Kai — Quantitative rule research

Educational payoff ladder only. No verified walk-forward data supplied. Recalculation of the source probabilities gives E[R] = -0.11350 R, not the positive expectancy claimed by the original; no profitable edge is established.

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## 2. Mathematical Definition of the 5-State Ladder

Let total base risk on the setup be $1R$ (e.g., $150 on a $10,000 balance at 1.5% risk).  
Each leg represents $0.25R$ of risk ($w_i = 0.25$ for $i \in \{1, 2, 3, 4\}$).

$$\text{Initial Stop Loss} = -1,000 \text{ pts across all legs} = 4 \times (-250 \text{ pts}) = -1,000 \text{ pts} = -1.00R$$

| State | Event Description | Points Captured / Leg | Calculation & Payoff | Net R-Multiple |
|---|---|---|---|---|
| **State 0** | Full Stop-Out (All 4 legs hit SL) | $L_{1..4}: -1000$ | $4 \times (-1000 \times 0.25) / 1000$ | **$-1.000R$** |
| **State 1** | TP1 Hit (+300 pts), Legs 2–4 hit SL | $L_1: +300, L_{2..4}: -1000$ | $(+300 - 3000) / 4000$ | **$-0.675R$** |
| **State 2** | TP1 & TP2 Hit (+900 pts), Legs 3–4 hit SL | $L_1: +300, L_2: +600, L_{3..4}: -1000$ | $(+900 - 2000) / 4000$ | **$-0.275R$** |
| **State 3** | TP1–3 Hit (+1800 pts), Leg 4 hit SL | $L_1: +300, L_2: +600, L_3: +900, L_4: -1000$ | $(+1800 - 1000) / 4000$ | **$+0.200R$** |
| **State 4** | Full 4-TP Completion | $L_1: +300, L_2: +600, L_3: +900, L_4: +1200$ | $(+3000) / 4000$ | **$+0.750R$** |