UNNS Laboratory — Chamber XXXV (INLINE)

Ensemble Configuration

Ensemble Size M Inherited from XXXIV

Node Count n Inherited from XXXIV

Generator Family Locked from XXXIV

RNG Seed

Ω4b Configuration

Keep Fraction f 0.30 (LOCKED)

V Target Mode Canonical Ω4b

τ Operator Selection

τ Family

Iterations T

Step Size λ 0.05

Constraint Strength μ 0.02

εᵥ Lock (V tolerance)

Admissibility Guardrails

Δ (min residual improvement)

δ (max invariant drift)

Acceptance Band [a, b]

Run Controls

Pipeline Flow

Size: —

V̄: —

R_Λ: —

→

Ω4b(E)

Size: —

V̄: —

R_Λ: —

Accept: —

→

τ(E_Ω)

V̄: —

R_Λ: —

CR: —

Visualization

V Distribution Evolution

R_Λ Contraction Path

Core Metrics

Baseline R_Λ

—

Post-Ω R_Λ

—

Post-τ R_Λ

—

Contraction Ratio

—

Acceptance Rate

—

Max Invariant Drift

—

Protected Macro-Invariants

Invariant	Baseline	Post-τ	Drift %	Status
Scaled Spectral Radius	—	—	—	—
Energy per Node	—	—	—	—
Degree Entropy	—	—	—	—

Run Log

Chamber XXXV initialized. Awaiting run...

▶

📚 Laboratory Guide

Chamber XXXV: Ω→τ Coupled Pipeline

Purpose: This chamber tests whether τ-operators become admissible only after Ω4b selection by measuring residual contraction, macro-invariant protection, and τ-locking windows.

✅ Validated Passing Seeds

The following seeds have been validated with these exact settings:

⚙️ Configuration Used

τ Family:	τB: Spectral Band-Limiter
Iterations T:	10
Step Size λ:	0.05
Constraint Strength μ:	0.02
εᵥ Lock (V tolerance):	0.05
Δ (min residual improvement):	0.002
δ (max invariant drift):	0.05
Acceptance Band [a, b]:	0.2, 0.5

💡 Use these exact settings to reproduce the results below

🟢 Small Seeds (Direct Mapping)

Seed	CR	Status
137042	—	Canonical (Phase F default)
1640	6.3%	✓ PASS
1671	5.6%	✓ PASS
1811	8.9%	✓ PASS
4850	20.2%	✓ PASS
137044	28.1%	✓ PASS (high CR)
588148	51.8%	✓ PASS (marginal)

CR = Contraction Ratio (RL_tau / RL_baseline). Lower is better.

🔷 Large Seeds (FNV-1a Hashed)

Seeds larger than 2^32-1 are hashed deterministically. Tested with same configuration above:

Raw Input	→ Hash	CR
5125458795456987457895426700	1306975018	13.0%
4569874525895424700	3199416829	5.9%
5125458795456987457895429900	602815471	23.9%

All large seeds passed with the configuration above. Each produces a unique, reproducible ensemble.

💡 Quick Start (Default Settings Loaded):

Default settings above are already loaded in the UI
Just change the seed to 1811 → expect PASS with CR ≈ 8.9%
Or try 137044 for strong contraction → expect CR ≈ 28%
Experiment with large seeds to test hash mapping
Compare τ_B vs τ_E performance on same seed

Theoretical Foundation

The Ω→τ coupling hypothesis asserts that certain geometric operators (τ) act as stabilizers only when applied to ensembles pre-selected by Ω4b. The pipeline evaluates:

E → Ω4b(E): Canonical bandpass selection (fixed from Chamber XXXIV)
Ω4b(E) → τ(E_Ω): Geometric stabilization under test
Admissibility criteria: Residual contraction + invariant protection + nondegeneracy

τ-Operator Families

τA: Laplacian Heat-Flow — Diffusive smoothing via discrete heat equation

τB: Spectral Band-Limiter — Mode filtering without target shift (100% pass rate tested)

τC: Divergence Minimizer — Constrained flux optimization

τD: Curvature Equalizer — Local curvature variance reduction

τE: Multi-Scale — Coarse-grain + refine pipeline (75% pass rate tested)

Admissibility Criteria

A τ-operator is admissible post-Ω4b if it satisfies ALL of:

Ω-fixedness: Does not modify V_target or re-run Ω
Residual contraction: R_Λ(τ(E_Ω)) ≤ R_Λ(E_Ω) - Δ
Macro-invariant protection: All protected drifts ≤ δ
Nondegeneracy: Ω acceptance rate ∈ [a, b]
Structural well-posedness: τ is deterministic and maps structures to structures

Seed Specifications

Supported Range: 0 to 2^32-1 (direct) or any integer (hashed)

Safe Input: Integers < 2^53-1 for optimal precision

Large Seeds: >2^32-1 automatically hashed via FNV-1a

Reproducibility: Same seed → identical results (verified)

Example	Behavior	Status
137042	Direct uint32 mapping	✓ Good
5125458...429900	Hashed to 602815471	⚠️ Hashed
3.14159	String-hashed (non-integer)	⚠️ Hashed
NaN / Infinity	Invalid input rejected	✗ Avoid

Verdict Logic

Verdict	Meaning
PASS	τ is admissible — genuine post-Ω stabilizer
FAIL (residual)	No residual contraction beyond Δ
FAIL (invariant)	Protected invariant drift exceeds δ
FAIL (no lock)	No stable parameter window found
INVALID	Upstream Ω conditions not met

Operational Protocol

Verify Ω4b lock: Ensemble configuration from XXXIV (read-only)
Select τ family: Choose operator and configure parameters
Set guardrails: Δ, δ, [a,b] thresholds
Run pipeline: E → Ω4b(E) → τ(E_Ω)
Evaluate verdict: Explicit PASS/FAIL with diagnostics
Export results: Full JSON with reproducibility signature

⚠️ Critical Constraints:

Ω4b parameters are LOCKED — τ cannot modify or re-run Ω
V_target is canonical — must remain fixed throughout
Verdicts are binary — no probabilistic or averaged outcomes
All τ randomness must be seed-controlled
Protected invariants are guarded — violations trigger FAIL

🔬 Scientific Significance

Why This Matters:

Coupling validation: Tests whether Ω→τ is a genuine two-layer architecture
Admissibility framework: Establishes rigorous criteria for post-selection operators
Falsifiability: Explicit rejection modes prevent circular validation
Phase F foundation: Connects Ω-stratum (XXXIV) to geometry operators
Complementarity discovery: τ_B and τ_E show regime-dependent performance

Version: 1.0.1 (seedfix) | Engine: GraphEnsembleEngine | Phase: F (Coupled Operators) | Status: Production Ready