Edge Persistence Through Expected Path Coherence

(Probabilistic Path Match)

(Edge Persistence, Not Trend Persistence)

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Abstract

Directional bias alone does not define a tradable edge. What determines realized performance is the persistence of statistical alignment between a model and observed price behavior under prevailing structural conditions. This paper introduces edge persistence as a first-class modeling object and formalizes how regime, scale alignment, and path geometry reshape the duration profile of an edge without requiring constant profitability or monotonic price movement.

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1. Direction Is Not the Edge

Most trend-following formulations implicitly conflate direction, performance, and validity.

This is a modeling error.

An edge may remain statistically valid even while producing drawdowns. Conversely, short-lived profits can occur during periods of misalignment.

We therefore separate:

• Directional bias — a projection of expected movement
• Edge validity — alignment between expected and realized structure
• Performance — the realized P&L path during traversal

Only the second determines persistence.

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2. Edge Persistence as a Random Variable

Let an edge ( $\mathcal{E}$ ) be defined by a statistical hypothesis about path structure, not outcome direction.

Define:

• ( $A_t$ ): alignment score between expected path geometry and realized price path
• ( K ): number of bars for which the edge remains statistically valid

Then:

$$[ K \sim P(K \mid \mathcal{E}, \mathcal{C}) ]$$

This framework models alignment survival, not directional continuation.

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3. Alignment Does Not Imply Profitability

Edge validity and performance must be decoupled.

A strong edge may exhibit:

• high regression fit
• low residual error
• strong geometric similarity

while simultaneously undergoing drawdown if price traverses an unfavorable region of the expected path.

Such drawdowns do not represent edge decay — they represent expected traversal cost.

Invalidation occurs only when alignment degrades beyond tolerance.

Note: Full path dependence introduces a dynamically evolving state space in which each realized transition redefines the comparison baseline. For tractability, this framework starts with absolute path-coherence measures: over a fixed window, we treat the expected path and realized path as vectors, normalize their movements (e.g., via z-scored returns), and measure similarity with a simple regression or correlation-based score. Selecting the exact normalization and similarity metric, and extending this to fully state-dependent dynamics, is deferred to future work.

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4. Structural Conditions ( $\mathcal{C}$ )

Conditions are observable properties that affect alignment durability, not price sign.

Examples:

• multi-timeframe coherence
• volatility regime
• distance traveled along expected path
• recent dominant opposing structure

These act as filters on edge persistence, not predictors of outcome.

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5. Conditional Expectation of Edge Persistence

We now consider:

$$[ E[K \mid \mathcal{E}, \mathcal{C}] ]$$

and the full shape of:

$$[ P(K \mid \mathcal{E}, \mathcal{C}) ]$$

Different conditions reshape:

• the half-life of validity
• the probability of early invalidation
• the tolerance window for drawdowns

(Placeholder — Matplotlib: edge-survival distributions under varying conditions)

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6. Hazard Rate of Invalidation

Define the hazard function:

$$[ \lambda(k \mid \mathcal{E}, \mathcal{C}) = P(\text{edge invalidates at } k \mid \text{edge valid up to } k) ]$$

This answers the operational question:

“Given the edge has remained aligned up to this point, how likely is it to degrade now?”

High hazard implies:

• shortened expected patience
• tighter invalidation thresholds
• faster exit discipline

Low hazard permits:

• extended drawdowns
• scaling behavior
• looser traversal costs

(Placeholder — Matplotlib: hazard curves by structural regime)

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7. Multi-Timeframe Coherence as Alignment Stability

Define slope or shape estimates across scales:

$$[ S(\tau) = \text{alignment statistic at time scale } \tau ]$$

for $( \tau \in [\tau_{\min}, \tau_{\max}] )$.

Define coherence:

$$[ \kappa = f({S(\tau)}) ]$$

High coherence indicates:

• strong structural agreement
• slower alignment decay
• broader tolerance for performance variance

Low coherence implies fragility.

(Placeholder — Matplotlib: alignment distributions from 1m–1000m)

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8. Distance Traversed as Alignment Stress

Distance traveled along an expected path is not predictive — it is stress.

Let ( D ) denote cumulative traversal distance.

Then:

$$[ P(K \mid \mathcal{E}, \mathcal{C}, D) ]$$

Long-running edges are not flawed — they are more exposed.

Persistence modeling must account for this incremental risk.

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9. Filtering Without Directional Pollution

Let ( S ) be a base strategy.

Filtering does not change its directional logic:

$$[ \text{sign}(S') = \text{sign}(S) ]$$

It changes:

• allowable drawdown depth
• expected holding duration
• exit geometry

This preserves statistical rigor while adapting to structural stress.

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10. What This Framework Explicitly Rejects

This model does not:

• predict tops or bottoms
• equate drawdowns with failure
• require monotonic positivity
• use narrative interpretation

It models when alignment decays, not when price “should” reverse.

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11. Implications for System Design

• patience becomes measurable
• exits become alignment-aware
• filters replace intuition
• performance is contextualized, not judged prematurely

This is edge control, not discretionary override.

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12. Open Questions

• empirical estimation of alignment decay
• tolerance envelope calibration
• interaction with execution friction
• extension to spreads and delta-neutral edges

(Placeholder — empirical validation charts)

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Conclusion

Edges do not fail when performance deteriorates — they fail when alignment collapses.

By modeling edge persistence explicitly, systems can distinguish between expected traversal cost and genuine invalidation, preserving robustness without sacrificing discipline.