⚙️ CHAMBER XXIX: PROTO-CLOSURE

Chamber XXIX: Proto-Closure Evaluation

1. Purpose & Scope

Chamber XXIX evaluates mechanisms for proto-closure via structural recurrence under refinement.

Out of scope (hard exclusions):

• Numerical constants evaluation
• Numeric field evolution (τ-fields, etc.)
• Value convergence, divergence, or stability
• Matching against known constants (φ, π, e, √2)

2. Mechanism Definition

Each mechanism M = (Σ, R, C, O) consists of:

• Σ: Finite symbolic alphabet (opaque tokens)
• R: Rewrite/generation rules (symbolic transformations only)
• C: Admissibility constraints (decidable on symbolic structures)
• O: Operator schedule (ordering of rule application)

3. Structural Equivalence Metric (v0.2)

Two-level metric on windowed trajectories:

1. Canonical Hash (primary): Deterministic hash of canonicalized trajectory
   • Window = last W configurations from generation (default W=8)
   • Alpha-renaming normalization across window
   • Period detection within window
   • Symbol order normalization
   • If hashes match → d = 0
2. Feature Vector Distance (secondary): When hashes differ
   • Extract from entire window: symbol histograms, diversity, patterns
   • Compute L1 distance between feature vectors
   • Includes: avg config length, inter-config variation, compressibility
   • Normalized to avoid scale bias

Key Advantage: Windowed approach captures temporal structure (oscillations, cycles) that single-snapshot canonicalization would miss. Essential for detecting proto-closure in mechanisms with rule-level periodicity.

4. Refinement Process

For each mechanism, generates sequence S₀ → S₁ → S₂ → ... where each step applies refinement:

• Increased generation depth (multiplicative: 20 → 40 → 80 → ...)
• Tightened constraints (if applicable)
• Reduced equivalence tolerance

5. Closure Detection

Structural Closure Criterion: A mechanism exhibits structural closure iff:

• ∃ ε > 0 and k ≥ 1 such that for sufficiently large n: d(Sₙ, Sₙ₊ₖ) < ε
• AND recurrent orbit contains > 1 equivalence class (non-trivial)
• AND recurrence persists to terminal refinement level

Excluded from closure:

• Fixed-point freezing (single state orbit)
• Transient recurrence (detected but collapses under refinement)
• Null collapse
• Exhaustion termination

6. Verdict System (v0.2.5 – Binary Classification)

Verdict	Meaning
PROTO-CLOSED	Structural closure detected and persists under refinement
STRUCTURAL-COLLAPSE	No persistent recurrence detected, or structure degenerates

7. Reference Mechanism Library (Phase 1)

Control Mechanisms:

• CTRL-BINARY-FLIP: Expected verdict: PROTO-CLOSED (period-2 orbit persists)
• CTRL-UNBOUNDED-GROWTH: Expected verdict: STRUCTURAL-COLLAPSE (unbounded growth, no recurrence)
• CTRL-META-UNSTABLE: Expected verdict: STRUCTURAL-COLLAPSE (transient period-3, then growth dominates)
  ⚠ Shows transient recurrence annotation

Test Mechanisms:

• TEST-TERNARY-CYCLE: Expected verdict: PROTO-CLOSED (period-3 orbit persists)
• TEST-BALANCED-PARENS: Expected verdict: STRUCTURAL-COLLAPSE (constraints accelerate collapse)
  📌 Important negative result: constraints ≠ closure guarantee

8. Output Contract (v0.2.5)

Each mechanism receives JSON output with:

{
  "mechanism_id": "...",
  "alphabet_signature": "...",
  "rule_signature": "...",
  "constraint_signature": "...",
  "operator_schedule": "...",
  "metric_used": "canonical_hash + feature_vector_L1 (windowed)",
  "window_size": 8,
  "refinement_ladder": [...],
  "orbit_detected": true | false,
  "orbit_period": k | null,
  "orbit_signature": "...",
  "distinct_states": n | null,
  "verdict": "PROTO-CLOSED" | "STRUCTURAL-COLLAPSE",
  "failure_modes": [...],
  "meta_instability": {
    "detected": false | true,
    "first_level": null | n,
    "last_level": null | m,
    "max_duration": 0 | (m - n),
    "period": null | k
  },
  "elapsed_ms": ...,
  "expected_verdict": "..." | null,
  "validation_passed": true | false | null,
  "timestamp": "..."
}

9. Compliance Checklist

Chamber XXIX is compliant iff:

• ✅ Never evaluates numbers
• ✅ Never tests values
• ✅ Never searches for constants
• ✅ Never ranks mechanisms by numeric proximity
• ✅ Reports only structural behavior
• ✅ Binary verdict system (no ambiguous third class)
• ✅ Meta-instability as annotation, not verdict override

Implementation Notes

Mechanism DSL

Rules (R):

{
  "pattern": ["A", "B"],
  "replacement": ["B", "A", "C"],
  "condition": null
}

Constraints (C):

[
  {"type": "max_length", "value": 100},
  {"type": "forbidden_substring", "value": ["A", "A", "A"]},
  {"type": "symbol_balance", "symbols": ["(", ")"], "balanced": true}
]

Feature Vector Components

• Symbol frequency histogram (normalized)
• Unique symbol count
• Total length (windowed average)
• Pattern diversity (unique 2-grams)
• Compressibility estimate (run-length encoding ratio)
• Inter-config variation (length variance across window)

Theoretical Alignment

This implementation strictly follows:

• Definition 3 (Mechanism)
• Definition 4 (Structural State)
• Definition 5 (Refinement)
• Definition 7 (Structural Closure)
• Definition 9 (Proto-Closure)
• Definition 11 (Structural Equivalence Metric)
• Definition 12 (Refined Structural Closure)

Key Theoretical Findings

Version: 0.2.6 FINAL | Layer: L2 (Structural) | Mode: Deterministic Seeded RNG | Status: Production Ready