Manifold Singularity Engine

Manifold Singularity Engine: Catastrophe Theory Detection Through Multi-Dimensional Topology Analysis

The Manifold Singularity Engine applies catastrophe theory from mathematical topology to multi-dimensional price space analysis, identifying potential reversal conditions by measuring manifold curvature, topological complexity, and fractal regime states. Unlike traditional reversal indicators that rely on price pattern recognition or momentum oscillators, this system reconstructs the underlying geometric surface (manifold) that price evolves upon and detects points where this topology undergoes catastrophic folding—mathematical singularities that correspond to forced directional changes in price dynamics.

The indicator combines three analytical frameworks: phase space reconstruction that embeds price data into a multi-dimensional coordinate system, catastrophe detection that measures when this embedded manifold reaches critical curvature thresholds indicating topology breaks, and Hurst exponent calculation that classifies the current fractal regime to adaptively weight detection sensitivity. This creates a geometry-based reversal detection system with visual feedback showing topology state, manifold distortion fields, and directional probability projections.

What Makes This Approach Different
Phase Space Embedding Construction
The core analytical method reconstructs price evolution as movement through a three-dimensional coordinate system rather than analyzing price as a one-dimensional time series. The system calculates normalized embedding coordinates: X = normalize(price_velocity, window), Y = normalize(momentum_acceleration, window), and Z = normalize(volume_weighted_returns, window). These coordinates create a trajectory through phase space where price movement traces a path across a geometric surface—the market manifold.

This embedding approach differs fundamentally from traditional technical analysis by treating price not as a sequential data stream but as a dynamical system evolving on a curved surface in multi-dimensional space. The trajectory's geometric properties (curvature, complexity, folding) contain information about impending directional changes that single-dimension analysis cannot capture. When this manifold undergoes rapid topological deformation, price must respond with directional change—this is the mathematical basis for catastrophe detection.

Statistical normalization using z-score transformation (subtracting mean, dividing by standard deviation over a rolling window) ensures the coordinate system remains scale-invariant across different instruments and volatility regimes, allowing identical detection logic to function on forex, crypto, stocks, or indices without recalibration.

Catastrophe Score Calculation
The catastrophe detection formula implements a composite anomaly measurement combining multiple topology metrics: Catastrophe_Score = 0.45×Curvature_Percentile + 0.25×Complexity_Ratio + 0.20×Condition_Percentile + 0.10×Gradient_Percentile. Each component measures a distinct aspect of manifold distortion:

Curvature (κ) is computed using the discrete Laplacian operator: κ = √[(d²X/dt²)² + (d²Y/dt²)²], which measures how sharply the manifold surface bends at the current point. High curvature values indicate the surface is folding or developing a sharp corner—geometric precursors to catastrophic topology breaks. The Laplacian measures second derivatives (rate of change of rate of change), capturing acceleration in the trajectory's path through phase space.

Topological Complexity counts sign changes in the curvature field over the embedding window, measuring how chaotically the manifold twists and oscillates. A smooth, stable surface produces low complexity; a highly contorted, unstable surface produces high complexity. This metric detects when the geometric structure becomes informationally dense with multiple local extrema, suggesting an imminent topology simplification event (catastrophe).

Condition Number measures the Jacobian matrix's sensitivity: Condition = |Trace| / |Determinant|, where the Jacobian describes how small changes in price produce changes in the embedding coordinates. High condition numbers indicate numerical instability—points where the coordinate transformation becomes ill-conditioned, suggesting the manifold mapping is approaching a singularity.

Each metric is converted to percentile rank within a rolling window, then combined using weighted sum. The percentile transformation creates adaptive thresholds that automatically adjust to each instrument's characteristic topology without manual recalibration. The resulting 0-100% catastrophe score represents the current bar's position in the distribution of historical manifold distortion—values above the threshold (default 65%) indicate statistically extreme topology states where reversals become geometrically probable.

This multi-metric ensemble approach prevents false signals from isolated anomalies: all four geometric features must simultaneously indicate distortion for a high catastrophe score, ensuring only true manifold breaks trigger detection.

Hurst Exponent Regime Classification
The Hurst exponent calculation implements rescaled range (R/S) analysis to measure the fractal dimension of price returns: H = log(R/S) / log(n), where R is the range of cumulative deviations from mean and S is the standard deviation. The resulting value classifies market behavior into three fractal regimes:

Trending Regime (H > 0.55): Persistent price movement where future changes are positively correlated with past changes. The manifold exhibits directional momentum with smooth topology evolution. In this regime, catastrophe signals receive 1.2× confidence multiplier because manifold breaks in trending conditions produce high-magnitude directional changes.

Mean-Reverting Regime (H < 0.45): Anti-persistent price movement where future changes tend to oppose past changes. The manifold exhibits oscillatory topology with frequent small-scale distortions. Catastrophe signals receive 0.8× confidence multiplier because reversal significance is diminished in choppy conditions where the manifold constantly folds at minor scales.

Random Walk Regime (H ≈ 0.50): No statistical correlation in returns. The manifold evolution is geometrically neutral with moderate topology stability. Standard 1.0× confidence multiplier applies.

This adaptive weighting system solves a critical problem in reversal detection: the same geometric catastrophe has different trading implications depending on the fractal regime. A manifold fold in a strong trend suggests a significant reversal opportunity; the same fold in mean-reversion suggests a minor oscillation. The Hurst-based regime filter ensures detection sensitivity automatically adjusts to market character without requiring trader intervention.
The implementation uses logarithmic price returns rather than raw prices to ensure

stationarity, and applies the calculation over a configurable window (default 5 bars) to balance responsiveness with statistical validity. The Hurst value is then smoothed using exponential moving average to reduce noise while maintaining regime transition detection.

Multi-Layer Confirmation Architecture

The system implements five independent confirmation filters that must simultaneously validate
before any singularity signal generates:

1. Catastrophe Threshold: The composite anomaly score must exceed the configured threshold (default 0.65 on 0-1 scale), ensuring the manifold distortion is statistically extreme relative to recent history.

2. Pivot Structure Confirmation: Traditional swing high/low patterns (using ta.pivothigh and ta.pivotlow with configurable lookback) must form at the catastrophe bar. This ensures the geometric singularity coincides with observable price structure rather than occurring mid-swing where interpretation is ambiguous.

3. Swing Size Validation: The pivot magnitude must exceed a minimum threshold measured in ATR units (default 1.5× Average True Range). This filter prevents signals on insignificant price jiggles that lack meaningful reversal potential, ensuring only substantial swings with adequate risk/reward ratios generate signals.

4. Volume Confirmation: Current volume must exceed 1.3× the 20-period moving average, confirming genuine market participation rather than low-liquidity price noise. Manifold catastrophes without volume support often represent false topology breaks that don't translate to sustained directional change.

5. Regime Validity: The market must be classified as either trending (ADX > configured threshold, default 30) or volatile (ATR expansion > configured threshold, default 40% above 30-bar average), and must NOT be in choppy/ranging state. This critical filter prevents trading during geometrically unfavorable conditions where edge deteriorates.

All five conditions must evaluate true simultaneously for a signal to generate. This conjunction-based logic (AND not OR) dramatically reduces false positives while preserving true reversal detection. The architecture recognizes that geometric catastrophes occur frequently in noisy data, but only those catastrophes that align with confirming evidence across price structure, participation, and regime characteristics represent tradable opportunities.

A cooldown mechanism (default 8 bars between signals) prevents signal clustering at extended pivot zones where the manifold may undergo multiple small catastrophes during a single reversal process.

Direction Classification System
Unlike binary bull/bear systems, the indicator implements a voting mechanism combining four

directional indicators to classify each catastrophe:
Pivot Vote: +1 if pivot low, -1 if pivot high, 0 otherwise

Trend Vote: Based on slow frequency (55-period EMA) slope—+1 if rising, -1 if falling, 0 if flat

Flow Vote: Based on Y-gradient (momentum acceleration)—+1 if positive, -1 if negative, 0 if neutral

Mid-Band Vote: Based on price position relative to medium frequency (21-period EMA)—+1 if above, -1 if below, 0 if at

The total vote sum classifies the singularity: ≥2 votes = Bullish, ≤-2 votes = Bearish, -1 to +1 votes = Neutral (skip). This majority-consensus approach ensures directional classification requires alignment across multiple timeframes and analysis dimensions rather than relying on a single indicator. Neutral signals (mixed voting) are displayed but should not be traded, as they represent geometric catastrophes without clear directional resolution.
Core Calculation Methodology
Embedding Coordinate Generation
Three normalized phase space coordinates are constructed from price data:
X-Dimension (Velocity Space):

price_velocity = close - close[1]
X = (price_velocity - mean) / stdev over hurstWindow

Y-Dimension (Acceleration Space):

momentum = close - close[embedLength]
momentum_accel = momentum - momentum[1]
Y = (momentum_accel - mean) / stdev over hurstWindow

Z-Dimension (Volume-Weighted Space):

vol_normalized = (volume - mean) / stdev over embedLength
roc = (close - close[hurstWindow]) / close[hurstWindow]
Z = (roc × vol_normalized - mean) / stdev over hurstWindow

These coordinates define a point in 3D phase space for each bar. The trajectory connecting these points is the reconstructed manifold.
Gradient Field Calculation
First derivatives measure local manifold slope:

dX/dt = X - X[1]
dY/dt = Y - Y[1]
Gradient_Magnitude = √[(dX/dt)² + (dY/dt)²]

The gradient direction indicates where the manifold is "pushing" price. Positive Y-gradient suggests upward topological pressure; negative Y-gradient suggests downward pressure.
Curvature Tensor Components
Second derivatives measure manifold bending using discrete Laplacian:

Laplacian_X = X - 2×X[1] + X[2]
Laplacian_Y = Y - 2×Y[1] + Y[2]
Laplacian_Magnitude = √[Laplacian_X² + Laplacian_Y²]

This is then normalized:

Curvature_Normalized = (Laplacian_Magnitude - mean) / stdev over embedLength

High normalized curvature (>1.5) indicates sharp manifold folding.
Complexity Accumulation
Sign changes in curvature field are counted:

Sign_Flip = 1 if sign(Curvature[0]) ≠ sign(Curvature[1]), else 0
Topological_Complexity = sum(Sign_Flip) over embedLength window

This measures oscillation frequency in the geometry. Complexity >5 indicates chaotic topology.
Condition Number Stability Analysis
Jacobian matrix sensitivity is approximated:

dX/dp = dX/dt / (price_change + epsilon)
dY/dp = dY/dt / (price_change + epsilon)
Jacobian_Determinant = (dX/dt × dY/dp) - (dX/dp × dY/dt)
Jacobian_Trace = dX/dt + dY/dp
Condition_Number = |Trace| / (|Determinant| + epsilon)

High condition numbers indicate numerical instability near singularities.
Catastrophe Score Assembly
Each metric is converted to percentile rank over embedLength window, then combined:

Curvature_Percentile = percentrank(abs(Curvature_Normalized), embedLength)
Gradient_Percentile = percentrank(Gradient_Magnitude, embedLength)
Condition_Percentile = percentrank(abs(Condition_Z_Score), embedLength)
Complexity_Ratio = clamp(Topological_Complexity / embedLength, 0, 1)

Final score:

Raw_Anomaly = 0.45×Curvature_P + 0.25×Complexity_R + 0.20×Condition_P + 0.10×Gradient_P
Catastrophe_Score = Raw_Anomaly × Hurst_Multiplier

Values are clamped to [0, 1] range.
Hurst Exponent Calculation
Rescaled range analysis on log returns:

Calculate log returns: r = log(close) - log(close[1])
Compute cumulative deviations from mean
Find range: R = max(cumulative_dev) - min(cumulative_dev)
Calculate standard deviation: S = stdev(r, hurstWindow)
Compute R/S ratio
Hurst = log(R/S) / log(hurstWindow)
Clamp to [0, 1] and smooth with 5-period EMA

Regime Classification Logic
Volatility Regime:

ATR_MA = SMA(ATR(14), 30)
Vol_Expansion = ATR / ATR_MA
Is_Volatile = Vol_Expansion > (1.0 + minVolExpansion)

Trend Regime (Corrected ADX):

Calculate directional movement (DM+, DM-)
Smooth with Wilder's RMA(14)
Compute DI+ and DI- as percentages
Calculate DX = |DI+ - DI-| / (DI+ + DI-) × 100
ADX = RMA(DX, 14)
Is_Trending = ADX > (trendStrength × 100)

Chop Detection:

Is_Chopping = NOT Is_Trending AND NOT Is_Volatile

Regime Validity:

Regime_Valid = (Is_Trending OR Is_Volatile) AND NOT Is_Chopping

Signal Generation Logic
For each bar:

Check if catastrophe score > topologyStrength threshold
Verify regime is valid
Confirm Hurst alignment (trending or mean-reverting with pivot)
Validate pivot quality (price extended outside spectral bands then re-entered)
Confirm volume/volatility participation
Check cooldown period has elapsed
If all true: compute directional vote
If vote ≥2: Bullish Singularity
If vote ≤-2: Bearish Singularity
If -1 to +1: Neutral (display but skip)

All conditions must be true for signal generation.
Visual System Architecture
Spectral Decomposition Layers
Three harmonic frequency bands visualize entropy state:
Layer 1 (Surface Frequency):

Center: EMA(8)
Width: ±0.3 × 0.5 × ATR
Transparency: 75% (most visible)
Represents fast oscillations

Layer 2 (Mid Frequency):

Center: EMA(21)
Width: ±0.5 × 0.5 × ATR
Transparency: 85%
Represents medium cycles

Layer 3 (Deep Frequency):

Center: EMA(55)
Width: ±0.7 × 0.5 × ATR
Transparency: 92% (most transparent)
Represents slow baseline

Convergence of layers indicates low entropy (stable topology). Divergence indicates high entropy (catastrophe building). This decomposition reveals how different frequency components of price movement interact—when all three align, the manifold is in equilibrium; when they separate, topology is unstable.
Energy Radiance Fields
Concentric boxes emanate from each singularity bar:
For each singularity, 5 layers are generated:

Layer n: bar_index ± (n × 1.5 bars), close ± (n × 0.4 × ATR)
Transparency gradient: inner 75% → outer 95%
Color matches signal direction

These fields visualize the "energy well" of the catastrophe—wider fields indicate stronger topology distortion. The exponential expansion creates a natural radiance effect.
Singularity Node Geometry
N-sided polygon (default hexagon) at each signal bar:

Vertices calculated using polar coordinates
Rotation angle: bar_index × 0.1 (creates animation)
Radius: ATR × singularity_strength × 2
Connects vertices with colored lines

The rotating geometric primitive marks the exact catastrophe bar with visual prominence.
Gradient Flow Field
Directional arrows display manifold slope:

Spawns every 3 bars when gradient_magnitude > 0.1
Symbol: "↗" if dY/dt > 0.1, "↘" if dY/dt < -0.1, "→" if neutral
Color: Bull/bear/neutral based on direction
Density limited to flowDensity parameter

Arrows cluster when gradient is strong, creating intuitive topology visualization.
Probability Projection Cones
Forward trajectory from each singularity:

Projects 10 bars forward
Direction based on vote classification
Center line: close + (direction × ATR × 3)
Uncertainty width: ATR × singularity_strength × 2
Dashed boundaries, solid center

These are mathematical projections based on current gradient, not price targets. They visualize expected manifold evolution if topology continues current trajectory.

Dashboard Metrics Explanation
The real-time control panel displays six core metrics plus regime status:

H (Hurst Exponent):
Value: Current Hurst (0-1 scale)
Label: TREND (>0.55), REVERT (<0.45), or RANDOM (0.45-0.55)
Icon: Direction arrow based on regime
Purpose: Shows fractal character—only trade when favorable

Σ (Catastrophe Score):
Value: Current composite anomaly (0-100%)
Bar gauge shows relative strength
Icon: ◆ if above threshold, ○ if below
Purpose: Primary signal strength indicator

κ (Curvature):
Value: Normalized Laplacian magnitude
Direction arrow shows sign
Color codes severity (green<0.8, yellow<1.5, red≥1.5)
Purpose: Shows manifold bending intensity

⟳ (Topology Complexity):
Value: Count of sign flips in curvature
Icon: ◆ if >3, ○ otherwise
Color codes chaos level
Purpose: Indicates geometric instability

V (Volatility Expansion):

Value: ATR expansion percentage above 30-bar average
Icon: ● if volatile, ○ otherwise
Purpose: Confirms energy present for reversal

T (Trend Strength):

Value: ADX reading (0-100)
Icon: ● if trending, ○ otherwise
Purpose: Shows directional bias strength

R (Regime):

Label: EXPLOSIVE / TREND / VOLATILE / CHOP / NEUTRAL
Icon: ✓ if valid, ✗ if invalid
Purpose: Go/no-go filter for trading

STATE (Bottom Display):

Shows: "◆ BULL SINGULARITY" (green), "◆ BEAR SINGULARITY" (red), "◆ WEAK/NEUTRAL" (orange), or "— Monitoring —" (gray)
Purpose: Current signal status at a glance

How to Use This Indicator
Initial Setup and Configuration
Apply the indicator to your chart with default settings as a starting point. The default parameters (21-bar embedding, 5-bar Hurst window, 2.5σ singularity threshold, 0.65 topology confirmation) are optimized for balanced detection across most instruments and timeframes. For very fast markets (scalping crypto, 1-5min charts), consider reducing embedding depth to 13-15 bars and Hurst window to 3 bars for more responsive detection. For slower markets (swing trading stocks, 4H-Daily charts), increase embedding depth to 34-55 bars and Hurst window to 8-10 bars for more stable topology measurement.

Enable the dashboard (top right recommended) to monitor real-time metrics. The control panel is your primary decision interface—glancing at the dashboard should instantly communicate whether conditions favor trading and what the current topology state is. Position and size the dashboard to remain visible but not obscure price action.

Enable regime filtering (strongly recommended) to prevent trading during choppy/ranging conditions where geometric edge deteriorates. This single setting can dramatically improve overall performance by eliminating low-probability environments.

Reading Dashboard Metrics for Trade Readiness
Before considering any trade, verify the dashboard shows favorable conditions:

Hurst (H) Check:
The Hurst Exponent reading is your first filter. Only consider trades when H > 0.50. Ideal conditions show H > 0.60 with "TREND" label—this indicates persistent directional price movement where manifold catastrophes produce significant reversals. When H < 0.45 (REVERT label), the market is mean-reverting and catastrophes represent minor oscillations rather than substantial pivots. Do not trade in mean-reverting regimes unless you're explicitly using range-bound strategies (which this indicator is not optimized for). When H ≈ 0.50 (RANDOM label), edge is neutral—acceptable but not ideal.

Catastrophe (Σ) Monitoring:
Watch the Σ percentage build over time. Readings consistently below 50% indicate stable topology with no imminent reversals. When Σ rises above 60-65%, manifold distortion is approaching critical levels. Signals only fire when Σ exceeds the configured threshold (default 65%), so this metric pre-warns you of potential upcoming catastrophes. High-conviction setups show Σ > 75%.

Regime (R) Validation:
The regime classification must read TREND, VOLATILE, or EXPLOSIVE—never trade when it reads CHOP or NEUTRAL. The checkmark (✓) must be present in the regime cell for trading conditions to be valid. If you see an X (✗), skip all signals until regime improves. This filter alone eliminates most losing trades by avoiding geometrically unfavorable environments.

Combined High-Conviction Profile:
The strongest trading opportunities show simultaneously:
H > 0.60 (strong trending regime)
Σ > 75% (extreme topology distortion)
R = EXPLOSIVE or TREND with ✓
κ (Curvature) > 1.5 (sharp manifold fold)
⟳ (Complexity) > 4 (chaotic geometry)
V (Volatility) showing elevated ATR expansion

When all metrics align in this configuration, the manifold is undergoing severe distortion in a favorable fractal regime—these represent maximum-conviction reversal opportunities.
Signal Interpretation and Entry Logic

Bullish Singularity (▲ Green Triangle Below Bar):
This marker appears when the system detects a manifold catastrophe at a price low with bullish directional consensus. All five confirmation filters have aligned: topology score exceeded threshold, pivot low structure formed, swing size was significant, volume/volatility confirmed participation, and regime was valid. The green color indicates the directional vote totaled +2 or higher (majority bullish).

Trading Approach: Consider long entry on the bar immediately following the signal (bar after the triangle). The singularity bar itself is where the geometric catastrophe occurred—entering after allows you to see if price confirms the reversal. Place stop loss below the singularity bar's low (with buffer of 0.5-1.0 ATR for volatility). Initial target can be the previous swing high, or use the probability cone projection as a guide (though not a guarantee). Monitor the dashboard STATE—if it flips to "◆ BEAR SINGULARITY" or Hurst drops significantly, consider exiting even if target not reached.

Bearish Singularity (▼ Red Triangle Above Bar):
This marker appears when the system detects a manifold catastrophe at a price high with bearish directional consensus. Same five-filter confirmation process as bullish signals. The red color indicates directional vote totaled -2 or lower (majority bearish).

Trading Approach: Consider short entry on the bar following the signal. Place stop loss above the singularity bar's high (with buffer). Target previous swing low or use cone projection as reference. Exit if opposite signal fires or Hurst deteriorates.

Neutral Signal (● Orange Circle at Price Level):
This marker indicates the catastrophe detection system identified a topology break that passed catastrophe threshold and regime filters, but the directional voting system produced a mixed result (vote between -1 and +1). This means the four directional components (pivot, trend, flow, mid-band) are not in agreement about which way the reversal should resolve.
Trading Approach: Skip these signals. Neutral markers are displayed for analytical completeness but should not be traded. They represent geometric catastrophes without clear directional resolution—essentially, the manifold is breaking but the direction of the break is ambiguous. Trading neutral signals dramatically increases false signal rate. Only trade green (bullish) or red (bearish) singularities.

Visual Confirmation Using Spectral Layers
The three colored ribbons (spectral decomposition layers) provide entropy visualization that helps confirm signal quality:

Divergent Layers (High Entropy State):
When the three frequency bands (fast 8-period, medium 21-period, slow 55-period) are separated with significant gaps between them, the manifold is in high entropy state—different frequency components of price movement are pulling in different directions. This geometric tension precedes catastrophes. Strong signals often occur when layers are divergent before the signal, then begin reconverging immediately after.

Convergent Layers (Low Entropy State):
When all three ribbons are tightly clustered or overlapping, the manifold is in equilibrium—all frequency components agree. This stable geometry makes catastrophe detection more reliable because topology breaks clearly stand out against the baseline stability. If you see layers converge, then a singularity fires, then layers diverge, this pattern suggests a genuine regime transition.

Signal Quality Assessment:
High-quality singularity signals should show:

Divergent layers (high entropy) in the 5-10 bars before signal
Singularity bar occurs when price has extended outside at least one of the spectral bands (shows pivot extended beyond equilibrium)
Close of singularity bar re-enters the spectral band zone (shows mean reversion starting)
Layers begin reconverging in 3-5 bars after signal (shows new equilibrium forming)

This pattern visually confirms the geometric narrative: manifold became unstable (divergence), reached critical distortion (extended outside equilibrium), broke catastrophically (singularity), and is now stabilizing in new direction (reconvergence).
Using Energy Fields for Trade Management
The concentric glowing boxes around each singularity visualize the topology distortion

magnitude:
Wide Energy Fields (5+ Layers Visible):
Large radiance indicates strong catastrophe with high manifold curvature. These represent significant topology breaks and typically precede larger price moves. Wide fields justify wider profit targets and longer hold times. The outer edge of the largest box can serve as a dynamic support/resistance zone—price often respects these geometric boundaries.

Narrow Energy Fields (2-3 Layers):
Smaller radiance indicates moderate catastrophe. While still valid signals (all filters passed), expect smaller follow-through. Use tighter profit targets and be prepared for quicker exit if momentum doesn't develop. These are valid but lower-conviction trades.

Field Interaction Zones:
When energy fields from consecutive signals overlap or touch, this indicates a prolonged topology distortion region—often corresponds to consolidation zones or complex reversal patterns (head-and-shoulders, double tops/bottoms). Be more cautious in these areas as the manifold is undergoing extended restructuring rather than a clean catastrophe.

Probability Cone Projections
The dashed cone extending forward from each singularity is a mathematical projection, not a

price target:
Cone Direction:
The center line direction (upward for bullish, downward for bearish, flat for neutral) shows the expected trajectory based on current manifold gradient and singularity direction. This is where the topology suggests price "should" go if the catastrophe completes normally.

Cone Width:
The uncertainty band (upper and lower dashed boundaries) represents the range of outcomes given current volatility (ATR-based). Wider cones indicate higher uncertainty—expect more price volatility even if direction is correct. Narrower cones suggest more constrained movement.

Price-Cone Interaction:
Price following near the center line = catastrophe resolving as expected, geometric projection accurate
Price breaking above upper cone = stronger-than-expected reversal, consider holding for larger targets
Price breaking below lower cone (for bullish signal) = catastrophe failing, manifold may be re-folding in opposite direction, consider exit
Price oscillating within cone = normal reversal process, hold position

The 10-bar projection length means cones show expected behavior over the next ~10 bars. Don't confuse this with longer-term price targets.

Gradient Flow Field Interpretation
The directional arrows (↗, ↘, →) scattered across the chart show the manifold's Y-gradient (vertical acceleration dimension):

Upward Arrows (↗):
Positive Y-gradient indicates the momentum acceleration dimension is pushing upward—the manifold topology has upward "slope" at this location. Clusters of upward arrows suggest bullish topological pressure building. These often appear before bullish singularities fire.

Downward Arrows (↘):
Negative Y-gradient indicates downward topological pressure. Clusters precede bearish singularities.

Horizontal Arrows (→):
Neutral gradient indicates balanced topology with no strong directional pressure.

Using Flow Field:
The arrows provide real-time topology state information even between singularity signals. If you're in a long position from a bullish singularity and begin seeing increasing downward arrows appearing, this suggests manifold gradient is shifting—consider tightening stops. Conversely, if arrows remain upward or neutral, topology supports continuation.
Don't confuse arrow direction with immediate price direction—arrows show geometric slope, not price prediction. They're confirmatory context, not entry signals themselves.
Parameter Optimization for Your Trading Style
For Scalping / Fast Trading (1m-15m charts):

Embedding Depth: 13-15 bars (faster topology reconstruction)
Hurst Window: 3 bars (responsive fractal detection)
Singularity Threshold: 2.0-2.3σ (more sensitive)
Topology Confirmation: 0.55-0.60 (lower barrier)
Min Swing Size: 0.8-1.2 ATR (accepts smaller moves)
Pivot Lookback: 3-4 bars (quick pivot detection)

This configuration increases signal frequency for active trading but requires diligent monitoring as false signal rate increases. Use tighter stops.
For Day Trading / Standard Approach (15m-4H charts):

Keep default settings (21 embed, 5 Hurst, 2.5σ, 0.65 confirmation, 1.5 ATR, 5 pivot)
These are balanced for quality over quantity
Best win rate and risk/reward ratio
Recommended for most traders

For Swing Trading / Position Trading (4H-Daily charts):

Embedding Depth: 34-55 bars (stable long-term topology)
Hurst Window: 8-10 bars (smooth fractal measurement)
Singularity Threshold: 3.0-3.5σ (only extreme catastrophes)
Topology Confirmation: 0.75-0.85 (high conviction only)
Min Swing Size: 2.5-4.0 ATR (major moves only)
Pivot Lookback: 8-13 bars (confirmed swings)

This configuration produces infrequent but highly reliable signals suitable for position sizing and longer hold times.

Volatility Adaptation:
In extremely volatile instruments (crypto, penny stocks), increase Min Volatility Expansion to 0.6-0.8 to avoid over-signaling during "always volatile" conditions. In stable instruments (major forex pairs, blue-chip stocks), decrease to 0.3 to allow signals during moderate volatility spikes.
Trend vs Range Preference:

If you prefer trading only strong trends, increase Min Trend Strength to 0.5-0.6 (ADX > 50-60). If you're comfortable with volatility-based trading in weaker trends, decrease to 0.2 (ADX > 20). The default 0.3 balances both approaches.
Complete Trading Workflow Example

Step 1 - Pre-Session Setup:
Load chart with MSE indicator. Check dashboard position is visible. Verify regime filter is enabled. Review recent signals to gauge current instrument behavior.

Step 2 - Market Assessment:
Observe dashboard Hurst reading. If H < 0.45 (mean-reverting), consider skipping this session or using other strategies. If H > 0.50, proceed. Check regime shows TREND, VOLATILE, or EXPLOSIVE with checkmark—if CHOP, wait for regime shift alert.

Step 3 - Signal Wait:
Monitor catastrophe score (Σ). Watch for it climbing above 60%. Observe spectral layers—look for divergence building. If you see curvature (κ) rising above 1.0 and complexity (⟳) increasing, catastrophe is building. Do not anticipate—wait for the actual signal marker.

Step 4 - Signal Recognition:
▲ Bullish or ▼ Bearish triangle appears at a bar. Dashboard STATE changes to "◆ BULL/BEAR SINGULARITY". Energy field appears around the signal bar. Check signal quality:

Was Σ > 70% at signal? (Higher quality)
Are energy fields wide? (Stronger catastrophe)
Did layers diverge before and reconverge after? (Clean break)
Is Hurst still > 0.55? (Good regime)

Step 5 - Entry Decision:
If signal is green/red (not orange neutral), all confirmations look strong, and no immediate contradicting factors appear, prepare entry on next bar open. Wait for confirmation bar to form—ideally it should close in the signal direction (bullish signal → bar closes higher, bearish signal → bar closes lower).

Step 6 - Position Entry:
Enter at open or shortly after open of bar following signal bar. Set stop loss: for bullish signals, place stop at singularity_bar_low - (0.75 × ATR); for bearish signals, place stop at singularity_bar_high + (0.75 × ATR). The buffer accommodates volatility while protecting against catastrophe failure.

Step 7 - Trade Management:
Monitor dashboard continuously:

If Hurst drops below 0.45, consider reducing position
If opposite singularity fires, exit immediately (manifold has re-folded)
If catastrophe score drops below 40% and stays there, topology has stabilized—consider partial profit taking
Watch gradient flow arrows—if they shift to opposite direction persistently, tighten stops

Step 8 - Profit Taking:
Use probability cone as a guide—if price reaches outer cone boundary, consider taking partial profits. If price follows center line cleanly, hold for larger target. Traditional technical targets work well: previous swing high/low, round numbers, Fibonacci extensions. Don't expect precision—manifold projections give direction and magnitude estimates, not exact prices.

Step 9 - Exit:
Exit on: (a) opposite signal appears, (b) dashboard shows regime became invalid (checkmark changes to X), (c) technical target reached, (d) Hurst deteriorates significantly, (e) stop loss hit, or (f) time-based exit if using session limits. Never hold through opposite singularity signals—the manifold has broken in the other direction and your trade thesis is invalidated.

Step 10 - Post-Trade Review:
After exit, review: Did the probability cone projection align with actual price movement? Were the energy fields proportional to move size? Did spectral layers show expected reconvergence? Use these observations to calibrate your interpretation of signal quality over time.

Best Performance Conditions

This topology-based approach performs optimally in specific market environments:
Favorable Conditions:

Well-Developed Swing Structure: Markets with clear rhythm of advances and declines where pivots form at regular intervals. The manifold reconstruction depends on swing formation, so instruments that trend in clear waves work best. Stocks, major forex pairs during active sessions, and established crypto assets typically exhibit this characteristic.

Sufficient Volatility for Topology Development: The embedding process requires meaningful price movement to construct multi-dimensional coordinates. Extremely quiet markets (tight consolidations, holiday trading, after-hours) lack the volatility needed for manifold differentiation. Look for ATR expansion above average—when volatility is present, geometry becomes meaningful.

Trending with Periodic Reversals: The ideal environment is not pure trend (which rarely reverses) nor pure range (which reverses constantly at small scale), but rather trending behavior punctuated by occasional significant counter-trend reversals. This creates the catastrophe conditions the system is designed to detect: manifold building directional momentum, then undergoing sharp topology break at extremes.

Liquid Instruments Where EMAs Reflect True Flow: The spectral layers and frequency decomposition require that moving averages genuinely represent market consensus. Thinly traded instruments with sporadic orders don't create smooth manifold topology. Prefer instruments with consistent volume where EMA calculations reflect actual capital flow rather than random tick sequences.

Challenging Conditions:
Extremely Choppy / Whipsaw Markets: When price oscillates rapidly with no directional persistence (Hurst < 0.40), the manifold undergoes constant micro-catastrophes that don't translate to tradable reversals. The regime filter helps avoid these, but awareness is important. If you see multiple neutral signals clustering with no follow-through, market is too chaotic for this approach.

Very Low Volatility Consolidation: Tight ranges with ATR below average cause the embedding coordinates to compress into a small region of phase space, reducing geometric differentiation. The manifold becomes nearly flat, and catastrophe detection loses sensitivity. The regime filter's volatility component addresses this, but manually avoiding dead markets improves results.

Gap-Heavy Instruments: Stocks that gap frequently (opening outside previous close) create discontinuities in the manifold trajectory. The embedding process assumes continuous evolution, so gaps introduce artifacts. Most gaps don't invalidate the approach, but instruments with daily gaps >2% regularly may show degraded performance. Consider using higher timeframes (4H, Daily) where gaps are less proportionally significant.

Parabolic Moves / Blowoff Tops: When price enters an exponential acceleration phase (vertical rally or crash), the manifold evolves too rapidly for the standard embedding window to track. Catastrophe detection may lag or produce false signals mid-move. These conditions are rare but identifiable by Hurst > 0.75 combined with ATR expansion >2.0× average. If detected, consider sitting out or using very tight stops as geometry is in extreme distortion.

The system adapts by reducing signal frequency in poor conditions—if you notice long periods with no signals, the topology likely lacks the geometric structure needed for reliable catastrophe detection. This is a feature, not a bug: it prevents forced trading during unfavorable environments.

Theoretical Justification for Approach
Why Manifold Embedding?

Traditional technical analysis treats price as a one-dimensional time series: current price is predicted from past prices in sequential order. This approach ignores the structure of price dynamics—the relationships between velocity, acceleration, and participation that govern how price actually evolves.

Dynamical systems theory (from physics and mathematics) provides an alternative framework: treat price as a state variable in a multi-dimensional phase space. In this view, each market condition corresponds to a point in N-dimensional space, and market evolution is a trajectory through this space. The geometry of this space (its topology) constrains what trajectories are possible.

Manifold embedding reconstructs this hidden geometric structure from observable price data. By creating coordinates from velocity, momentum acceleration, and volume-weighted returns, we map price evolution onto a 3D surface. This surface—the manifold—reveals geometric relationships that aren't visible in price charts alone.

The mathematical theorem underlying this approach (Takens' Embedding Theorem from dynamical systems theory) proves that for deterministic or weakly stochastic systems, a state space reconstruction from time-delayed observations of a single variable captures the essential dynamics of the full system. We apply this principle: even though we only observe price, the embedded coordinates (derivatives of price) reconstruct the underlying dynamical structure.

Why Catastrophe Theory?

Catastrophe theory, developed by mathematician René Thom (Fields Medal 1958), describes how continuous systems can undergo sudden discontinuous changes when control parameters reach critical values. A classic example: gradually increasing force on a beam causes smooth bending, then sudden catastrophic buckling. The beam's geometry reaches a critical curvature where topology must break.
Markets exhibit analogous behavior: gradual price changes build tension in the manifold topology until critical distortion is reached, then abrupt directional change occurs (reversal). Catastrophes aren't random—they're mathematically necessary when geometric constraints are violated.

The indicator detects these geometric precursors: high curvature (manifold bending sharply), high complexity (topology oscillating chaotically), high condition number (coordinate mapping becoming singular). These metrics quantify how close the manifold is to a catastrophic fold. When all simultaneously reach extreme values, topology break is imminent.

This provides a logical foundation for reversal detection that doesn't rely on pattern recognition or historical correlation. We're measuring geometric properties that mathematically must change when systems reach critical states. This is why the approach works across different instruments and timeframes—the underlying geometry is universal.

Why Hurst Exponent?

Markets exhibit fractal behavior: patterns at different time scales show statistical self-similarity. The Hurst exponent quantifies this fractal structure by measuring long-range dependence in returns.

Critically for trading, Hurst determines whether recent price movement predicts future direction (H > 0.5) or predicts the opposite (H < 0.5). This is regime detection: trending vs mean-reverting behavior.

The same manifold catastrophe has different trading implications depending on regime. In trending regime (high Hurst), catastrophes represent significant reversal opportunities because the manifold has been building directional momentum that suddenly breaks. In mean-reverting regime (low Hurst), catastrophes represent minor oscillations because the manifold constantly folds at small scales.

By weighting catastrophe signals based on Hurst, the system adapts detection sensitivity to the current fractal regime. This is a form of meta-analysis: not just detecting geometric breaks, but evaluating whether those breaks are meaningful in the current fractal context.

Why Multi-Layer Confirmation?

Geometric anomalies occur frequently in noisy market data. Not every high-curvature point represents a tradable reversal—many are artifacts of microstructure noise, order flow imbalances, or low-liquidity ticks.

The five-filter confirmation system (catastrophe threshold, pivot structure, swing size, volume, regime) addresses this by requiring geometric anomalies to align with observable market evidence. This conjunction-based logic implements the principle: extraordinary claims require extraordinary evidence.

A manifold catastrophe (extraordinary geometric event) alone is not sufficient. We additionally require: price formed a pivot (visible structure), swing was significant (adequate magnitude), volume confirmed participation (capital backed the move), and regime was favorable (trending or volatile, not chopping). Only when all five dimensions agree do we have sufficient evidence that the geometric anomaly represents a genuine reversal opportunity rather than noise.

This multi-dimensional approach is analogous to medical diagnosis: no single test is conclusive, but when multiple independent tests all suggest the same condition, confidence increases dramatically. Each filter removes a different category of false signals, and their combination creates a robust detection system.

The result is a signal set with dramatically improved reliability compared to any single metric alone. This is the power of ensemble methods applied to geometric analysis.

Important Disclaimers

This indicator applies mathematical topology and catastrophe theory to multi-dimensional price space reconstruction. It identifies geometric conditions where manifold curvature, topological complexity, and coordinate singularities suggest potential reversal zones based on phase space analysis. It should not be used as a standalone trading system.

The embedding coordinates, catastrophe scores, and Hurst calculations are deterministic mathematical formulas applied to historical price data. These measurements describe current and recent geometric relationships in the reconstructed manifold but do not predict future price movements. Past geometric patterns and singularity markers do not guarantee future market behavior will follow similar topology evolution.

The manifold reconstruction assumes certain mathematical properties (sufficient embedding dimension, quasi-stationarity, continuous dynamics) that may not hold in all market conditions. Gaps, flash crashes, circuit breakers, news events, and other discontinuities can violate these assumptions. The system attempts to filter problematic conditions through regime classification, but cannot eliminate all edge cases.

The spectral decomposition, energy fields, and probability cones are visualization aids that represent mathematical constructs, not price predictions. The probability cone projects current gradient forward assuming topology continues current trajectory—this is a mathematical "if-then" statement, not a forecast. Market topology can and does change unexpectedly.

All trading involves substantial risk. The singularity markers represent analytical conditions where geometric mathematics align with threshold criteria, not certainty of directional change. Use appropriate risk management for every trade: position sizing based on account risk tolerance (typically 1-2% maximum risk per trade), stop losses placed beyond recent structure plus volatility buffer, and never risk capital needed for living expenses.
The confirmation filters (pivot, swing size, volume, regime) are designed to reduce false signals but cannot eliminate them entirely. Markets can produce geometric anomalies that pass all filters yet fail to develop into sustained reversals. This is inherent to probabilistic systems operating on noisy real-world data.

No indicator can guarantee profitable trades or eliminate losses. The catastrophe detection provides an analytical framework for identifying potential reversal conditions, but actual trading outcomes depend on numerous factors including execution, slippage, spreads, position sizing, risk management, psychological discipline, and market conditions that may change after signal generation.

Use this tool as one component of a comprehensive trading plan that includes multiple forms of analysis, proper risk management, emotional discipline, and realistic expectations about win rates and drawdowns. Combine catastrophe signals with additional confirmation methods such as support/resistance analysis, volume patterns, multi-timeframe alignment, and broader market context.

The spacing filter, cooldown mechanism, and regime validation are designed to reduce noise and over-signaling, but market conditions can change rapidly and render any analytical signal invalid. Always use stop losses and never risk capital you cannot afford to lose. Past performance of detection accuracy does not guarantee future results.

Technical Implementation Notes
All calculations execute on closed bars only—signals and metric values do not repaint after bar close. The indicator does not use any lookahead bias in its calculations. However, the pivot detection mechanism (ta.pivothigh and ta.pivotlow) inherently identifies pivots with a lag equal to the lookback parameter, meaning the actual pivot occurred at bar[lookback] but is recognized at bar[0]. This is standard behavior for pivot functions and is not repainting—once recognized, the pivot bar never changes.

The normalization system (z-score transformation over rolling windows) requires approximately 30-50 bars of historical data to establish stable statistics. Values in the first 30-50 bars after adding the indicator may show instability as the rolling means and standard deviations converge. Allow adequate warmup period before relying on signals.

The spectral layer arrays, energy field boxes, gradient flow labels, and node geometry lines are subject to TradingView drawing object limits (500 lines, 500 boxes, 500 labels per indicator as specified in settings). The system implements automatic cleanup by deleting oldest objects when limits approach, but on very long charts with many signals, some historical visual elements may be removed to stay within limits. This does not affect signal generation or dashboard metrics—only historical visual artifacts.

Dashboard and visual rendering update only on the last bar to minimize computational overhead. The catastrophe detection logic executes on every bar, but table cells and drawing objects refresh conditionally to optimize performance. If experiencing chart lag, reduce visual complexity: disable spectral layers, energy fields, or flow field to improve rendering speed. Core signal detection continues to function with all visual elements disabled.

The Hurst calculation uses logarithmic returns rather than raw price to ensure stationarity, and implements clipping to [0, 1] range to handle edge cases where R/S analysis produces invalid values (which can occur during extended periods of identical prices or numerical overflow). The 5-period EMA smoothing reduces noise while maintaining responsiveness to regime transitions.
The condition number calculation adds epsilon (1e-10) to denominators to prevent division by zero when Jacobian determinant approaches zero—which is precisely the singularity condition we're detecting. This numerical stability measure ensures the indicator doesn't crash when detecting the very phenomena it's designed to identify.

The indicator has been tested across multiple timeframes (5-minute through daily) and multiple asset classes (forex majors, stock indices, individual equities, cryptocurrencies, commodities, futures). It functions identically across all instruments due to the adaptive normalization approach and percentage-based metrics. No instrument-specific code or parameter sets are required.

The color scheme system implements seven preset themes plus custom mode. Color assignments are applied globally and affect all visual elements simultaneously. The opacity calculation system multiplies component-specific transparency with master opacity to create hierarchical control—adjusting master opacity affects all visuals proportionally while maintaining their relative transparency relationships.

All alert conditions trigger only on bar close to prevent false alerts from intrabar fluctuations. The regime transition alerts (VALID/INVALID) are particularly useful for knowing when trading edge appears or disappears, allowing traders to adjust activity levels accordingly.

— Dskyz, Trade with insight. Trade with anticipation.