Siglet-Qubits — Part 2: Mathematical Foundations & Simulation Methodology

In this second entry of the Siglet-Qubits series, we leave philosophical framing behind and move into formal structures—grounding the concept of the siglet through precise definitions, loss functions, and simulation flows. This is where symbolic form meets temporal evolution, and ethics begins to emerge rather than be imposed.

What Is a Siglet?

At its core, a siglet is a dynamic symbolic unit described by five essential properties:

• Resonance weight (r): symbolic salience bounded in [-1, 1]
• Temporal coherence (τ): the persistence of signal over time
• Ethical alignment vector (ε): a direction in latent ethical space
• Modality coherence score (μ): consistency across expressive forms (text, audio, gesture, etc.)
• Compression efficiency (c): the symbolic information per unit complexity

Together, these form the identity of a siglet, which evolves in quantum-symbolic space under epistemically humble, physically grounded rules.

Siglet Truth Potential Function

We model a siglet’s symbolic coherence across time via its truth potential:

def truth_potential(siglet, t):
    r, tau, epsilon_norm, mu, c = siglet
    noise = np.random.normal(0, 0.01)
    return r * (tau + noise)**t * epsilon_norm * mu * c

This function integrates decay, resonance, and multimodal integrity under perturbation. High values indicate preserved coherence and ethical salience. Low values suggest symbolic decay or ethical drift.

RealitySeekerNet Loss

To simulate siglet evolution under symbolic pressure, we introduce the RealitySeekerNet loss:

• Penalizes weak resonance and low persistence
• Tracks divergence from emergent ethical anchors (ε̂)
• Measures modality mismatch (μ̂)

L(s_i) = α(1 - r) + β(decay(τ)) + γ||ε - ε̂|| + δ|μ - μ̂|

This is not a traditional loss—it guides evolution, allowing local chaos while favoring global convergence toward ethical attractors.

From Classical Sim to Quantum Readiness

The system initializes siglets from semantically agnostic Pauli operator spaces:

ε ∈ {I, σ_x, σ_y, σ_z}^⊗n

These vectors serve as seeds for ethical emergence—not labels. Future versions will embed these directly into quantum circuits using Hermitian operators, paving the way for quantum-native ethical cognition.

Simulation Architecture

We simulate siglet trajectories over time using:

• Truth-decay computation over discrete timesteps
• Dynamic Time Warping (DTW) to compare shape-based similarity
• Spectral Clustering to find emergent ethical groupings

These clusters are evaluated for:

• Noise robustness (±2% perturbation)
• Shape separability
• Ethical axis drift tolerance

Hybrid Enumeration + Bayesian Refinement

To identify stable ethical primitives:

1. Enumerate combinatorial siglet groups
2. Use Optuna for Bayesian refinement
3. Maximize cluster stability and separability

study.optimize(objective, n_trials=trial_count)

The goal: discover ethical observables without semantic preloading—just raw symbolic behavior.

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In Part 3, we’ll explore how these clusters form, behave, and stabilize—charting the birth of emergent ethical modalities. The experimental results section reveals the first glimpse into ethically meaningful dynamics… discovered, not declared.