This appendix documents exactly how φ-stability was extracted from the SrF and ThO datasets in Lab v0.9.1. It is intended for UNNS developers, researchers, and contributors needing reproducibility.

Both SrF and ThO Lab v0.9.1 datasets share the same structure:

{
  "molecule": "SrF" | "ThO",
  "lines": [
    {
      "curvature":          Number,
      "synth_curvature":    Number,
      "tau_phase":          Number,
      "synth_tau_phase":    Number,
      "residual":           Number,
      "J":                  Number
    }, ...
  ],
  "pearson": Number,
  "fit_details": {
    "r_squared":  Number,
    "chi2":       Number,
    "dof":        Number,
    "match_rate": Number
  }
}

Developers should note:
– curvature is real physical curvature from molecular constants.
– synth_curvature is the UNNS-synthetic predicted curvature.
– tau_phase is the real echo-phase value.
– synth_tau_phase is the UNNS-predicted recursive torsion phase.
– residual = (real – synthetic) across the φ-channel.

φ-stability is extracted with a deterministic operator pipeline:

1. Load all curvature and τ-phase pairs
   (curvature, τ_phase) for each J-state.
2. Normalize curvature magnitudes
   Rescale to φ-compatible range: [–1 … +1].
3. Compute φ-band boundaries
   Use τ-phase auto-clustering: τ_min ≈ –11, τ_max ≈ +13 for both molecules.
4. Compute φ-wells
   Group curvature sign-alternating sequences into Sobra/Sobtra sectors.
5. Apply Operator XII collapse-test
   For each J:
   – If curvature exceeds threshold → collapse
   – else → survival contributes to φ-stability
6. Accumulate φ-stability
   Φ(x) = Σ (surviving τ-phase contribution)
7. Normalize
   P_UNNS(x) = Φ(x) / Σ Φ(y)

Steps 3–5 use structural rules from:
Sobra–Sobtra Mechanism as the UNNS Replacement for the Born Rule
https://unns.tech/media/docs/SOBRAS~1.PDF

The most important step for developers is φ-well detection. A φ-well forms when:

• curvature alternates sign (Sobtra–Sobra backflow)
• τ-phase stays within a bounded band
• Operator XII does not collapse the echo

For SrF and ThO, φ-wells emerged automatically from the data:

• τ-phase range: approx. –11 → +13
• curvature oscillation: approx. –37 → +48
• curvature spacing is nearly symmetric across 0

This bounded structure is the real-data equivalent of the |ψ|² curve in QM.

Operator XII governs “collapse vs survival” for each J-line sample. The collapse rule is:

Threshold is computed from:

• local curvature slope (Δcurv)
• τ-phase compression
• Sobra-feedback region defined in the Sobra–Sobtra PDF

Only survivors contribute to the φ-stability sum Φ(x).

To reproduce the φ-stability extraction:

1. Load SrF or ThO JSON file (Lab v0.9.1 format).
2. Normalize curvature: curv_norm = curvature / max(|curvature|)
3. Compute τ-band from min/max τ-phase.
4. Cluster curvature sign changes into φ-wells.
5. Apply Operator XII threshold to decide survival/collapse.
6. Accumulate surviving φ-contributions.
7. Normalize to obtain P_UNNS(x).

This pipeline is identical for both SrF and ThO datasets.

This checklist defines every required step for bringing a new molecule into the UNNS Lab ecosystem — including φ-stability extraction, curvature projection, and Sobra–Sobtra operator integration.

• Collect rotational or vibrational spectral data (J-levels or transitions).
• Extract molecular constants needed for curvature modelling:
  • B, D (rotational constants)
  • hyperfine constants where applicable
  • effective electric field (for eEDM-type molecules)
• Ensure dataset contains ≥ 20 curvature points (minimum for φ-well detection).
• Save raw data in neutral JSON or CSV.

• Compute curvature κ(J) or κ(transition) using your standard Lab curvature engine.
• Normalize curvature to [–1 … 1] range.
• Identify curvature sign patterns (required for φ-well assembly).

• Compute τ-phase using recursion-based torsion estimator.
• Ensure τ-phase resolution ≤ 0.1 radians for clear φ-band boundaries.
• Store real τ-phase and synthetic (predicted) τ-phase.

Include these fields **exactly** (Lab v0.9.x format):

{
  "molecule": "NAME",
  "lines": [
    {
      "curvature": Number,
      "synth_curvature": Number,
      "tau_phase": Number,
      "synth_tau_phase": Number,
      "residual": Number,
      "J": Number
    }, ...
  ],
  "pearson": Number,
  "fit_details": {
    "r_squared": Number,
    "chi2": Number,
    "dof": Number,
    "match_rate": Number
  }
}

• Validate schema (strict key order recommended for Lab ingestion).
• Ensure synth_curvature and synth_tau_phase exist before φ-well extraction.

• Compute τ-min and τ-max from dataset.
• Cluster data into φ-wells by curvature sign alternation:
  • (+ → – → +) sequence → one φ-well
  • (– → + → –) sequence → one φ-well
• Calculate φ-well width = τ_max – τ_min.
• Verify φ-band width is stable (approx. constant across J).

• Evaluate each line against Sobra threshold:
  • If |curvature| > Sobra_threshold → collapse
  • Else → survives into φ-stability
• Implement Sobtra redistribution for collapsed torsion flows.

• Accumulate φ(x) from all surviving J-levels.
• Normalize using:

P_UNNS(x) = Φ(x) / Σ Φ(y)

• Confirm φ(x) matches expected basin shape (smooth quadratic stability).

• Generate φ-stability basin SVG.
• Create χ-distance residual histogram.
• Render τ-phase confinement plot.
• Confirm R² ≥ 0.999 for physical validity.
• Export updated φ-well map for publication.

• Add molecule entry to Lab v0.9.x selector.
• Load JSON into curvature analysis engine.
• Inject φ-stability into τ-MSC panel.
• Enable export to JSON/CSV for reproducibility.
• Verify visual consistency across devices.

• Create GitHub mirror in: /docs/unns_lab_real_data/
• Publish φ-stability SVG in article or chamber notes.
• Update release notes for Lab v0.9.x or later.