Home » Posts tagged 'free energy principle'

Tag Archives: free energy principle

Free Will as Attractor Autonomy: A Dynamical Account of Agency

May 25, 2026 4:20 pm / Leave a comment

Author: Robert Galida https://fantasyattractor.com/
Date: May 2026

Abstract

Free will is often seen as either a magical mystery (libertarianism) or an illusion (hard determinism).
This paper offers a third view using the attractor framework.

In this framework, your mind is a dissipative, self‑referential attractor of your whole body.
Free will is redefined as attractor autonomy:

The ability to generate behaviour from your own internal dynamics.
To keep yourself stable over time.
To model yourself.
And to reshape your own attractor landscape over time.

Agency comes in degrees – it is not a simple yes/no.
We give a mathematical formula for an agency index $A$ A that combines three factors:

Attractor dimensionality $D$ D (complexity of your brain’s activity)
Recursive self‑modification $R$ R (your ability to change your own habits)
Self‑reference strength $S$ S (how well you have a persistent self‑model)

The paper makes a falsifiable prediction: an inverted‑U relationship between attractor dimensionality and sense of agency – too low or too high reduces agency.
We describe how to test this with EEG, intentional binding tasks, and statistical methods. We also engage with classic compatibilist philosophers (Frankfurt, Dennett) and address Pereboom’s manipulation argument.
We even provide an explicit rule to avoid the “liver problem” (a false positive for self‑reference).

1. Introduction

The attractor framework says that persistence under disturbance is the basic mark of reality.
Minds are dissipative attractors – patterns that need constant energy flow, integrating the whole body.
In this view, free will cannot be a supernatural break from cause and effect. Instead, it must be a dynamical property of certain attractors.

We do not claim to solve the ancient free will debate. We offer a naturalistic, testable redefinition that adds new empirical content to compatibilism.

2. What Free Will Is Not – And What It Is

2.1 Rejecting supernatural libertarianism

Libertarian free will requires an uncaused choice – a break in the chain of cause and effect.
The attractor framework rejects this: there is no evidence for it, and it contradicts physical laws.

2.2 The error of hard determinism

Hard determinism says freedom is an illusion because everything is determined. But it confuses “determined” with “externally coerced”.
A system can be internally determined – by its own attractor – yet still be free. That is the core of compatibilism.

2.3 Free will as attractor autonomy

We define free will (or agency) as the degree to which a system has four properties:

Dissipative persistence – it stays alive by using energy and exporting waste (measured by energy use and recovery speed).
Self‑reference – it has an internal subsystem (an “indexical locus”) that models the whole system and is stable.
Trajectory selection – it can choose among different possible futures (measured by policy entropy $H (π)$ H(π)).
Recursive self‑engineering – it can change its own attractor shape (measured by learning‑to‑learn or metacognitive accuracy).

These four are jointly necessary. If any is missing, agency is at best primitive.

Because they are necessary, we combine them with a multiplicative formula (if any factor is zero, agency is zero). $A = {(\frac{D - D_{\min}}{D_{\max} - D_{\min}})}^{α} {(\frac{R}{R_{\max}})}^{β} {(\frac{S - S_{\min}}{S_{\max} - S_{\min}})}^{γ}$ A=(Dmax−DminD−Dmin)α(RmaxR)β(Smax−SminS−Smin)γ

Where:

$D$ D = attractor dimensionality (e.g., from EEG)
$R$ R = recursive modification capacity (e.g., improvement in a meta‑learning task)
$S$ S = self‑reference strength (normalised mutual information)

The constants ( $D_{\min}, D_{\max}$ Dmin,Dmax, etc.) are set from a reference population.
The exponents $α, β, γ$ α,β,γ are estimated from data (e.g., comparing healthy people with patients).
A threshold $A_{crit}$ Acrit (e.g., the 5th percentile of healthy humans) decides where agency begins.

Agency is graded:

Rock: $A \approx 0$ A≈0
Thermostat: $A \approx 0$ A≈0
Worm: $A \approx 0.1$ A≈0.1 (some learning, little self‑model)
Human: $A \approx 0.8$ A≈0.8

3. The Indexical Locus: Defining the “Self” and Avoiding the “Liver Problem”

The indexical locus $L$ L is the part of the system that acts as a persistent self‑model.
To avoid trivial cases (like a liver having high mutual information with the rest of the body), we add three extra conditions:

Top‑down causal influence – $L$ L can change the rest of the body in ways that serve the body’s goals (measured by variance explained beyond bottom‑up effects).
Informational closure – $L$ L’s own dynamics are relatively independent of the rest over short timescales (conditional mutual information > 0).
Self‑referential loop – $L$ L influences the body, and the body influences $L$ L back (bidirectional Granger causality).

These criteria rule out livers, pacemakers, and simple homeostats. The indexical locus is a recursive self‑model, not just a predictive subsystem.

4. Active Inference and Policy Entropy

In active inference (Friston), agents try to minimise “free energy” – they pick policies (sequences of actions).
Each policy is a trajectory through the agent’s attractor landscape.

Policy entropy $H (π) = - \sum p (π) \log p (π)$ H(π)=−∑p(π)logp(π) measures how many different policies are available.

Low entropy → rigid, one‑track mind.
High entropy → flexible, but possibly noisy.

Free will is the ability to access many low‑energy policies. The agent’s choices are not random; they are constrained by the attractor geometry. But if several attractor basins are open, the agent can choose among them – that is what we feel as free choice.

Policy entropy can be measured in behavioural tasks where multiple choices are equally good (e.g., probabilistic reversal learning, two‑armed bandit tasks).

5. The Inverted‑U Prediction and Falsification

5.1 Core prediction

We predict an inverted‑U relationship between attractor dimensionality $D$ D and the subjective sense of agency (e.g., from intentional binding experiments).

Very low $D$ D → chaotic, unstable (like schizophrenia) → low agency.
Very high $D$ D → rigid, stuck (like OCD) → low agency.
In the middle → flexible but stable → high agency.

The agency index $A$ A also includes $R$ R and $S$ S, which we think increase agency across the board. So to test the inverted‑U for $D$ D alone, you need to control for $R$ R and $S$ S (e.g., study people matched on those, or use partial correlation).

5.2 How to measure and test

Attractor dimensionality DD – use the Grassberger‑Procaccia algorithm on 5‑min resting‑state EEG/MEG.
Sense of agency – use the intentional binding paradigm: press a key, then a tone sounds; participants estimate the time between action and tone. Stronger binding means higher agency.
Statistical test – fit a quadratic regression: agency = $β_{0} + β_{1} D + β_{2} D^{2}$ β0+β1D+β2D2.
If $β_{2} < 0$ β2<0 and the vertex lies inside the observed range of $D$ D, the inverted‑U is supported. Use bootstrap (1000 resamples) to check confidence intervals.

5.3 Falsification condition

The framework is falsified if:

The quadratic coefficient $β_{2}$ β2 is not negative (no inverted‑U).
Or, in a clinical experiment (e.g., increasing $D$ D in OCD patients with NMDA drugs), agency does not decrease but keeps increasing.

6. Experimental Proxies – Summary Table

Construct	Measure	How to record	Expected relation to agency
Attractor dimensionality $D$ D	Correlation dimension (Grassberger‑Procaccia)	Resting‑state EEG/MEG (5 min)	Inverted‑U
Policy entropy $H (π)$ H(π)	Entropy of choice distribution	Probabilistic reversal learning (200 trials)	Inverted‑U
Sense of agency	Intentional binding magnitude	Action‑outcome interval compression (50 trials)	Max at intermediate $D$ D
Recursive self‑modification $R$ R	Learning‑to‑learn improvement	Meta‑learning task (pre‑post difference)	Positive (more is better)
Self‑reference strength $S$ S	Normalised mutual info $I_{n} (L; S)$ In(L;S)	Resting‑state fMRI or MEG	Threshold > θ

7. Hierarchical Constraints and Social Attractors

Free will is nested inside larger attractors – society, culture, laws, economy. Your range of choices is partly set by these.
This is not an objection; it is just the fact that freedom is always constrained autonomy.
We predict that societies with more cultural diversity (higher “cultural entropy”) allow more individual agency, other things being equal. This can be tested by cross‑cultural comparisons of policy entropy in decision tasks.

8. Engagement with Compatibilist Literature

8.1 Standard compatibilists (Frankfurt, Dennett)

Frankfurt (1971): freedom is about your will aligning with your own desires. Our framework adds that those desires must be encoded in a persistent self‑referential attractor. The recursive self‑engineering component $R$ R maps directly to Frankfurt’s “second‑order volitions”.
Dennett (1984): freedom is about being able to respond to reasons. Our framework adds that this requires a certain basin geometry and recursive plasticity.

8.2 Addressing Pereboom’s manipulation argument

Pereboom argues: if a neuroscientist engineers your brain, you are not free – even if your behaviour comes from internal dynamics.
Our reply: agency requires recursive self‑modification ( $R > 0$ R>0) at some point in your history.

A perfectly manipulated agent that never changed its own attractor would have $R \approx 0$ R≈0 and thus $A \approx 0$ A≈0.
A healthy human who learned and adapted has $R > 0$ R>0 and genuine agency.

The origin of the initial attractor does not matter – only the presence of self‑modification over time.

9. Open Questions and Limitations

Calibrating exponents – $α, β, γ$ α,β,γ and the threshold $θ$ θ need to be estimated from large‑scale data (e.g., Human Connectome Project) using maximum likelihood.
The liver problem – our exclusion criteria need empirical validation; we must show that organs like the liver do not satisfy them.
Inverted‑U for policy entropy – the same shape is predicted but may be hidden by decision noise.
Moral responsibility – the framework gives a basis for responsibility (if $A > A_{crit}$ A>Acrit), but it does not settle all normative questions – it only gives a scientific starting point.

10. Conclusion

Free will is not a supernatural escape from physics. It is a dynamical property of certain dissipative, self‑referential attractors:

The ability to act from your own internal dynamics.
To keep a stable self‑model over time.
And to reshape your own attractor landscape.

This account is compatibilist, testable, and graded.
The inverted‑U prediction, with a specified statistical test, gives a clear falsification criterion.
The dance of free will is the dance of a self that persists under perturbation.

Suggested citation: Galida, R. S. (2026). Free Will as Attractor Autonomy: A Dynamical Account of Agency in the Attractor Framework (Reader‑Friendly Version). Fantasy Attractor.

Attractor Dynamics in Belief Formation, Correction, and Mental Health: A Research Programme

May 25, 2026 4:19 pm / Leave a comment

Author: Robert Galida https://fantasyattractor.com/
Date: May 2026

Abstract

This paper applies the attractor framework (persistence under disturbance) to belief systems and mental health.

We introduce three measurable concepts:

Attractor depth – how rigid or unstable a belief is.
Error half‑life – how long it takes for a false belief to fade after correction.
Coupling strength to error signals – how open a belief is to reality checks.

We contrast two disorders:

OCD (obsessive‑compulsive disorder) may involve overly deep (rigid) attractors.
Schizophrenia may involve too shallow (unstable) attractors – with appropriate caution.

We propose experiments to measure error half‑life, detect early warning signs of belief shifts (while managing false alarms), and find the optimal pace for correction (“critical damping”).

We also outline:

N=1 attractor engineering (self‑experimentation)
Wearable early‑warning systems for relapse prevention (discussing lag time and false positives)
Cross‑coupling as a measure of resilience (distinguishing healthy from brittle coupling)

This paper is a research roadmap, not a finished theory.

1. Introduction

In the attractor framework, your mind is a dissipative attractor of your whole body – a pattern that needs energy, can be disturbed, and can adapt (Galida, 2026, Persistence Under Perturbation).
Beliefs are smaller attractors inside that landscape. Their stability determines how easily you update when faced with contradictory evidence.

This paper turns attractor concepts into testable ideas about how beliefs form, stick, and change – and how to help them change. It is a roadmap, not the final word.

2. Attractor Depth and Mental Disorders

Neurocomputational models suggest a contrast between OCD and schizophrenia, but we must be careful.

Disorder	Attractor Property	Behavioural Sign	Example Task
OCD	Too deep (rigid)	Stuck, hard to switch	Reversal learning (changing rules)
Schizophrenia	Too shallow (unstable)	Jumpy, over‑sensitive to noise	Delayed match‑to‑sample with distractions

Evidence:

Unmedicated OCD patients make many perseverative errors on reversal‑learning tasks; this correlates with symptom severity (Remijnse et al., 2006).
Reduced NMDA/GABA function in schizophrenia makes attractor networks unstable, leading to cognitive slips and delusions (Rolls, 2021).

Caveats:

Mental disorders are complex, with multiple attractors. We are talking about symptom clusters, not whole‑disorder diagnoses.
Disorders like anxiety, depression, and personality disorders lie in the middle – their attractors are domain‑specific (e.g., depression has deep negative‑belief basins but shallow positive ones).

Prediction: Attractor depth could be measured from behaviour (switching rates, reaction time variability) by fitting a two‑state hidden Markov model to reversal‑learning data – a hypothesis for future work.

3. Error Half‑Life: A New Measure of Belief Rigidity

Error half‑life $T_{1 / 2}$ T1/2 is the time it takes for a false belief’s confidence to drop by half after you present corrective evidence.

How to measure it

Give people a false belief (e.g., a made‑up fact).
Give them correct information (text, video) every day for a while.
Ask them to rate their belief confidence (0–100) at intervals.
Assume a simple exponential decay model $C (t) = C_{0} e^{- t / τ}$ C(t)=C0e−t/τ as a starting point (real decay could be sigmoidal or power‑law).
Then $T_{1 / 2} = τ \ln 2$ T1/2=τln2.

What we expect in different conditions

Delusional disorders → very long half‑life (deep attractor).
Depression → long half‑life for negative self‑beliefs, but normal for positive ones (asymmetric updating).
Anxiety → short half‑life, but possible overshoot (shallow basin → oscillation).

Therapeutic application

The goal is to shorten error half‑life. Methods like spaced repetition and active recall (quizzing) could help – they strengthen corrective memory traces, similar to memory reconsolidation.

Relationship to attractor depth

Attractor depth is a static measure (inertia). Error half‑life is a dynamic measure (recovery speed). They are related but not the same: depth gives initial resistance, half‑life gives the time course. We need both.

4. Critical Slowing Down Before Belief Shifts

Before a sudden change of belief (e.g., leaving a cult, political conversion, therapy breakthrough), you may see early warning signals – rising variance, higher autocorrelation, slower recovery from small disturbances. This is called critical slowing down (Scheffer et al., 2009).

How to detect it

Collect daily belief ratings, mood scores, or social media sentiment.
Compute rolling variance and autocorrelation with a moving window.
If they exceed a baseline threshold, a shift may be coming.

False positive problem

Rising variance can be caused by other things (seasonal mood, life events). To reduce false alarms:

Use control periods (compare with a stable trait belief).
Combine multiple signals (HRV, sleep, activity) with self‑report.
Use a conservative threshold (e.g., 3 standard deviations above baseline).

This is a research tool, not a clinical diagnostic yet.

Prediction: You can detect these signals in diaries before a person deconverts, changes politics, or relapses into depression. A well‑timed prompt might help, but false positives must be managed.

5. Optimal Correction Dosing (Critical Damping)

From control theory, there is an optimal pace for delivering corrections: not too slow (oscillates), not too fast (overshoot/backfire). This is called critical damping.

N=1 protocol

Vary the gap between corrections (massed vs. spaced).
Track belief confidence over time.
Measure how quickly and smoothly it changes.

Hypothesis: Spaced correction (e.g., daily micro‑doses) works better than one big confrontation – a well‑known finding in memory research (Ebbinghaus, spaced repetition). The twist is applying it to beliefs, which are more emotional and identity‑linked. The mechanism may be similar, but emotional valence may change the optimal schedule.

6. Fantasy vs. Shared Reality Attractors – Operational Metrics

Metric	Low Corrective Permeability (Fantasy)	High Corrective Permeability (Shared Reality)
Coupling to error signals	Low (few fact‑checks, no update)	High (active correction)
Basin depth	Deep (needs large evidence)	Shallow (small anomalies work)
Error‑correction latency	Long (days/weeks)	Short (hours/days)
Information diversity tolerated	Low (echo chamber)	High (multiple sources)

Double‑bind computational model

In conspiracy cultures, contradictory evidence gets reinterpreted as confirmation (“cover‑up”). We can model this as an asymmetric Bayesian update: $P (belief ∣ contrary evidence) \geq P (belief ∣ supporting evidence)$ P(belief∣contrary evidence)≥P(belief∣supporting evidence)

Example: Start with belief probability 0.9. A contrary piece of evidence that would normally lower it to 0.3 is instead interpreted as evidence of suppression, so the new probability stays at 0.85. The belief drifts only slowly.

Breaking the loop: Indirect interventions work better than direct refutation:

Point out internal inconsistencies.
Seed doubt through trusted messengers.
Use graduated reality‑testing.

7. Wearable Early Warning of Attractor Shifts

Protocol: Use consumer wearables (HRV, skin conductance, actigraphy, sleep) plus daily self‑reports (mood, belief rigidity). Compute rolling variance and autocorrelation in real time.

Evidence: Drops in nocturnal HRV preceded a depressive relapse in a case study (Tonge et al., 2024).
Prediction: Rising variance/autocorrelation in HRV, plus mood volatility, can predict an imminent crisis.

Latency and false alarms

Useful lead time is days, not hours. HRV changes can appear 1–2 weeks before relapse.
False positives are a concern. Use a two‑stage alert: first detect statistical anomaly, then confirm with a brief self‑report (EMA).
Specificity needs to be established in longitudinal N=1 studies.

Intervention: When thresholds are crossed, trigger a micro‑intervention (mindfulness, therapist call) – a closed‑loop prevention system.

8. N=1 Attractor Engineering – Minimal Perturbation Protocol

Goal: Find the smallest intervention that shifts a maladaptive attractor (phobia, obsessive thought) without causing oscillation or backfire.

Procedure:

Define the target (e.g., fear rating 0–10).
Start with very low‑intensity perturbations (e.g., brief exposure, mild counter‑evidence).
Measure change after each step.
When a threshold shift is detected (say, 30% reduction – a provisional starting point; adjust based on baseline variability), record the dose.
Back off slightly and check stability.

Principle: Never collapse an attractor faster than reality can correct. Use fine steps (5–10% increments) and frequent monitoring. This is precision self‑regulation. Generalisability from N=1 to populations is an open question (see Section 12).

9. Cross‑Coupling as a Resilience Metric

Hypothesis: High cross‑domain coupling (e.g., HRV ↔ mood ↔ sleep) indicates adaptive resilience – the system is coordinated and self‑correcting. Low coupling or unidirectional cascades indicate brittle coupling (a disturbance in one area spreads uncontrollably).

Measurement: Collect simultaneous time series (HRV, sleep, activity, mood). Compute cross‑correlation or Granger causality.

Adaptive = bidirectional, with negative feedback (e.g., poor sleep → lower HRV → mood drop → social support → sleep improves).
Brittle = unidirectional, amplifying (e.g., sleep loss → stress → more sleep loss).

Prediction: Good recovery from stress shows strong bidirectional influences. Low coupling or unidirectional cascades will precede breakdowns.

Intervention: Improve adaptive coupling with synchrony exercises (e.g., daily breathing with light exposure, yoga, social rhythm therapy). Testable in an N=1 self‑tracking experiment.

10. Philosophical Extensions (Brief)

Are attractors real? Yes, as structural patterns (process metaphysics). They have causal power – like the path of a river.
Free will as attractor autonomy – acting according to your own attractor is compatibilist freedom. Our framework adds that freedom is about basin width and flexibility, not a binary.
Cosmic attractor – speculative. The universe might have a global attractor (e.g., heat death), but it’s untestable now.
Darwinian problem of evil – animal suffering is a strong challenge to theism; the “deep harmonies” hypothesis is hard to falsify.

11. Open Questions and Next Steps

Can error half‑life be measured reliably from smartphone‑based belief tracking? What decay model fits best?
What is the dose‑response curve for corrective interventions? Linear, exponential, or threshold? How does it vary with attractor depth?
Can wearables detect early warning signs before a psychiatric relapse? What are the false‑positive rates and lead times?
Does adaptive cross‑coupling improve after synchrony‑based therapies?
How are error half‑life and attractor depth related? Same thing at different timescales, or different constructs?
How can N=1 findings be aggregated into population‑level knowledge? One approach: meta‑analysis of single‑subject time series using hierarchical Bayesian models.

12. Conclusion

This research programme puts attractor dynamics to work on beliefs and mental health.

We have proposed testable metrics (attractor depth, error half‑life, coupling strength) and experimental protocols for N=1 self‑engineering and early warning.

The framework provides a naturalistic language for understanding why some beliefs resist correction and how to intervene optimally.

We acknowledge our limitations – the exponential decay assumption, false positives in early warning, and the generalisability of N=1 results – and treat them as open questions for future work.

This extends the attractor trilogy into actionable health and epistemology.

Suggested citation: Galida, R. S. (2026). Attractor Dynamics in Belief Formation, Correction, and Mental Health: A Research Programme (Reader‑Friendly Version). Fantasy Attractor.

The Persistence Functional: A Mathematical Measure of Attractor Resilience

May 25, 2026 4:18 pm / Leave a comment

Author: Robert Galida
Date: May 2026

Abstract

The attractor framework says that persistence under disturbance is the basic mark of reality.
To turn this idea into a formal science, we introduce the persistence functional $P (x)$ P(x).

$P (x)$ P(x) is a single number that measures:

How deep a state is inside an attractor basin.
How quickly it returns after a knock.

We define $P (x)$ P(x) for three different kinds of systems:

Deterministic dissipative systems – here $P$ P is linked to Lyapunov exponents and basin stability.
Stochastic systems – here $P$ P is linked to escape time and quasipotential.
Information‑theoretic systems – here $P$ P is linked to negative free energy or mutual information.

The recovery rate $- \dot{P} / P$ −P˙/P is a universal sign of critical slowing down – a warning that a system is about to tip.

We also discuss limitations: resilience may depend on direction (“anisotropic”), and multiple timescales may need vector or tensor persistence. We list open mathematical problems.
This paper is a roadmap, not a finished theory.

1. Introduction

In the attractor framework, persistence under disturbance is central. But we have not had a single number to say how persistent a state is.
The persistence functional $P (x)$ P(x) aims to fill that gap.

What $P (x)$ P(x) should do:

$P (x) > 0$ P(x)>0 for states inside an attractor basin.
For a conservative attractor (like a free electron), $P$ P is maximal (normalised to 1).
For a dissipative attractor, PP drops after a disturbance and then recovers.
The recovery rate −P˙/P−P˙/P equals:
- the negative of the largest Lyapunov exponent (for deterministic systems)
- the inverse return time (for stochastic systems)
- the rate of information loss (for informational systems)
$P$ P falls as the system approaches a bifurcation, giving early warning.

We do not give one universal formula. Instead, we give a family of definitions, each suited to a different type of system, all united by the same purpose – measuring resilience.

2. Deterministic Dissipative Systems

Consider a smooth system $\dot{x} = f (x)$ x˙=f(x) with a stable attractor $A$ A and its basin $B (A)$ B(A).

A natural candidate for $P (x)$ P(x) uses a Lyapunov function $V (x)$ V(x) – a kind of energy that always decreases inside the basin ( $\dot{V} < 0$ V˙<0).

We define: $P (x) = 1 - \frac{V (x) - V_{A}}{V_{max} - V_{A}}$ P(x)=1−Vmax−VAV(x)−VA

This gives $P = 1$ P=1 on the attractor and $P \to 0$ P→0 at the basin boundary.
Near the attractor, the recovery rate is related to the largest Lyapunov exponent $λ_{1}$ λ1: $- \dot{P} / P \approx - λ_{1}$ −P˙/P≈−λ1

When the system approaches a tipping point, $λ_{1} \to 0^{-}$ λ1→0−, so the recovery rate slows down – this is critical slowing down.

Conclusion: For deterministic systems, $P$ P can be built from a Lyapunov function. The recovery rate equals the negative of the largest Lyapunov exponent.

3. Stochastic Systems

When noise is present, persistence is about how long it takes to escape from the basin.
The mean first passage time $τ (x)$ τ(x) – the average time to leave – is a natural measure.

We define: $P (x) = \frac{τ (x)}{τ_{max}}$ P(x)=τmaxτ(x)

where $τ_{max}$ τmax is the value at the attractor.

For weak noise, $τ (x)$ τ(x) grows exponentially with the quasipotential $U (x)$ U(x) (Freidlin–Wentzell theory): $τ (x) \sim e^{U (x) / ϵ}$ τ(x)∼eU(x)/ϵ

So: $P (x) \propto e^{- (U_{max} - U (x)) / ϵ}$ P(x)∝e−(Umax−U(x))/ϵ

The recovery rate is the inverse of the return time. As a tipping point is approached, the return time diverges, and the recovery rate goes to zero. This again gives critical slowing down – rising variance and autocorrelation.

Conclusion: For stochastic systems, $P$ P is proportional to the mean exit time (or the exponential of the quasipotential). This connects persistence to large deviation theory.

4. Information‑Theoretic Systems

For systems where information matters (neural, cognitive, social), we can define persistence using mutual information between past and future.

Let $I_{past,future}$ Ipast,future be the predictive information. Then: $P (t) = I (past; future at time t) or P = e^{- surprisal}$ P(t)=I(past;future at time t)orP=e−surprisal

The decay of $P (t)$ P(t) over time measures memory loss.
Landauer’s principle connects information loss to entropy production: $\dot{P} / P \leq - \frac{\dot{S}}{k_{B} \ln 2}$ P˙/P≤−kBln2S˙

Alternatively, in the free energy principle (Friston), the negative free energy $- F$ −F acts like a Lyapunov function. We can set: $P = e^{- F / k T} or P = - F$ P=e−F/kTorP=−F

Then $- \dot{P} / P$ −P˙/P is the rate of free energy minimisation, which slows near bifurcations.

Conclusion: For information‑theoretic systems, $P$ P can be defined via mutual information decay or negative free energy, linking persistence to entropy production and predictive coding.

5. Unifying Recovery Rate and Critical Slowing Down

Across all types of systems, the recovery rate $λ_{rec} = - \dot{P} / P$ λrec=−P˙/P (just after a small disturbance) is a universal indicator:

Deterministic dissipative: $λ_{rec} = ∣ λ_{1} ∣$ λrec=∣λ1∣ (absolute value of the largest Lyapunov exponent)
Stochastic: $λ_{rec}$ λrec = inverse of the return time, related to the quasipotential’s curvature
Information‑theoretic: $λ_{rec}$ λrec = rate of free energy minimisation or information loss

As the system approaches a bifurcation, $λ_{rec} \to 0$ λrec→0. This is critical slowing down.
It shows up as rising lag‑1 autocorrelation and variance (Scheffer et al., 2009).
So $P$ P and its recovery rate give early warnings.

6. Normalisation for Conservative Attractors

For a perfect conservative attractor (e.g., an electron in its ground state, no decay), the persistence functional should be constant and maximal: $P_{cons} = 1 for all times$ Pcons=1for all times

No recovery rate is defined (or it is zero). This anchors the scale.

For emergent approximate conservative systems (like atomic clocks), $P$ P is very close to 1 and decays extremely slowly.

7. Limitations – Scalar Collapse and Anisotropic Resilience

A single scalar $P (x)$ P(x) may not be enough for systems where resilience is anisotropic – that is, recovery speed depends on the direction of the perturbation.
High‑dimensional systems can have multiple timescales (fast and slow modes). A scalar average can miss important structure.

Future work may need:

Vector persistence – a list of recovery rates along different directions.
Tensor persistence – a metric that captures the full shape of the basin.
Persistence manifold – the geometry of the basin in state space.

We accept this limitation. The scalar $P$ P is a useful first approximation for systems with isotropic resilience or for early‑warning applications where a single number is enough. For complex systems, a multidimensional generalisation is an open research problem.

8. Open Mathematical Problems

Derive P(x)P(x) from first principles for a given class of systems (e.g., from a variational principle).
Prove that −P˙/P=∣λ1∣−P˙/P=∣λ1∣ for a wide class of dissipative systems.
Extend the definition to systems with multiple attractors and chaotic basins (where basin stability is fractal).
Establish a rigorous relationship between PP and the mutual information decay rate for non‑equilibrium processes.
Formulate a universal persistence functional that works across all regimes – or prove it’s impossible.
Test the predictive power of PP in controlled experiments (e.g., ecological microcosms, neural cultures, social media sentiment).
Develop vector/tensor persistence for anisotropic resilience.

9. Conclusion

The persistence functional $P (x)$ P(x) gives a mathematical language for attractor resilience.

We have given operational definitions for three regimes:

Deterministic dissipative → Lyapunov / basin stability
Stochastic → escape time / quasipotential
Information‑theoretic → mutual information / free energy

The recovery rate $- \dot{P} / P$ −P˙/P unifies critical slowing down across all these domains.

We have explicitly noted limitations (scalar collapse, anisotropy) as open problems.

This paper is a roadmap, not a final theory. The framework now has a quantitative step.

Suggested citation: Galida, R. S. (2026). The Persistence Functional: Towards a Mathematical Measure of Attractor Resilience (Reader‑Friendly Version). Fantasy Attractor.

Fantasy Attractor

Tag Archives: free energy principle

Free Will as Attractor Autonomy: A Dynamical Account of Agency

Abstract

1. Introduction

2. What Free Will Is Not – And What It Is

2.1 Rejecting supernatural libertarianism

2.2 The error of hard determinism

2.3 Free will as attractor autonomy

3. The Indexical Locus: Defining the “Self” and Avoiding the “Liver Problem”

4. Active Inference and Policy Entropy

5. The Inverted‑U Prediction and Falsification

5.1 Core prediction

5.2 How to measure and test

5.3 Falsification condition

6. Experimental Proxies – Summary Table

7. Hierarchical Constraints and Social Attractors

8. Engagement with Compatibilist Literature

8.1 Standard compatibilists (Frankfurt, Dennett)

8.2 Addressing Pereboom’s manipulation argument

9. Open Questions and Limitations

10. Conclusion

Attractor Dynamics in Belief Formation, Correction, and Mental Health: A Research Programme

Abstract

1. Introduction

2. Attractor Depth and Mental Disorders

3. Error Half‑Life: A New Measure of Belief Rigidity

How to measure it

What we expect in different conditions

Therapeutic application

Relationship to attractor depth

4. Critical Slowing Down Before Belief Shifts

How to detect it

False positive problem

5. Optimal Correction Dosing (Critical Damping)

N=1 protocol

6. Fantasy vs. Shared Reality Attractors – Operational Metrics

Double‑bind computational model

7. Wearable Early Warning of Attractor Shifts

Latency and false alarms

8. N=1 Attractor Engineering – Minimal Perturbation Protocol

9. Cross‑Coupling as a Resilience Metric

10. Philosophical Extensions (Brief)

11. Open Questions and Next Steps

12. Conclusion

The Persistence Functional: A Mathematical Measure of Attractor Resilience

Abstract

1. Introduction

What P(x)P(x) should do:

2. Deterministic Dissipative Systems

3. Stochastic Systems

4. Information‑Theoretic Systems

5. Unifying Recovery Rate and Critical Slowing Down

6. Normalisation for Conservative Attractors

7. Limitations – Scalar Collapse and Anisotropic Resilience

8. Open Mathematical Problems

9. Conclusion

Subscribe to "Fantasy Attractor" via Email

What $P (x)$ P(x) should do: