Geometría Radial como Base del Espacio Físico: Una Hipótesis sobre la Reducción Dimensional desde el Origen

This paper introduces the Radial Unitary Geometry (RUG) Hypothesis, proposing that the effective dynamics of homogeneous quantum cosmological models can be reparametrized using a single scalar radial variable r ≥ 0. This approach simplifies the effective Hamiltonian, removes the discrete ℤ₂³ redundancy in triad space, and provides precise falsification criteria for quantum gravity candidates. Key updates in v1.8.2 (Spherical 4D Geometry and Dimensional Flow): §4.2b corrected: The λ-scan toy model was implicitly 4-dimensional from the start. BFS shells are discrete S² hypersurfaces — the structure already contained r + S²(θ,φ) + t = 4 spacetime dimensions. λ=0 collapses a full 4D spacetime to its single irreducible degree of freedom: the radial coordinate r. This is not a reduction from flat 2D to a line — it is the collapse of a spherical spacetime to its radial geodesic. §4.2c (new): Convergences with established physics: (1) Einstein’s closed geodesic S¹ without boundaries; (2) the center of the universe as an observer operator — a geometric reason for the Cosmological Principle; (3) Bohr’s atom as microscopic validation — orbital shells are BFS shells, n=1 is the atomic analogue of d_s=1. §4.2d (new): Quantum indeterminacy as a consequence of spherical symmetry. The electron is not “everywhere by probability” — it is everywhere because the complete shell is its state. All nodes of a shell are equivalent under rotational symmetry. Position determination requires breaking that symmetry through interaction with another foliation — the geometric analogue of wavefunction collapse. New Appendix D.4: Complete dimensional flow table: λ=0 gives d_s=1.000 invariantly from 600 to 60,000 nodes (UV and IR); λ=1 with N_LAYERS≥50 recovers d_s→2 (Giasemidis 2012); CDT formal with S²+t would reach d_s→4 (Carlip 2017). New Appendix E: Conjectural extensions — relativistic time dilation, multiscale fractal dimensionality, holographic principle from first principles, quantum indeterminacy as spherical symmetry. Explicitly marked as open research lines, not results. Scripts A5 and A6 (new): Real spherical BFS graph with 3 verifiable demos (A5); complete dimensional flow micro→macro with Giasemidis recovery at N_LAYERS=50 (A6). New in v1.8.3 (Topological Crossover and UV Equivalence): §4.2e (new): Topological crossover S¹ vs open chain. For NRADIAL≥65, both implementations produce identical d_s=1.0007±0.0321 (Δd_s=0.000). For NRADIAL<65, the open chain exhibits double-regime behaviour (Δα≈0.261), confirming the random walker reaches the boundary within the fit window [6:50]. The S¹ topology eliminates this crossover entirely. Central argument: convergence at large scale reflects that a large S¹ is locally indistinguishable from ℝ — the global topology remains compact. The result d_s=1.0007±0.0321 is not a statement about an open radial chain — it is a statement about the single irreducible degree of freedom of the compact radial geodesic S¹, measured in the UV regime where boundary effects are negligible. New GRU_A7_crossover_topology.py: Three-experiment verification script — (1) NRADIAL sweep 20–480 comparing S¹ and open chain; (2) double-regime verification via early [4:20] and late [30:50] fit windows (Δα=0.261 for N=20, Δα=0.019 for N=120); (3) S¹ scale invariance confirmation from NRADIAL=60 to 960. Verified in four independent environments (numpy 2.4.4). Table §3.2 corrected: The “—” dash in the “Solo r≥0” row (Information lost column) now renders correctly in both PDF and HTML versions. Statistical validation:d_s(λ=0) = 1.0007 ± 0.0321 → interval [0.969, 1.033] — contains 1.0 ✓d_s(λ=1) = 1.9818 ± 0.1020 → interval [1.880, 2.084] — contains 2.0 ✓Separation: 9.2σ — categorical difference (discovery threshold in physics: 5σ) Quantitative demarcation criterion:α = 0.5 ± 0.1 → d_s → 1 (confirms radial reduction)α = 1.0 ± 0.1 → d_s → 2 (refutes extension to CDT) Repository Content:GRU_v1.8.3_completo.pdf: Full preprint with §4.2b–e, Appendices A–E, corrected figures.GRU_v1_8_3_web.html: Interactive web version with embedded figures and scripts (831 KB).GRU_A1_cdt_radial_demos.py: CDT radial graph + exact mathematical demos.GRU_A2_lambda_scan_v3.py: λ-scan Variant 3 (reference, numpy 2.x compatible).GRU_A3_lambda_scan_v1_legacy.py: λ-scan Variant 1 (numpy 1.x, legacy generator).GRU_A4_robustez_escala_ventana.py: Scale and UV-window robustness verification.GRU_A5_spherical_bfs_demos.py: Real spherical BFS graph — S² geometry confirmed.GRU_A6_dimensional_flow.py: Complete dimensional flow d_s=1→2→4 micro and macro.GRU_A6_v2_1_final.py: S¹ circular geometry v2.1 — closed topology verification, 9.2σ statistical validation.GRU_A7_crossover_topology.py: Topological crossover S¹ vs open chain — UV equivalence threshold NRADIAL≥65, double-regime verification, scale invariance confirmation. DOI: 10.5281/zenodo.20399477