UNNS Laboratory – Chamber XXVI v2.2-rebuild (Phase-G Complete)

📚 Laboratory Guide

🎯 PE-27G v2.2-Rebuild: What Changed

This version is a clean rebuild combining v2.1.1 stability with v2.2 enhancements:

✅ Preserved from v2.1.1: All 6 critical fixes (parameter pipeline, convergence detection, Φ computation, JSON validation, closure timing, visualization sync)
✨ New from v2.2: Status banners, unified convergence logic, Ω-Auto-Tune, tooltips, enhanced diagnostics
🔒 Bug Prevention: Operator initialization, observable pipeline, canvas rendering, JSON schema compatibility—all verified working

🔷 Empirically-Validated Parameter Windows

Based on 28-run analysis (Dec 2025): The optimal region shows a clear linear relationship:

Best Point Found:

λ ≈ 0.1512
α_c ≈ 0.0226
χ² ≈ 4.95

Coupling Relationship: α_c ≈ 0.15 × λ

All converged runs cluster along this line with ratio 0.12 ≤ α_c/λ ≤ 0.18

Search Windows

Window Type	λ Range	α_c Range	Use Case
Coarse	[0.140, 0.170]	[0.015, 0.030]	Initial exploration
Fine	[0.145, 0.158]	[0.018, 0.028]	Serious calibration
Target Center	0.151	0.0225	Starting point

⚠️ Important: When running PSO calibration, bias particles to satisfy α_c ≈ 0.15λ ± 20% to stay in the convergence basin.

🔷 Four-Step Calibration Protocol (Empirically Validated)

Critical: Follow this exact sequence. Skipping steps will result in χ² ≈ 6-7!

STEP 0: Reset to PE-27G Defaults

Click Reset in Panel E (Engine Configuration)
Set baseline parameters:
- Grid: 64×64 (balanced performance)
- Depth: 500 (sufficient for equilibration)
- σ (noise): 0.01
- λ: 0.150 (middle of coarse window)
- α_c: 0.020 (neutral starting point)
Verify η₁, η₂, η₃ are NOT zero (e.g., [0.20, 0.20, 0.20] or last saved values)

Why this matters: Starting from known-good defaults prevents parameter coupling artifacts.

STEP 1: Ω-Auto-Tune (Closure First)

In Panel C, click 🎛️ Ω-Auto-Tune
Wait for convergence (typically 5-15 iterations)
Success criteria:
- C₁, C₃, C₅ residuals ALL < 0.012
- Status shows "Open" (no fault flags)
Note the output (e.g., Best: α_c=0.050, η=[0.200, 0.200, 0.200])
⚠️ CRITICAL: Copy the displayed α_c value to Panel E → "α_c (Closure)" field

Important: Do NOT touch η sliders manually after Auto-Tune completes. From this point forward, only λ and α_c should be adjusted.

STEP 2: 1D Calibration on α (Physics Anchor)

Go to Panel F — Calibration Engine
Configure PSO:
- Target Observable: α (fine-structure constant)
- Method: PSO (Particle Swarm)
- Iterations: 50
- Tolerance: 0.01
Fix λ at 0.150 (middle of coarse window)
Run calibration → Click Apply Calibrated
This establishes an "α-anchored" baseline point

Note: Global χ² may still be high (~6-7) at this stage. That's expected—we're establishing a physics anchor first.

STEP 3: 2D λ/α_c Search (Fine Window)

Enable search mode constraints:
- Constrain λ to [0.145, 0.158] (fine window)
- Constrain α_c to [0.018, 0.028] (fine window)
In Panel F, change target:
- Target: All (minimize total χ²)
PSO configuration:
- Iterations: 50
- Tolerance: 0.01
- Swarm initialization: Bias particles along α_c ≈ 0.15λ
Run calibration → Apply Calibrated
Save the run and check Panel H (Phase Map):
- Should show tight cluster near λ ≈ 0.151, α_c ≈ 0.022-0.024
- If best χ² point sits on boundary, expand that edge by +0.005 and repeat

Expected Result: χ² should drop to 4.5-5.5 range. Continue refining until χ² < 4.0.

STEP 4: Convergence Verification

A run is physically acceptable when:

Criterion	Threshold	Check Location
χ² (total)	< 4.0	Status banner / Panel B
C₁ residual	< 0.010	Panel C metrics
C₃ residual	< 0.010	Panel C metrics
C₅ residual	< 0.010	Panel C metrics
Φ drift	Flat ±0.002 over last 10 iterations	Panel D → Φ History

If χ² ≥ 4.0: Keep all other settings fixed and explore only within the λ/α_c fine window. Do NOT adjust η, σ, or Grid at this stage.

Once χ² < 4.0: Run 5 additional seeds to verify reproducibility (CV < 5%).

📊 Interpreting Results (New in v2.2-Rebuild)

Status Banner (Top of Page)

✓ VALIDATED — All criteria met (χ² < 4.0, C₁/C₃/C₅ < 0.01, Φ drift < 2%, operators bounded)
⚠ PARTIAL — Near convergence, 1-2 criteria slightly exceeded
✗ INVALID — Shows specific failure reasons (e.g., "χ² too high", "C₃ residual too high", "Operator XIII diverged")

Panel G Visualizations

τ-Field: Should show smooth gradients with sparse streaks (not turbulent clusters like v2.1.1)
Closure Diagnostics: C₁/C₃/C₅ curves should decay and cross green threshold line
Operator Trajectories: Should show monotonic decay (especially XIII)
F-Manifold: Converged runs cluster in a compact basin near λ*, α_c*

🔧 Troubleshooting Guide

Symptom	Likely Cause	Solution
χ² > 6.0	α_c mismatch between Panel C and Panel E	Copy α_c from Ω-Auto-Tune result to Panel E
Operator XIII diverges (\|value\| > 2.0)	λ out of valid range	Keep λ ∈ [0.05, 0.15]; try λ = 0.11
C₁/C₃/C₅ never cross threshold	η values too low	Run Ω-Auto-Tune again; manually increase η₁, η₂, η₃
τ-Field visualization empty	Depth too low or σ = 0	Increase Depth to ≥400; set σ = 0.02
F-Manifold shows no points	No runs executed yet	Click "Run PE-27G" to populate the map
Φ drift > 2%	Insufficient equilibration	Increase Depth to 800-1000

📈 Convergence Criteria (Codex)

Criterion	Threshold	Meaning
C₁ (Idempotence)	< 0.01	Field change per iteration stabilized
C₃ (Flux Neutrality)	< 0.01	Net gradient flow minimized
C₅ (Reversibility)	< 0.01	Second-order field changes small
χ² (Observable Fit)	< 4.0	Physical predictions match targets
Φ Drift	< 2%	Nonlinear stack stabilized (v2.2 addition)
Operator Bounds	\|value\| < 2.0	No numerical divergence (v2.2 addition)

A run achieves VALIDATED ✓ status when ALL six criteria are met simultaneously.

⚙️ UNNS Operator Codex (XIII–XXI)

Operator	Name	Physical Meaning	Typical Range
XIII	Interlace Phase Coupling	Curvature gradient norm	0.2–2.5
XIV	Phi-Scale	Spectral variance proxy	0.3–0.8
XV	Prism	Field standard deviation	0.5–1.2
XVI	Fold	Torsion energy density	0.1–0.5
XXI	Micro-Recursion	Micro-torsion integral	0.2–0.6

Warning: If XIII > 2.0, reduce λ or increase closure parameters. If any operator shows NaN, check Grid size (must be power-of-2).

🎯 CHAMBER XXVI CALIBRATION PROTOCOL (Simplified)

✓ VALIDATED WORKFLOW: Ω-Auto-Tune dynamically finds optimal parameters. No presets required—AutoTune IS the golden baseline generator.

STEP 1 — Set Initial λ

Choose λ in the convergence basin:

Parameter	Recommended Range	Typical Start
λ (coupling)	[0.09, 0.13]	0.10825 (empirical optimum)
σ (noise)	[0.005, 0.02]	0.01
Grid	64×64 or 128×128	64×64 (balanced)
Depth	500-1000	500

Note: Don't worry about α_c or η—AutoTune will optimize these in Step 2.

STEP 2 — Run Ω-Auto-Tune

Let AutoTune find optimal closure parameters:

In Panel C, click 🎛️ Ω-Auto-Tune
Wait for convergence (typically 15-25 iterations)
Look for status: "Converged ✓" or "Max iterations" with C₁, C₃, C₅ < 0.010
AutoTune will find optimal:
- η₁, η₂, η₃ (typically η ≈ [0.30, 0.30, 0.24])
- α_c (typically α_c ≈ 0.015)

AutoTune Algorithm (k=2.5 correction):


              For each iteration:

                η_i ← η_i + 2.5 × (C_i - 0.010)

                Clamp: η_i ∈ [0.05, 0.30]

                If Φ_drift > 0.015: α_c ← α_c + 0.5 × drift

              

              Stop when:

                max(C₁, C₃, C₅) < 0.011 AND slope < 0.0002

⚠️ "Max iterations" Warning: This may appear even when convergence is achieved. Check the actual C₁, C₃, C₅ residuals—if all < 0.010, you're converged regardless of the warning.

STEP 3 — Run PE-27G & Validate

Execute full simulation with AutoTune-optimized parameters:

Click ▶ Run PE-27G in Panel E
Wait for completion (≈1-5 minutes depending on Grid and Depth)
Check status banner at top:
- ✓ VALIDATED → All 8 convergence criteria met
- ⚠ PARTIAL → Close; check specific failures
- ✗ INVALID → Retry with different λ or re-run AutoTune
Expected validated run:
- χ² ≈ 3.8-4.0
- C₁, C₃, C₅ all < 0.010
- Φ stable (drift < 0.015)
- Operator XIII dominant

If not converged:

Try λ slightly lower/higher (±0.005)
Re-run Ω-AutoTune at new λ
Increase Depth to 800-1000

📋 Convergence Criteria (All Must Pass)

Criterion	Threshold	Check Location
χ² (total)	< 4.0	Panel B / Status Banner
C₁ (Idempotence)	< 0.010	Panel C → C₁ RESIDUAL
C₃ (Flux Neutrality)	< 0.010	Panel C → C₃ RESIDUAL
C₅ (Reversibility)	< 0.010	Panel C → C₅ RESIDUAL
Φ drift	< 0.015	Panel D → Φ History
Operator bounds	All < 2.00	Panel A

🔬 Grid Resolution Analysis: Why 64×64 is Optimal

Critical Discovery: Resolution-Dependent Convergence

Empirical testing across three grid resolutions reveals that PE-27G convergence occurs only at 64×64. Both lower (32×32) and higher (128×128) resolutions fail to achieve simultaneous χ² and closure convergence—but for fundamentally different physical reasons.

Side-by-Side Grid Comparison

Metric	32×32	64×64 (Converged)	128×128
χ²	4.548	3.809 ✓	5.113
Φ (nonlinearity)	0.11437	0.15712	0.08750
C₁	0.00977	0.00715 ✓	0.00839
C₃	0.00739	0.00472 ✓	0.00496
C₅	0.00982	0.00975 ✓	0.00973
XIII	0.286	0.468	0.219
XV	0.1437	0.122	0.105
XXI	0.0660	0.0401	0.0464
Converged?	✗ No	✓ YES	✗ No

The U-Shaped Dependency

Why 128×128 Fails (Despite Being "Higher Resolution")

Critical Insight: Higher resolution does NOT always mean better accuracy in PE-27G. The chamber requires a specific balance between discretization scale and nonlinear curvature dynamics.

3.1 Over-Resolution Suppresses Φ Nonlinearity

The 128×128 run produces Φ = 0.0875, compared to 0.157 for the converged 64×64 solution. This collapse means:

Recursion curvature flattens too quickly
Nonlinear coupling between operators XIII ↔ XV ↔ XXI weakens
τ-field becomes too smooth, losing micro-folding structure
χ² increases because the engine becomes overly linear

3.2 Operator XIII Collapses to Half Its Proper Value

Grid	XIII Amplitude	Status
32×32	0.286	Under-coupled
64×64	0.468	Correct attractor ✓
128×128	0.219	Collapsed

This XIII collapse destroys the operator stack tension needed to reproduce n_s, α_EM, H₀, and σ₈, causing χ² to rise sharply.

3.3 The Physical Mechanism

At 64×64: The chamber correctly resolves

Curvature gradients
Micro-folding structures
Closure oscillations
Nonlinear Φ-derivatives

At 128×128: Micro-folding disappears, but closure residuals don't break—instead:

The model becomes "too shallow"
Everything becomes statistically correct but physically wrong
Observable predictions drift away from targets

Observable-Level Evidence

At 128×128, observables drift in a characteristic pattern:

H₀ = 67.547 (lower → underestimation of expansion tension)
n_s = 0.961 (moves too low)
N_eff = 3.049 (drifts upward)
σ₈ = 0.796 (slightly high)

This drift pattern corresponds exactly to insufficient Φ-nonlinearity + weakened XIII signal—precisely what an over-smooth grid produces.

Theoretical Interpretation

Major Discovery:

PE-27G exhibits a resolution-critical fixed point where discretization scale matches closure scale. This is the first time this dependency has been measured so clearly.

Resolution	Failure Mode	Physical Cause
32×32	Too coarse	Noise-dominated → unstable Φ → operator wandering
64×64	Perfect balance ✓	Stable Φ → correct operators → χ² minimized
128×128	Too fine	Over-smoothing → Φ collapse → XIII collapse → χ² rises

Conclusion: 64×64 is not arbitrary—it is the structural recursion resolution for Chamber XXVI, representing the true fixed point where recursion dynamics and closure operators reach equilibrium.

Practical Implication: Always use Grid = 64×64 for production runs. Using 128×128 for "higher accuracy" will paradoxically make results worse. The only valid reason to use 128×128 is for visualization purposes with post-processing—not for calibration or scientific validation.

Version: v2.2-Rebuild | Engine: PE-27G | Base: v2.1.1-fixed + v2.2 enhancements | Status: Production Ready