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). Table §3.2 corrected: The “—” dash in the “Solo r≥0” row now renders correctly. 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 (error S¹=0.0130 vs open=0.0387). §4.2e extended — Topological echo (discrete constructive interference): The P(σ) peak is analogous to constructive interference of a wavefunction on a closed ring. Peak scales linearly with NRADIAL (σpeak≈N, not N²) — 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. Verified in four independent environments, 68s runtime. §4.3 (new) — Chronological narrative A1–A11: Full documentation of how and why each script evolved. The strongest conclusion is not that all regimes are equivalent, but that the correct topology must be chosen according to the physical scale being measured. In the UV regime of GRU (NRADIAL=120, window [6:50]), the compact radial geometry S¹ is the appropriate description. New GRU_A7_v184_crossover_topology.py: Updated with 4th experiment (A7-4): S¹ reduces error by factor 3× at N=60. Full autosuperposition explanation. New GRU_A9_truncamiento_IR.py: IR truncation: d_s(λ=0)=1.0007 identical in A2 and A6 — dispersion numerically compatible with zero. A2 and A6 are consistent. New GRU_A10_eco_topologico.py: Topological echo verification. Linear threshold N>σmax confirmed. Retrospective validation of A1–A6. New GRU_A11_recurrencia_periodica.py: Periodic recurrence: 5 echoes at σ≈kN, stable amplitude ~0.47·A1 for k≥4. Window [6:50] optimal — free of topological contamination for N=120. New in v1.8.5 (Consolidated Script, Laplacian Spectrum, Topological Robustness and External Validation Framework): Bug fix — GRU_principal_consolidado.py: The return G statement was incorrectly placed inside the for t loop, producing an incomplete graph for λ>0. Corrected in v1.8.5: return G moved outside all loops. Results for λ=0 were unaffected; intermediate λ values now produce correct dimensional flow. UTF-8 encoding fix added for Colab compatibility. New GRU_A12_espectro_radial.py: Laplacian spectrum of the S¹ graph. The spectral gap λ₁ = 2(1−cos(2π/N)) scales as 1/N², with numerical error <0.01% vs analytic formula for N=20–120. Fit: k=39.17 (vs 4π²=39.48, difference 0.77%). Connects topological echo (A.10) with discrete energy levels: small λ₁ for N≥65 explains why the echo falls outside the UV window [6:50]. §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. Instead, in the relevant UV regime both topologies produce compatible d_s values within numerical errors, indicating that the observable is robust against the choice of topology. The controlled divergence appears only in the interval 40<N<65, where S¹ produces a clean crossover and a well-defined topological echo (A.10), while the open chain generates a contaminated d_s with two distinct slopes (Δα≈0.261, A.7-2). This pattern — UV convergence plus intermediate-scale divergence — is exactly what GRU predicts from topological closure: not a numerical artifact, but the expected experimental signature of a compact geometry. Sections 11–14 (new): (11) Connection with current physics — CDT dimensional reduction, de-redundantization under spherical symmetry; (12) Synthesis of consolidated script results with full λ-flow table; (13) Possible implementations in CDT, LQC, LQG and central-force systems; (14) Chronological congruence table A1–A12 + Consolidado. Executive Summary (EN): One-page technical summary in English, ready for referees and CDT collaborators. Quick-start guide (GRU_quickstart.md): Guide for CDT researchers — what to run first, parameters to always report, 30-minute reading path, CDT adaptation protocol with S¹ closed topology. 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 — backward compatible with standard CDT codes (A.7).• d_s(λ=0) = 1.0007 ± 0.0321 invariant: NRADIAL=60→960, N_WALKS, window [±reasonable] (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_5_print.html: Print-ready version — open in Chrome, Ctrl+P, save as PDF.GRU_v1_8_5_web.html: Interactive web version with all scripts A.1–A.12 + Consolidado embedded (953 KB).GRU_A1_cdt_radial_demos.py: CDT radial graph + exact mathematical demos.GRU_A2_lambda_scan_v3.py: λ-scan Variant 3 (numpy 2.x compatible).GRU_A3_lambda_scan_v1_legacy.py: λ-scan Variant 1 (numpy 1.x, legacy).GRU_A4_robustez_escala_ventana.py: Scale and UV-window robustness.GRU_A5_spherical_bfs_demos.py: Real spherical BFS graph — S² confirmed.GRU_A6_dimensional_flow.py: Complete dimensional flow d_s=1→2→4.GRU_A6_v2_1_final.py: S¹ circular geometry v2.1 — 9.2σ validation.GRU_A7_v184_crossover_topology.py: Crossover + autosuperposition, 4 experiments.GRU_A9_truncamiento_IR.py: IR truncation — A2 vs A6 consistency.GRU_A10_eco_topologico.py: Topological echo — linear threshold, retrospective validation.GRU_A11_recurrencia_periodica.py: Discrete standing wave — periodic recurrence, optimal window.GRU_A12_espectro_radial.py: Laplacian spectrum — λ₁ ∝ 1/N², <0.01% error vs analytic.GRU_principal_consolidado.py: Full λ-scan S¹ CLOSED — bug fix (return G corrected), UTF-8 fix, dimensional flow λ=0→1.GRU_quickstart.md: Quick-start guide for CDT researchers — S¹ closed topology protocol.GRU_colab.ipynb: Ready-to-run Google Colab notebook — 3 cells, S¹ closed topol