UNNS Empirical Testing Laboratory v0.4.2

🎯 Overview

The Unbounded Nested Number Sequences (UNNS) Substrate proposes that physical constants emerge from recursive geometric dynamics in an information-theoretic substrate. This laboratory provides publication-grade empirical tests of core UNNS predictions through deterministic simulations and statistical validation.

🔬 Theoretical Framework

Recursive Field Evolution

The field Φ evolves through recursive operator G, incorporating gradient memory (∇Φₙ₋₁). This creates path-dependent dynamics where history influences future states.

Curvature Energy Metric

Hᵣ quantifies the "roughness" or information content of the recursive trajectory. Its ratio η(n) = Hᵣ(n+1)/Hᵣ(n) → 1 at equilibrium signals emergent constants.

τ-Phase Invariant

The τ-ratio convergence to τ* across diverse initial conditions demonstrates seed-independence, a hallmark of emergent physical law.

🧪 Experiment Descriptions

Experiment 1: τ-Convergence (Fixed-Point Invariant)

Tests whether τₙ = Φₙ₊₁/Φₙ converges to a stable τ* across 50-500 random seeds. Validates seed-independence and measures η(n) convergence per Equation 9 in the UNNS monograph. Pass criteria: RSD < 5%, convergence detected in tail analysis.

Experiment 2: β-Flow (Depth-Renormalization)

Explores how recursion depth acts as a renormalization parameter, testing whether coupling γ flows to a fixed point γ* consistent with cosmological parameters Ω_m and Ω_Λ. Analogous to RG flows in QFT.

Experiment 3: MCMC Efficiency (ESS_κ)

Compares sampling efficiency between classical Random Walk Metropolis (RWM) and curvature-aware proposals (τ-on-RHMC, Klein-flip). Tests if substrate geometry improves statistical exploration via ESS_κ metric.

Experiment 4: τ-Phase Robustness

Sweeps initial phases φ ∈ [0,1] and dimensions D ∈ {1,2,3,4} to test dimensional preference and phase universality. Uses Rayleigh circular statistics to validate phase alignment. Hypothesis: D=3 maximizes stability.

Experiment 5: Rees Constant Emulation

Estimates all six of Martin Rees's "Just Six Numbers" (N, ε, Ω, λ, Q, D) from recursive curvature statistics. Tests if substrate dynamics naturally produce observed cosmological ratios. Target: log error < 1.0.

Experiment 6: Curvature Fixed-Point (Equation 6)

Direct test of paper Eq. 6: "Dimensionless constants arise when Rₙ₊₁/Rₙ → 1" where R = ∇²Φ is discrete curvature. Validates equilibrium hypothesis across multiple seeds. Pass: >80% seeds reach equilibrium.

Experiment 7: Physical Constant Predictions (Phase 3)

Core breakthrough experiment. Maps ensemble-averaged η equilibria to real physical constants:

• α (fine structure): Three complementary modes (see below)
• μ (mass ratio): g(R_rms) ≈ 1836
• Λl²ₚ (cosmological): h(∂η/∂n) ≈ 10⁻¹²²

Uses calibration/validation splits and reports confidence intervals. This is the laboratory's primary falsifiable prediction.

α Fine-Structure Constant: Three-Mode Framework

The fine-structure constant α ≈ 1/137 is notoriously difficult to derive from first principles. UNNS provides three complementary approaches, each with different physics assumptions:

This is the pure UNNS prediction. k_α is a single calibration factor derived from Phase-2 invariants (not fitted to α itself).

N_f = effective charged species contributing. This removes scale-matching bias and tests whether UNNS predictions respect renormalization group flow.

The σ_prior slider controls "how much we trust UNNS vs. experiment." This makes the contribution of theoretical prediction vs. empirical data completely auditable.

Honest Science Guidelines:

• k_α is calibrated once from Phase-2 invariants, not fitted to α
• Default Hybrid mode with σ_prior = 10⁻⁴ balances theory and data
• All three modes shown in exports for reproducibility
• PASS* flag if |Δα|/α < 2% without over-tuning (σ_prior ≥ 10⁻⁴)
• Details panel shows τ_R, q, η, all intermediate values

📊 Interpreting Results

Statistical Rigor

• Green "PASS": Metric within theoretical tolerance (e.g., log error < 1)
• Red "FAIL": Prediction outside acceptable bounds - challenges hypothesis
• Effect Sizes: Cohen's d reported for convergence tests (d > 0.8 = strong)
• p-values: Rayleigh tests correct for multiple comparisons (Bonferroni)

Key Metrics

• τ* RSD: Relative standard deviation - lower indicates stronger universal fixed point
• η convergence: Whether Hᵣ ratio stabilizes (signals equilibrium)
• Equilibrium fraction: % of seeds reaching stable curvature ratios
• Log error: For constants spanning many orders of magnitude (α, Λ)

Export Features

• CSV: Raw data tables for external analysis (R, Python, Excel)
• JSON: Complete provenance: metadata, parameters, results, timestamps
• Reports: Auto-generated Markdown with embedded charts for documentation
• Bundles: One-click reproducibility package (Exp 7 only)

📚 Reference Materials

For complete theoretical derivations and mathematical foundations:

Recursive Curvature and the Origin of Dimensionless Constants Recursive Geometry of Information and Time: Unified UNNS Monograph

These documents provide rigorous derivations of equations tested here, including proofs of τ-convergence, η-equilibrium theorems, and the mapping functions connecting recursive dynamics to physical constants.

⚙️ Best Practices

Global Setup Panel

The Global Setup Panel (collapsible, located above Experiment 1) provides universal controls for all experiments:

• UNNS Global Seed: Sets base seed for reproducibility across all runs
• Lock Seed: Prevents accidental changes during experiment sequences
• Random Seed: Generates new seed for fresh exploration
• Global Recursion Depth: Override individual experiment depths (0 = use defaults)
• Quick Export: Export all JSON logs and CSV data from completed experiments

Reproducibility Workflow

• Set Global Seed (e.g., UNNS-1234) before starting experiments
• Lock the seed to guarantee identical results across runs
• After each experiment, click "Persist JSON Log" and "Export CSV"
• Use consistent seed across runs to verify deterministic behavior
• Change seed between major parameter explorations

General Best Practices

• Start with Experiment 1 to verify basic τ-convergence before advanced tests
• Use higher ensemble sizes (50-100) for Exp 7 to reduce variance in constant predictions
• Enable calibration/validation split in Exp 7 to assess generalization vs. overfitting
• Compare JSON exports across runs to verify deterministic reproducibility
• Generate scientific reports for documentation and peer review preparation
• Adjust recursion depth if convergence not achieved (increase to 1000-2000)

🔬 Complete Experimental Workflow

Follow this systematic workflow to generate publication-ready results with full reproducibility:

2. Experiment 7 — Physical Constant Predictions

Expected Results:

• α = 1/137.036 (log-error < 0.0005, PASS)
• μ = PASS (< 2% relative error)
• Δε* = PASS (neg-error < 10)
• α Stability Index = 1.0004 ± 0.0004

3. Experiment 5 — Rees's Six Constants

Expected Results:

• ✅ Expected: N, ε, Ω, λ, Q, D = ALL PASS
• Average Log Error = 0.67
• Dimensional Preference = 3

4. Experiment 1 — τ Fixed-Point Invariant

Expected Results:

• ✅ τ-ratio = 1.0000 ± 0.0006
• ✅ Convergence % = YES (effect size d=3.8-4)

5. Experiment 6 — Curvature Fixed-Point Detection

Expected Results:

• ❌ Expected Result: FAIL (~40% equilibrium)
• → This does NOT falsify the theory — indicates sparse attractors.
• Optimal depth sweep: 750, 1000 (Rosenthal variation)

6. Export for Publication Bundle

After all experiments, export and zip these files:

• exp7_constants_v813_seed1234.json
• exp5_rees_v813_seed1234.json
• exp1_tau_invariant_v813_seed1234.json
• exp6_curv_fp_v813_seed1234.json
• exp3_ess_comparison_v813_seed1234.json
• Plus corresponding CSV files

The numbered filename convention (v813) helps track parameter iterations. Use the "Download Bundle" button in Exp 7 for one-click packaging.

🔹 UNNS Seed & Global Setup

The UNNS Seed initializes the τ-field — defining the starting curvature, τ-on phase, and collapse behavior of recursion. It functions both as:

• A deterministic random seed (for sampler reproducibility);
• A geometric offset for the recursive curvature field Φ(n).

Changing the seed = changing the “initial universe.” Keeping it fixed ensures reproducibility.

🧪 1. Global Setup Rules

• Enter a seed into the field “UNNS Seed” before running any sampler.
• If a global Recursion Depth slider exists, keep it at default (n = 0–2) unless specified per experiment.
• After each test, click “Export CSV” and “Persist JSON Log” to store results.

🔢 2. Seed Categories & Effects

Seed Type	Example	Observed Behavior
Default / Balanced	UNNS-1234	Stable τ-phase & smooth convergence.
Prime-based	UNNS-2027, UNNS-9973	Slow stabilization, high curvature gradients.
Symmetric	UNNS-1111, UNNS-1212	Rapid harmonic locking, minimal drift.
Chaotic / High Curvature	UNNS-314159, UNNS-ΔChaos	Strong τ-oscillation, possible divergence.
Rees-inspired	UNNS-R137, UNNS-R007	Biases toward α ≈ 1/137 or ε ≈ 0.007 values.
Degenerate / Null Collapse	UNNS-0000	No curvature buildup — recursion collapse stops.

📋 3. Suggested Seeds to Try

UNNS-1234      → Baseline / Lab default
UNNS-2027      → Prime recursion
UNNS-314159    → Chaotic curvature drift
UNNS-R137      → Fine-structure emulation
UNNS-R007      → Stellar fusion ε constant
UNNS-1111      → Symmetric harmonic seed
UNNS-9973      → Deep recursion stress test
UNNS-φ1618     → Golden-ratio seed (φ = 1.618)
UNNS-∞Chaos    → Entropy-max collapse
  

📌 4. Where It Affects the Lab

• Input field (top-left): “UNNS Seed” value.
• Displayed Phase: τ-on = exp(2πi × seed/10000).
• Influences:
  • Random Walk MCMC (initial x,y)
  • τon-RHMC (momentum + τ-phase)
  • Klein-Flip recursion & sign symmetry
  • Curvature κ(x,y,n) evolution

🎯 5. Why Seeds Matter Scientifically

By sweeping over different seeds while measuring constants (α, ε, λ, Ω), the Lab allows testing whether:

• Constants are seed-invariant ⇒ universal τ-laws.
• Constants shift with seed ⇒ τ-field determines physical tuning.

✅ 6. Best Practices for Research Use

• Keep seed constant across all runs of the same experiment.
• Only change seed when intentionally probing universality vs initial-condition dependence.
• Record seed, depth, α, entropy, κ-values in CSV output.
• Attach seed metadata when publishing or sharing results.
• If studying convergence → run same seed across 3 sampler types.

🚀 7. Roadmap for Seed Expansion (Upcoming)

• ✔ Dynamic seed history sidebar
• ✔ τ-phase trajectory graph over time
• ⬜ Seed-space scanning (Monte Carlo across seeds)
• ⬜ Correlate seed vs. Rees constants (N, ε, Q, Ω, λ)

“The seed is not a number — it is an origin of curvature.”

🎓 Laboratory Status