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# John Ehlers' Laguerre Stochastic Slow Oscillator Pro (v3.00)
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Quantitative Time-Warped Stochastic & Dual-Smoothed Momentum Oscillator Suite
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---
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## 1. Summary (Introduction)
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**Laguerre Stoch Slow Pro** is an institutional-grade non-linear momentum oscillator developed by aerospace engineer and quantitative trading pioneer John Ehlers.
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In classical technical analysis, George Lane's Stochastic Oscillator evaluates the closing price relative to a fixed $N$-bar price range ($H_N - L_N$), introducing significant phase lag and false boundary pinning during strong trends. **John Ehlers resolved this by replacing historical price lookback with four orthogonal Laguerre state registers ($L_0, L_1, L_2, L_3$)**:
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* **Zero Historical Lookback Overhead:** The Stochastic envelope is derived dynamically from the instantaneous dispersion of the four Laguerre polynomial stages.
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* **Dual-Stage Moving Average Smoothing:** Filters raw stochastic micro-noise by applying a Slowing Moving Average to generate **Slow %K** and a Signal Moving Average to generate **Signal %D**.
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* **Near-Zero Phase Lag Inflections:** The time-warped all-pass transfer function enables the oscillator to recognize market cyclical turns with virtually zero phase delay.
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```text
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┌────────────────────────────────────────────────────────────────────────┐
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│ LAGUERRE STOCHASTIC SLOW SYSTEM │
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├──────────────────────┬────────────────────────┬────────────────────────┤
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│ Line Plot │ Color Code │ Core Role │
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├──────────────────────┼────────────────────────┼────────────────────────┤
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│ Slow %K Main Line │ clrDodgerBlue (Width:2)│ Adaptive Momentum Line │
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│ Signal %D Line │ clrCoral (Width:1) │ Smoothed Trigger Line │
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└──────────────────────┴────────────────────────┴────────────────────────┘
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```
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### Key Capabilities
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* **Time-Warped 4-Element Stochastic Normalization:** Computes %K strictly from the internal envelope formed by $L_0, L_1, L_2,$ and $L_3$.
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* **Dual-Engine Smoothing Flexibility:** Independent moving average algorithms for %K Slowing and %D Signal smoothing (supports SMA, EMA, SMMA, LWMA, TMA, DEMA, TEMA, and Volume-Weighted VWMA).
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* **Harmonic Fibonacci Damping Control ($\gamma$ – Gamma):** Aligns filter sensitivity with golden ratio proportions from ultra-sensitive scalping ($\gamma = 0.236$) to macro trend smoothing ($\gamma = 0.882$).
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* **Unified 2026 MTF Framework:** Higher-timeframe Laguerre Stochastic curves (e.g., H1 or H4) map onto lower-timeframe execution charts (M1, M5, M15) with flat, non-warping steps via `DataSync_Tools.mqh`.
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* **Synthetic Heikin Ashi Support:** Computes smoothed momentum from filtered Heikin Ashi candles via `CLaguerreStochSlowCalculator_HA` composition.
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---
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## 2. Mathematical Foundations & Laguerre Stochastic Mechanics
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```text
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RAW PRICE (Market Highs, Lows & Closes)
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│
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▼
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┌────────────────────────────────────────────────────────┐
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│ 4-Stage Orthogonal Laguerre State Engine │
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│ L0(t) = (1 - γ)·P(t) + γ·L0(t - 1) │
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│ L1(t) = -γ·L0(t) + L0(t - 1) + γ·L1(t - 1) │
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│ L2(t) = -γ·L1(t) + L1(t - 1) + γ·L2(t - 1) │
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│ L3(t) = -γ·L2(t) + L2(t - 1) + γ·L3(t - 1) │
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└────────────────────────────┬───────────────────────────┘
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│
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▼
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Highest = max(L0, L1, L2, L3)
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Lowest = min(L0, L1, L2, L3)
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Raw %K = [ (L0 - Lowest) / (Highest - Lowest) ] × 100
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│
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▼
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Slow %K = MA(Raw %K, Period = Slowing, Method = Slow_MA)
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Signal %D = MA(Slow %K, Period = Signal, Method = Sig_MA)
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```
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### 2.1. The 4-Element Laguerre State Engine
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Given input price $P_t$ and damping coefficient $\gamma = \text{InpGamma}$ ($0.0 \le \gamma \le 1.0$):
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$$L_0(t) = (1 - \gamma) P_t + \gamma L_0(t-1)$$
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$$L_1(t) = -\gamma L_0(t) + L_0(t-1) + \gamma L_1(t-1)$$
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$$L_2(t) = -\gamma L_1(t) + L_1(t-1) + \gamma L_2(t-1)$$
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$$L_3(t) = -\gamma L_2(t) + L_2(t-1) + \gamma L_3(t-1)$$
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---
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### 2.2. Dynamic Laguerre Envelope Extrema & Raw %K
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The instantaneous highest and lowest boundaries across the 4 orthogonal stages are defined as:
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$$\text{Highest}_t = \max \left( L_0(t), L_1(t), L_2(t), L_3(t) \right)$$
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$$\text{Lowest}_t = \min \left( L_0(t), L_1(t), L_2(t), L_3(t) \right)$$
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$$\text{Range}_t = \text{Highest}_t - \text{Lowest}_t$$
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The position of the primary stage ($L_0$) within this envelope yields the Raw %K:
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$$\text{Raw \%K}_t = \begin{cases} \left( \frac{L_0(t) - \text{Lowest}_t}{\text{Range}_t} \right) \times 100, & \text{if } \text{Range}_t > 10^{-9} \\ \text{Raw \%K}_{t-1}, & \text{if } \text{Range}_t = 0 \end{cases}$$
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---
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### 2.3. Slow %K (Main Line) & %D (Signal Line) Dual Smoothing
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$$\text{Slow \%K}_t = \mathcal{MA}\left( \text{Raw \%K}, \text{Period} = \text{InpSlowingPeriod}, \text{Type} = \text{InpSlowingMethod} \right)$$
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$$\text{Signal \%D}_t = \mathcal{MA}\left( \text{Slow \%K}, \text{Period} = \text{InpSignalPeriod}, \text{Type} = \text{InpSignalMethod} \right)$$
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---
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### 2.4. Harmonized Fibonacci Gamma ($\gamma$) Spectrum Matrix
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| Fibonacci Gamma | Damping Depth | Phase Latency (Lag) | Target Market Regime | Equivalent EMA Benchmark | Quantitative Concept & Application |
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| :---: | :---: | :---: | :--- | :---: | :--- |
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| **`0.236`** | Ultra-Light | Near-Zero | High-Frequency Scalping / Momentum | $\approx 5\text{ EMA}$ | **Extreme Sensitivity.** High-frequency turning points and micro-cycle reversals. |
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| **`0.382`** | Light | Very Low | Day Trading / Intraday Execution | $\approx 10\text{ EMA}$ | **Optimal Execution Baseline.** Fast cycle turns with minimal noise. |
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| **`0.500`** | Balanced | Medium-Low | Swing Trading / Volatility Pivots | $\approx 20\text{ EMA}$ | **Balanced Corridor Center.** Standard baseline for medium swing setups on M15/H1 charts. |
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| **`0.618`** | Medium-Strong | Medium | Medium-Term Trend Following | $\approx 50\text{ EMA}$ | **The Golden Ratio Anchor.** Smooth cycle transitions without whipsaws. |
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| **`0.700`** | Default | Medium-High | Default Trend-Cycle Balance | $\approx 65\text{ EMA}$ | **Ehlers Default Damping.** Optimal separation of primary cycle waves. |
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| **`0.764`** | Strong | High | Macro Cycle Filtering | $\approx 100\text{ EMA}$ | **Structural Support.** Identifies macro turning points on H4/D1 charts. |
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| **`0.882`** | Ultra-Strong | Very High | Secular Cycle Smoothing | $\approx 200\text{ EMA}$ | **Absolute Noise Elimination.** Identifies macro secular bull/bear cycle bottoms. |
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---
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## 3. MQL5 Architecture & Engineering Standards
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```text
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┌────────────────────────────────────────────────────────┐
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│ Laguerre_Engine.mqh │
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│ (Core DSP Math: Computes L0..L3 Orthogonal States) │
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└──────────────────────────┬─────────────────────────────┘
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│ Feeds L0..L3 State Buffers
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▼
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┌────────────────────────────────────────────────────────┐
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│ Laguerre_Stoch_Slow_Calculator.mqh │
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│ (Envelope Extrema & Dual-Stage Smoothing Engine) │
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├──────────────────────────┬─────────────────────────────┤
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│ MovingAverage_Engine │ Composition Engine │
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│ • Slow %K Smoothing │ • CLaguerreEngine_HA │
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│ • Signal %D Smoothing │ • Full VWMA Support │
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└──────────────────────────┴─────────────────────────────┘
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│ Outputs Slow %K and Signal %D in O(1)
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▼
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┌────────────────────────────────────────────────────────┐
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│ Laguerre_Stoch_Slow_Pro.mq5 │
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│ (Unified Wrapper: Native Timeframe & MTF Engine) │
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├──────────────────────────┬─────────────────────────────┤
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│ Direct Mode (O(1)) │ Synchronized MTF Pipeline │
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│ • Current Timeframe │ • DataSync_Tools Daemon │
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│ • 2 Output Plots │ • Staircase Flat-Force │
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└──────────────────────────┴─────────────────────────────┘
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```
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1. **4-Tier Modular Hierarchy:** Isolates raw polynomial physics (`Laguerre_Engine.mqh`) from stochastic calculation (`Laguerre_Stoch_Slow_Calculator.mqh`) and moving average smoothing (`MovingAverage_Engine.mqh`).
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2. **Leak-Free Pointer Protection:** Factory methods safely verify pointer validity (`CheckPointer`) before re-allocation on parameter updates.
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3. **2026 MTF Framework with Staircase Solution:**
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* Asynchronous 1-second timer daemon (`OnTimerUpdate`) ensures higher-timeframe data synchronization without UI lag.
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* Dynamic staircase anchor (`first_bar_of_forming_htf`) synchronizes all lower-timeframe sub-bars belonging to the active higher-timeframe candle.
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---
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## 4. Parameters Reference
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### Timeframe Settings
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* `InpTimeframe` (*default: `PERIOD_CURRENT`*): Calculation timeframe. When set to `PERIOD_CURRENT`, it operates in native zero-lag mode. When set to a higher timeframe (e.g., `PERIOD_H1`, `PERIOD_D1`), it activates the synchronized MTF engine.
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### Laguerre Settings
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* `InpGamma` (*default: `0.7`*): Damping factor ($\gamma$). Controls the time-warp compression ratio ($0.0 \le \gamma \le 1.0$). Supports 3-decimal Fibonacci tuning (`0.236`, `0.382`, `0.500`, `0.618`, `0.700`, `0.764`, `0.882`).
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* `InpSourcePrice` (*default: `PRICE_CLOSE_STD`*): Price series source (Supports all 7 Standard and 7 Heikin Ashi modes).
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### Stochastic Slowing & Signal Settings
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* `InpSlowingPeriod` (*default: `3`*): Smoothing period for the Raw %K line.
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* `InpSlowingMethod` (*default: `SMA`*): Smoothing method for Slow %K (`SMA`, `EMA`, `SMMA`, `LWMA`, `TMA`, `DEMA`, `TEMA`, `VWMA`).
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* `InpSignalPeriod` (*default: `3`*): Smoothing period for the Signal %D line.
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* `InpSignalMethod` (*default: `SMA`*): Smoothing method for Signal %D.
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### Indicator Levels (0–100 Range)
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* `InpLevelExtrHigh` (*default: `90.0`*): Extreme Overbought Climax boundary.
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* `InpLevelHigh` (*default: `80.0`*): Overbought Warning threshold.
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* `InpLevelMid` (*default: `50.0`*): Directional Equilibrium threshold.
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* `InpLevelLow` (*default: `20.0`*): Oversold Warning threshold.
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* `InpLevelExtrLow` (*default: `10.0`*): Extreme Oversold Climax boundary.
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* `InpLevelColor` (*default: `clrSilver`*): Color of horizontal level lines.
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* `InpLevelStyle` (*default: `STYLE_DOT`*): Line style of horizontal level lines.
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### Visual Settings
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* `InpColorK` (*default: `clrDodgerBlue`*): Slow %K line color (Width: 2, Solid).
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* `InpColorD` (*default: `clrCoral`*): Signal %D line color (Width: 1, Solid).
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---
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## 5. Quantitative Trading Playbooks
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```text
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┌────────────────────────────────────────────────────────────────────────┐
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│ LAGUERRE STOCHASTIC TRADING PLAYBOOKS │
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├────────────────────────────────────────────────────────────────────────┤
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│ 1. 50-Line Equilibrium Cross: Slow %K crossing > 50 = Bullish Flow. │
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│ Slow %K crossing < 50 = Bearish Flow. │
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│ 2. Signal Crossover Trigger: Slow %K crosses Signal %D in direction │
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│ of higher-timeframe trend. │
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│ 3. Deep Boundary Reversals: Slow %K exiting > 90 / < 10 extreme │
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│ boundaries signals cycle exhaustion. │
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└────────────────────────────────────────────────────────────────────────┘
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```
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### 5.1. The 50.0 Equilibrium Directional Shift
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* **Bullish Momentum Dominance:** Slow %K crossing decisively **above 50.0** confirms that the primary Laguerre stage ($L_0$) is dominating upper envelope states ($L_1 \dots L_3$). Favor long positions.
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* **Bearish Momentum Dominance:** Slow %K crossing decisively **below 50.0** confirms downward dominance.
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### 5.2. Trend-Following Signal Line Crossovers (%K / %D)
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* **Bullish Continuation Trigger:** Price is in an established uptrend, Slow %K pulls back toward the 50.0 line, and crosses **above Signal %D** $\rightarrow$ Enter Long.
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* **Bearish Continuation Trigger:** Price is in a downtrend, Slow %K rallies toward 50.0, and crosses **below Signal %D** $\rightarrow$ Enter Short.
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### 5.3. Multi-Timeframe Macro Confluence
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* Attach an **H1-calculated Laguerre Stoch Slow ($\gamma=0.700$)** onto an **M5 execution chart**.
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* **Rule:** Only take M5 long breakout/pullback entries when **H1 Slow %K is above 50.0 and rising**.
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---
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## 6. Indicator Buffer Map (For Developers & EA Integration)
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| Buffer Index | Name | Type | Description |
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| :---: | :---: | :--- | :--- |
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| **0** | `BufferSlowK` | `INDICATOR_DATA` | Laguerre Stochastic Slow %K (Main Line) |
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| **1** | `BufferSignalD` | `INDICATOR_DATA` | Smoothed Signal %D Line |
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*Both buffers strictly maintain non-series chronological order (`ArraySetAsSeries = false`), ensuring instant compatibility with Expert Advisors and scanner dashboards via `iCustom()`.*
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