Eigenstructure Instability and Wavelet Energy

Research Status

Causal research diagnostic. Not part of the current public production report. Not a validated directional trading signal.

The current Vyreon system estimates expectation and uncertainty across four maturity horizons.

Those four horizon states can be studied individually. They can also be treated as one coupled geometric object.

The eigenstructure-instability research asks:

Does the relationship among the four horizon states reorganize before periods of larger market movement?

This is different from asking whether the expected return is positive or negative.

The research measures whether the dominant organization of the horizon system is stable or rotating.

The Four-Dimensional Expectation State

For each date, define one standardized state per horizon:

near-term state
short-term state
medium-term state
long-term state

The exploratory implementation used a signal similar to:

mu / sigma

for each horizon.

This produces a four-dimensional vector:

x(t) = [
    near mu/sigma,
    short mu/sigma,
    medium mu/sigma,
    long mu/sigma
]

The vector does not imply that the horizons are independent.

They are four observational slices through one maturity surface. Each slice has different sensitivity, inertia, and noise characteristics.

The research uses their joint behavior to describe cross-horizon organization.

Rolling Covariance

A rolling covariance matrix is calculated from the four-dimensional state history.

The covariance matrix measures:

• Which horizons are moving together.
• Which horizons are separating.
• How many independent directions are active.
• Whether one common mode dominates the system.
• Whether the relationship among maturities is stable.

The matrix is recomputed causally from data available up to the current date.

No future price label is required to form the structural measurement.

Dominant Eigenstructure

The covariance matrix is decomposed into eigenvalues and eigenvectors.

The dominant eigenvector describes the strongest common organization mode within the four-horizon system.

Its entries show how the horizon states contribute to that mode.

The eigenvector sign is arbitrary in ordinary PCA. A vector and its negative describe the same axis.

Therefore, the implementation aligns the sign of each new dominant eigenvector with the previous one. Without this step, harmless sign flips would appear as large structural rotations.

Loading Rotation Speed

The primary instability measurement is the day-to-day Euclidean change in the dominant loading vector:

loading_rotation_speed(t)
= ||v1(t) - v1(t - 1)||

where v1(t) is the dominant eigenvector at date t.

Interpretation:

• Low rotation means the dominant cross-horizon organization is stable.
• High rotation means the relationship among the horizons is changing rapidly.
• A spike means the latent geometry reorganized, not that price must move in one direction.

This is the physically meaningful object in the current research.

Earlier notes also described a stability score and its inverse. The raw rotation speed is simpler and more direct.

[Insert dominant loading rotation chart]

Suggested caption: Dominant loading rotation measures the day-to-day change in the principal cross-horizon organization mode. High values indicate rapid structural reorganization. They do not identify the direction of the later market move.

Why Wavelets Were Applied To Instability

The raw rotation signal is sparse and irregular.

The next question was whether a transition has multiple time scales.

A stationary wavelet transform was applied to the instability series rather than directly to price.

The exploratory implementation used a Coiflet wavelet and multiple SWT levels.

The transform separates the signal into:

• Detail coefficients.
• Approximation coefficients.

Detail Energy

Detail energy captures localized, faster perturbations in the instability process.

A detail-energy spike can be interpreted as:

• Local turbulence.
• A rapid change in the organization mode.
• A short-lived burst of structural disagreement.

It is not automatically an early directional warning.

Approximation Energy

Approximation energy captures slower background movement in the instability process.

An increase can be interpreted as:

• Broader structural reorganization.
• Persistence of the transition state.
• Slow relocking of the horizon system.

Observed Sequence

The live exploratory charts repeatedly showed a sequence similar to:

detail energy expands
    -> loading rotation rises
    -> approximation energy increases
    -> instability later falls

The working interpretation was:

localized turbulence
    -> active structural reorganization
    -> slower regime consolidation
    -> coherent relocking

This is a descriptive hypothesis.

It is not a complete transition model.

[Insert instability plus SWT energy chart]

Suggested caption: The chart compares raw eigenstructure instability with faster detail energy and slower approximation energy. The series describe different time scales of reorganization. The chart is diagnostic and does not provide directional trade instructions.

Historical Causal Reconstruction

The pattern was first noticed in live telemetry.

To test whether it existed historically, a causal proxy was reconstructed from the walk-forward model archive.

The proxy used prior expectation and prior uncertainty, similar to:

historical state proxy = y_prior / sqrt(S)

This allowed the four-horizon state vector, rolling covariance, dominant eigenvector, loading rotation, and wavelet energy to be reconstructed over 2016 through 2024 without using future price outcomes inside the signal.

This research path matters.

The signal was not created by first scanning price labels for the best-looking indicator. It was observed in live system behavior, formalized, and then reconstructed historically.

That reduces one form of data-mining risk.

It does not eliminate overfitting or selection bias.

Threshold Event Study

Exploratory thresholds were applied to loading rotation speed.

The main working levels were approximately:

Rotation threshold	Working interpretation
0.10	Broad turbulence or mild reorganization
0.15	Meaningful structural reorganization
0.20	Rare major transition

At a threshold near 0.15, one internal event study produced approximately 49 events over 2016 through 2024.

The reported average movement-magnitude lift relative to baseline was approximately:

• 1.19 times at 5 trading days.
• 1.13 times at 10 trading days.
• 1.21 times at 20 trading days.

The immediate one-day effect was weak.

The result suggested:

Large loading rotation is associated more with elevated medium-horizon movement than with an immediate price shock.

These figures are exploratory.

They are affected by:

• Threshold selection.
• Overlapping event windows.
• Serial dependence.
• Market-regime composition.
• The method used to define baseline movement.
• The reconstructed historical proxy.

The event study is useful evidence, not a final statistical claim.

What Survived

Structural Instability Is Measurable

The signal produces sparse periods of cross-horizon reorganization.

The high-rotation events align with many known stress and transition periods.

Movement Probability Is More Supported Than Direction

After instability events, future realized volatility and absolute movement were often elevated.

Positive and negative directional outcomes both occurred.

The Hierarchy Degrades Gradually

Lower thresholds identified broader turbulence. Higher thresholds isolated rarer events.

The behavior did not collapse immediately under small threshold changes.

This is encouraging, although it does not replace formal robustness testing.

Wavelet Energy Adds A Time-Scale Description

Detail and approximation energies offer a useful descriptive split between fast perturbation and slower structural reorganization.

They were more informative as diagnostics than as trading rules.

What Failed

Simple Spike Trade

The rule:

loading rotation spike -> enter directional trade

did not survive.

Spikes preceded both positive and negative moves.

Paired-Spike Trade

A second spike within a fixed window did not reliably identify a favorable directional setup.

Paired spikes described repeated instability, not a consistent price path.

Downward-Resolution Plus Second-Spike Hypothesis

A small subset looked promising after conditioning on an earlier downside resolution.

The sample fell to approximately four events.

That is too small for a deployable conclusion.

Wavelet Exit Rules

Several exit rules based on detail peaks, approximation decay, or instability peaks depended heavily on the 2020 period.

When 2020 was removed, many results weakened materially.

Double-Loading Bullish Hypothesis

A repeated live pattern suggested that a second loading period might precede an upward resolution after the first period failed.

Historical testing did not support a reliable directional rule.

The idea is not retained.

Hazard-Rate Interpretation

The most defensible interpretation is not directional.

It is closer to a hazard or transition measurement.

A high value says:

• The dominant cross-horizon organization is changing.
• The current state is less stable.
• Larger future movement is more likely than during ordinary periods.
• Path risk is elevated.

It does not say:

• The market will rise.
• The market will fall.
• A crash is imminent.
• A rebound is imminent.
• A specific option trade has positive expectancy.

This distinction matches a broader result across the Vyreon research:

Market state and transition risk are more observable than future path.

Relationship To The Volatility Signal

Eigenstructure instability and innovation RMS describe different objects.

Innovation RMS measures the mismatch between prior expectations and realized behavior.

Loading rotation measures the change in the relationship among horizon states.

They can rise together during stress, but they are not interchangeable.

A useful conceptual split is:

• Innovation RMS measures realized model surprise.
• Loading rotation measures structural reorganization within the expectation system.
• Wavelet detail energy measures localized turbulence in that reorganization.
• Wavelet approximation energy measures slower organization of the transition.

Current Use

The signal is used as a research diagnostic.

It can support questions such as:

• Is cross-horizon organization becoming less stable?
• Is a transition broadening from local turbulence into slower reorganization?
• Are volatility and structural instability confirming one another?
• Is a quiet price period hiding a changing expectation geometry?

It is not currently part of the weekly public report.

Next Experiments

Pre-Register Event Rules

Fix:

• Rolling covariance window.
• Minimum observations.
• Standardization method.
• Rotation thresholds.
• Cooldown.
• Forward horizons.
• Baseline definition.

Then run the study without changing those choices after seeing the results.

Evaluate Dependence-Aware Uncertainty

Overlapping forward windows violate independent-event assumptions.

Use block bootstrap, non-overlapping events, or another dependence-aware method.

Separate Transition Type From Direction

Test whether instability predicts:

• Realized volatility.
• Absolute return.
• Drawdown probability.
• Trend break probability.
• Volatility term-structure stress.

Do not force a directional label if the signal does not contain one.

Test Additional Assets

The four-state representation can be applied to other assets only after their underlying horizon models are separately validated.

Compare With Simpler Alternatives

The signal should be compared with:

• Changes in innovation RMS.
• Realized-volatility acceleration.
• Cross-horizon dispersion.
• Simple covariance-condition metrics.
• PCA eigenvalue concentration.

If a simpler diagnostic performs as well, the simpler one should be preferred.

Eigenvalues And Eigenvectors Answer Different Questions

The dominant eigenvalue measures how much of the rolling cross-horizon variance is explained by the first mode.

The dominant eigenvector describes the shape of that mode.

A system can have:

• A stable dominant eigenvalue and rotating loadings.
• A changing dominant eigenvalue and stable loadings.
• Several similar eigenvalues with an unstable principal direction.
• One overwhelmingly dominant mode.

Loading rotation focuses on organization shape.

Eigenvalue concentration focuses on dimensionality and dominance.

Both may be useful.

The current instability signal should not be described as a complete eigenstructure measurement when it uses only loading rotation.

Sign Alignment And Degeneracy

PCA loading direction becomes unstable when the largest eigenvalues are close.

Small data changes can rotate the dominant eigenvector substantially inside a nearly degenerate subspace.

That rotation may represent:

• Real structural reorganization.
• Numerical sensitivity because no one mode is dominant.
• Both.

A fuller diagnostic can include:

• Gap between the first and second eigenvalues.
• Fraction of variance explained by PC1.
• Condition number of the covariance matrix.
• Subspace distance across the first two principal components.

This is an important limitation.

A large loading rotation is not always a unique physical event.

It can occur because the dominant mode itself became poorly defined.

Alternative Structural Metrics

The research should compare loading rotation with simpler or related quantities.

Candidates include:

• Cross-horizon standard deviation.
• Average absolute pairwise correlation.
• Change in correlation matrix.
• Frobenius norm of covariance change.
• First-to-second eigenvalue ratio.
• Principal-angle change.
• Entropy of normalized eigenvalues.

If one simpler metric provides the same movement-risk information, it may be easier to explain and operate.

Public Communication

The phrase "bifurcation" can imply a formal dynamical-system result.

The current research has not established a complete bifurcation model with governing equations and branch stability.

For public writing, safer terms include:

• Structural reorganization.
• Cross-horizon instability.
• Dominant loading rotation.
• Transition-risk diagnostic.

The historical bifurcation language can remain in the case-study record, with its limits explained.

Current Conclusion

Dominant loading rotation is a causal measurement of changing cross-horizon organization.

Its strongest evidence is as a structural-transition and future-movement diagnostic.

The wavelet decomposition provides a useful description of fast and slow transition energy.

The research does not support a simple directional trading strategy.

That boundary is not a disappointment.

It is the central result.