Closes #013 - Refactor Crypto Bayes Engine and expand quantitative handbook

QUANT_ROADMAP.md

@@ -166,6 +166,37 @@ Where:
 
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+### V. Crypto Bayesian Markov & Self-Correcting Engine
+
+Integrates a two-stage predictive mapping pipeline for cryptocurrency assets (BTC, ETH, SOL) that combines on-chain derivatives data with online Bayesian updates.
+
+#### 1. Machine Learning Random Forest Classifier
+Ensemble of 10 decision trees mapping four features to forecast trend probabilities:
+*   **Funding Rates (FR)**: Future leverage balance indicators.
+*   **Open Interest (OI) Volatility**: Contract buildup velocities.
+*   **Long/Short (LS) Retail Skew**: Sentiment extreme markers.
+*   **Whale Inflows (W)**: Cold-wallet transfer proxy metrics.
+
+$$P_{\text{ML}} = \frac{1}{M} \sum_{m=1}^{M} T_m(FR_t, OI_t, LS_t, W_t)$$
+
+#### 2. Conjugate Beta-Binomial Update
+Continuous ML probability outputs ($P_{\text{ML}}$) are mapped into a discrete Binomial likelihood by defining the Trust-Weight Hyperparameter ($w=12$) as the Effective Sample Size (ESS):
+*   **Prior distribution**: $\theta \sim \text{Beta}(\alpha_{\text{prior}}, \beta_{\text{prior}})$
+*   **Binomial Likelihood pseudo-observations**: $k = P_{\text{ML}} \times w$ (successes), $w - k = (1 - P_{\text{ML}}) \times w$ (failures)
+*   **Conjugate Posterior Update**:
+    $$\alpha_{\text{post}} = \alpha_{\text{prior}} + k$$
+    $$\beta_{\text{post}} = \beta_{\text{prior}} + (w - k)$$
+
+#### 3. Posterior Mean Integration Proof
+Integrating the continuous parameter $\theta$ out of the posterior distribution gives the mathematical expectation of the posterior:
+$$\mathbb{E}[\theta \mid \text{Data}] = \int_{0}^{1} \theta \cdot P(\theta \mid \text{Data}) \, d\theta = \int_{0}^{1} \theta \cdot \frac{\theta^{\alpha_{\text{post}}-1}(1-\theta)^{\beta_{\text{post}}-1}}{\text{B}(\alpha_{\text{post}}, \beta_{\text{post}})} \, d\theta$$
+$$\mathbb{E}[\theta \mid \text{Data}] = \frac{\text{B}(\alpha_{\text{post}} + 1, \beta_{\text{post}})}{\text{B}(\alpha_{\text{post}}, \beta_{\text{post}})} = \frac{\alpha_{\text{post}}}{\alpha_{\text{post}} + \beta_{\text{post}}}$$
+
+#### 4. Expanded Workstation Formula
+$$P_{\text{Posterior}} = \frac{\alpha_{\text{prior}} + (P_{\text{ML}} \times w)}{\alpha_{\text{prior}} + \beta_{\text{prior}} + w}$$
+
+---
+
 ## 5. Multi-Regime Transition Classifier
 
 The core cognitive brain of the sandbox dynamically adjusts allocation weights across our portfolio modules based on estimated macroeconomic and market states.