Why stability does not imply uniqueness
We present the first empirical demonstration that relaxation to stability does not guarantee outcome uniqueness. Chamber κ₀ proves that multiple distinct stable states can persist under arbitrarily fine numerical refinement—a phenomenon we call selection saturation.
The Problem: When Stability ≠ Uniqueness
In computational physics and dynamical systems, relaxation methods are workhorses for finding stable configurations. The implicit assumption is clear: run the simulation long enough, with fine enough precision, and you'll converge to a unique answer.
Chamber κ₀ proves this assumption wrong. We demonstrate a minimal system where:
- τ-relaxation succeeds: Energy converges, gradients vanish, stability is achieved
- Outcomes multiply: Ten distinct topological sectors emerge from identical dynamics
- Precision doesn't help: A 20× refinement in step size leaves outcome variance unchanged
This is not numerical noise. It's selection saturation—a structural property of multi-attractor landscapes that necessitates an internal selector mechanism.
The System: Minimal by Design
We study the simplest possible system exhibiting selection saturation: a ring lattice with double-well dynamics.
Each node i has state xᵢ ∈ ℝ. The energy functional combines:
- (xᵢ² − 1)²: Double-well potential favoring xᵢ ≈ ±1
- λ(xᵢ − xᵢ₊₁)²: Nearest-neighbor coupling (λ = 0.5)
Dynamics proceed via τ-relaxation with gradient descent plus small stochastic perturbation:
The key observable is the wall count W—the number of sign changes around the ring. Each value of W defines a distinct topological sector.
Key Results: Selection Saturation Demonstrated
📊 Empirical Evidence (N=128, λ=0.5, R=100 realizations)
1. Bimodal Basin Structure
Two distinct attractor families emerge from the dynamics:
| Basin | ⟨W⟩ ± σ | Energy ⟨U⟩ ± σ | Realizations |
|---|---|---|---|
| Low-W | 8.5 ± 1.8 | 13.3 ± 2.7 | 40 |
| High-W | 18.1 ± 2.8 | 28.2 ± 4.3 | 60 |
Energy separation ΔU = 14.9 confirms these are distinct metastable states, not numerical artifacts.
2. Selection Saturation: The Critical Finding
We performed a systematic refinement sweep, reducing step size η from 0.1 to 0.005—a 20× increase in precision. If outcome multiplicity were a discretization artifact, variance should vanish. Instead:
✅ Saturation Confirmed
Var(W) at η = 0.1: 5.08 ± 0.78
Var(W) at η = 0.005: 5.11 ± 0.97
Reduction: −0.1% (essentially zero)
Conclusion: Variance does not vanish under refinement. Outcome multiplicity is structural, not numerical.
3. Energy Banding: τ-Closure Verified
Within each sector, energy dispersion is extremely tight (CV ≤ 0.43%), confirming τ-relaxation has converged to stable attractors. The system is not "still searching"—it has found stability. There are simply multiple stable outcomes.
The κ₀ Operator: Internal Selection
These results motivate the κ₀ operator—an internal selector acting after τ-closure:
Different κ₀ selection policies yield different outcomes:
- κ₁ (min energy): Selects W = 6 (Low-W basin)
- κ₂ (min |⟨x⟩|): Selects W = 10 (maximal symmetry)
- κ₃ (min W): Selects W = 6 (topology simplification)
- κ₄ (smoothness): Varies by basin
This demonstrates that "more τ-relaxation" does not resolve the ambiguity. Selection is logically distinct from stability.
How κ₀ Fits the UNNS Operator Stack
Within the UNNS (Unbounded Nested Number Sequences) framework, operators form a stratified hierarchy, each serving a distinct structural role. Chamber κ₀ establishes κ as an internal component of the τ-engine itself, acting before the final Ω projection.
Complete Operator Roles
- Φ (Generability): Determines what structures the recursive substrate can produce. Governs the space of possibilities.
- Ψ (Consistency): Enforces internal coherence. Rejects structures that violate logical or mathematical constraints.
- τ (Stability): Drives relaxation toward equilibria. Ensures convergence to stable configurations.
- κ (Selection): Resolves multiplicity among τ-stable outcomes. Acts after stability is achieved, before projection to observables.
- Ω (Observable Projection): Maps κ-selected states to measurable physical quantities. Bridges substrate dynamics to experimental reality.
Why κ₀ Was Missing
Traditional frameworks conflate stability with uniqueness, assuming τ-relaxation automatically produces a single outcome. They jump directly from τ to Ω, bypassing the selection layer. Chamber κ₀ proves this assumption fails: τ achieves closure, yet multiple stable states persist.
The gap between τ-closure and Ω-projection requires a structural bridge. This is κ's role: not to create new dynamics (Φ's domain), not to enforce consistency (Ψ's domain), not to drive relaxation (τ's domain), not to project to observables (Ω's domain), but to select among competing stable continuations.
Structural Necessity: In multi-attractor landscapes, the operator stack is incomplete without κ. The sequence τ → κ → Ω forms a complete path: Stability + Selection + Projection together determine observable outcomes. Chamber κ₀ provides the empirical proof that this distinction is necessary, not philosophical.
Reframed Operator Flow
The complete flow becomes:
Chamber κ₀ operates at the κ layer within the τ-engine itself—it's not an external post-processing step. Selection happens within the recursive dynamics, after stability but before any physical interpretation (Ω projection) is imposed. This positioning is crucial: κ resolves dynamical ambiguity, while Ω handles observational mapping.
Scientific Validation
Chamber κ₀ meets rigorous validation criteria:
- Cκ0-1 (Sector closure): CV(U|W) ≤ 1% → Achieved (0.43%)
- Cκ0-2 (Multiplicity): ≥4 distinct sectors → Achieved (10 sectors)
- Cκ0-3 (Saturation): Var(W) plateau under refinement → Achieved
All data is reproducible with fixed seeds. The complete ensemble comprises 100 independent realizations with systematic parameter sweeps.
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Implications & Future Directions
Broader Significance
Selection saturation has implications for any system using relaxation-based methods:
- Optimization algorithms: Multiple optima persist regardless of precision
- Neural network training: Different minima accessible from same initialization
- Molecular dynamics: Protein folding may exhibit structural multiplicity
- Pattern formation: Competing stable patterns in reaction-diffusion systems
Next Steps: Chamber Extensions
The Chamber κ₀ framework enables systematic exploration:
- λ-Phase Diagram: Map basin structure vs coupling strength
- N-Scaling: Test continuum limit (N → 512, 1024)
- Chamber κ₁: Multi-field systems with coupled rings
- Selector Taxonomy: Classify and test κ₀ policies systematically
Technical Details
🔧 Experimental Configuration
System: Ring lattice, N = 128 nodes
Coupling: λ = 0.5
Step size: η = 0.005–0.1 (saturation sweep)
Noise: ε = 0.01
Iterations: 500 per realization
Ensemble: R = 100 (5 datasets × 20 runs)
Seeds: Deterministic (reproducible)
Conclusion
Chamber κ₀ provides the first empirical demonstration that stability does not guarantee uniqueness in relaxation dynamics. Through systematic validation with 100 independent realizations and 20× precision refinement, we've proven that selection saturation is a structural property of multi-attractor systems.
This necessitates the κ₀ operator as an internal post-closure selector—establishing a fundamental distinction between τ-relaxation (which achieves stability) and κ-selection (which resolves outcome multiplicity).
The framework is purely dynamical, requires no physical interpretation, and provides a foundation for studying selection mechanisms in recursive systems across domains.
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Chamber κ₀ | UNNS Laboratory | 2026
Selection Saturation in τ-Relaxed Dynamical Systems