Degeneracy is not Coincidence: The Regime Collapse under Unified Quantum Layers

AbstractFor centuries, physics has faced an open problem: why do identical geometricpatterns appear at every scale, from atoms to galaxies? Quantum Mechanics andGeneral Relativity describe these regimes separately, but provide no unifying operator.We propose Regimetime as a scale-invariant geometric framework governed by auniversal regime parameter κ. The framework is defined byR[κ] = η · F (κ) + ∆ · G(κ), (1)where η and ∆ are geometric coefficients and F, G are regime functions.We show that under the condition κ → κ0, distinct geometric layers Rni of anysystem collapse to a single manifold R0, providing a deterministic geometric mechanismfor quantum degeneracy and orbital organization.To describe the emergence of diversity, we introduce Decomposition Mapping:R0 ∆κ−−→ {Rn1 , Rn2 , ..., Rnk }, (2)as the bifurcation of unity into diversity under regime perturbation.In the local spacetime limit (x, u), the operator becomesR(x, u) = η(x, u) + ∆(x, u) + D(x, u), (3)which recovers General Relativity for D → 0.Thus, Regimetime provides a single geometric framework that unifies QuantumMechanics and Cosmology, and offers a candidate solution to a centuries-old openproblem.