11  From Structure to Time: FEP and Attractors

Status: Draft

v0.3

11.1 Introduction

Graph structure, invariances, and abstract relations constrain the dynamics a model can represent, but they do not constitute the dynamics themselves. This chapter defines admissible state spaces and mechanism classes, then defines attractors relative to a specified flow. The Free Energy Principle (FEP) is an optional specialised interpretation, not a prerequisite for CDMs.

Their working meanings and limits are collected in the glossary. Markov blankets below are optional conditional-independence models, not CDM scope rules.

11.2 Structural Summary: What We’ve Learned

11.2.1 What We Learned in Structural

In Part I (Structural), we established:

  1. The Causal Hierarchy and Three Strata (Chapter 1): strata × levels, and a minimal CDM spine
  2. The Primary Unit: The Dyad (Chapter 2): fundamental unit of directed dependence
  3. Graphs, SCMs, identification, do-calculus, counterfactuals, transport (Chapters 3–8): invariant structure before time
  4. Causal discovery (Chapter 05b): centripetal interlude (Observable → Structural), not a substitute for a user DAG

The Structural stratum is characterised by:

  • Idealised structure: Causal structure expressed as assumptions (e.g., graphs/mechanisms) that are treated as time-invariant within a modelling context
  • Repeatable patterns: Relations or organisations that may recur across different particulars or histories; their mathematical description is not itself an event-like cause
  • Graph Structure: Directed dependencies (edges) encoding structural assumptions and invariances
  • Abstract Relations: Relations are still abstract; this chapter opens the door to time

11.2.2 Structural Constraints Before Attractors

Structure may specify a state space \(\mathcal X\), a feasible set \(\mathcal C\subseteq\mathcal X\), conservation laws, symmetries or a mechanism class \(\mathcal F\). None is an attractor by itself. Attraction is a dynamical property: it requires a flow, iterated map or transition kernel.

For a deterministic system \(\dot X=f(X;\theta)\) with flow \(\Phi_t\), an invariant set \(\mathcal A\) is attracting only when a neighbourhood \(N(\mathcal A)\) exists such that \(d(\Phi_t(x),\mathcal A)\to0\) as \(t\to\infty\) for \(x\in N(\mathcal A)\). Objectives, preferred controller states and potential landscapes may motivate analysis, but are not automatically attractors or free-energy minima.

11.2.3 Constraint, Mechanism, Attractor, Preference and Lure

Several terms are often run together in discussions of organised biological systems. Here they must remain distinct.

Term Modelling role What it does not establish
Formal constraint A declared invariance, feasibility condition, viability condition, or cross-embodiment relation An efficient cause or an additional variable in the graph
Mechanism An assignment or transition law that realises changes under stated conditions A proof that every admissible state is reached
Attractor An invariant set approached under a specified deterministic or stochastic dynamics A goal, preference, or explanation independent of the dynamics
Basin The set of initial conditions from which the specified dynamics approach an attractor A final cause or a position in a separate realm
Preferred state density / policy A component of an explicit FEP or control model used to evaluate prospective trajectories Agency, cognition, or a guarantee of viability without empirical assessment
Lure1 Relevance of a possibility to a concrete becoming Remains distinct from a basin, a force, an efficient cause, or a disembodied agent

A basin is geometry induced by a particular \(f\), whereas a lure concerns the relevance of possibility within a concrete becoming. An operational model may represent empirically tractable aspects of organised orientation through a declared constraint, a preference density \(p^*(o)\), an expected-free-energy objective \(G(\pi)\), a policy, and the reachable-set geometry of embodied dynamics. These objects can guide measurement and comparison without exhausting the lure’s intensive relevance or converting final causation into an efficient force (Friston and Ao 2012; Segall 2025, 2026). The model therefore makes its empirical commitments answerable to evidence while leaving the ontology of possibility explicit.

11.2.4 Regenerative morphology as a constraint question

Regeneration and synthetic morphology separate a surprising realised trajectory from a new event-like cause. Anthrobots assembled from adult human tracheal cells and reconfigurable Xenobots can exhibit collective behaviour outside their cells’ ordinary developmental setting (Gumuskaya et al. 2024; Kriegman et al. 2021). Such observations justify a demanding question: which evolved local competencies, physical constraints, boundary conditions, and collective mechanisms make the trajectory reachable? They do not by themselves show that a pattern has entered the system from a non-physical domain (Levin 2026, 2022).

For a regeneration experiment, write a viability claim as \(\mathcal V\subseteq\mathcal X\), damage as \(do(D=d)\), and recovery as a family of trajectories \(X^{do(D=d)}_{0:T}\). Evidence for a target morphology is stronger when different perturbations return to a bounded viable region with measured recovery time, control cost, and failure modes. The substantive work is then to test the proposed mechanism and its portability across embodiments, rather than to relabel the unexplained remainder as an external causal parent.

The toy trajectory below makes the logic testable without making a biological claim: a scalar state relaxes towards a declared viable interval after a damage intervention. The mechanism, assay, and biological interpretation remain open (Figure 11.1).

T_recovery = 32
damage_time = 9
viable_interval = (0.9, 1.1)
recovery_trajectory = ones(Float64, T_recovery)
recovery_trajectory[damage_time] = 0.2
for t in damage_time:(T_recovery - 1)
    recovery_trajectory[t + 1] = recovery_trajectory[t] + 0.24 * (1.0 - recovery_trajectory[t])
end
nothing
Figure 11.1: Schematic recovery under a declared damage intervention. The viable interval and relaxation rule are modelling choices; the plot is not biological evidence or an FEP derivation.

11.2.5 Structural Dependencies (Edges)

In the Structural stratum, edges encode both:

  • Idealised structure: Which variables can directly affect which others (model assumptions)
  • Invariant patterns: Stable, environment-independent relations (invariances)

Edge structure remains constant, ignoring environments and contexts; edges remain abstract and timeless, not yet temporal.

11.3 The Free Energy Principle as Bridge

11.3.1 FEP as an Optional Specialised Module

The Free Energy Principle (FEP) can supply a specialised interpretation for systems meeting additional assumptions; it does not establish CDM boundaries, attractors or edge evolution by itself (Friston 2010; Friston et al. 2006). Under suitable commitments, variational free energy can be understood as:

  • Surprise minimization: Systems avoid surprising states
  • Prediction error minimization: Systems minimise prediction errors
  • Active inference: Systems act to maintain expected states

11.3.2 Assumption Ladder

Claim Additional requirements
Markov-blanket conditional independence A joint distribution and a valid partition
Persistent boundary Approximate stability over a stated interval
NESS-based FEP Appropriate steady-state density and regularity conditions
Bayesian-mechanics interpretation A mapping from internal states to densities over external states
Active inference A generative model, preferences and an action-selection process
Biological agency Independent empirical and functional justification

Thus \(\text{CDM}+\text{FEP assumptions}\Rightarrow\text{active-inference CDM}\); FEP is not a universal explanation of causal topology or attracting behaviour.

11.3.3 The Markov Blanket: System Boundaries in FEP

In an explicitly assumed FEP model, a Markov blanket is a conditional-independence partition that can be interpreted as an interface between modelled internal and external states (Friston 2010, 2013; Whitehead 1978). It does not, by itself, identify an organism, an environment, or an ontological boundary. Those are modelling and scientific interpretations layered on top of the stated partition.

FEP partitions the generative story as follows:

  1. Internal states (\(\mu\)): The organism, latent variables the model evolves (e.g. \(X_t\) in a state-space model). Each \(\mu_t\) is the settled state that becomes input for the next step.
  2. External states (\(\eta\)): The environment in the strict sense, everything outside the modelled organism whose direct influence is mediated by the blanket (or left unmodelled).
  3. Markov blanket (\(s\), \(a\)): The interface:
  • Sensory states (\(s\)): What is given in observation (\(Y_t\)), not the full latent process.
  • Active states (\(a\)): Endogenous actions generated within the system. They are distinct from external interventions that replace assignments.

Whitehead distinguishes a society—a nexus of actual occasions ordered by the inheritance of a defining character—from an enduring object, a society whose order takes the form of a historical route of successive occasions. More complex physical objects may themselves be societies of enduring objects (Whitehead 1978). In CDCS, a referent may be interpreted as an enduring object when its continuity and constitution support that account; the graph representation alone establishes neither.

A DAG records which kinds of influence we treat as admissible. A model of what persists as “the system” therefore declares its referent and continuity criterion; a Whiteheadian interpretation additionally asks whether successive occasions form the ordered society constituting an enduring object. Neither claim is equivalent to an invariant graph \(G\), mechanism class \(F\), attractor regime, or statistical population. Variables mark roles at a declared scale; the steps \(t\) index represented times, not occasions (a timestamp does not individuate an occurrence). The book’s passage from Structural through Dynamical to Observable is how we analyse a CDM, not a claim that the graph exhausts the process it represents.

Simondon’s account of individuation sharpens this point: an individual may be a continuing achievement of organisation rather than a fixed bearer of properties (Simondon 2020). CDCS uses the thought where it helps to specify constitution, causal inheritance, and identity through transformation. Its dynamical counterpart remains precise: metastability, bifurcation, and regime change are claims about a declared system and must be established through its transition law and perturbation behaviour.

Within a temporal DAG, occasion and enduring name ontological character only. They do not select how many nodes appear. Within a TimeUnrolledGraph, declared temporal_support determines multiplicity: pointwise support expands across represented times; FromOnsetSupport(t₀) and GlobalSupport() reuse a single node; PointSupport and IntervalSupport name an explicitly bounded extent. graph_kind declares which representation is in view and guards whether such unrolling is admissible. Always show temporal support separately when discussing ontology.

Contrasting examples keep the axes apart:

  • An enduring sheep (continuity under an identity criterion) may still have several point-supported weight variables.
  • One occasion can extend across an interval that contains several observations.
  • One trajectory-valued node can describe many occasions.

Prefer placing identity_criterion and optional ontological_character on a ReferentSpec, with variables pointing via referent / referent_id. Declare ontology only when the example makes an ontological claim or teaches the distinction. For ordinary estimation, referent, value representation, and temporal support are enough. An unspecified ontology is not a defective causal model.

Book time indexes take \(t = 0\) as the initial state or baseline and \(t = 1\) as the first subsequent transition or decision stage. That pedagogy is distinct from Julia array indexing. FromOnsetSupport(t₀) still requires an explicit onset that follows the mechanism: if diagnosis[0] generates an attribute through a lag-one assignment, that attribute has FromOnsetSupport(1); an attribute already present at baseline has FromOnsetSupport(0).

In figures, DAGMakie glyphs follow value_representation. Rounded rectangles denote interval summaries under an explicit visual convention; labels or support bars identify the intervals summarised. Shape does not indicate enduring identity, temporal support, or node count. A summary indexed or reported at a point need not be point-supported merely because the graph places a node at one timestamp. Show referent identity through labelled grouping or lanes, separately from causal arrows (Concept Reference, Table 8; Figure 11.2).

Figure 11.2: Temporal unrolling for one enduring sheep referent. Circles are pointwise states (diagnosis[t], weight[t]) and the from-onset pasture attribute; the rounded rectangle is the interval summary week_mean_weight. Labels or support bars identify the interval summarised. Shape does not indicate enduring identity.

11.3.4 Declared contracts and refusals

Package validators check declared contracts and supported conditions. They do not certify that those declarations are empirically true. Exhaustive signatures live in the package manuals; the book needs only the methodological lesson. For each gate below, one accepted pattern and one rejected pattern.

11.3.4.1 Causal projection

A semantic diagram may mix constitutive, participatory, and causal edges. Ordinary identification acts on a causal projection, and that projection requires justification.

Accepted. Influence edges remain after projection and feed a backdoor query.

Rejected. Treating a constitutive part–whole edge as if it were already a causal arrow in \(G\), without projecting or justifying the residual graph.

11.3.4.2 Feasibility and interval-summary gates

A formally expressible assignment may still lack admissible intervention semantics.

Accepted. Scalar do on a point-supported treatment, or an interval summary after an explicit macro-intervention justification.

Rejected. Silent scalar do on an :interval_summary node, or an intervention that violates a declared :feasibility constraint without justification.

11.3.4.3 Information availability, \(\mathcal{H}_t\)

A policy may use only what is available at decision time. Temporal support is not availability: a quantity can be realised earlier yet remain unavailable to \(\mathcal{H}_t\).

Accepted. Policy(...; information_set = [:w]) when w is available at \(t\), or an ObservationBridge whose availability clock admits the observation into \(\mathcal{H}_t\).

Rejected. Letting a future assay, or a quantity whose availability clock is later than \(t\), enter the decision rule merely because it appears in the same panel row.

11.3.4.4 Mediation relation-kind refusal

Constitution or participation must not silently become a mediation path.

Accepted. A mediator path whose edges are declared :causal_influence (or otherwise admitted by the mediation planner).

Rejected. Routing NDE / NIE through a :constitutive_dependence or :participation edge as if it were already a causal mediator.

Manuals document the corresponding APIs (causal_projection, validate_intervention_semantics, information_set / ObservationBridge.availability, mediation relation-kind checks).

11.3.5 Conditional Independence Structure

The blanket’s statistical content is conditional independence:

\[ \mu ⫫ \eta \mid (s, a) \]

Once we hold fixed what the organism presents (\(s\)) and what it does (\(a\)), internal and external states are informationally separated. We do not observe \(\eta\) directly; we sense the environment only through \(s\), and act on it only through \(a\). That asymmetry is why data must not be confused with mechanism.

11.3.6 FEP and System Boundaries

Under additional generative-model and variational assumptions, FEP analyses may relate variational free energy to an assumed blanket structure (Friston 2010; Friston et al. 2006). A CDM or blanket alone does not entail free-energy minimisation, organismal persistence, cognition, or a stable interface. In a CDM, specifying \((f, h, G)\) states modelling assumptions; it does not establish a society or an FEP account.

Stratum context: The Markov blanket is a Structural/Dynamical modelling object. Structurally, it states a conditional-independence restriction; dynamically, its persistence must be assessed over a specified time interval. Neither claim establishes an invariant causal graph or a stable organismal boundary.

11.3.7 Connection to Three Levels of Reason

The blanket mediates all three levels of Reason:

  • Level 1 (Association / Seeing): Learn from observations, \(P(Y_t \mid Y_{1:t-1})\) and related objects use only \(s\), not full \(\mu\) or \(\eta\).
  • Level 2 (Intervention / Doing): \(do(\cdot)\) on \(a\) fixes what the organism may take up, propagating through \(f\) into future \(\mu\).
  • Level 3 (Counterfactual / Imagining): A declared cross-world coupling supports alternative modelled actualisations under a different action on \(a\); it does not supply an identity criterion.

11.3.8 Summary: process terms and CDM notation

The glossary gives the concise definitions and scope boundaries. The table below records only how the terms are used in this chapter.

Process term CDM / FEP role
Organism Internal \(\mu_t\) (e.g. \(X_t\)); trajectory under \(f\)
Environment External \(\eta\); parents and \(U\) outside \(\mu\)
Society Whiteheadian nexus of occasions ordered by inherited defining character; personal order constitutes an enduring object
Enduring object Society with personal order and a historical route; CDCS requires an explicit referent and identity criterion
Enduring Ontological character of a referent; not node multiplicity
Occasion Ontological character of an occurrence at the chosen scale; not “one node per observation step”
Temporal support Declared extent of a variable’s value (PointwiseSupport, FromOnsetSupport, …); determines unrolling with graph_kind
Markov blanket Sensory \(s\) (\(Y_t\)), active \(a\) (\(A_t\)); \(\mu ⫫ \eta \mid (s,a)\)
Prehension Recipient-specific incorporation into an occurrence
Creative advance Novelty in actualisation; not synonymous with \(\mathbf{u}\)
Alternative modelled actualisations Counterfactual trajectories under a declared cross-world coupling (L3)

A statistical population is not automatically a society: transport and cohort generalisation require a shared defining characteristic (see Transportability). Full CDM notation and the host-immunity worked example are in Chapter 28. Operational detail appears in State-Space Models and Causal Decision-Making; Concept Reference, Table 4.

11.4 Stratum comparison: Structural vs Dynamical

Aspect Structural role Dynamical role
Primary object State space, constraints and mechanism class Flow, transition kernel and realised trajectory
Invariant concept Symmetry, conservation law or admissible set Invariant manifold or set under specified dynamics
Attractor Not defined before dynamics Defined relative to the flow and its basin
Examples Graph, parent sets and mechanism constraints ODE/SDE trajectories, basins and bifurcations

11.5 From Constraints to Attractors

The proper sequence is structural constraints \(\rightarrow\) specified dynamics \(\rightarrow\) invariant sets and attractors \(\rightarrow\) realised trajectory. An attractor need not maximise fitness or any scalar objective, and a changing environment may require non-autonomous or stochastic notions of stability.

11.6 Time Enters: The Dynamical Dimension

11.6.1 What Changes: Structural Relations → Temporal Relations

With the transition to Dynamical, time enters the framework:

  • Structural constraints → Temporal mechanisms: a declared mechanism class is instantiated as a transition law
  • Admissible relations → Lagged or state-dependent relations: dependence is indexed by time and state
  • Feasible sets → Reachable sets, invariant sets and attractors: stability is assessed only after dynamics are specified

11.6.2 Realised Processes and Their Models

An ODE or SDE is a mathematical representation, while the process represented by \(X_t\) is already realised whether or not it is observed. The abstract/concrete distinction therefore cuts across structural, dynamical, and observable roles; it is not a ladder on which observation creates reality.

11.6.3 Structural Edges → Dynamic Edges

The transition in terms of edges:

  • Invariant relations → Dynamic relations
  • Invariant structural relations (repeatable patterns and mechanism constraints) → Dynamic edge semantics (time-dependent realisations)
  • Timeless but structured → Temporal but abstract

The graph may remain fixed while transition functions, parameters or environmental inputs change. Evolving topology is an additional modelling claim, not a consequence of adding time.

11.6.4 Heraclitean Change

Heraclitus’s river retains a recognisable form while different waters flow through it (Kirk et al. 1983). The relevant modelling question is which patterns and mechanisms persist through change. In stochastic dynamics, this question extends from individual trajectories to the flow of probability over possible states: a distribution may remain stationary while the trajectories it describes continue to fluctuate.

11.7 Setting Up Part II: Dynamical

Part II (Dynamical) develops:

  1. Deterministic Dynamics (ODEs): Dynamic processes as causal mechanisms (dynamic edges)
  2. Stochastic Dynamics (SDEs): Time-dependent processes with environmental variation (stochastic edges)
  3. Regime Switching: Multiple dynamic attractors (switching edge mechanisms)
  4. Network Models: Spatial causal structure and lagged influence
  5. Dynamics on Networks: Collective dynamic attractors (network edge dynamics)
  6. Resilience and Robustness: Stability of dynamic attractors (edge stability)

The Dynamical stratum introduces:

  • Time: Temporal dimension enters (edges become time-dependent)
  • Dynamic Processes: Time-dependent mechanisms (edge mechanisms evolve over time)
  • Dynamical attractors: Stability properties of a specified non-autonomous or stochastic model; not optima
  • Temporal but Abstract: Processes are time-dependent but still abstract, not yet material/physical

11.7.1 Emphasising Edges in the Dynamical Stratum

The Dynamical stratum then specifies how:

  • Edges become temporal: From invariant relations to dynamic, time-indexed dependence
  • Edge structure becomes time-dependent: The graph structure evolves over time
  • Edge mechanisms are dynamic: The functions on those edges change with time
  • Edge topology remains abstract: Still abstract, but now temporal

Edges remain graphical summaries of direct relevance; assignments and their parent sets remain the causal mechanisms. Measurement is a separate record-generating interaction.

11.8 Pedagogical attractors and intervention timing

These examples bridge Attractor Progression to Part II: multistability on a fixed graph, epidemic thresholds under vaccination rollout, and ecosystem health under restoration effort. Each uses the organism–society vocabulary introduced in The Markov Blanket.

The same interaction graph (who is linked to whom) can still support more than one stable long-run pattern. An ecosystem may sit in a healthy configuration or a degraded one; a regulatory network may settle into two expression regimes. Which pattern appears often depends on where the system starts, not on rewiring the graph (Scheffer et al. 2009; Strogatz 2014). For policy, that distinction matters: some interventions only shake the system within its current pattern; others push it across a threshold into a different attractor (a tipping point).

Figure 11.3 is a small pedagogical cartoon: five coupled units on a wheel (hub plus four neighbours), the sort of layout you might sketch for a hub gene and four regulated targets, or a keystone species and four strongly interacting partners. It sits in the spirit of multistable attractor networks on fixed regulatory graphs (Wang et al. 2016).

For the pedagogical simulation, we read each node as a small dynamical unit and what it receives along incoming links as dependence on neighbours. The shared wheel topology and coupling rule could support a society interpretation if the units inherit a defining character through an ordered succession of occasions. The simulation itself establishes only a graph, a mechanism, and trajectories, so that interpretation remains an additional assumption.

We run the same model twice with different starting values:

  1. Each unit is pulled toward one of two local tendencies (high or low), think “on/off” or “high/low” expression.
  2. Neighbours gently encourage one another to agree (coupling along the wheel).
  3. The two runs end in different rim patterns, two stable expression (or abundance) patterns on one wiring diagram.

Node size and colour show how strongly each unit has settled into its role; edge thickness highlights links between strongly settled neighbours. The equations and simulation are in the figure caption and code; the causal lesson is: shared structure, different histories, different equilibria.

Figure 11.4 gives the same idea as a landscape picture: two valleys (markers) separated by a ridge. Motion tends downhill into one valley or the other, two basins of attraction. The plot is a toy surface for intuition only; it is not the literal energy landscape of the five-node simulation, and many real systems need richer summaries than a single two-dimensional hillscape (Strogatz 2014; Leonov and Kuznetsov 2016). It still makes multistability easy to see: more than one place the dynamics can settle, with a barrier between them.

Figure 11.3: Wheel coupled-units topology (pedagogical; attractor-network motivation (Wang et al. 2016)): two simulated trajectories from \(u_i' = u_i - u_i^3 - \lambda(Lu)_i\); node size and colour show intensity \(\iota_i = \mathrm{clip}((u_i+1)/2)\) at the terminal state; edge thickness uses \(\max(\iota_i,\iota_j)\).
Figure 11.4: Toy double-well potential \(V(x,y)=0.22(x^2-1)^2+0.55y^2\), shown as filled contours in the \(x\)–\(y\) plane (plan view of the usual basin landscape); overlaid contours mark level sets, and markers sit near the two minima. Gradient flow \(\dot{x}=-\partial V/\partial x\), \(\dot{y}=-\partial V/\partial y\) has stable equilibria in those valleys. A general multistable ODE need not be gradient; high-dimensional systems are often summarised by quasi-potentials (Scheffer et al. 2009; Leonov and Kuznetsov 2016).

11.9 Example: Epidemic thresholds and rollout timing

Achieving herd immunity requires vaccinating a critical threshold of the population (typically 70–90% depending on transmissibility) (Anderson and May 1992; Keeling and Rohani 2011). Below that threshold, an outbreak can grow large before fading; above it, transmission is damped from the start. The generative model below is a weekly SIR compartment model on population fractions: a seeded infection, mass-action transmission, mean infectious period of about ten days (\(\gamma^{-1}\)), and a vaccination campaign that moves susceptibles into \(R\) at a scenario-specific rate (slow vs fast rollout, plus baseline immunity).

First rows of the generative panel (weekly fractions):

Table 11.1
6×10 DataFrame
Row week scenario S I R cumulative_incidence baseline_immunity herd_immunity_threshold peak_infected final_attack_rate
Int64 String Float64 Float64 Float64 Float64 Float64 Float64 Float64 Float64
1 0 slow rollout (~32% immune at t=0) 0.679 0.001 0.32 0.32 0.32 0.740741 0.166693 0.982145
2 1 slow rollout (~32% immune at t=0) 0.677703 0.00176097 0.320536 0.320536 0.32 0.740741 0.166693 0.982145
3 2 slow rollout (~32% immune at t=0) 0.675347 0.00309357 0.321559 0.321559 0.32 0.740741 0.166693 0.982145
4 3 slow rollout (~32% immune at t=0) 0.671254 0.00541152 0.323334 0.323334 0.32 0.740741 0.166693 0.982145
5 4 slow rollout (~32% immune at t=0) 0.664289 0.00939737 0.326313 0.326313 0.32 0.740741 0.166693 0.982145
6 5 slow rollout (~32% immune at t=0) 0.652631 0.0161192 0.331249 0.331249 0.32 0.740741 0.166693 0.982145

Figure 11.5 summarises the two campaigns. With \(R_0 \approx 3.9\) the herd-immunity threshold is \(1 - 1/R_0 \approx 74\%\) immune. The slow rollout scenario peaks at roughly 17% currently infected and an attack rate near 98%; the fast rollout keeps the peak near 0.5%, same epidemic mechanism, different timing and coverage of the intervention.

Figure 11.5: Weekly SIR with vaccination campaign (\(R_0 \approx 3.9\), threshold \(\approx 74\%\) immune): slow vs fast rollout. Top: infected fraction \(I(t)\); bottom: cumulative incidence \(1-S-I\). Dashed line: 10% peak prevalence reference.

This illustrates different simulated epidemic burdens under one specified SIR mechanism and different campaign schedules. It does not establish a shift between attractors: the conclusion follows from the stated finite-horizon model and intervention schedules.

11.10 Example: Resilience, resistance, and restoration effort

Restoring a degraded ecosystem requires different intervention effort depending on how far the system sits from a healthy attractor. The generative model below tracks an ecosystem health index \(H_t \in [0,1]\) (1 = reference healthy state) on a lake or rangeland slowly stressed by nutrient loading. Intrinsic dynamics are bistable (healthy and degraded basins); restoration effort \(u_t\) (stocking, rewilding, reduced grazing) can push the index across a tipping band near \(H \approx 0.5\).

First rows of the generative panel:

Table 11.2
6×6 DataFrame
Row year scenario health_index environmental_stress restoration_effort tipping_threshold
Int64 String Float64 Float64 Float64 Float64
1 1 early + maintenance 0.85 0.000167927 0.062 0.5
2 2 early + maintenance 0.881984 0.000428067 0.062 0.5
3 3 early + maintenance 0.910032 0.000740005 0.062 0.5
4 4 early + maintenance 0.933423 0.00109119 0.062 0.5
5 5 early + maintenance 0.951992 0.00147479 0.062 0.5
6 6 early + maintenance 0.966074 0.00188636 0.062 0.5

Figure 11.6 plots the three scenarios. No restoration declines from \(H_0 \approx 0.85\) toward a degraded basin (about 0.4 by year 70). Early + maintenance (strong effort years 1–22, then light upkeep) holds health near 0.8–0.99. Late intensive effort (from year 29, after health has slipped to \(\approx 0.85\) and would fall further without action) recovers toward 1.0, but only with stronger, longer forcing, illustrating resistance when control is delayed.

Figure 11.6: Ecosystem health \(H_t\) under rising environmental stress (nutrient-loading proxy) and three restoration policies. Dashed line: tipping threshold \(H_c=0.5\). Right axis: stress and effort (scenario with early intervention).

Early intervention (when the state is still reachable with modest effort) differs from reversing a shift after deep degradation, when the same process organisation may require sustained forcing.

11.11 Worked Example: Sheep System Evolving Through Strata

11.12 Worked Example: Sheep System from Structural to Dynamical

Structural View (Part I):

  • Idealised causal structure: predator–prey relations
  • Invariant patterns: stable, environment-independent interactions
  • Abstract relations: graph structure of interactions
  • No time: timeless, spaceless causal structure

Dynamical View (Part II):

  • Dynamic mechanisms: time-dependent predator-prey dynamics
  • Dynamic attractors: population states that change with environment
  • Dynamic processes: actual population trajectories over time

The same system, now with time: from abstract structural relations to dynamic processes.

11.13 Transition to Dynamical

As we move to Part II, we transition from:

  • Constraints and mechanism classes → Specified transition laws
  • Specified transition laws → Invariant sets, basins and attractors where stability is established
  • Modelled process → Measurement interaction and record

The Dynamical role introduces temporal evolution while preserving explicit causal assumptions; Measurement then models how records arise from an already realised process.

11.14 Stratum context

This chapter bridges the Structural and Dynamical modelling roles. It separates constraints from stability claims and presents FEP only under its additional assumptions.

11.15 Summary

  1. Structural summary: Invariant patterns and admissible parents live in the Structural stratum
  2. Organism–environment–society: Markov blanket as the sensory/active interface; see The Markov Blanket
  3. FEP as conditional module: FEP interpretations require assumptions beyond a blanket partition
  4. Attractors follow dynamics: constraints precede, but do not constitute, attractors
  5. Measurement records: observations register selected aspects of an already realised process
  6. This transition prepares us for Part II: Dynamical

11.16 Further Reading

  • Whitehead (1978): Process and Reality: organism, environment, and society (Part II; optional process background)
  • Friston (2010): “The free-energy principle: a unified brain theory?”
  • Friston et al. (2006): “A free energy principle for the brain”
  • The next chapter begins Part II: Dynamical
  • See Deterministic Dynamics: ODEs as Causal Processes to continue

  1. Whitehead uses this term for a possibility’s relevance to a concrete becoming. CDCS uses it as an interpretive gloss for final causation while bracketing the wider cosmological account of valuation (Whitehead 1978; Segall 2025, 2026).↩︎