The Three Metronomes: Criteria for the Apparently Eternal Skeleton [F] (2026) Robert Galida – June 2026

Abstract

The attractor framework distinguishes conservative attractors (eternal skeleton) from dissipative attractors (transient dance). The most fundamental conservative attractors are the electron, proton, and neutrino class – collectively the three metronomes. This paper defines explicit criteria for a “metronome”: (1) apparent immortality (no observed decay), (2) effective indivisibility under ordinary perturbations, (3) conservation‑law protection, and (4) possession of a rest frame (non‑zero rest mass). It shows that electrons, protons, and neutrinos (the three mass eigenstates treated as a single class) are the best‑supported examples under current physics. The number three is empirical, not derived; the framework is corrigible. The three metronomes form the apparently eternal skeleton – a pragmatic substrate for measuring the transient dance of dissipative systems.

1. Introduction

The attractor framework divides persistent structures into two classes:

Conservative attractors (eternal skeleton) – persist without energy input, without observed decay, without internal change. They are mindless, time‑symmetric, and invariant.
Dissipative attractors (transient dance) – persist only by consuming energy, export entropy, and eventually decay.

(The conservative/dissipative dichotomy is a framework stipulation, not a physical law; it is defended in the broader attractor framework literature, e.g., Persistence Under Perturbation and Basin Defense and Stable Addition.)

The most fundamental conservative attractors are the three metronomes: the electron, proton, and the class of neutrino mass eigenstates (ν₁, ν₂, ν₃). Their name evokes their role as invariant reference entities – they provide a stable substrate against which all change can be measured. This paper defines explicit criteria for a metronome and applies them to each candidate.

2. Criteria for a Metronome

A metronome in the attractor framework must satisfy four criteria:

Criterion	Meaning	Operational check
1. Apparent immortality	No observed decay; no lighter state exists for it to decay into under known laws	Lifetime lower bounds >> age of universe; no allowed decay channel
2. Effective indivisibility under ordinary perturbations	Behaves as a stable, indivisible unit under all perturbations relevant to the framework (scattering, binding, chemical reactions)	Remains the same particle after typical disturbances; does not spontaneously change identity
3. Conservation‑law protection	Protected by an exact conservation law or an accidental symmetry that is effectively exact in the Standard Model	Lightest carrier of a conserved quantum number (electric charge, baryon number, lepton number)
4. Possession of a rest frame	Has non‑zero rest mass, hence a proper time and the ability to serve as a reference clock in its own rest frame	Invariant mass > 0

Rationale for Criterion 4: Measurement requires a local frame. A massless particle has no rest frame, no proper time, and cannot be used as a persistent local reference. While photons are extremely long‑lived, they serve as signal carriers, not as the invariant substrate. The framework prioritises rest‑frame existence because the “eternal skeleton” is meant to be the background against which change is measured – a background must have a local perspective to anchor measurements. This is a definitional choice, not a consequence of particle physics, and it is consistently applied.

Note on Criterion 3: Baryon number and lepton number are accidental symmetries, not gauge symmetries. The paper treats them on equal footing because both provide effective stability for the proton and neutrinos under Standard Model physics. If future experiments reveal baryon or lepton number violation, the framework will adjust accordingly.

3. Why the Electron Is a Metronome

Apparent immortality: Lightest negatively charged particle; no decay channel.
Effective indivisibility: Remains an electron after scattering, binding, etc.
Conservation protection: Electric charge and lepton number conservation.
Rest frame: Non‑zero rest mass.

→ The electron is a metronome.

4. Why the Proton Is a Metronome (Despite Being Composite)

Apparent immortality: No observed decay; experimental lower limit on half‑life > 10³⁴ years (Super‑Kamiokande, 2020).
Effective indivisibility: For all practical purposes (chemistry, nuclear physics, stellar processes), the proton behaves as a stable, indivisible unit.
Conservation protection: Baryon number is an accidental symmetry; it protects the proton from decay in the Standard Model.
Rest frame: Non‑zero rest mass.

→ The proton is a metronome. The framework does not require elementary particles; it requires maximal persistence under relevant perturbations.

5. Why the Neutrino Class (ν₁, ν₂, ν₃) Is a Metronome

The three neutrino mass eigenstates are treated as a single metronome class because they share the same stability argument, differ only in mass, and are grouped for the framework’s hierarchical classification.

Apparent immortality: No observed decay; cosmological and astrophysical lower bounds on neutrino lifetimes are orders of magnitude longer than the age of the universe. Neutrino oscillation is flavour mixing, not decay – the mass eigenstates are stable.
Effective indivisibility: Once a neutrino is in a mass eigenstate, it propagates without changing identity. (Weak interactions produce flavour eigenstates – superpositions of mass eigenstates – but the mass eigenstates themselves are stable and travel freely.)
Conservation protection: Lepton number is an accidental symmetry; in the Standard Model it protects neutrinos from decay. (If future experiments confirm that neutrinos are Majorana particles – violating lepton number – the framework will adjust; this is part of its corrigibility.)
Rest frame: Neutrinos have non‑zero rest mass (confirmed by oscillation experiments), albeit very small.

→ The neutrino class is a metronome. The three mass eigenstates count as one metronome type for the framework’s hierarchical classification.

6. Why Not Other Candidates?

Candidate	Fails criterion	Explanation
Free neutron	1 (apparent immortality)	Decays in ~15 minutes.
Neutron in a nucleus	2 (effective indivisibility)	Stability is environment‑dependent; not an irreducible attractor.
Photon	4 (rest frame)	Massless; no proper time. Excluded by definition (see rationale for Criterion 4).
Muon, tau	1	Decay rapidly.
Dark matter candidates	Not yet identified	If discovered and shown to be stable, massive, and effectively indivisible, they could become additional metronomes.
Composite stable structures (nuclei, atoms)	2	Not effectively indivisible; they are built from metronomes and are dissipative or emergent attractors, not part of the invariant skeleton.

7. The Number Three: Empirical, Not Derived

The paper’s title uses “three metronomes” as a convenient label for the electron, proton, and the neutrino class (the three mass eigenstates grouped together). The number three is not derived from first principles; it reflects current best empirical knowledge. If new stable particles are discovered (e.g., dark matter), the list will expand. The framework is corrigible by design.

8. The Apparently Eternal Skeleton

The term “apparently eternal” is strictly empirical: these particles have never been observed to decay or be transient, and for all practical purposes they behave as if they have no end. The three metronomes form the eternal skeleton – a pragmatic substrate against which the transient dance of dissipative systems (life, mind, society) is measured. This is a framework‑internal construct, not a metaphysical claim.

9. Stable Resonances and the Grounding of Dissipative Time Metrics

Each of the three metronomes possesses an invariant quantum frequency – its Compton frequency, given by $f = m c^{2} / h$ f=mc2/h. For the electron, this is ~1.24 × 10²⁰ Hz; for the proton, ~2.27 × 10²³ Hz; for neutrinos, the frequencies are very small but non‑zero. These frequencies are invariant, universal, and identical for every identical particle in the universe. They are stable resonances of the eternal skeleton.

Why this matters for dissipative systems:

Every dissipative system (a living cell, a brain, a society) is composed of or continuously interacts with electrons, protons, and neutrinos. The time constant τ that appears in corrective permeability (κ = 1/τ) can, in principle, be expressed as a multiple of these fundamental resonance periods. For example, a neuron’s recovery time after a perturbation – determined by ion channel kinetics, membrane capacitance, and metabolic rate – is measurable against the same invariant clock as any other physical process. The metronome provides the unit of time, not the mechanism.

Thus, κ is a genuine physical variable, not a mere metaphor. It refers to a ratio of measurable durations, anchored in the invariant frequencies of the metronomes.

Cross‑domain comparability:

The framework’s ability to compare κ values across vastly different domains (e.g., a thermostat’s seconds‑scale τ and a political movement’s months‑scale τ) does not follow from shared Compton‑frequency units alone. It follows from the framework’s definitional choice to treat κ as a domain‑general variable – a diagnostic that measures the same functional property (speed of return to baseline) in every system, regardless of scale or substrate. The metronomes ensure that such measurements are, in principle, commensurable; they do not guarantee that the comparison is meaningful in every case. That is a framework commitment, not a physics claim.

Caveat: The expression of τ as a multiple of Compton periods is a conceptual grounding, not a practical measurement protocol. No one will measure a society’s reaction time in electron oscillations. The importance is that κ is not an arbitrary label; it is a dimensionless ratio of durations, and durations are defined by the invariant resonances of the three metronomes.

10. κ and Basin Depth as Heuristics

The attractor framework introduces corrective permeability (κ = 1/τ) and basin depth (B) as conceptual heuristics. For the metronomes:

κ for decay is vanishingly small (effectively zero) on all observable timescales.
Basin depth is the energy barrier required to change the particle’s identity – effectively infinite for all practical purposes.

These are qualitative descriptors; they are not operational quantities in particle physics. They are included here for completeness of the framework’s vocabulary. For the application of κ and B to dissipative systems (e.g., belief updating, neural recovery), see the papers Basin Defense and Stable Addition and Why Clockwork Interventions Fail.

11. Corrigibility and Falsifiability

The framework explicitly invites revision:

If proton decay is observed, the proton will be downgraded to “very long‑lived” (or removed).
If neutrino decay or Majorana nature is confirmed, the neutrino class’s status will be revised.
If new stable particles are discovered, they will be added.

The attractor framework is a philosophical taxonomy and diagnostic tool, not a predictive physical theory. Its value lies in providing a unified language for persistence across domains.

12. Conclusion

The electron, proton, and neutrino class satisfy the attractor framework’s four criteria for metronomes: apparent immortality, effective indivisibility under ordinary perturbations, conservation‑law protection, and possession of a rest frame. They are the best‑supported examples of the apparently eternal skeleton under current physics. The framework is corrigible, the number three is empirical, and the language of “eternal skeleton” is pragmatic. The three metronomes anchor the distinction between conservative and dissipative persistence.

Suggested citation: Galida, R. S. (2026). The Three Metronomes: Criteria for the Apparently Eternal Skeleton. Fantasy Attractor.

Free Will as Attractor Autonomy: A Dynamical Account of Agency

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.