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# John Ehlers' Laguerre Relative Strength Index (Laguerre RSI) Pro (v3.00)
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Quantitative Time-Warped Zero-Lag Momentum & Relative Strength Oscillator Suite
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---
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## 1. Summary (Introduction)
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**Laguerre RSI 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, J. Welles Wilder Jr.'s standard Relative Strength Index (RSI) averages price gains and losses over a fixed 14-bar lookback window, introducing unavoidable phase lag and slow signal resolution. **John Ehlers resolved this by applying the RSI mathematical concept across four orthogonal Laguerre polynomial state registers ($L_0, L_1, L_2, L_3$)**:
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* **Zero Historical Lookback Overhead:** Rather than calculating across $N$ price bars, Laguerre RSI evaluates the instantaneous directional displacement between the four Laguerre filter stages.
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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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* **Bounded 0–100 Scale with Clear Extremes:** Provides precise, non-lagging overbought ($\ge 80.0 / 90.0$) and oversold ($\le 20.0 / 10.0$) reversal thresholds.
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```text
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┌────────────────────────────────────────────────────────────────────────┐
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│ LAGUERRE RSI OSCILLATOR SUITE │
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├──────────────────────┬────────────────────────┬────────────────────────┤
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│ Line Plot │ Color Code │ Core Role │
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├──────────────────────┼────────────────────────┼────────────────────────┤
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│ Laguerre RSI Line │ clrMediumTurquoise (W:2│ Main Momentum Curve │
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│ Signal Line │ clrLightCoral (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 RSI Synthesis:** Computes relative strength strictly from internal $L_0 \dots L_3$ polynomial stage differences.
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* **Harmonic Fibonacci Damping Control ($\gamma$ – Gamma):** Aligns filter sensitivity with golden ratio proportions (`0.236`, `0.382`, `0.500`, `0.618`, `0.764`, `0.882`).
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* **Integrated Signal Smoothing Engine:** Supports 8 moving average algorithms (including Volume-Weighted VWMA) for the signal trigger line.
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* **Unified 2026 MTF Framework:** Higher-timeframe Laguerre RSI 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 `CLaguerreRSICalculator_HA` composition.
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---
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## 2. Mathematical Foundations & Laguerre RSI 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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CU = (L0 - L1 if > 0) + (L1 - L2 if > 0) + (L2 - L3 if > 0)
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CD = (L1 - L0 if > 0) + (L2 - L1 if > 0) + (L3 - L2 if > 0)
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│
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▼
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Laguerre RSI = [ CU / (CU + CD) ] × 100
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│
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▼
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Signal Line = MA(Laguerre RSI, Period = Signal, Type = MA_Type)
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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. Cumulative Upward ($\text{CU}$) and Downward ($\text{CD}$) Displacements
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The directional displacement across adjacent stages is accumulated:
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$$\text{Stage 1: } \text{if } L_0(t) \ge L_1(t) \implies \text{CU}_1 = L_0(t) - L_1(t), \quad \text{else } \text{CD}_1 = L_1(t) - L_0(t)$$
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$$\text{Stage 2: } \text{if } L_1(t) \ge L_2(t) \implies \text{CU}_2 = L_1(t) - L_2(t), \quad \text{else } \text{CD}_2 = L_2(t) - L_1(t)$$
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$$\text{Stage 3: } \text{if } L_2(t) \ge L_3(t) \implies \text{CU}_3 = L_2(t) - L_3(t), \quad \text{else } \text{CD}_3 = L_3(t) - L_2(t)$$
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$$\text{CU}_t = \text{CU}_1 + \text{CU}_2 + \text{CU}_3$$
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$$\text{CD}_t = \text{CD}_1 + \text{CD}_2 + \text{CD}_3$$
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---
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### 2.3. Normalized Laguerre RSI Formulation
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$$\text{LRSI}_t = \begin{cases} \left( \frac{\text{CU}_t}{\text{CU}_t + \text{CD}_t} \right) \times 100, & \text{if } (\text{CU}_t + \text{CD}_t) > 10^{-9} \\ \text{LRSI}_{t-1}, & \text{if } (\text{CU}_t + \text{CD}_t) = 0 \end{cases}$$
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$$\text{Clamped Bounds: } 0.0 \le \text{LRSI}_t \le 100.0$$
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---
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### 2.4. Signal Line Smoothing
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$$\text{Signal}_t = \mathcal{MA}\left( \text{LRSI}, \text{Period} = \text{InpSignalPeriod}, \text{Type} = \text{InpSignalMAType} \right)$$
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---
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### 2.5. 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.** Instantaneous cycle inflection and micro-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 momentum transitions without whipsaws. |
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| **`0.764`** | Strong | High | Macro Cycle Filtering | $\approx 100\text{ EMA}$ | **Structural Support.** Identifies macro momentum exhaustion 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_RSI_Calculator.mqh │
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│ (CU/CD Extraction & Normalized Laguerre RSI Engine) │
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├──────────────────────────┬─────────────────────────────┤
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│ MovingAverage_Engine │ Composition Engine │
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│ • Signal MA Smoothing │ • CLaguerreEngine_HA │
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│ • Full VWMA Support │ • Clean Pointer Safety │
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└──────────────────────────┴─────────────────────────────┘
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│ Outputs LRSI and Signal in O(1)
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▼
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┌────────────────────────────────────────────────────────┐
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│ Laguerre_RSI_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. **Modular 3-Tier Hierarchy:** Isolates polynomial tracking (`Laguerre_Engine.mqh`) from ratio calculation (`Laguerre_RSI_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 RSI Settings
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* `InpGamma` (*default: `0.5`*): 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`, `0.764`, `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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### Signal Line Settings
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* `InpDisplayMode` (*default: `DISPLAY_LRSI_AND_SIGNAL`*): Display mode (`DISPLAY_LRSI_ONLY` or `DISPLAY_LRSI_AND_SIGNAL`).
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* `InpSignalPeriod` (*default: `3`*): Smoothing period for the Signal line.
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* `InpSignalMAType` (*default: `EMA`*): Smoothing method for Signal line (`SMA`, `EMA`, `SMMA`, `LWMA`, `TMA`, `DEMA`, `TEMA`, `VWMA`).
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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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* `InpColorLRSI` (*default: `clrMediumTurquoise`*): Laguerre RSI line color (Width: 2, Solid).
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* `InpColorSignal` (*default: `clrLightCoral`*): Signal 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 RSI TRADING PLAYBOOKS │
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├────────────────────────────────────────────────────────────────────────┤
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│ 1. 50-Line Equilibrium Cross: LRSI crossing > 50 = Bullish Momentum. │
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│ LRSI crossing < 50 = Bearish Momentum. │
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│ 2. Signal Crossover Trigger: LRSI crosses Signal line in direction of │
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│ higher-timeframe macro trend baseline. │
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│ 3. Deep Boundary Reversals: LRSI 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:** Laguerre RSI crossing decisively **above 50.0** confirms that upward stage displacements ($\text{CU}$) are dominating downward displacements ($\text{CD}$). Favor long continuation setups.
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* **Bearish Momentum Dominance:** Laguerre RSI crossing decisively **below 50.0** confirms downward dominance.
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### 5.2. Trend-Following Signal Line Crossovers (LRSI / Signal)
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* **Bullish Continuation Trigger:** Price is in an established uptrend, Laguerre RSI pulls back toward the 50.0 line, and crosses **above the Signal Line** $\rightarrow$ Enter Long.
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* **Bearish Continuation Trigger:** Price is in a downtrend, Laguerre RSI rallies toward 50.0, and crosses **below the Signal Line** $\rightarrow$ Enter Short.
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### 5.3. Multi-Timeframe Macro Confluence
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* Attach an **H1-calculated Laguerre RSI ($\gamma=0.618$)** onto an **M5 execution chart**.
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* **Rule:** Only take M5 long breakout/pullback entries when **H1 Laguerre RSI is above 50.0 and rising**. This ensures you never trade against higher-timeframe institutional momentum.
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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** | `BufferLRSI` | `INDICATOR_DATA` | Main Laguerre Relative Strength Index Line |
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| **1** | `BufferSignal` | `INDICATOR_DATA` | Smoothed Signal Trigger 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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