⚖️ CHAMBER XXXIII: κ-OPERATOR v0.2.3

Pre-Projective Coherence — Certified Production Ready
Mode: Gatekeeper Stage: Pre-Projective Domain: Abstract Status: Experimental
Structure Source
κ Family
κ₃ Depth Balance: Enforces bounded depth asymmetry. For trees: prunes deepest branches when D(S) > Δ.
τ Projection & Residual
Protected metrics count toward ID threshold.
Diagnostic metrics are informational only.
Thresholds
Structure View: Before / After
S (Baseline)
κ(S) (Regulated)
Projection Output
Metric τ(S) τ(κ(S)) Δ% Status
Run comparison to populate
Gatekeeper Scatter
Quadrants: ■ Pass (high RR, low ID) | ■ Fail regions
Run Status
IDLE — Ready to run
κ Converged
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Iterations
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Verdict
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Structure Δ
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Convergence Plot
‖κᵗ⁺¹(S) − κᵗ(S)‖
Depth Analysis
Success Criteria
Residual reduction ≥ 20%
Protected τ-invariants ≤ 0.1%
κ converges idempotent
No instabilities stable
⚠️ v0.2 Update: ID now computed only on protected τ-invariants. Diagnostic metrics may change.
📚 Quick Guide
v0.2.1 Update: κ-iteration now updates live! Watch the convergence plot animate, iteration counter increment, and green indicator (●) pulse during execution. No more UI freezing.
v0.2.0 Major Update: Tree structure support + Protected/Diagnostic τ-metric split. κ₃ can now be tested properly on depth-imbalanced structures.

Testing κ₃ (Depth Balance)

  1. Select Tree structure type
  2. Choose Skew Tree for max imbalance
  3. Set L=80, side branches=0.15
  4. κ₃ family with Δ=10
  5. Use Depth Residual mode
  6. Run and observe depth imbalance reduction

τ-Metric Roles

  • Protected (P): Must remain stable (≤0.1% drift). Counts toward ID threshold.
  • Diagnostic (D): Allowed to change. Informational only.
  • For trees: tree_height/paths are diagnostic (κ₃ will change them)
  • entropy/spectral radius are typically protected
Version: 0.2.3 | Chamber: XXXIII | Operator: κ (Pre-Projective Coherence) | Status: ✓ Certified Production Ready
v0.2.3 (2026-01-08) — Semantic & Robustness Fixes:
  • Fixed parent edge detection in κ₃ (robust to edge direction)
  • Renamed 'diameter' → 'tree_height' for semantic accuracy
  • Added defensive guards to residual reduction calculation
  • Clarified avg_path_length is root-based proxy
  • Added note: κ₂ spectral cap on trees is approximate
v0.2.2 (2026-01-08) — Critical Bug Fixes: Fixed node ID reindexing, adjacency dimension mismatch, truncated function, added structure type warnings.
v0.2.1 (2026-01-08): Added live UI updates during κ-iteration. v0.2.0: Tree generators + Protected/Diagnostic τ-split.