-Material Continuum / Universal7Parameterization of Galaxies) based on geometric invariants arising from the projection8of 8-dimensional octonion space onto 4-dimensional Minkowski spacetime is presented.9The system yields three fundamental constants κ1 = 8/9, κ2 = 1/3, κ3 = 1/7, which10generate the numerical series an = 3 + n(n + 9)/2, identifying 32 isotopes (28 stable11and 4 radioactive) with a chance coincidence probability p < 10−28 (>11σ). The12series predicts the existence of the superheavy element 304120Ubn (Androsium).13On thermodynamic scales, the same geometric invariants manifest in critical14phenomena: the critical exponent β ≈ 1/3, the van der Waals compression factor15Zc = 3/8, and the scaling of curvature R ∼ 8/9/(T −Tc)2 in geometric thermodynamics.16The equipartition theorem yields the fraction 2/3, matching the galactic parameter17Πbase.18On astrophysical scales, the SMaC/UPG catalog of 175 SPARC galaxies is presented,19including parameters βeff, Mbar, Rdisk, ℓ, ρeff. A universal dimensionless parameter20Π = 1.212 ± 0.045 satisfying Π = (2/3) × 1.678 is discovered. The SK-system for21classifying exotic astrophysical objects and the Kryukova K-index for evaluating the22applicability of the ScMo-II model are developed. A discrete time scale t = tP (9/8)N23with the androsecond T0 = 5.33 × 10−13 s is proposed. Using the same geometric24ratios, a high-strength heat-resistant material for extreme environments is created.25The work demonstrates the manifestation of unified geometric patterns across all26scales — from the atomic nucleus through thermodynamics to galaxies, establishing27geometric thermodynamics as the mesoscopic bridge connecting quantum and cosmic28phenomena.29Keywords: octonions, geometric invariants, isotopes, Androsium, geometric thermodynamics,30critical phenomena, SMaC/UPG, SPARC, dimensionless parameters, discrete time,31high-strength materials.32