Objective: Demonstrate that τ-relaxation alone cannot uniquely determine final state. Multiple stable outcomes persist despite increased precision, requiring an internal selector κₙ.
System: N nodes in a ring, each with state xᵢ ∈ ℝ. Energy U(x) = Σᵢ[(xᵢ²−1)² + λ(xᵢ−xᵢ₊₁)²]
Key Metric: Wall count W = number of sign changes around ring. Selection saturation = Var(W) plateaus > 0 as precision increases.
Reframed Operator Stack:
Φ → Ψ → τ → κ (internal) → τ (re-closure)
Key Insight: κₙ is NOT "between τ and observables." κₙ is an internal selector that resolves competing τ-stable attractors when τ-flow alone cannot pick a unique continuation.
Each node i has state xᵢ ∈ ℝ. Total energy:
U(x) = Σᵢ[(xᵢ² − 1)² + λ(xᵢ − xᵢ₊₁)²]
Gradient descent with stochastic perturbation:
xᵢ ← xᵢ − η·∂U/∂xᵢ + ε·ξᵢ
where ξᵢ ~ N(0,1) and update order randomized each step.
A domain wall occurs when sign(xᵢ) ≠ sign(xᵢ₊₁). Wall count:
W = |{i : sign(xᵢ) ≠ sign(xᵢ₊₁)}|
Possible sectors: W ∈ {0, 2, 4, 6, ...} (must be even on a ring).
Selection Saturation Criterion:
System exhibits saturation if Var(W) across realizations does NOT vanish when:
Each κₙ implements a different selection rule over competing sectors:
| Selector | Criterion J | Meaning |
|---|---|---|
| κ₁ | min U(s) | Minimal energy |
| κ₂ | min |⟨x⟩| | Maximal symmetry |
| κ₃ | min W | Topology simplification |
| κ₄ | min Σ(xᵢ−xᵢ₊₁)² | Smoothness maximization |
| κ₅ | max basin | Robustness under perturbation |