Project: Relational Time Geometry (RTG) / Emergent Simplicial Manifolds (ESM)
Version: 12.2 (Freeze + Reproducibility + Cosmology Clarification)
Date: January 14, 2026
Status: Core static/algebraic framework validated and frozen | Cosmology engine functional (mass2 threshold confirmed; signed model mismatch documented)
Contents
- 1. Executive Summary
- 2. Core Principle: Chronological Consistency
- 3. Vacuum Structure & Gauge Emergence
- 4. Topological Matter (β1) and Artifact Control
- 5. Proto-Gravity Probe (Multi-Channel, β1=8)
- 6. Cosmological Dynamics (Drift Engine)
- 7. Reproducibility Guide
- 8. Theoretical Significance
- 9. Lessons from Failed Approaches
- 10. Open Questions & Next Steps
- Appendix A. Change Log
1. Executive Summary
We have numerically validated a geometric framework where gauge screening, topological matter, hidden non-Abelian curvature (κ3),
and proto-gravitational attraction emerge from a frustrated simplicial vacuum. These results are reproducible on a grown cluster (N=175, β1=8)
and on geometric baselines (face-centered cubic, FCC) that represent a “hot foam” frustration floor.
- ✓ PROVEN: Universal gauge screening (structure equation holds at ≈10^-11–10^-12 relative error on grown and FCC geometries).
- ✓ PROVEN: Vacuum cooling (η_irr ≈ 0.75–0.80 “hot foam floor” on FCC → η_irr ≈ 0.403 on grown cluster).
- ✓ PROVEN: Topological matter (β1 = 8 on grown cluster with a clean spectral gap ≈10^9; FCC β1 values are artifactual with no gap).
- ✓ PROVEN: Proto-gravity in the probe (attractive harmonic interaction; quadratic amplitude law E_int,h = κ·φ^2 with κ ≈ −0.175; vacuum and κ3 channels ~10^-14–10^-15).
- ⚠ PARTIAL: Cosmology drift (expansion + gravity engine is stable; mass2 coupling shows a reproducible binding threshold near G≈50; signed coupling is unstable/mismatched to probe physics).
1.1 Freeze Decision
We stop additional cosmology parameter trials at this stage and document what is reproducible and what is not.
The static/algebraic layer (Hodge + screening + β1 + κ3 + probe gravity) is ready for a first paper.
Cosmology is documented as a toy dynamics engine with one phenomenological coupling (mass2) and one currently invalid coupling (signed).
1.1.1 Why Signed Cosmology Differs from the Probe
The issue: The gravity probe measures attraction between correlated defects (high harmonic overlap; e.g., cosine similarity ≈ +0.3489)
with opposite signs chosen deliberately to produce destructive interference in the harmonic sector. This is best understood as a
mode-space overlap interaction (within the β1-dimensional harmonic subspace).
What cosmology currently implements: signed coupling uses a scalar “charge” on each selected edge,
q_i = z_h[i], and applies it to all pairwise forces in real space:
- Like-sign pairs: attract (++ or –)
- Unlike-sign pairs: repel (+-)
- Net effect: strong cancellations and mode-mixing because z_h contains mixed signs from multiple β1=8 modes
Why this mismatches the probe:
- The probe effectively couples defects within aligned harmonic structure (mode-space overlap selection).
- The cosmology signed model couples all edge pairs regardless of which harmonic modes they project onto (real-space mixing).
- Mixing orthogonal modes causes destructive interference and/or repulsive substructures; at high G this can drive rapid deconcentration (“blow-up”).
Correct signed implementation would require:
- Decompose harmonic content into β1 mode coefficients: z_h = Σ_j c_j φ_j (j=1..β1).
- Use mode-space charges/vectors and couple via overlap: F_AB ∝ Σ_j (c_j^A · c_j^B).
- This is conceptually different and computationally heavier than scalar edge charges; it is not implemented yet.
Freeze rationale: We do not proceed to implement unvalidated mode-space cosmology in this phase.
Instead, we record that mass2 provides a phenomenological “universal attraction” toy model that demonstrates a clear clustering threshold (G≈50+).
1.2 Gravity Interaction Strength Reporting (Normalization Clarification)
The same proto-gravity physics appears in three common reporting formats:
- Absolute interaction energy: E_int,h = −0.7008 at φ=2.0 (probe measurement).
- Relative interaction strength (canonical for v5.3): −14.15% = E_int / (|dE_A| + |dE_B|) at φ=2.0 (normalizes by injected single-defect energies).
- Coupling constant (canonical parameter): κ = −0.175 from the fit E_int,h = κ·φ^2 (fundamental measured parameter).
Earlier probe variants reported values like “−83%” due to using a different baseline normalization (e.g., comparing against a different single-defect reference or geometric mean).
The underlying physics (attractive harmonic coupling with quadratic scaling) is consistent; only the normalization changed.
Use κ = −0.175 as the canonical number.
1.3 Cosmology Phase Diagram (mass2 coupling)
The cosmology drift sweeps start from the same baseline (conc0 ≈ 19.0%, normRg0 ≈ 4.282) and measure final concentration and radius of gyration.
Rg and normRg are in arbitrary embedding units (dimensionless for now; no physical unit mapping is claimed).
| Mode | G | Cutoff set | conc0 → conc1 | normRg0 → normRg1 | Status | Verdict |
|---|---|---|---|---|---|---|
| mass2 | 5 | 0.1 / 0.05 / 0.01 / auto:q=0.9 / auto:q=0.8 | 19.0% → 3.8% | 4.282 → 5.006–5.022 | FREE | WEAK/FREE |
| mass2 | 20 | 0.1 / 0.05 / 0.01 / auto:q=0.9 / auto:q=0.8 | 19.0% → 3.7–3.8% | 4.282 → 4.476–4.487 | MARGINAL | BALANCED |
| mass2 | 50 | 0.1 / 0.05 / 0.01 / auto:q=0.9 / auto:q=0.8 | 19.0% → 21.7% | 4.282 → 4.020–4.035 | BOUND++ | MODERATE CLUSTERING |
| mass2 | 100 | 0.1 / 0.05 / 0.01 / auto:q=0.9 / auto:q=0.8 | 19.0% → 21.2–21.5% | 4.282 → 3.953–3.965 | BOUND++ | MODERATE CLUSTERING |
| mass2 | 200 | 0.1 / 0.05 / 0.01 / auto:q=0.9 / auto:q=0.8 | 19.0% → 20.9–21.4% | 4.282 → 3.813–3.856 | BOUND / BOUND++ | MODERATE CLUSTERING |
Conclusion: mass2 coupling exhibits a clear binding threshold near G≈50 under this initialization.
Across the tested cutoffs and initialization relaxation iterations (initI = 100/200/500), results are nearly unchanged, indicating that in the current model
gravity strength dominates over cutoff selection in this range.
1.4 Publication Readiness Assessment
✓ Ready for Publication (Tier 1)
- Gauge screening theorem and universality (Sections 2–3)
- Vacuum cooling benchmark (FCC hot foam floor vs grown cooled vacuum) (Section 3)
- Topological matter identification (β1=8 with ≈10^9 spectral gap) (Section 4)
- Proto-gravity probe on β1=8 with multi-channel neutrality and quadratic fit κ≈−0.175 (Section 5)
- κ3 dark state (holonomy deficit) measurement and flux decorrelation (Section 3.4)
⚠ Needs Additional Work (Tier 2)
- Distance-dependent gravity: measure E_int(r) as a function of separation/overlap, not only a single overlap-selected pair.
- Handle localization: map the 8 harmonic eigenvectors spatially; measure localization length scales.
- κ3 vs discrete Ricci: compute Forman or Ollivier Ricci curvature and test correlation with Δ (holonomy deficit).
❌ Not Ready (Tier 3)
- Cosmological structure formation as physics: current mass2 is phenomenological; signed is mismatched. Cosmology must be reframed as “toy model” until a probe-faithful mode-space coupling is implemented and validated.
- Continuum limit: systematic N-scaling and renormalization are not studied.
- Quantum corrections: current framework is classical on a fixed simplicial complex.
1.5 Core Equations (Quick Reference)
| Observable | Equation | Measured / Validated |
|---|---|---|
| Time-delay frustration | z = ln(r_em / (s r_geo)) | Computed on edges from geometry + frequency map |
| Hodge decomposition | z = z_exact + z_coexact + z_harmonic | Exact (by construction/solver) |
| Gauge screening | D1 B = D1 z_coexact | Error ≈ 10^-11–10^-12 |
| Irreducible fraction | η_irr = (Var(z_co) + Var(z_h)) / Var(z) | ≈0.75–0.80 (FCC), ≈0.403 (grown) |
| Holonomy deficit (κ3) | Δ_f = Re(Tr(H_f))/2 − cos(F_target/2) | Δ_RMS ≈ 0.52 (grown) |
| Proto-gravity coupling | E_int,h = κ · φ^2 | κ ≈ −0.175 (probe fit) |
| Spectral protection | λ_9 / λ_8 | ≈10^9 (β1=8 gap) |
2. Core Principle: Chronological Consistency
In RTG, proper time is relational: dτ ≈ dφ / ω. On a frustrated simplicial network, embedding distances r_geo
mismatch the frequency-derived distances r_em, producing edge time-delay frustration z. Closed loops would accumulate
path-dependent Δτ (“time warp”) unless a compensator field emerges.
Compensator field B: restores loop consistency by canceling coexact curvature while preserving harmonic topology.
3. Vacuum Structure & Gauge Emergence
3.1 Hot foam floor vs cooled vacuum
We benchmarked the grown cluster against a geometric “hot foam” baseline: a face-centered cubic (FCC) lattice with tetrahedral frustration.
| System | N | η_irr | Screening error | β1 | Interpretation |
|---|---|---|---|---|---|
| FCC tetra baseline | 14 | 0.769 | 5.2×10^-12 | 30 (no gap) | Hot foam floor (generic geometric frustration) |
| FCC tetra baseline | 140 | 0.798 | 3.3×10^-11 | 197 (no gap) | |
| FCC tetra baseline | 1400 | 0.764 | 4.7×10^-11 | 104 (no gap) | |
| Grown cluster | 175 | 0.403 | 7.7×10^-12 | 8 (gap ≈10^9) | Cooled vacuum (organized ground state) |
Conclusion: η_irr ≈ 0.75–0.80 is the empirical hot-foam frustration floor for unoptimized geometric baselines.
The growth algorithm produces a cooled vacuum with η_irr ≈ 0.403 (≈46% reduction relative to this floor).
3.2 Screening theorem (universal)
Validated identity: z_total − z_exact − B ≡ z_harmonic, with B solving the screening equation D1B = D1 z_coexact.
- Structure error ≈ 10^-11–10^-12 (grown and FCC baselines)
- Interpretation: gauge screening is substrate-independent (hot or cooled)
3.3 Vacuum composition (example audit fractions)
| Component | Grown (example) | FCC (typical) | Physical meaning |
|---|---|---|---|
| Exact | ≈60.4% | ≈20–24% | Redefinable / gauge-like |
| Coexact | ≈36.1% | ≈60–75% | Vacuum foam (screened) |
| Harmonic | ≈4.1% | ≈1–20% | Topological matter (unscreenable) |
Linear screening removes Abelian curvature, but SU(2) holonomy contains commutator curvature that remains as a holonomy deficit Δ.
- Grown: Δ_RMS ≈ 0.52
- FCC: Δ_RMS ≈ 0.65–0.78
- Decorrelated from flux magnitude: corr(|F|,|Δ|) ≈ −0.016 (grown), ≈ −0.28 (FCC sample)
Interpretation: The vacuum can appear Abelian-flat (screened) while remaining non-Abelian-stiff (κ3 dark state).
4. Topological Matter (β1) and Artifact Control
4.1 β1=8 on the grown cluster
The 1-form Laplacian spectrum shows 8 near-zero eigenvalues (~10^-15) separated by a ≈10^9 gap from the first massive mode.
| Modes | Eigenvalue scale | Interpretation |
|---|---|---|
| 1–8 | ≈10^-15 | Harmonic (topological matter modes) |
| 9 | ≈5×10^-3 | First massive mode (gap ≈10^9) |
4.2 FCC topology is artifactual
FCC baselines show large β1 counts (30 / 197 / 104) with no spectral gap and strong dependence on sampling and face construction.
This demonstrates that “matter topology” is only meaningful on manifold-like, optimized geometries (the grown cluster).
5. Proto-Gravity Probe (Multi-Channel, β1=8)
5.1 Probe design
- Select two edges with strong harmonic overlap and opposite injection signs (anti-phase).
- Sweep amplitudes φ ∈ {0.25, 0.5, 1.0, 2.0}.
- Measure interaction energy decomposed into harmonic (matter), vacuum (coexact), and κ3 channels.
5.2 Results (canonical table)
| Amplitude φ | E_int,h (harmonic) | Relative % | E_int,v (vacuum) | E_int,κ3 | Verdict |
|---|---|---|---|---|---|
| 0.25 | −1.095×10^-2 | −2.74% | −2.84×10^-14 | −7.11×10^-15 | ATTRACTION (all amplitudes) |
| 0.50 | −4.380×10^-2 | −5.09% | −2.84×10^-14 | 0 | |
| 1.00 | −1.752×10^-1 | −8.88% | +2.84×10^-14 | −7.11×10^-15 | |
| 2.00 | −7.008×10^-1 | −14.15% | −5.68×10^-14 | −1.42×10^-14 |
5.3 Quadratic coupling law
Fit across all amplitudes:
E_int,h = κ · φ^2 with κ ≈ −0.175
Key discovery: Vacuum and κ3 channels remain at machine scale (~10^-14–10^-15) across the sweep.
This is evidence (in this probe setting) that vacuum energy is screened from gravitational coupling, and κ3 is not the mediator of the measured attraction.
6. Cosmological Dynamics (Drift Engine)
6.1 What the cosmology engine does
We simulate an expanding target geometry with scale a(t): 1 → 2 over 20 steps, relaxing node coordinates each frame.
We track Rg, normRg, a spatial concentration metric, virial diagnostics, and force decomposition (spring vs gravity).
6.2 Critical bugs fixed (must be present)
- Self-force bug: diagonal terms in pairwise force matrix created spurious self-forces. Fix: set diagonal distances to infinity.
- Center-of-mass (COM) drift: without recentering each step, the cluster drifts and inflates Rg artificially. Fix: subtract COM each relaxation step.
- Frozen harmonic content: earlier versions could keep z_h effectively fixed, making percentile-style concentration misleading. Fix: recompute Hodge/harmonic content each step and use a spatial concentration metric.
6.3 mass2 vs signed: what we observed
- mass2 (q = z_h^2): always-attractive phenomenological gravity; shows a clear transition:
FREE at G=5, MARGINAL around G=20, BOUND++ around G≥50 for the same initial conditions. - signed (q = z_h): phase-dependent attraction/repulsion; in the current real-space formulation it cancels/mixes modes and becomes unstable at high G.
6.4 Signed blow-up runs (documented examples)
From v6 outputs using --gravity-cutoff auto:q=0.9 (top 10% |z_h| edges; cutoff_used ≈ 0.0890; matter_edges=36/356 at baseline):
- signed, G=100: Final normRg ≈ 8.4866 (init 4.2820); conc 19.0% → 0.2%; Status FREE; Verdict WEAK/FREE.
- signed, G=200: Final normRg ≈ 12.2889 (init 4.2820); conc 19.0% → 0.0%; Status FREE; Verdict WEAK/FREE.
Interpretation: These runs do not indicate “stronger gravity”; they indicate that the signed model is not probe-faithful and can create net deconfinement under expansion.
6.5 v3 “success” caveat
Earlier v3 “binding” results likely reflected (i) a more compact initialization and (ii) a concentration metric artifact.
We do not treat v3 as evidence of robust clustering emergence from diffuse initial conditions.
7. Reproducibility Guide
All results are produced by scripts in the RTG repository. Default seed is typically 42 (verify by reading script headers or CLI output).
The canonical grown dataset used here is:
runs/locked_dipole_0/grown_cluster_v12_2_gauge_nobreak_auditpack.npz
7.1 Core validation (expected to work deterministically)
# Gauge + Hodge + κ3 audit on grown cluster python esm_gauge_audit_v12_1_kappa3.py runs/locked_dipole_0/grown_cluster_v12_2_gauge_nobreak_auditpack.npz # FCC hot-foam benchmark (example sizes) python esm_fcc_tetra_auditpack.py --n-nodes 14 --face-mode delaunay --seed 42 --run-audit python esm_fcc_tetra_auditpack.py --n-nodes 140 --face-mode delaunay --seed 42 --run-audit python esm_fcc_tetra_auditpack.py --n-nodes 1400 --face-mode delaunay --seed 42 --run-audit # Proto-gravity probe (β1=8, amplitude sweep) python gravity_probe_genus8_v5_3.py --npz runs/locked_dipole_0/grown_cluster_v12_2_gauge_nobreak_auditpack.npz --amps 0.25 0.5 1.0 2.0 --vac-mode raw_curl
7.2 Cosmology (parameter-sensitive; use for replication of sweeps)
# mass2 sweep exemplars python cosmology_drift_v6.py --gravity-mode mass2 --gravity-g 5 python cosmology_drift_v6.py --gravity-mode mass2 --gravity-g 20 python cosmology_drift_v6.py --gravity-mode mass2 --gravity-g 50 python cosmology_drift_v6.py --gravity-mode mass2 --gravity-g 100 python cosmology_drift_v6.py --gravity-mode mass2 --gravity-g 200 # signed blow-up exemplars (documented) python cosmology_drift_v6.py --gravity-mode signed --gravity-g 100 --gravity-cutoff auto:q=0.9 python cosmology_drift_v6.py --gravity-mode signed --gravity-g 200 --gravity-cutoff auto:q=0.9
7.3 Expected headline outputs (tolerances)
- Screening (grown): structure error ≈ 10^-11–10^-12.
- η_irr floor (FCC): ≈ 0.75–0.80.
- η_irr (grown): ≈ 0.403.
- β1 (grown): 8 with ≈10^9 spectral gap.
- Probe gravity: κ ≈ −0.175; E_int,h(φ=2) ≈ −0.70; vacuum and κ3 channels ~10^-14–10^-15.
- Cosmology mass2 threshold: FREE at G=5, MARGINAL at G=20, BOUND++ at G≥50 for this initialization.
8. Theoretical Significance
8.1 Why this matters
1) Gauge symmetry is emergent
Gauge screening (B) arises automatically from the requirement of chronological consistency on frustrated geometry.
It works on both “hot foam” and “cooled vacuum”, indicating gauge symmetry is a consequence of geometric constraints rather than an input axiom.
2) A toy mechanism for cosmological constant screening
The probe shows the vacuum (coexact) sector contributes ≈10^-14 to interaction energy while still carrying frustration energy.
This demonstrates a concrete toy mechanism where geometric vacuum energy can exist without gravitating (in the tested setting).
3) Matter is topologically protected
β1=8 harmonic modes cannot be screened and persist with a ≈10^9 spectral gap on the grown cluster.
This provides a geometric reason matter-like degrees of freedom are not gauge-removable: they are locked by topology.
4) Gravity emerges as harmonic stress minimization
The quadratic coupling E_int,h ∝ −φ^2 with vacuum neutrality indicates attraction emerges from harmonic interference (stress reduction).
This aligns with thermodynamic/entropic gravity ideas, but here it is measured directly on an explicit network model.
8.2 What this framework cannot yet explain
- Why β1=8 specifically (no Standard Model mapping yet).
- Connection to SU(3)×SU(2)×U(1) in a continuum limit.
- Fermionic statistics and spin-1/2 emergence.
- Lorentz invariance (whether it emerges approximately in the cooled vacuum).
- Quantum dynamics (current framework is classical).
9. Lessons from Failed Approaches
9.1 What we tried that did not work (or was misleading)
| Approach | Expectation | Outcome | Lesson |
|---|---|---|---|
| Signed coupling (real-space scalar q=z_h) | Probe-like phase dependence | Strong cancellations/mode mixing; high-G blow-up; conc → 0% | Need mode-space overlap coupling, not scalar edge charges |
| Changing cutoff (0.1 → 0.01; auto:q=0.8/0.9) | Increase “matter participation” and rescue clustering at low G | At G=5, still FREE; at G≥50, clustering appears regardless of cutoff | In current model, G dominates over cutoff choice in this range |
| Increasing init relaxation iterations (initI=100→500) | Start more compact and increase binding | Minimal change in outcomes in sweep summaries | Initializer does not materially change harmonic spatial distribution under these settings |
| Percentile-only concentration metrics (older variants) | Track clustering reliably | Can be insensitive / misleading if selection is definitional | Use spatial concentration metric tied to geometry and COM |
| Interpreting v3 “BOUND” as emergence from diffuse states | Claim structure formation at low G | Likely compact init + metric artifact | Do not use v3 as robust cosmology evidence |
9.2 Critical implementation fixes (historical)
- Self-force removal (diagonal pair terms)
- Center-of-mass recentering each relaxation step
- Dynamic recomputation of Hodge/harmonic content in drift
- Spatial concentration metric (geometry-dependent)
10. Open Questions & Next Steps
- Distance-dependent gravity: measure E_int as a function of separation/overlap and test whether an effective law emerges.
- Harmonic mode localization: map eigenvectors for β1=8; quantify localization length.
- κ3 geometry interpretation: test correlation with discrete Ricci curvature measures.
- Probe-faithful cosmology: implement mode-space coupling (vector charges in β1 space), validate against probe, then revisit cosmology.
- Scaling/continuum: N-scaling study to identify stable observables and possible continuum limit.
Appendix A. Change Log
- v12.2: Added signed mismatch explanation; added mass2 phase diagram table; added publication readiness tiers; added core equations quick reference; clarified gravity normalization (−83% vs −14.15% vs κ); expanded reproducibility commands; added theoretical significance and failure analysis; removed image placeholders.
- v12.1: Introduced freeze decision; incorporated mass2/signed sweeps; documented signed blow-up runs; tightened claims around cosmology.
- v12.0: Consolidated core validation results (screening, η_irr baseline, β1=8, κ3, probe gravity) and early cosmology lessons (bugs/metrics).
End of document.