τ-Filtered Observability — Foundations Chamber

This Chamber operationalizes the constructions introduced in "UNNS as an ∞-Operadic Substrate" by providing a controlled computational environment in which operadic recursion, τ-filtering, and collapse-selected dynamics can be examined step-by-step.

The Chamber is designed to produce explicit sequences, operator traces, admissibility masks, and phase-space partitions derived from deterministic recursion rules. All outputs are generated directly from the configured model parameters and exported in structured form.

1. State Generator (Scalar Drift)

The generator evolves a scalar state sequence according to:

Where:

• δ is a constant drift parameter
• ε_n is bounded noise
• the random number generator is deterministic when seeding is enabled

The generator produces the raw state trajectory x_n, which is then processed by subsequent operators.

2. Sobtra Threshold Clamp

Sobtra is the threshold-based projection operator used in the Chamber.

It is defined as:

Where:

• θ is the Sobtra threshold parameter

Sobtra partitions each step into:

• a retained component (s_n)
• a rejected component captured by the residue

The residue is defined as:

This decomposition allows the Chamber to distinguish between locally retained structure and suppressed contributions at each step.

3. Residue

The residue r_n records the portion of the state excluded by Sobtra. Residue magnitude directly contributes to curvature and collapse detection.

Residue values are visualized in:

• κ(n) diagnostics
• admissibility barcodes
• collapse overlays

4. Curvature κ and Admissibility

Curvature at step n is defined as:

A step is admissible if:

Where:

• Λ is the admissibility threshold (τ-filter parameter)

Admissibility is a binary, local property evaluated independently at each step.

5. Operator Assignment

Each step is assigned an operator symbol based on its local behavior:

• Φ — generative continuation
• Ψ — structural modulation
• τ — admissible evolution
• XII — collapse (inadmissible step)

Operator assignment is recorded explicitly in the exported data and summarized in dominance charts.

6. τ-Filtering

τ-filtering applies the admissibility criterion after generation, without re-executing the generator.

This allows:

• re-classification of steps under different Λ values
• construction of Λ × δ phase diagrams
• comparative analysis across thresholds

τ-filtering operates purely on recorded state data.

7. Phase Diagrams (Λ × δ)

The Chamber constructs phase diagrams in the Λ × δ plane, classifying regions according to:

• fraction of admissible steps
• collapse prevalence
• transitional regimes

These diagrams are computed directly from admissibility masks and are independent of visual scaling.

8. Diagnostics

Diagnostics summarize run-level properties, including:

• total steps
• admissible steps
• survival ratio
• collapse onset location
• operator frequencies

All diagnostic values are derived from per-step data and are exported verbatim.

9. Multi-Run Comparison

The Compare module aligns multiple imported runs by step index and evaluates:

• survival ratio stability
• path persistence
• operator dominance drift
• collapse consistency

Comparisons do not recompute any dynamics; they operate solely on imported run data.

Appendix A — How to Read the Outputs

A.1 Time-Series Plots

• x(n) shows raw state evolution
• overlays highlight admissible vs collapsed regions

A.2 κ(n) Diagnostic

• spikes indicate sharp residue variation
• κ exceeding Λ marks inadmissible steps

A.3 Admissibility Barcode

• binary visualization of admissible vs collapsed steps
• useful for detecting clustering and regime shifts

A.4 Trajectory with Overlay

• combines x(n) with admissibility shading
• visually correlates collapse with state growth

A.5 Phase Diagram (Λ × δ)

• horizontal slices correspond to fixed δ
• vertical transitions mark admissibility boundary crossings

A.6 Operator Dominance Charts

• display relative operator frequencies over time or across runs

A.7 Structural Verdicts

• summarize invariance or divergence across compared runs
• derived strictly from computed metrics

Appendix B — Testing & Reproducibility

Two JSON files are provided as reference inputs for testing, validation, and regression checks.