⚛️ CHAMBER XXXIV: Ω-ONLY EXPLORATORY

Global Mode Selection & Vacuum Admissibility
PHASE F: EXPLORATORY
Configuration

Ensemble Generation

Ω Selection Operator

Vacuum Proxy Weights

Visualization
V(S) Distribution: Baseline vs Ω-Filtered
Z₀ vs Gap: Accept/Reject Scatter
Results Summary

Baseline vs Ω-Filtered

Acceptance Rate
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RΛ Baseline
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RΛ Filtered
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RΛ Reduction
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V̄ Baseline
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V̄ Filtered
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Var(V) Baseline
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Var(V) Filtered
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Protected Macro Invariants (τ-derived)

Invariant Baseline Ω-Filtered Drift % Status
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📚 Chamber Guide

Purpose

Chamber XXXIV tests whether a global selection layer (Ω) can measurably reduce vacuum residuals (RΛ) while keeping τ-scale macro invariants stable. This is an Ω-only exploratory chamber—no internal confinement operators (σ/κ), just pure ensemble selection.

Key Question

Does Ω signature exist? Can global mode selection reduce vacuum residual proxies without destabilizing protected metrics?

Operational Model

Ensemble E = {S₁, S₂, ..., Sₘ}: Collection of graph structures generated under controlled constraints

Vacuum Proxy V(S): Global observable combining:

  • Z₀(S) = Zero-mode mass (near-null eigenvalue count / n)
  • Gap(S) = Spectral gap (Fiedler value λ₂ of Laplacian)
  • CycleT(S) = Cycle tension (|E| - |V| + components) / n

Formula: V(S) = α·Z₀(S) + β·(1/(Gap(S)+ε)) + γ·CycleT(S)

Λ Residual Proxy: RΛ = |V̄ - V_target| / max(|V_target|, ε)

Measures how far the ensemble-averaged V̄ deviates from expected stationary vacuum band.

Ω Operators

Operator Mechanism Interpretation
Ω₁ Zero-Mode Culling Reject if Z₀(S) > z₀_max
Ω₂ Holonomy Bandpass Keep only if c_min ≤ CycleT(S) ≤ c_max
Ω₃ Spectral Gap Floor Reject if Gap(S) < gap_min
Ω₄a Min-V (Exploratory) Select lowest V(S) structures (may worsen RΛ)
Ω₄b V-Bandpass (RΛ-Aligned) Select structures closest to V_target (guaranteed RΛ reduction)
⚠️ Ω₄a vs Ω₄b — Important Distinction

Ω₄a (Minimum-V): Exploratory operator that selects structures with lowest V(S) values regardless of baseline V_target. This can reduce V̄ but may increase RΛ (vacuum residual), which is scientifically valid behavior for an exploratory selector. PASS/FAIL verdicts both meaningful.

Ω₄b (V-Bandpass): Canonical operator that selects structures closest to V_target (baseline median). Guaranteed to reduce RΛ by construction. This is the RΛ-aligned operator — PASS verdict indicates true Ω→τ compatibility.

Key Insight: Only Ω₄b is aligned with the vacuum residual metric RΛ. Ω₄a serves as an exploratory selector testing a different hypothesis (minimum absolute vacuum proxy) and may legitimately worsen RΛ while still being admissible.

Success Criteria

Ω Signature Detected if:

  • RΛ_Ω < RΛ₀ - δ (meaningful reduction, δ > 5%)
  • Macro invariant drift ≤ 10% (stability preserved)
  • Acceptance rate: 10–60% (coherent band, not degenerate)
  • Stable across seeds (reproducible)

Protected Metrics (ADM Framework)

These τ-derived macro invariants must remain stable (drift ≤ 10%):

  • scaled_spectral_radius: Ensemble mean of ρ(A)/n
  • energy_per_node: Tr(L)/n averaged over ensemble
  • entropy_degree: -Σ pᵢ log(pᵢ) from degree distribution (total entropy, not per-edge)

Interpretation

If Ω signature is detected with stable protected metrics, this suggests:

  • Global selection can structure vacuum admissibility
  • Vacuum residuals are sensitive to global mode constraints
  • Ω layer is a legitimate addition to UNNS operator suite
  • Foundation for Chamber XXXV: Coupled Ω→τ pipeline

⚠️ Important Notes:

  • XXXIV is exploratory—null results are valuable
  • If acceptance rate → 0% or 100%, Ω parameters need tuning
  • Variance reduction alone is not sufficient—RΛ must decrease
  • Protected metrics must stay stable for valid Ω signature

Recommended Workflow

  1. Quick Test: M=50, n=30, Ω₄ with default weights (~30sec)
  2. Balanced Run: M=100, n=50, test all Ω modes (~2-3min)
  3. Production Run: M=200, n=100, Ω₄ optimal parameters (~8-10min)
  4. Multi-Seed: Repeat with seeds [137042, 137043, 137044] for reproducibility
  5. Generator Comparison: Test across lattice-lr, small-world, scale-free
  6. Export & Analyze: Save JSON for each configuration

Version: 1.1.0 (Bug Fixes) | Phase: F (Exploratory) | Status: Production Ready

v1.1.0 Changelog (Critical Fixes):
  • Fixed Z₀: Now uses exact component count / n (was incorrect eigenvalue-based)
  • Fixed Gap: Proper Fiedler value approximation via Rayleigh quotient (was largest eigenvalue)
  • Fixed RΛ: Shared baseline V_target for filtered ensemble (was incomparable)
  • Fixed energy_per_node: Exact trace(L) = 2|E| / n (was partial eigenvalue sum)
  • Fixed entropy: Total degree entropy with robust drift calculation (denominator floor)
  • Fixed lattice generator: Uses actualN to avoid isolated nodes when n not perfect square
  • Fixed small-world: True rewiring (removes old edge before adding new)