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# John Ehlers' Laguerre Slope Pro (v3.00)
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Quantitative Time-Warped First-Derivative Momentum & Velocity Subwindow Suite
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---
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## 1. Summary (Introduction)
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**Laguerre Slope Pro** is an institutional-grade kinetic momentum oscillator that calculates the mathematical **first derivative (rate of change / slope)** of John Ehlers' Time-Warped Laguerre Filter.
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Because the underlying Laguerre Filter eliminates high-frequency market noise using an orthogonal all-pass transfer function with near-zero phase lag, its discrete first difference produces one of the most responsive, low-noise velocity histograms available in technical analysis.
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```text
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┌────────────────────────────────────────────────────────────────────────┐
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│ LAGUERRE SLOPE MOMENTUM ENGINE │
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├──────────────────────┬────────────────────────┬────────────────────────┤
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│ Output Plot │ Color Mapping │ Core Kinetic Role │
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├──────────────────────┼────────────────────────┼────────────────────────┤
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│ Laguerre Slope Hist │ 5-Zone Thermal Palette │ Instantaneous Velocity │
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│ Signal MA Line │ clrMaroon (Width:1) │ Smoothed Trigger Line │
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└──────────────────────┴────────────────────────┴────────────────────────┘
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```
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### Key Capabilities
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* **Discrete First-Derivative Velocity Engine:** Computes exact price-velocity points ($\Delta \text{Laguerre}$) with zero latency.
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* **Symmetrical 5-Zone Momentum Matrix:** Distinguishes between accelerating trend impulses, decelerating pullbacks, and flat consolidation noise.
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* **Integrated Signal Smoothing Engine:** Supports 8 moving average smoothing algorithms (including Volume-Weighted VWMA) directly over the slope histogram.
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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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* **Unified 2026 MTF Framework:** Higher-timeframe Laguerre Slope histograms (e.g., M15, H1, H4) map onto lower-timeframe execution charts (M1, M5) with flat, non-warping steps via `DataSync_Tools.mqh`.
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* **Natural Zero-Anchor Subwindow:** Features a clean, uncluttered visual layout where histogram bars naturally project from the $0.0$ baseline.
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---
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## 2. Mathematical Foundations & Slope Dynamics
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```text
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Δ Laguerre = Laguerre(t) - Laguerre(t - 1)
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▲ (Bullish Acceleration: MediumSeaGreen)
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│
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──────┼──────────────────────────────────────────── 0.0 (Neutral Chop: Gray)
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│
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▼ (Bearish Acceleration: Crimson)
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```
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### 2.1. The Discrete First Derivative (Velocity / Slope)
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Given the Laguerre Filter baseline $\text{Laguerre}_t = \frac{L_0(t) + 2 L_1(t) + 2 L_2(t) + L_3(t)}{6}$:
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$$\text{Slope}_t = \text{Laguerre}_t - \text{Laguerre}_{t-1}$$
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---
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### 2.2. Symmetrical 5-Zone Momentum Matrix
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To classify momentum velocity and second-derivative acceleration, the slope is categorized into 5 quantitative zones evaluated against a user-defined neutral deadband threshold ($\epsilon = \text{InpThreshold}$):
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| State Index | Color | Classification | Mathematical Condition | Market Interpretation |
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| :---: | :---: | :--- | :--- | :--- |
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| **0.0** | `clrGray` | **Neutral / Consolidation** | $\|\text{Slope}_t\| \le \epsilon$ | Low kinetic energy; consolidation chop. |
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| **1.0** | `clrMediumSeaGreen` | **Strong Bullish Acceleration** | $\text{Slope}_t > \epsilon \quad \text{and} \quad \text{Slope}_t > \text{Slope}_{t-1}$ | Buyers in aggressive control; upward velocity speeding up. |
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| **2.0** | `clrPaleGreen` | **Weak Bullish Deceleration** | $\text{Slope}_t > \epsilon \quad \text{and} \quad \text{Slope}_t \le \text{Slope}_{t-1}$ | Uptrend slowing down; potential pullback or exhaustion forming. |
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| **3.0** | `clrCrimson` | **Strong Bearish Acceleration** | $\text{Slope}_t < -\epsilon \quad \text{and} \quad \text{Slope}_t < \text{Slope}_{t-1}$ | Sellers in aggressive control; downward velocity speeding up. |
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| **4.0** | `clrLightCoral` | **Weak Bearish Deceleration** | $\text{Slope}_t < -\epsilon \quad \text{and} \quad \text{Slope}_t \ge \text{Slope}_{t-1}$ | Downtrend losing downward thrust; short-covering bounce potential. |
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---
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### 2.3. Signal Moving Average Smoothing
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An optional moving average smoothing line filters high-frequency slope micro-noise:
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$$\text{Signal}_t = \mathcal{MA}\left( \text{Slope}, \text{Period} = \text{InpSignalPeriod}, \text{Type} = \text{InpSignalType} \right)$$
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*When `InpSignalType = VWMA`, real or tick volume is incorporated:*
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$$\text{Signal}_{\text{VWMA}, t} = \frac{\sum_{j=0}^{M-1} \text{Slope}_{t-j} \cdot V_{t-j}}{\sum_{j=0}^{M-1} V_{t-j}}$$
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---
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### 2.4. Harmonized Fibonacci Gamma ($\gamma$) Spectrum Matrix
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| Fibonacci Gamma | Smoothing 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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| **`0.764`** | 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 Laguerre Filter Baseline
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▼
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┌────────────────────────────────────────────────────────┐
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│ Laguerre_Slope_Calculator.mqh │
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│ (Slope Difference & 5-Zone Momentum Matrix Engine) │
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└──────────────────────────┬─────────────────────────────┘
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│ Outputs Slope & Color Index in O(1)
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▼
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┌────────────────────────────────────────────────────────┐
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│ Laguerre_Slope_Pro.mq5 │
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│ (Unified Wrapper: Native Timeframe & MTF Engine) │
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├──────────────────────────┬─────────────────────────────┤
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│ Buffer Layer (3) │ Centralized Framework │
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│ • BufferSlope (DATA) │ • DataSync_Tools.mqh │
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│ • BufferSlopeColor(IDX)│ • MovingAverage_Engine │
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│ • BufferSignalMA (DATA)│ • Staircase Flat-Force │
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└──────────────────────────┴─────────────────────────────┘
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```
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1. **Modular 3-Tier Engine Hierarchy:** Isolates raw polynomial state tracking (`Laguerre_Engine.mqh`) from first-derivative velocity calculation (`Laguerre_Slope_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_M15`, `PERIOD_H1`), it activates the synchronized MTF engine.
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### Laguerre 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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* `InpThreshold` (*default: `0.00005`*): The deadband threshold ($\epsilon$) below which slope velocity is classified as neutral consolidation (`clrGray`).
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### Signal MA Settings
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* `InpShowSignal` (*default: `true`*): Toggle visibility of the Signal Moving Average line.
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* `InpSignalPeriod` (*default: `5`*): Lookback period for the signal line.
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* `InpSignalType` (*default: `EMA`*): Smoothing algorithm (`SMA`, `EMA`, `SMMA`, `LWMA`, `TMA`, `DEMA`, `TEMA`, `VWMA`).
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* `InpColorSignal` (*default: `clrMaroon`*): Color applied to the signal line plot.
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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 SLOPE TRADING PLAYBOOKS │
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├────────────────────────────────────────────────────────────────────────┤
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│ 1. Acceleration Expansion: Enter in direction of strong momentum when │
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│ bars turn MediumSeaGreen / Crimson. │
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│ 2. Early Warning Fade: When bars change to PaleGreen / LightCoral, │
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│ momentum is decaying; tighten stop-loss. │
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│ 3. Signal Line Crossover: Slope crosses Signal MA in direction of │
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│ higher-timeframe macro trend baseline. │
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└────────────────────────────────────────────────────────────────────────┘
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```
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### 5.1. Velocity Acceleration vs. Deceleration
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* **Trend Confirmation (Aggressive Entry):** A histogram bar transitioning from `clrGray` or `clrPaleGreen` to **`clrMediumSeaGreen`** confirms that bullish momentum is accelerating with high efficiency.
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* **Early Take-Profit (Deceleration Warning):** When the histogram remains positive but changes from `clrMediumSeaGreen` to **`clrPaleGreen`**, the trend is losing momentum. This often precedes candle structure pullbacks or consolidations.
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* **Bearish Continuation:** A transition into **`clrCrimson`** signals expanding downward selling velocity.
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### 5.2. Signal MA Line Crossovers
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* **Bullish Crossover:** The Laguerre Slope histogram crosses **above** the Signal MA line from below $\rightarrow$ Early buy trigger.
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* **Bearish Crossover:** The Laguerre Slope histogram crosses **below** the Signal MA line from above $\rightarrow$ Early sell / exit trigger.
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### 5.3. Multi-Timeframe Macro Bias Alignment
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* Attach an **H1-calculated Laguerre Slope ($\gamma=0.618$)** onto an **M5 execution chart**.
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* **Rule:** Only take intraday long pullbacks on M5 when **H1 Laguerre Slope is positive and Green (`MediumSeaGreen`)**.
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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** | `BufferSlope` | `INDICATOR_DATA` | Discrete Laguerre Slope 1st Derivative Values ($\Delta \text{Laguerre}$) |
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| **1** | `BufferSlopeColor` | `INDICATOR_COLOR_INDEX` | Symmetrical 5-Zone Palette Index ($0.0 \dots 4.0$) |
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| **2** | `BufferSignalMA` | `INDICATOR_DATA` | Smoothed Signal Moving Average Plot |
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*All buffers strictly maintain non-series chronological order (`ArraySetAsSeries = false`), ensuring instant compatibility with Expert Advisors and scanner dashboards via `iCustom()`.*
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