01-09-2026, 01:29 AM
(This post was last modified: 01-09-2026, 01:34 AM by Leejohnston.)
Spectral Feedback Criticality (SFC) — An Emerging Diagnostic Law
Status:
Exploratory diagnostic framework with mathematical formulation and preliminary physical-domain validation.
Not yet peer-reviewed or formally published.
⸻
Overview
Spectral Feedback Criticality (SFC) is a diagnostic framework for identifying when delayed feedback systems approach instability — not through sudden divergence or oscillation, but through the emergence of slow modes and loss of damping.
The framework is designed to answer a specific question:
“How can we detect hidden fragility in systems that still appear stable?”
SFC does not attempt to replace classical stability theory. Instead, it provides a compact diagnostic lens for detecting *pre-failure regimes* where conventional indicators often remain silent.
⸻
The problem SFC addresses
Many real-world systems remain apparently stable until failure occurs:
• power grids
• control systems
• supply chains
• financial and economic systems
• large-scale engineered networks
In these systems:
• feedback delays mask growing risk
• fast modes remain stable while slow modes degrade
• instability emerges gradually, not catastrophically
SFC focuses specifically on this **quiet approach to criticality**.
⸻
The governing parameter
SFC is organised around a single dimensionless feedback parameter:
K = g · rⁿ
Where:
• g is feedback gain
• r is the response or amplification factor
• n is the effective feedback delay (or lag order)
This parameter compresses the combined effect of feedback strength and delay into one quantity that governs proximity to instability.
⸻
Critical threshold
The system approaches criticality when:
K ≈ 1 − b₀
Where:
• b₀ represents intrinsic damping or baseline stability
As K approaches this threshold:
• dominant eigenvalues drift toward unity
• damping weakens
• recovery times lengthen
• slow modes begin to dominate
Importantly, the system may still satisfy conventional stability criteria while becoming increasingly fragile.
⸻
Reduced dynamical form (technical)
The behaviour diagnosed by SFC is captured by the following reduced-form dynamical equation:
xₜ₊₁ = b₀ xₜ + K xₜ₋ₙ + εₜ
Where:
• xₜ is the system state deviation (e.g. frequency offset, imbalance, error signal)
• b₀ ∈ (0,1) is intrinsic damping
• K = g rⁿ is delayed feedback gain
• n is the feedback delay
• εₜ represents noise or perturbations
This reduced model isolates the essential dynamics: intrinsic damping competing with delayed feedback.
From this equation, the SFC critical condition K ≈ 1 − b₀ follows directly, along with the emergence of slow modes as the dominant eigenvalue approaches unity.
⸻
Slow-mode onset
The defining signature of SFC is not oscillation or divergence, but **slowing**.
As criticality is approached:
• perturbations decay more slowly
• autocorrelation increases
• low-frequency power grows
• recovery from shocks becomes sluggish
This phenomenon — often called *critical slowing down* — appears here as a natural consequence of delayed feedback approaching the stability threshold.
⸻
Physical-domain demonstration
SFC has been tested against real-world power-grid frequency data.
In the UK GB grid (Dec 2025):
• rolling-window analysis identified a consistent feedback delay (~5 s)
• estimated parameters showed proximity to the SFC threshold
• slow-mode pressure increased during stress periods
• no failure occurred — but fragility was detectable
This demonstrates SFC’s role as an **early warning diagnostic**, not a failure predictor.
⸻
What SFC does not claim
SFC does not claim:
• universality across all systems
• prediction of exact failure times
• replacement of full dynamical modelling
• that threshold crossing guarantees collapse
It is a diagnostic indicator, not a deterministic prophecy.
⸻
Why this framework exists
Many catastrophic failures are preceded by long periods of invisible fragility.
SFC exists to make that fragility measurable — before failure, not after.
⸻
Open questions and future work
SFC remains an evolving framework. Open directions include:
• broader physical and synthetic validation
• domain-specific parameter estimation
• formal links to critical slowing-down theory
• coupling with belief and decision frameworks
• stress-testing across engineered and natural systems
Discussion, critique, and refinement are welcome.
⸻
One-sentence summary
Spectral Feedback Criticality (SFC) is a diagnostic law for detecting when delayed feedback systems approach instability through the emergence of slow modes — before failure occurs.
Status:
Exploratory diagnostic framework with mathematical formulation and preliminary physical-domain validation.
Not yet peer-reviewed or formally published.
⸻
Overview
Spectral Feedback Criticality (SFC) is a diagnostic framework for identifying when delayed feedback systems approach instability — not through sudden divergence or oscillation, but through the emergence of slow modes and loss of damping.
The framework is designed to answer a specific question:
“How can we detect hidden fragility in systems that still appear stable?”
SFC does not attempt to replace classical stability theory. Instead, it provides a compact diagnostic lens for detecting *pre-failure regimes* where conventional indicators often remain silent.
⸻
The problem SFC addresses
Many real-world systems remain apparently stable until failure occurs:
• power grids
• control systems
• supply chains
• financial and economic systems
• large-scale engineered networks
In these systems:
• feedback delays mask growing risk
• fast modes remain stable while slow modes degrade
• instability emerges gradually, not catastrophically
SFC focuses specifically on this **quiet approach to criticality**.
⸻
The governing parameter
SFC is organised around a single dimensionless feedback parameter:
K = g · rⁿ
Where:
• g is feedback gain
• r is the response or amplification factor
• n is the effective feedback delay (or lag order)
This parameter compresses the combined effect of feedback strength and delay into one quantity that governs proximity to instability.
⸻
Critical threshold
The system approaches criticality when:
K ≈ 1 − b₀
Where:
• b₀ represents intrinsic damping or baseline stability
As K approaches this threshold:
• dominant eigenvalues drift toward unity
• damping weakens
• recovery times lengthen
• slow modes begin to dominate
Importantly, the system may still satisfy conventional stability criteria while becoming increasingly fragile.
⸻
Reduced dynamical form (technical)
The behaviour diagnosed by SFC is captured by the following reduced-form dynamical equation:
xₜ₊₁ = b₀ xₜ + K xₜ₋ₙ + εₜ
Where:
• xₜ is the system state deviation (e.g. frequency offset, imbalance, error signal)
• b₀ ∈ (0,1) is intrinsic damping
• K = g rⁿ is delayed feedback gain
• n is the feedback delay
• εₜ represents noise or perturbations
This reduced model isolates the essential dynamics: intrinsic damping competing with delayed feedback.
From this equation, the SFC critical condition K ≈ 1 − b₀ follows directly, along with the emergence of slow modes as the dominant eigenvalue approaches unity.
⸻
Slow-mode onset
The defining signature of SFC is not oscillation or divergence, but **slowing**.
As criticality is approached:
• perturbations decay more slowly
• autocorrelation increases
• low-frequency power grows
• recovery from shocks becomes sluggish
This phenomenon — often called *critical slowing down* — appears here as a natural consequence of delayed feedback approaching the stability threshold.
⸻
Physical-domain demonstration
SFC has been tested against real-world power-grid frequency data.
In the UK GB grid (Dec 2025):
• rolling-window analysis identified a consistent feedback delay (~5 s)
• estimated parameters showed proximity to the SFC threshold
• slow-mode pressure increased during stress periods
• no failure occurred — but fragility was detectable
This demonstrates SFC’s role as an **early warning diagnostic**, not a failure predictor.
⸻
What SFC does not claim
SFC does not claim:
• universality across all systems
• prediction of exact failure times
• replacement of full dynamical modelling
• that threshold crossing guarantees collapse
It is a diagnostic indicator, not a deterministic prophecy.
⸻
Why this framework exists
Many catastrophic failures are preceded by long periods of invisible fragility.
SFC exists to make that fragility measurable — before failure, not after.
⸻
Open questions and future work
SFC remains an evolving framework. Open directions include:
• broader physical and synthetic validation
• domain-specific parameter estimation
• formal links to critical slowing-down theory
• coupling with belief and decision frameworks
• stress-testing across engineered and natural systems
Discussion, critique, and refinement are welcome.
⸻
One-sentence summary
Spectral Feedback Criticality (SFC) is a diagnostic law for detecting when delayed feedback systems approach instability through the emergence of slow modes — before failure occurs.