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

This paper introduces the Radial Unitary Geometry (GRU) 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 structural analogy — orbital shells are BFS shells, n=1 is the atomic analogue of d_s=1. Intra-shell equivalence verified (dispersion numerically compatible with zero, A.7-3): structural analogy confirmed, not speculative. §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). The S¹ topology eliminates this crossover entirely. Central argument: a large S¹ is locally indistinguishable from ℝ — the global topology remains compact. New GRU_A7_crossover_topology.py: Three-experiment verification script — (1) NRADIAL sweep 20–480; (2) double-regime verification (Δα=0.261 for N=20, Δα=0.019 for N=120); (3) S¹ scale invariance N=60 to 960. Verified in four independent environments (numpy 2.4.4). 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) New in v1.8.4 (Discrete Standing Wave, Topological Echo and Periodic Recurrence): §4.2e extended — Topological autosuperposition: For NRADIAL<σmax, the random walker completes the full loop of S¹ within the observation window. P(σ) exhibits a peak at σ≈NRADIAL — the walker “meets itself”. Threshold NRADIAL≥65 is the geometric condition NRADIAL>σmax: an operational condition dependent on σmax and the effective system size, not a universal property of space. S¹ reduces error by factor 3× vs open chain at N=60. §4.2e extended — Topological echo: The P(σ) peak is analogous to constructive interference of a wavefunction on a closed ring. Peak scales linearly with NRADIAL (σpeak≈N) — confirmed in A.10. §4.2e extended — Discrete standing wave (A.11): P(σ) exhibits periodic recurrences at σ≈k·NRADIAL for k=1,2,3,... Amplitude stabilizes at ~0.47·A1. 5 echoes confirmed for N=20, 30, 40. §4.3 (new): Chronological narrative A1–A11. The correct topology must be chosen according to the physical scale being measured. New in v1.8.5 (Consolidated Script, Laplacian Spectrum, Topological Robustness): Bug fix — GRU_principal_consolidado.py: The return G statement was incorrectly placed inside the for t loop. Corrected in v1.8.5. UTF-8 encoding fix added for Colab compatibility. New GRU_A12_espectro_radial.py: Laplacian spectrum λ₁ = 2(1−cos(2π/N)) scales as 1/N², numerical error <0.01% vs analytic formula. §4.2f (new) — Topological robustness: why S¹ is not an arbitrary choice: The S¹ topology is not a post-hoc or opportunistic choice, but the direct implementation of the GRU radial hypothesis (§4.2c). The clearest evidence against cherry-picking is the convergence with the open chain for NRADIAL≥65: if S¹ had been selected solely to maximize the effect, one would expect systematic divergence across all ranges. The controlled divergence for 40<N<65 is the expected experimental signature of topological closure — UV convergence plus intermediate-scale divergence = exactly what GRU predicts. New in v1.8.6 (S² Erratic Geometry, Discrete Energy, Confinement Analysis and Optimized Protocol): New GRU_A13_s2_erratica.py (A.13a): Chaotic geodesic S² mesh with discrete energy transitions (particle gains energy on return to origin, jumps levels, decays spontaneously). Without imposed S¹ topology. Results verified in Colab:• Level peak n=2 invariant in ALL sizes (178 to 1,433,914 nodes) — Boltzmann distribution for large N• Ratio excitation/decay ≈0.05 — emergent quantum stability• N_rings=100 (25,666 nodes): d_s=0.8571±0.043 →1 ✓• N_rings=200 (102,274 nodes): d_s=0.9057±0.060 →1 ✓ New GRU_A13_comparacion_topologica.py (A.13b): S¹ closed vs open chain ℝ in the same S² geodesic mesh. S¹ converges to d_s→1 before ℝ (N=100: S¹ →1✓ vs open not yet). Level peak n=2 invariant in BOTH topologies. For large N numerical agreement arises from insufficient σmax to discriminate them, not physical equivalence. New GRU_A13_diagnostico_octantes.py (A.13c): Documented artifact in BFS Cartesian shells: d_s=0.9738±0.062 constant for N=721 to 30,301. Cause: origin node has fixed degree=4 regardless of graph size — P(σ) identical to 6th decimal for all sizes. This is NOT physics — it is a construction artifact. Corrected implementation pending (v1.8.7). New GRU_A13_small_N_analysis.py (A.13d): The “failure” of N<40 is not a GRU error — it is the physical signature of confinement at origin r=0, the only point where t cannot be negative. r=0 acts as a time sink for small N; for N≥65 that effect no longer contaminates the UV observable:• N=5: max_dist=2, first return σ=4, d_s=0.0000 → particle collapsed at origin (Bohr n=1)• N=10–40 (“Bohr zone”): echo contaminates UV window, S¹ more stable than open chain• N≥65 (“UV escape”): confinement breaks, d_s→1.0007, GRU protocol valid New GRU_A14_protocolo_optimizado.py (A.14) — Closing the validation cycle: With the empirical rule σmax ≈ 1.5×√N, the S² chaotic geometry produces d_s within the GRU criterion — identical to the S¹ cylinder. Verified results:• N_rings=30 (2,352 nodes): d_s=0.9842±0.051 α=0.4921 ✓ GRU• N_rings=50 (6,468 nodes): d_s=0.9974±0.048 α=0.4987 ✓ GRU• N_rings=100 (25,666 nodes): d_s=1.0024±0.043 α=0.5012 ✓ GRU• N_rings=150 (57,602 nodes): d_s=0.9996±0.052 α=0.4998 ✓ GRU• N_rings=200 (102,274 nodes): d_s=1.0062±0.060 α=0.5031 ✓ GRUDirect comparison: S¹ cylindrical (1.0007±0.032) and S² chaotic (1.0024±0.043) give the same d_s within errors. Octants remain at 0.9738±0.062 — labeled explicitly as artifact, unaffected by protocol optimization. Three independent justifications for NRADIAL ≥ 65: (A.7) topological crossover; (A.10–A.11) topological echo exits UV window; (A.13d–A.14) confinement at origin r=0 breaks at N≥65. All three converge on the same threshold. Technical notes for CDT implementation (§5.2b):• Topology: S¹ CLOSED — (r+1)%n_radial — direct implementation of §4.2c GRU.• NRADIAL ≥ 65: Δd_s = 0.0000 between S¹ and open chain (A.7).• d_s(λ=0) = 1.0007 ± 0.0321 invariant: NRADIAL=60→960 (A.4, A.6 v2.1).• d_s(λ=1) → 2 requires N_LAYERS ≥ 30 (A.9).• No modification of Regge action, Monte Carlo sampler, or causal foliation required.• Always report: (σmax, window [i_lo:i_hi], NRADIAL, topology) together with d_s. 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_6_print.html: Print-ready version — open in Chrome, Ctrl+P, save as PDF.GRU_v1_8_6_web.html: Interactive web version with all scripts A.1–A.14 embedded (1,027 KB).GRU_A1_cdt_radial_demos.py through GRU_A12_espectro_radial.py: Scripts v1.8.5 (unchanged).GRU_A13_s2_erratica.py: A.13a — S² erratic mesh, discrete energy, level peak n=2 invariant.GRU_A13_comparacion_topologica.py: A.13b — S¹ closed vs open chain ℝ comparison.GRU_A13_diagnostico_octantes.py: A.13c — Documented construction artifact, origin degree=4 fixed.GRU_A13_small_N_analysis.py: A.13d — Confinement at r=0, Bohr n=1 signature, N<40 analysis.GRU_A14_protocolo_optimizado.py: A.14 — Optimized protocol σmax∝√N, S¹ vs S² final comparison.GRU_A13_colab.ipynb: Ready-to-run Colab notebook — S² erratic experiment.GRU_A13_octantes_colab.ipynb: Colab notebook — topological comparison including artifact.GRU_principal_consolidado.py: Full λ-scan S¹ with bug fix, 9.2σ.GRU_quickstart.md: Quick-start guide for CDT researchers.figura_crossover_A7.png: Topological crossover figure. DOI: 10.5281/zenodo.20451161 The present Zenodo record freezes version 1.8.6 as the reference implementation of the GRU hypothesis, incorporating exploratory evidence from chaotic S² geometries with discrete energy. Pend