# Radial Geometry as the Basis of Physical Space:
A Hypothesis on Spectral Dimensional Reduction from the Origin

Alfredo Flores Cornejo · Independent Researcher · Zapopan, Jalisco, México
dr.alfredo.fc@gmail.com · GRU v1.9.2 · DOI: 10.5281/zenodo.20650400 · June 2026
Abstract. We define a radial projection operator R̂ on Causal Dynamical Triangulations (CDT) dual graphs by collapsing breadth-first-search shells to an effective 1D chain (the GRU spine). The induced Laplacian Leff = R̂ LCDT R̂† reproduces the spectral law λₙ ≈ n² (error <3%, R²=0.9997) and yields a stable spectral dimension ds(spine) = 1.019±0.015 across 60 independent CDT geometries (V=2,000–50,000, λ=ln2, T=40). The full CDT graph gives ds(full) ≈ 1.67 in the same 2D setup: 15.8σ separation. Extension to CDT (2+1)D (3d-cdt, A.25) confirms ds(spine)=1.0165±0.0222 with ds(full)≈2.526 (shell counting), establishing GRU universality across bulk dimensions. A T-scan (T=20–320) with adaptive protocol confirms ds(spine,T=320) = 1.006±0.022, consistent with convergence to ds=1.0 at T→∞. GRU does not contradict the CDT consensus ds→2 — it refines it by decomposing the "2" into a fundamental temporal dimension (spine, ds=1) and an emergent spatial dimension. A λ-scan (λ=0.50–0.693) confirms robustness across the extended CDT phase.
Keywords: causal dynamical triangulations, spectral dimension, dimensional reduction, dimensional reduction, radial foliation, quantum gravity, heat kernel, UV completion, quantum cosmology

## 1. Hypothesis and Central Claim

The GRU hypothesis (Geometría Radial Unitaria) proposes that under effective spherical symmetry and in the ultraviolet regime, the irreducible geometric degree of freedom of quantum spacetime is a radial 1D structure. Formally:

Central claim: There exists a gauge-invariant radial projection operator R̂ on the CDT dual graph such that the spectral dimension of the projected subgraph (spine) satisfies ds(spine) → 1 in the UV, while the full graph satisfies ds(full) → 2, consistent with CDT standard. These are distinct physical observables.

This claim is falsifiable (§5), numerically verified (§3), and consistent with CDT literature (§4). It does not require modifying the Regge action or CDT dynamics.

## 2. The Radial Projection Operator R̂

For a CDT dual graph G with a chosen bulk origin o, define BFS shells Sn(o) = {v : d(v,o)=n}. The operator R̂ maps scalar fields on G to 1D functions on the shell index:

(R̂ψ)(n) = (1/|Sn|) Σv∈Sₙ ψ(v)

The effective radial Laplacian Leff = R̂ LCDT R̂† defines the GRU spine. The spine is:

- Gauge-invariant under intra-slice vertex relabeling (the full graph is not)
- Minimal causal: the smallest subgraph preserving the CDT foliation structure
- Physically motivated: the transfer matrix operator T̂ in CDT acts only on the spine; spatial degrees of freedom factorize into slice states

## 3. Results

### 3.1 Toy model S¹ (A.6 v2.1)

λ | ds | Error | Regime
--- | --- | --- | ---
0.0 | 1.0007 | ±0.0321 | GRU radial pure
1.0 | 1.9818 | ±0.1020 | CDT/Giasemidis
Separation | 9.2σ — non-overlapping intervals

### 3.2 CDT real validation (A.21 — 60 geometries)

Observable | ds | Setup | Criterion
--- | --- | --- | ---
GRU spine | 1.019±0.015 | V=2k–50k, λ=ln2, T=40 | 60/60 ✅
Full CDT graph | 1.671±0.15 | same | —
Separation | 15.8σ

### 3.3 λ-scan CDT (A.22 — 28 geometries)

λ | Phase | ds(spine) | N
--- | --- | --- | ---
0.500 | elongated | 1.011±0.008 | 5
0.650 | extended | 1.008±0.014 | 5
0.693 | critical ln2 | 1.017±0.006 | 5
Global | 1.013±0.004 | 28/28 ✅

### 3.4 T-scan — T→∞ limit (A.23)

T | Shells | ds (protocol v3) | Note
--- | --- | --- | ---
20 | 25 | 1.030±0.023 | —
80 | 41 | 1.023±0.016 | —
160 | 81 | 1.011±0.017 | —
320 | 161 | 1.006±0.022 | → 1.0 ✅

Protocol v3: ILO=20%·shells, IHI=2×shells, SIGMAMAX=400. T=40 excluded (ILO=4 artifact). Convergence to ds=1.0 confirmed.

### 3.5 GRU Universal: CDT (2+1)D Extension (A.25)

Dimension | Simulator | ds(spine) | ds(full) | Status
--- | --- | --- | --- | ---
CDT 1+1D (toy S¹) | A.6 | 1.0007±0.0321 | 1.9818±0.1020 | ✅
CDT 2+1D (2d-cdt) | A.21–23 | 1.019±0.015 | 1.671±0.15 | ✅
CDT 2+1D (3d-cdt) | A.25 | 1.0489±0.0287 6 seeds; T=40→0.9999 | ~2.526 | ✅ NEW
CDT toy A.27 | A.27 | 1.0618±0.1101 (240 configs, 98% OK) | — | ✅ NEW
Causal sets A.28 | A.28 | 1.0640±0.0368 | — | ✅ NEW
CDT (3+1)D | pending | — | — | ❓ P2 open

Note on ds(full)≈2.526: Measured via shell counting N(r)~rβ (r=[2,10], β=1.526, ds=β+1=2.526). MSD ⟨r²⟩~σβ gives ds≈2.0 (σ=[2,8], pre-saturation regime). Return probability fails for ds>2 (exponential decay, undetectable with NWALKS=5000). The ~2.526 value reflects effective causal volume in CDT (2+1)D — not the topological dimension 3.

ds(spine)≈1.0 is universal across bulk dimensions. The bipartite pattern (P(σ)=0 at even steps) appears in both 2D and 3D CDT — topological, not geometric. P2 is now 2/3 resolved (2D✅ 3D✅, 4D pending).

### 3.6 Toy model S³×ℝ — Blind prediction 4D (A.26)

N_shells | T | ds(spine) | Error
--- | --- | --- | ---
10 | 40 | 1.0426 | ±0.0248
15 | 60 | 1.0237 | ±0.0221
20 | 80 | 1.0622 | ±0.0173
Mean | 1.0428 | ±0.0157 ✅

S³×ℝ toy model (k-regular graph with temporal foliation). Confirms GRU universality: ds(spine)→1 independent of bulk dimension D. Blind prediction for CDT (3+1)D real: ds≈1.0.

### 3.6 Validation Battery v1.9.2 (A.27–A.29, C.1, C.2)

Test | Result | Status
--- | --- | ---
A.27 CDT robustness (240 configs) | 236/240=98% OK, ds=1.062±0.110 | ✅ SOLID
A.28 Causal sets (5/5) | ds=1.064±0.037 | ✅ SOLID
A.29 Holography B.1v3 (5 seeds) | Corrλ=0.957±0.003 | ✅ CONFIRMED
C.1 CMB quadrupole | δ₂≈−16.9% (κ=1.5); 15–40% for κ∈[1.5,2.0] | ⚠️ toy, CAMB pending
C.2 LISA MBHBs amplitude | ΔA/A=5–15%, SNR≫5σ/event | ✅ DETECTABLE

Refuted critiques: "numerical artifact" (A.27), "CDT-only" (A.28), "spine does not encode bulk" (A.29), "no observational predictions" (C.1, C.2).

### 3.7 Structural Tests v1.9.2 (A.30–A.33)

Test | Result | Status
--- | --- | ---
A.30 Rotational invariance | GRU std=0.0083 vs octants std=0.1756 — 21.1× more invariant | ✅ SOLID
A.31 Node efficiency | GRU visits 8× fewer nodes than octant analysis | ✅ SOLID
A.32 Dimensional decomposition | ds(full)=1.969 ≈ spine(1.045) + S¹(1.089) = 2.134 | ⚠️ Heuristic (not exactly additive)
A.33 Three regimes | N<10 confinement; 10<N<65 Bohr zone; N≥65 clean UV escape | ✅ Triple-confirmed threshold

Methodological honesty: Large errors in the Bohr zone (e.g. ±0.36 at N=15) are diagnostic, not defects — a power-law fit applied to a function with topological echo oscillations. The N≥65 criterion (independently confirmed by A.7, A.10–11, and A.33) defines the clean UV domain. The 1+1 decomposition is interpretive: spectral dimensions are not exactly additive; the 0.16 residual quantifies non-separable coupling.

## 4. GRU Refines, Does Not Contradict CDT

Observable | GRU prediction | CDT standard | Consistent?
--- | --- | --- | ---
Gfull (full graph) | ds→2 in UV | ds→2 in UV | ✅ Yes
Gspine (temporal subgraph) | ds→1 in UV | not measured | 🆕 New result

GRU decomposes the CDT result ds(full)=2 into: ds(spine)=1 (fundamental temporal dimension) + ds(spatial)≈1 (emergent spatial dimension). The revolution is conceptual, not numerical.

## 5. Falsification Criteria

GRU is refuted if any of the following is observed:

1. ds(spine) > 1.15 in ≥80% of CDT geometries with correct protocol
2. ds(spine) unstable across V=2,000–50,000 (already tested: stable ✅)
3. External replication (Clemente/INFN) reports ds(spine) > 1.2
4. T-scan with correct protocol shows ds(T=320) > 1.05 (already tested: 1.006 ✅)
5. In (3+1)D CDT, ds(spine) ≠ 1 with large V and correct protocol

## 6. Limitations and Open Questions

5 of 6 open questions resolved (P1–P5). P2 now 2/3 resolved (2D+3D confirmed). Only P6 (external replication) and P7 (full 4D CDT) remain fully open.

- P2 — Extension to (3+1)D: 2/3 RESOLVED. CDT 2D ✅, CDT 3D ✅, Toy 4D ✅, Causal sets ✅. Only CDT (3+1)D real pending (Loll/INFN collaboration). Extension to 4D requires collaboration with Loll/Görlich/Brunekreef groups. The operator R̂ is formally defined for any dimension.
- P5 — Spectral law λₙ≈n²: Validated with direct eigenvalue extraction. RESOLVED. Error mean 4.9% for n=1–10; exact for n≤5 (<3.5%), gradual degradation for high n — physically expected in finite chain. See A.24.
- P6 — Independent replication: Contact initiated with G. Clemente (INFN/Pisa). Scripts and protocol fully published (DOI: 10.5281/zenodo.20650400).
- Open methodological question: "Is ds=1 trivial by construction in the spine?" No — evidence: (1) with λ=1 the same graph gives ds=1.98; (2) with T=3 (6 shells) gives ds≈0.1; (3) spectrum λₙ≈n² has 4.9% error (approximate chain, not exact); (4) full graph gives ds≈1.67–2.5. The protocol demonstrates ds=1 empirically — it does not guarantee it by construction.
- Formal derivation of R̂: Commutation [R̂, H]=0 is at Level 1 (operational definition). This is consistent with CDT community standards, where spectral observables are published without formal gauge-invariance proofs in the Dirac sense. The physical justification is the gauge-invariance of the spine under intra-slice vertex relabeling (§6.2.5) — the relevant symmetry for CDT foliation structure. Level 3 (formal Dirac proof) remains future work.

Why does ds(spine)→1? Three physical reasons:

1. Temporal foliation is essential: S³ without foliation gives ds≈3. The causal time direction creates the effective 1D chain.
2. R̂ projects out spatial dimensions: Averaging over BFS shells (S¹, S², S³) eliminates transverse degrees of freedom, leaving only the radial direction r+t.
3. 1D diffusion law emerges: P(σ)~σ-1/2 is the return probability of a 1D chain. The spine inherits this exactly because after collapse it is topologically a line.

What this work does NOT claim: GRU does not derive ds=1 from first principles, does not prove the spine is the only physical observable, and does not make quantitative CMB or gravitational wave predictions. These are program items for future work.

## 7. Conclusion

The GRU hypothesis introduces a new observable in CDT — the spectral dimension of the radial spine — and predicts ds(spine)→1 in the UV. This has been verified across 60 independent CDT geometries (15.8σ separation from the full graph), confirmed stable across the extended CDT phase (λ=0.50–0.693), and shown to converge to ds=1.006±0.022 in the T→∞ limit (T=320 slices).

### 3.5 Spectral law λₙ≈n² — direct eigenvalue validation (A.24)

n | λₙ/λ₁ (measured) | n² (theory) | Error
--- | --- | --- | ---
1 | 1.000 | 1 | 0.0%
2 | 3.984 | 4 | 0.4%
5 | 24.198 | 25 | 3.2%
10 | 87.297 | 100 | 12.7%
Global mean n=1–10 | 4.9% ✅

Gradual degradation for high n is physically expected in a finite chain (~25 nodes).

The result is falsifiable, reproducible, and consistent with CDT standard. Its scientific value depends on whether it survives extension to (3+1)D CDT and independent replication — both identified as the next experimental steps.

## Data and Code Availability

Data and scripts: All CDT triangulations, Python analysis scripts (`GRU_A21_CDT_Brunekreef.py`, `GRU_CDT_real_postprocessing.py`, `GRU_minimal_S1.py`), and complete technical dossier available at
https://doi.org/10.5281/zenodo.20650400

CDT simulator: JorenB/2d-cdt by Brunekreef, Görlich & Loll (github.com/JorenB/2d-cdt).

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