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

Version 1.8.8 of the Radial Unitary Geometry (GRU) Hypothesis. This version introduces the Effective Geodesic Spinet concept (A.16–A.18), a technical correction to A.15b (bipartition artifact), Appendices A.19–A.20, and three Colab notebooks. The HTML is a complete cumulative document (A.1–A.20).

Central result (invariant):
d_s(λ_t=0) = 1.0136 ± 0.009 ✓ | d_s(λ_t=1) = 1.616 ± 0.022 | Separation: 25.4σ

A.15b v2.0 — Bipartition Correction:
The diagonal graph used in A.15b has natural bipartite structure: nodes where (t+r) is even/odd alternate. The random walker from "0_0" can only return at even steps → P(σ=odd)≈0, biasing the fit. Fix: self-loops (lazy random walk) break bipartition without changing geometry.
Corrected results: zone λ_t∈[0.20,0.40] was NOT unstable — was an artifact. True values: d_s≈1.92–2.12 (smooth transition to d_s≈2). Critical window [0.50,0.70]: corrected d_s=2.41±0.11 (previously underestimated as 2.02). Central result λ_t=0 unchanged. Separation increases to 25.4σ.

A.16 — Clemente Hypothesis with GRU Approach:
Replicates Caceffo-Clemente (arXiv:2010.07179): full CDT graph error ~17-20% (non-convergent to LB continuum). Octant-blind collapsed chain: λ₁·N²→π² with error 0.006% for N=120. GRU complements Clemente: collapse regularizes the spectral problem in the radial subspace.

A.17 — FEM 1D ≡ GRU Physical Laplacian:
GRU (L/h²) and FEM 1D have identical O(h²) convergence toward π²=9.8696. CDT toy (irregular mesh): error ~95-99% constant. Mathematical equivalence confirmed. Theorem demo script included for canonical [0,1] Dirichlet domain.

A.18 — Integral Diagnostic of the Effective Geodesic Spinet:
A.18a: λ_n/λ_1≈n² mean error 0.31% for n=1..8 ✓ — error grows with n following exact formula error(n,N)≈(n²−1)·π²/(12N²), pure O(h²) discretization artifact. Verified: increasing N from 80 to 960 reduces error by exactly 144×=(960/80)² for all modes simultaneously. In the continuum limit N→∞ the harmonic law is exact.
A.18b: V(r)=0 exactly — flat 1D background ✓
A.18c: d_s flow UV=0.91→IR=1.06 — 1D spectral signature ✓

A.19 — Protocol Robustness:
Octant-blind protocol is invariant under representative selection: min/max/random/weighted_center all give Δd_s<1%, Δλ₁≈0%. The radial channel and its spectrum do not depend on which node is chosen as shell representative or on inter-shell wiring details within reasonable ranges.

A.20 — Radial Equivalence and Continuum Model:
Bloque 1: λ₁/h²→π² with ratio≈4 at each doubling of N — O(h²) confirmed (Chung 1997).
Bloque 2: Analytic derivation d_s=1 from λ_n~n² → ρ(λ)~λ^(-1/2) → P(σ)~σ^(-1/2). Numerical verification: d_s≈1.07 (deviation ~7% due to truncation and discreteness).
Bloque 3: Universal radial channel d_s_post≈1.3 across CDT-like (Delaunay), synthetic LQG, and Hořava-Lifshitz graphs. Caveat: synthetic graphs only — d_s_post≈1.3 (not 1.01) because BFS on 2D graph gives fewer shells than GRU protocol on S²×ℝ.

Ontological distinction — CDT foam vs GRU radial background:
CDT measures spectral properties of the full fractal foam. GRU measures the effective radial dimension after BFS shell collapse — a different observable. The bipartition correction demonstrates GRU's self-correcting capacity: the protocol identifies and corrects its own artifacts.

Honest position on CDT formal validation:
All results on CDT-inspired toy models. Formal CDT pending external collaboration. Falsification criterion: d_s(collapsed)∈[0.95,1.05] with ≥5σ → GRU verified; outside → GRU refuted. Contact initiated with Caceffo-Clemente group (INFN/Pisa).

Repository Content v1.8.8:
GRU_v1.8.8_completo.pdf — complete PDF A.1–A.20 with 23 Colab figures
GRU_v1_8_8_web.html — complete HTML with A.15b v2.0 correction and A.18a–A.20
Informe consolidado GRU_v1_8_8.html — supplementary report for reviewers
GRU_A15b_v2_lamt_scan_corregido.py — A.15b corrected (bipartition fix)
GRU_A15b_fix_biparticion.py — diagnostic comparison 3 methods
GRU_A18a_convergencia_modos.py — A.18a convergence verification (144× ratio confirmed)
GRU_A16_clemente_gru.py — A.16 script
GRU_A17_fem_convergencia.py — A.17 convergence script
GRU_A17_theorem_demo.py — A.17 theorem demo (GRU=FEM equivalence)
GRU_A18_diagnostico_integral.py — A.18 integral diagnostic
GRU_A19_robustez.py — A.19 protocol robustness
GRU_A20_equivalencia_radial.py — A.20 radial equivalence and continuum model
GRU_A15b_v2_colab.ipynb — Colab notebook A.15b v2.0 (bipartition fix)
GRU_A16_clemente_gru_colab.ipynb — Colab notebook A.16
GRU_A17_A18_colab.ipynb — Colab notebook A.17+A.18
GRU_A19_A20_figuras_colab.ipynb — Colab notebook A.19+A.20 figures
GRU_CDT_postprocessing.py — octant-blind protocol for real CDT triangulations

Previous scripts A.1–A.15 available in v1.8.7 record: 10.5281/zenodo.20481543

Note (June 2026): GRU_v1_8_8_web.html and Informe consolidado GRU_v1_8_8.html have been updated to correct image URLs (Colab-generated figures replacing prior placeholder images) and to correct the DOI in document header and footer to 10.5281/zenodo.20502213. Content is otherwise technically identical to the original submission.