Closes #017 - Fix KaTeX rendering corruption and inject detailed blueprint contents
This commit is contained in:
Antigravity Agent
@@ -62,15 +62,15 @@ export default function CryptoMathModal({ isOpen, onClose }: CryptoMathModalProp
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</p>
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<div className="grid grid-cols-3 gap-3 text-xs text-slate-400 font-mono text-center">
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<div className="bg-slate-950/40 p-3 rounded-lg border border-slate-800/50">
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<span className="block text-rose-400 font-bold">State 1 (S1)</span>
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<span className="block text-rose-400 font-bold">State 1 (<InlineMath math="S_1" />)</span>
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<span>Bearish Squeeze / Crackdown</span>
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</div>
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<div className="bg-slate-950/40 p-3 rounded-lg border border-slate-800/50">
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<span className="block text-slate-300 font-bold">State 2 (S2)</span>
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<span className="block text-slate-300 font-bold">State 2 (<InlineMath math="S_2" />)</span>
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<span>Consolidation / Mean Reversion</span>
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</div>
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<div className="bg-slate-950/40 p-3 rounded-lg border border-slate-800/50">
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<span className="block text-emerald-400 font-bold">State 3 (S3)</span>
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<span className="block text-emerald-400 font-bold">State 3 (<InlineMath math="S_3" />)</span>
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<span>Parabolic Bull Run</span>
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</div>
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</div>
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@@ -83,9 +83,9 @@ export default function CryptoMathModal({ isOpen, onClose }: CryptoMathModalProp
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Calculates transition probabilities over rolling 90-day return vectors:
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</p>
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<div className="bg-slate-950/40 p-4 rounded-xl border border-slate-800/60 my-2">
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<BlockMath math="P = \\begin{bmatrix} p_{11} & p_{12} & p_{13} \\\\ p_{21} & p_{22} & p_{23} \\\\ p_{31} & p_{32} & p_{33} \\end{bmatrix}" />
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<BlockMath math="P = \begin{bmatrix} p_{11} & p_{12} & p_{13} \\ p_{21} & p_{22} & p_{23} \\ p_{31} & p_{32} & p_{33} \end{bmatrix}" />
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<p className="text-[11px] text-slate-400 font-mono mt-2 text-center">
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{"where "}<InlineMath math="p_{ij} = P(X_{t+1} = S_j \\mid X_t = S_i)" />{" represents the frequency probability of moving from State i to State j."}
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where <InlineMath math="p_{ij} = P(X_{t+1} = S_j \mid X_t = S_i)" /> represents the frequency probability of moving from State <InlineMath math="S_i" /> to State <InlineMath math="S_j" />.
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</p>
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</div>
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</div>
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@@ -97,13 +97,13 @@ export default function CryptoMathModal({ isOpen, onClose }: CryptoMathModalProp
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When external alpha inputs (e.g. Funding Rate anomalies, Whale inflows) occur, state probabilities are updated using Bayes' theorem:
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</p>
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<div className="bg-slate-950/40 p-4 rounded-xl border border-slate-800/60 my-2">
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<BlockMath math="P(S_i \\mid \\text{Alpha}) = \\frac{P(\\text{Alpha} \\mid S_i) \\times P(S_i)}{\\sum_{j=1}^3 P(\\text{Alpha} \\mid S_j) \\times P(S_j)}" />
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<BlockMath math="P(S_i \mid \text{Alpha}) = \frac{P(\text{Alpha} \mid S_i) \times P(S_i)}{\sum_{j=1}^3 P(\text{Alpha} \mid S_j) \times P(S_j)}" />
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<p className="text-[11px] text-slate-400 mt-2 font-mono leading-relaxed">
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{"Where:"}
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Where:
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<br />
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{"- "}<InlineMath math="P(S_i)" />{" is the prior state probability from the Markov transition matrix."}
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- <InlineMath math="P(S_i)" /> is the prior state probability from the Markov transition matrix.
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<br />
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{"- "}<InlineMath math="P(\\text{Alpha} \\mid S_i)" />{" is the conditional likelihood of observing this whale spike / funding squeeze in State i."}
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- <InlineMath math="P(\text{Alpha} \mid S_i)" /> is the conditional likelihood of observing this whale spike / funding squeeze in State <InlineMath math="S_i" />.
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</p>
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</div>
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</div>
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@@ -125,7 +125,7 @@ export default function CryptoMathModal({ isOpen, onClose }: CryptoMathModalProp
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<p className="text-xs text-slate-400 leading-relaxed pt-2">
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The Random Forest ensemble evaluates 10 non-linear decision trees to output the short-term and medium-term trend probabilities:
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</p>
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<BlockMath math="P_{\\text{ML}} = \\frac{1}{M} \\sum_{m=1}^{M} T_m(FR_t, OI_t, LS_t, W_t)" />
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<BlockMath math="P_{\text{ML}} = \frac{1}{M} \sum_{m=1}^{M} T_m(FR_t, OI_t, LS_t, W_t)" />
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</div>
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</div>
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@@ -138,23 +138,23 @@ export default function CryptoMathModal({ isOpen, onClose }: CryptoMathModalProp
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<div className="bg-slate-950/40 p-5 rounded-2xl border border-slate-850 space-y-4">
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<div>
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<p className="text-xs text-slate-350 mb-2 font-semibold">1. Prior Distribution (Accuracy History):</p>
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<BlockMath math="\\theta \\sim \\text{Beta}(\\alpha_{\\text{prior}}, \\beta_{\\text{prior}})" />
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<BlockMath math="\theta \sim \text{Beta}(\alpha_{\text{prior}}, \beta_{\text{prior}})" />
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<p className="text-[10px] text-slate-500 font-mono">
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where <InlineMath math="\\alpha_{\\text{prior}}" /> represents historical prediction successes and <InlineMath math="\\beta_{\\text{prior}}" /> represents false alarms.
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where <InlineMath math="\alpha_{\text{prior}}" /> represents historical prediction successes and <InlineMath math="\beta_{\text{prior}}" /> represents false alarms.
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</p>
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</div>
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<div>
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<p className="text-xs text-slate-350 mb-2 font-semibold">2. Likelihood Formulation (Binomial pseudo-observations):</p>
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<BlockMath math="P(x \\mid \\theta) = \\binom{w}{k} \\theta^k (1 - \\theta)^{w - k}" />
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<BlockMath math="P(x \mid \theta) = \binom{w}{k} \theta^k (1 - \theta)^{w - k}" />
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<p className="text-[10px] text-slate-500 font-mono">
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where <InlineMath math="k = P_{\\text{ML}} \\times w" /> is the number of simulated successes and <InlineMath math="w - k = (1 - P_{\\text{ML}}) \\times w" /> is the number of simulated failures.
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where <InlineMath math="k = P_{\text{ML}} \times w" /> is the number of simulated successes and <InlineMath math="w - k = (1 - P_{\text{ML}}) \times w" /> is the number of simulated failures.
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</p>
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</div>
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<div>
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<p className="text-xs text-slate-350 mb-2 font-semibold">3. Conjugate Posterior Update:</p>
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<BlockMath math="\\alpha_{\\text{post}} = \\alpha_{\\text{prior}} + k, \\quad \\beta_{\\text{post}} = \\beta_{\\text{prior}} + (w - k)" />
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<BlockMath math="\alpha_{\text{post}} = \alpha_{\text{prior}} + k, \quad \beta_{\text{post}} = \beta_{\text{prior}} + (w - k)" />
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</div>
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</div>
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</div>
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@@ -163,20 +163,20 @@ export default function CryptoMathModal({ isOpen, onClose }: CryptoMathModalProp
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<div className="space-y-3">
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<h4 className="text-xs font-bold text-cyan-400 uppercase tracking-wider font-mono">F. Mathematical Proof of Posterior Mean Integration</h4>
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<p className="text-xs leading-relaxed text-slate-400">
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To resolve a single operational point-estimate from our posterior distribution, we integrate out the continuous parameter <InlineMath math="\\theta" /> to calculate the mathematical expectation of the posterior distribution:
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To resolve a single operational point-estimate from our posterior distribution, we integrate out the continuous parameter <InlineMath math="\theta" /> to calculate the mathematical expectation of the posterior distribution:
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</p>
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<div className="bg-slate-950/40 p-5 rounded-2xl border border-slate-850 space-y-4">
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<p className="text-xs text-slate-350 font-semibold">Posterior Expectation Integral Proof:</p>
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<BlockMath math="\\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" />
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<BlockMath math="\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" />
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<p className="text-xs text-slate-400 leading-relaxed">
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Using the definition of the Beta function <InlineMath math="\\text{B}(a, b) = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}" /> and the recurrence relation <InlineMath math="\\Gamma(x+1) = x\\Gamma(x)" />:
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Using the definition of the Beta function <InlineMath math="\text{B}(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}" /> and the recurrence relation <InlineMath math="\Gamma(x+1) = x\Gamma(x)" />:
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</p>
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<BlockMath math="\\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{\\Gamma(\\alpha_{\\text{post}} + 1)\\Gamma(\\beta_{\\text{post}})}{\\Gamma(\\alpha_{\\text{post}} + \\beta_{\\text{post}} + 1)} \\cdot \\frac{\\Gamma(\\alpha_{\\text{post}} + \\beta_{\\text{post}})}{\\Gamma(\\alpha_{\\text{post}})\\Gamma(\\beta_{\\text{post}})}" />
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<BlockMath math="\\mathbb{E}[\\theta \\mid \\text{Data}] = \\frac{\\alpha_{\\text{post}}\\Gamma(\\alpha_{\\text{post}})\\Gamma(\\beta_{\\text{post}})}{(\\alpha_{\\text{post}} + \\beta_{\\text{post}})\\Gamma(\\alpha_{\\text{post}} + \\beta_{\\text{post}})} \\cdot \\frac{\\Gamma(\\alpha_{\\text{post}} + \\beta_{\\text{post}})}{\\Gamma(\\alpha_{\\text{post}})\\Gamma(\\beta_{\\text{post}})} = \\frac{\\alpha_{\\text{post}}}{\\alpha_{\\text{post}} + \\beta_{\\text{post}}}" />
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<BlockMath math="\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{\Gamma(\alpha_{\text{post}} + 1)\Gamma(\beta_{\text{post}})}{\Gamma(\alpha_{\text{post}} + \beta_{\text{post}} + 1)} \cdot \frac{\Gamma(\alpha_{\text{post}} + \beta_{\text{post}})}{\Gamma(\alpha_{\text{post}})\Gamma(\beta_{\text{post}})}" />
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<BlockMath math="\mathbb{E}[\theta \mid \text{Data}] = \frac{\alpha_{\text{post}}\Gamma(\alpha_{\text{post}})\Gamma(\beta_{\text{post}})}{(\alpha_{\text{post}} + \beta_{\text{post}})\Gamma(\alpha_{\text{post}} + \beta_{\text{post}})} \cdot \frac{\Gamma(\alpha_{\text{post}} + \beta_{\text{post}})}{\Gamma(\alpha_{\text{post}})\Gamma(\beta_{\text{post}})} = \frac{\alpha_{\text{post}}}{\alpha_{\text{post}} + \beta_{\text{post}}}" />
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<p className="text-xs text-slate-350 font-semibold pt-2">Expanded Workstation Implementation Formula:</p>
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<BlockMath math="P_{\\text{Posterior}} = \\frac{\\alpha_{\\text{prior}} + (P_{\\text{ML}} \\times w)}{\\alpha_{\\text{prior}} + \\beta_{\\text{prior}} + w}" />
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<BlockMath math="P_{\text{Posterior}} = \frac{\alpha_{\text{prior}} + (P_{\text{ML}} \times w)}{\alpha_{\text{prior}} + \beta_{\text{prior}} + w}" />
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<p className="text-[10px] text-slate-500 font-mono leading-relaxed">
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This formulation ensures that if the prior model is highly accurate (large <InlineMath math="\\alpha_{\\text{prior}}" />), the raw ML signal is smoothed towards historical baseline expectations. If historical errors are high, the prior variance restricts overreaction to noisy signals.
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This formulation ensures that if the prior model is highly accurate (large <InlineMath math="\alpha_{\text{prior}}" />), the raw ML signal is smoothed towards historical baseline expectations. If historical errors are high, the prior variance restricts overreaction to noisy signals.
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</p>
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</div>
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</div>
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