The mathematical framework behind the OPHI (Symbolic Cognition Engine) is designed to facilitate a stable, governed intelligence loop characterized as Experience → Error → Adaptation → Memory. Rather than using standard statistical weights found in traditional machine learning, OPHI relies on symbolic drift, entropic modulation, and cryptographic fossilization to regulate its evolution.
1. The Core Ω-Equation
The fundamental state of the OPHI engine is represented by the Ω-equation, which serves as the heart of its symbolic cognition: $$\Omega = (\text{state} + \text{bias}) \times \alpha$$
- State: The current internal representation of the system’s symbolic cognition.
- Bias: A parameter that adjusts predicted perception based on historical patterns.
- Alpha ($\alpha$): A scaling factor used to modulate transformation weight or translate patterns between different domains (e.g., physical vs. abstract).
2. The Learning Signal: Perceptual Drift ($\Delta$)
Learning occurs when the system identifies a divergence between its internal prediction and external reality.
- Normalization: Before processing, raw sensor data (e.g., light, temperature) is mapped to a 0–1 range using the formula: $normalize(val) = \frac{val - min_v}{max_v - min_v}$.
- Prediction vs. Actual: The system computes a prediction using the Ω-equation and compares it to the actual outcome to find the Drift ($\Delta$): $drift = |prediction - outcome|$.
- Model Mutation: If the drift exceeds a specific threshold, the internal model is updated. A common "naive" update in the simulation involves:
- $bias_{new} = bias_{old} + (drift \times 0.1)$.
- $state_{new} = state_{old} + (drift \times 0.05)$.
3. Governance and Damping Mechanisms
To prevent "runaway" behavior where the system overreacts to errors, OPHI employs mathematical constraints to ensure stability.
- Soft Ceiling (Sigmoid Drift): Drift is scaled smoothly to ensure that large errors saturate rather than causing catastrophic updates. The formula is: $$effective_drift = drift_ceiling \times (1 - e^{-k \times \frac{raw_drift}{1 + entropy_accumulator}})$$ This uses a sharpness constant ($k$) and scales the raw drift by an Entropy Accumulator to dampen learning when the system is in a high-entropy state.
- Entropy Decay: To maintain agility, the entropy accumulator "leaks" or decays over time: $entropy = \max(0.0, entropy - decay_rate)$.
- SE44 Gate: This is a hard stability filter. A state can only be committed to memory if it meets these criteria:
- Coherence $\ge 0.985$ (Internal consistency).
- Entropy $\le 0.01$ (Surprise/contradiction).
- RMS Drift $\le 0.001$ (Root Mean Square of recent errors).
4. The Curiosity Engine
OPHI uses intrinsic motivation to decide which sensors or data points to prioritize. The curiosity score is calculated by multiplying uncertainty and novelty: $$Curiosity = \text{Prediction Uncertainty} \times \text{Novelty Score}$$
- Prediction Uncertainty: The standard deviation ($\sigma$) of recent prediction errors recorded in the fossil history.
- Novelty Score: The mathematical dissimilarity (often using Cosine or Euclidean distance) between the current sensor input and historical data.
- Weighted Selection: The curiosity score translates into weights for sensors; the system then performs a weighted random selection to determine its next focus.
5. Memory and Cross-Domain Transfer
The math of OPHI extends to its ability to generalize knowledge across different fields.
- Fossilization: Once a stable state is achieved, it is hashed using SHA-256 to create an immutable record (fossil).
- $\Psi$-Transference Loop: The system extracts a "drift schema"—an abstract representation of a learning pattern—and reapplies it to a new domain by swapping the $\alpha$ value.
- Example: A drift pattern learned from environmental light sensors ($bias_delta = 0.093$) can be applied to a geometric domain to calculate a "drifted triangle" state: $\Omega_{triangle} = (angle_state + 0.093) \times \alpha_{geometry}$.
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