Conditional Symmetry Emergence:
The κ-Series Discovery
How symmetry-based selection reveals that not all symmetries are created equal
The Problem: When Stability Isn't Enough
Imagine running a physical simulation to equilibrium. You expect a unique, stable outcome. Instead, you get 20 different stable states—all equally valid by the dynamics, yet fundamentally different in structure.
This is the phenomenon of selection saturation: the point where increasing computational precision no longer resolves outcome multiplicity. You've converged perfectly... to a set, not a point.
The question becomes: if dynamics can't decide, what does?
Traditional physics would suggest energy minimization. But what if multiple states have similar energies? What if the discriminator isn't energy at all, but structure?
Our first chamber, κ₀, systematically demonstrates this phenomenon using a minimal ring-lattice system with double-well potential. Across 100 independent realizations with identical dynamics, we observe persistent multiplicity: variance in final states plateaus despite increased precision.
Result: τ-relaxation alone cannot uniquely determine outcomes. An internal selector is necessary.
The Solution: Symmetry-Based Selection
If dynamics won't decide, we need a selection principle that operates after relaxation. Enter the κ₁ selector.
The key insight: don't modify the physics—measure what already exists. Every stable state has intrinsic structural properties we can quantify through symmetry measures:
The selector operates via a simple principle: κ[Σᵢ] = argmin Σᵢ(s)—select the state that minimizes symmetry measure Σᵢ.
Crucially, this selector:
- ✅ Does not modify τ-dynamics
- ✅ Does not generate new states
- ✅ Does not introduce energy preferences
- ✅ Operates purely on structural properties
Chamber κ₁ implements this selection framework and applies it to κ₀ ensembles. It computes all four symmetry measures across the stable states and tests selection policies—pure, lexicographic, and weighted combinations.
Result: Different symmetry measures select different states. But not all symmetries discriminate effectively...
Key Discoveries: What We Found
Across all 100 τ-stable states tested, chirality (Σ₄) showed complete degeneracy:
Unique values: 1/100 (all states identical)
Range: [0, ~10⁻¹⁵] (numerical noise only)
Interpretation: In symmetric recursive dynamics on a ring lattice, there is no spontaneous chiral symmetry breaking. In this symmetric ring system, Σ₄ remains null under the implemented update rules and boundary conditions. Chirality would require external forcing, asymmetric coupling, or multi-field interactions—none of which exist in this system.
This falsifies the assumption that chirality automatically emerges in relaxed systems.
While chirality vanished, reflection asymmetry (Σ₂) emerged as the sole strong discriminator:
| Measure | Gap | Status |
|---|---|---|
| Σ₂ (Reflection) | 75.6 – 130.3 | STRONG |
| Σ₁ (Mean Bias) | 0.498 | Weak |
| Σ₃ (Spectral) | 0.594 | Weak |
| Σ₄ (Chirality) | ~0 | NULL |
Crucially: Σ₂ is energy-agnostic. It selects states with similar energy to the minimum-energy state (ΔU ≈ 0), not higher or lower. This means structural discrimination is independent of energy minimization.
By detecting null symmetry axes empirically and filtering them out, we developed κ₁′ (kappa-one-prime)—an adaptive selector that auto-filters non-discriminative measures:
Result: +14.3% determinism improvement (85.7% → 100%) without modifying dynamics. The improvement replicates exactly across both R=20 and R=100 ensembles.
This demonstrates that determinism is conditional on symmetry relevance, not on dynamics.
Negative control experiments revealed an anti-intuitive energy correlation:
- Σ₁ (mean bias): Lower symmetry → higher energy (ΔU = +3.13)
- Σ₂ (reflection): No energy correlation (ΔU ≈ 0)
- Σ₃ (spectral): Lower symmetry → higher energy (ΔU = +3.20)
This falsifies the common assumption that "more symmetric = more stable." In double-well systems, asymmetric population of the wells can be energetically favorable. The minimum-energy state has non-zero mean bias.
Symmetry selection is a structural principle, not a thermodynamic one.
Experimental Validation
All findings replicate across ensemble sizes:
| Finding | R=20 | R=100 | Status |
|---|---|---|---|
| Σ₄ Null (chirality) | Var < 10⁻³¹ | Var < 10⁻³¹ | ✓ Replicated |
| Σ₂ Strong (reflection) | Gap = 75.6 | Gap = 130.3 | ✓ Replicated |
| κ₁ Determinism | 85.7% | 85.7% | ✓ Replicated |
| κ₁′ Determinism | 100% | 100% | ✓ Replicated |
| Active axes | Σ₂, Σ₃ | Σ₂, Σ₃ | ✓ Consistent |
| Null axes | Σ₁, Σ₄ | Σ₁, Σ₄ | ✓ Consistent |
Configuration: Ring size N=64, coupling λ=0.5, relaxation T=500 iterations, ensembles R=20 and R=100 realizations.
Implications & Significance
For Physics
This work challenges the assumption that symmetry emerges automatically from equilibration. Instead, we show that:
- Symmetry relevance is contextual—what emerges depends on system specifics, not universal laws
- Selection is separate from dynamics—stability doesn't determine which stable state is realized
- Null symmetries are meaningful outcomes—the absence of a symmetry is as informative as its presence
For Complex Systems
The κ-series methodology demonstrates a general principle: measure relevance before assuming it. This applies beyond physics:
- Machine learning: Not all features discriminate—adaptive feature selection improves determinism
- Optimization: Multiple local minima may differ structurally, not energetically
- Biology: Selection pressures may operate on structure independent of fitness
For UNNS Framework
This establishes the κ-layer as a distinct operator family in recursive dynamics:
- τ-layer: Relaxation dynamics → establishes stability
- κ-layer: Internal selection → resolves multiplicity
- Ω-layer: (Future) Observability constraints
Explore the Chambers
All experimental chambers are publicly accessible with full interactivity:
Publication
Authors: UNNS Research Collective
Published: January 2025
Type: Technical Report
The full paper presents the theoretical framework, experimental methodology, and complete validation results across 100+ τ-stable states.
Download PDFLooking Forward
The κ-series establishes a foundation for understanding selection mechanisms in complex systems. Future directions include:
- Multi-seed validation (seeds 41-45): Test universality of Σ₄-null across initial conditions
- 2D extension (κ₁.1): Test chirality emergence with boundary effects and topology changes
- κ₂ development: Conditional selectors using Σ₂ conditioned on topological parity
- Ω-layer integration: Connect selection to observability constraints
The discovery that chirality does not emerge spontaneously in symmetric recursive dynamics represents a falsification of an implicit assumption held across multiple domains. More broadly, the principle that symmetry relevance must be measured empirically applies whenever we encounter multiple stable outcomes.
Not all symmetries are created equal. Some emerge. Some don't. The difference matters.