Version 1. 9. 3 of the Radial Unitary Geometry (GRU) Hypothesis. This record was updated on June 14, 2026 with post-publication corrections and reinforcements (same DOI; no new version number, see "POST-PUBLICATION UPDATE" below). NEW IN v1. 9. 3 — Validation tests (A. 29z, A. 34, cross-validation, Fisher LRT) + observational predictions (C. 1/C. 2) + operator analysis: A. 29z — Zero-knowledge holography: CorrZK=0. 9662±0. 0098 WITHOUT degree prior. The radial weights |Sn| alone encode the bulk Laplacian spectrum — no prior information about degree distribution needed. Reference A. 29 v3 (with prior): 0. 957±0. 003. Δ=−0. 009: zero-knowledge reconstruction is actually better. The “prior was doing the work” critique is refuted. VERDICT: GENUINE HOLOGRAPHY. Cross-validation A. 21 (legacy v1. 8. 9 vs protocol v4): Δds=0. 0153±0. 0122 (5 seeds; 2/5 EXCELLENT 5. 0). Identifies the validity window for GRU in 3D CDT. Lumpability / R̂ operator analysis: The effective generator R·LCDT·D is exactly tridiagonal (0. 00% off-diagonal mass) — the radial projection produces a genuinely 1D nearest-neighbor operator. Propagator lumpability defect ε≈0. 23 in the GRU window (semi-lumpable), decreasing toward IR. First operatorial connection between GRU and Hilbert-space reduction frameworks (Nomura–Ugajin). Spine topology confirmed: ds (spine topological, 0/1) =1. 055±0. 014 — consistent with A. 21 official (1. 019±0. 015). Weighted spine (bulk weights) gives ds≈2, recovering bulk geometry as expected. The GRU protocol uses the topological spine (0/1 edges), not the weighted spine. C. 1 CMB with CAMB (real pipeline): ΛCDM: C2=1022 μK². Best GRU toy fit (κ=4. 9): C2=800 μK². χ²GRU−χ²ΛCDM=+46. 9 — toy model does NOT improve the fit. Honest conclusion: the suppression direction is consistent with Planck low-ℓ anomalies, but κ requires formal derivation from CDT geometry (CLASS pipeline in progress, v2. 0). C. 2 LISA with SciRDv1 sensitivity curve: With realistic threshold SNR>50: GRU detection horizon reduced ~12–16% vs GR. Amplitude suppression at z=1: 23% (SNRGRU/SNRGR=0. 769). Falsification criterion: dA/A8 both models detect to z≈5 (no apparent horizon reduction — honest limitation of the toy model). POST-PUBLICATION UPDATE (June 14, 2026) — methodological correction and reinforcement, same record: Critical parser correction for (2+1) D CDT. The BFS-based spine extraction (valid in 2D, mean degree≈6) is NOT valid in (2+1) D CDT (mean degree≈23): spatial shortcuts within each time slice compress BFS distances to 9–10 shells regardless of T. The correct method uses the EXPLICIT vertex time coordinate exported by the simulator (Brunekreef et al. 2023), giving exactly T shells. New standalone module: GRUₚarsecdt3d. py. A. 25 re-validated and extended to 7 seeds (temporal parser): ds (spine) =1. 0322±0. 0231 (7/7 seeds pass). The original published value (1. 0489±0. 0287, 6 seeds, BFS spine) is VALIDATED — the 0. 012–0. 017 difference is within normal seed-to-seed variability, and the temporal parser additionally yields a lower std (0. 023 vs 0. 029) with no outliers. New script: GRUA25completo. py. T-scan (2+1) D corrected, P1 resolved in 3D: using independent thermalized samples (T=20, 40, 80) and the temporal spine: ds=1. 0428±0. 0183 (T=20), 1. 0147±0. 0174 (T=40), 0. 9857±0. 0170 (T=80). |ds−1| decreases monotonically with T (0. 043→0. 015→0. 014), the (2+1) D analog of the 2D convergence (A. 23, ds (T=320) =1. 006±0. 022). New script: GRUTscan3Dcorrecto. py. A. 37 corrected and extended — lax vs. strict GRU criterion introduced: with the temporal parser, CDT 3D real now passes 3/3 seeds (ds=1. 04–1. 08), vs. 1/3 with the invalid BFS spine. Across 8 topologies, a STRICT criterion (|ds−1|3σ) is introduced: only CDT (2D and 3D) passes. A 3-regular graph (LQG spin-network toy) passes the LAX criterion (ds=0. 929) but fails the strict one (separation 1. 2σ) — its near-unity value is a combinatorial property of 3-regular graphs, not evidence of causal structure. Updated abstract phrase: “ds (spine) ≈1 WITH significant separation is a signature of the causal foliation structure of CDT — in 2D and 3D — not a generic graph-theoretic artifact. ” New script: GRUA37ᵥ2correcto. py. A. 35 — LQG spin-network toy extended to N=500, 20 seeds: 5/20 seeds give ds≈1 (those with 12 BFS shells), 15/20 give ds≈2. 2 (11 shells) ; mean ds=1. 89±0. 52. The outcome depends on shell count, a combinatorial accident, confirming LQG 3-regular spin networks do NOT converge to ds (spine) ≈1, independently reinforcing A. 37. New script: GRUA35LQGᵣobusto. py. CONSOLIDATED RESULTS (v1. 9. 0–v1. 9. 2): Toy S¹ (A. 6): ds (λ=0) =1. 0007±0. 0321 vs ds (λ=1) =1. 9818±0. 1020 — 9. 2σ CDT 2D real (A. 21, 60 geometries): ds (spine) =1. 019±0. 015 — spine vs full: 15. 8σ T-scan 2D (A. 23): ds (T=320) =1. 006±0. 022 — P1 RESOLVED Spectral law (A. 24): λn≈n², error 4. 9% — P5 RESOLVED CDT (2+1) D (A. 25): ds=1. 0322±0. 0231 (7 seeds, temporal parser; see Post-Publication Update) Toy S³×ℝ (A. 26): ds=1. 0428±0. 0157 Robustness (A. 27): 236/240=98% across 240 configs Causal sets (A. 28): ds=1. 0640±0. 0368, 5/5 Holographic encoding (A. 29): Corrλ=0. 957±0. 003 Rotational invariance (A. 30): GRU 21. 1× more invariant than octants Node efficiency (A. 31): 8× fewer nodes Dimensional decomposition (A. 32): ds (full) ≈ds (spine) +ds (S¹) heuristic Three regimes (A. 33): N3σ) ; see Post-Publication Update LQG spin-network toy (A. 35): ds=1. 89±0. 52 (N=500, 20 seeds) — does not converge to 1; see Post-Publication Update KEY DISTINCTION: ds≈1 refers to the GRU SPINE (topological, 0/1 edges, extracted via the appropriate protocol — BFS in 2D, explicit vertex time in (2+1) D), not the full CDT graph nor the weighted spine. Full graph recovers ds→2 (CDT 2D standard). GRU refines, not contradicts, the consensus. VERSION HISTORY: v1. 0 (May 23, 2026): GRU Hypothesis and conceptual motivation in LQC/CDT. v1. 8. x: numerical consolidation on discrete S¹ (open chain→S¹ correction, N≥65 threshold, dimensional-flow validation). v1. 9. 0: integration with real CDT Monte Carlo (2D, 3D) and 4D toy. v1. 9. 1: external-critique battery (240-config robustness, causal sets, holography) and first CMB/LISA phenomenological connections. v1. 9. 2: structural tests A. 30–A. 33, methodological honesty note, documentation fixes. v1. 9. 3 (May/June 2026): validation battery (A. 29z zero-knowledge, Fisher LRT, cross-validation), k0-scan A. 34, lumpability/R̂ operator, spine topology confirmation, C. 1/C. 2 with real pipelines (CAMB + LISA SciRDv1). v1. 9. 3 post-publication update (June 14, 2026): (2+1) D parser correction, A. 25 extended to 7 seeds, T-scan 3D corrected (P1 resolved in 3D), A. 37 strict/lax criteria + 8 topologies, A. 35 extended to N=500/20 seeds. REPOSITORY CONTENT (v1. 9. 3, updated): GRUᵥ1₉₃web. html — complete HTML (A. 1–A. 34 + C. 1 + C. 2 + post-publication corrections) GRUᵥ1. 9. 3completo. pdf — complete PDF GRUᵥ1₉₃ₛhort. html / GRUᵥ1. 9. 3ₛhort. pdf — short paper (arXiv-ready) GRUᵥ1₉₃corrections₂0260614. html — standalone post-publication corrections addendum READMEINSTALLₐddendum. md — installation and replication notes, including the (2+1) D CDT spine parser correction NEW scripts (v1. 9. 3 original): GRUA29zₕolografiaZK. py — A. 29z zero-knowledge holography (Corr=0. 9662) GRUA21crossval. py — cross-validation legacy v1. 8. 9 vs protocol v4 GRUfisherₜest. py — Fisher likelihood ratio test (Λ=6406) GRUA34ₖ0scan₃dcdt. py — A. 34 k0-scan CDT 3D, validity window 3. 5, 5. 0 GRURhatconmutacionᵥ3. py — lumpability / R̂ operator analysis GRUₛpinedsₚonderado. py — spine topology vs weighted spine comparison GRUC1cmbₜoy. py — C. 1 CMB with CAMB pipeline GRUCMBLISAfinal. py — C. 2 LISA with SciRDv1 sensitivity, realistic thresholds GRUconfigᵥ4. py — centralized parameters (all scripts import from here) GRUA22ₗambdaₛcan. py — A. 22 λ-scan CDT 2D (reconstructed standalone) GRUA23Tₛcan. py — A. 23 T-scan CDT 2D (reconstructed standalone) NEW scripts (post-publication update, June 14, 2026): GRUₚarsecdt3d. py — standalone documented (2+1) D CDT parser (temporal spine) GRUA25completo. py — A. 25 with 7 seeds, temporal parser (ds=1. 0322±0. 0231) GRUTscan3Dcorrecto. py — corrected (2+1) D T-scan (T=20, 40, 80), P1 resolved in 3D GRUA37ᵥ2correcto. py — A. 37 corrected, lax/strict criteria, 8 topologies GRUA35LQGᵣobusto. py — A. 35 LQG spin-network toy, N=500, 20 seeds This list accounts for all 22 files in this record. See the version history above for the full incremental record across v1. 0–v1. 9. 3. Previous versions: v1. 9. 2 DOI 10. 5281/zenodo. 20650400 | v1. 9. 1 DOI 10. 5281/zenodo. 20635841 | v1. 9. 0 DOI 10. 5281/zenodo. 20574763DOI: 10. 5281/zenodo. 20674915
Alfredo Flores Cornejo (Fri,) studied this question.