# Full Text: Active Inference -- A Synthetic Overview

> Extracted from `ActiveInferenceSynthetic_v0.5.0.pdf`

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Active Inference – A Synthetic Overview
An executable introduction to Active Inference and synthetic intelligences across equilibrium and
non-equilibrium settings
Daniel Ari Friedman
Active Inference Institute
daniel@activeinference.institute
ORCID: 0000-0001-6232-9096
and Alexandra Mikhailova
Active Inference Institute
alexandra@activeinference.institute
DOI: 10.5281/zenodo.23321281
2026-10-12

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Contents
1 Abstract 2
2 Introduction and Extended Abstract 3
3 Perception as Inference 5
4 Methods and Reproducibility 6
5 W orked Regimes: Equilibrium and Non-Equilibrium 8
6 Action: V ariational vs Expected F ree Energy 10
7 The Markov Blanket 11
7.1 A matched native Gaussian example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
8 The Steady State, Quantified 13
9 F ormal Reference and V erification 15
9.1 Selected results and hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.2 Numerical and formal evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.3 Qualified blanket and planning adapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
10 Planning: Policies as Action Sequences 16
10.1 An explicit upstream carrier bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
11 A Closed-Loop Synthetic Agent 17
12 When Information Improves a Decision 18
13 A T wo-Level Generative Hierarchy 20
14 Discussion: Informational and Thermodynamic Interpretation 21
14.1 Equilibrium and stationary circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
14.2 Statistical boundaries and synthetic agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
14.3 Reading and reusing the Institute repositories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
15 A Reader’s Route Beyond Synthetic 22

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1 Abstract
This tutorial connects Active Inference and synthetic intelligences to explicitly specified informational and dynamical
models. Finite categorical examples make Bayesian conditioning, variational free energy, expected free energy, policy
planning, and observation-contingent commitments executable. A decision-value example separates information gain
in nats from improved expected utility and observation cost. Gaussian examples distinguish deterministic relaxing
paths from stationary ensemble probability current, and marginal dependence from conditional independence. A two-
level model illustrates prediction and joint conditioning without assuming a neural implementation. The standalone
Python package supplies reproducible figures, independent analytical checks, and counterexamples; a separate Lean
layer imports selected real-valued results from a pinned revision of the broader FEP_Formal library. Generalized
Notation Notation and other Active Inference Institute resources provide further routes for model specification
and learning. The paper distinguishes computational correctness, theorem hypotheses, thermodynamic units, and
scientific interpretation. These controlled demonstrations explain particular models; they neither establish general
intelligence nor infer life from circulation.
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2 Introduction and Extended Abstract
This is a worked introduction to Active Inference and synthetic intelligences across equilibrium and non-equilibrium
thermodynamic and informational settings. The examples connect the Free Energy Principle and Bayesian mechan-
ics to inference, action, and self-organization. The broader fep_formal library provides the mathematical reference
[Friedman, 2026a,b]; this paper supplies selected executable models and their exposition. Variational, expected, and
thermodynamic free energy have distinct definitions and units. Persistent stationary current alone is not evidence
of life or intelligence.
Active Inference connects inference and action within a specified generative model [ Friston, 2010, Parr et al., 2022].
Updating a candidate belief changes its variational free energy after an observation. Choosing a future policy instead
requires an objective over predicted outcomes, with explicit preferences and information terms. Those objectives
are related by their model assumptions; they are not interchangeable quantities. Whether a biological or engineered
system realizes such computations is a further scientific question.
On the admitted support domain, variational free energy bounds surprise, the negative log evidence − ln 𝑝(𝑠). The
finite demonstration can compute both the exact posterior and that evidence independently, so the identity has a
direct numerical check. Relating this inference calculation to physical dynamics requires additional assumptions.
Likewise, a Markov blanket states conditional independence between external states 𝜂 and internal states 𝜇
given sensory and active blanket states. That screening condition alone does not establish causal direction or
a representational mapping. Bayesian mechanics investigates such mappings under specified dynamical and
statistical hypotheses [ Da Costa et al. , 2021, Ramstead et al. , 2023].
fig. 1 shows the move that this overview is built around, without a single equation of derivation. Both panels live
on the same free-energy landscape — a Gaussian bowl whose floor is the system’s expected state. At equilibrium
(panel a) the flow is pure descent, 𝑓 = −∇𝐹 : every trajectory falls straight to the mode and stops. Detailed balance
holds, the stationary probability current vanishes. This does not mean that stochastic microscopic motion ceases.
Panel (b) supplies a contrasting non-equilibrium example. There the flow gains a solenoidal term, 𝑓 = (𝑄 − Γ)∇𝐹 ,
that runs along the contours of 𝐹 rather than down them. The deterministic drift still lowers the quadratic potential,
while stationary probability mass circulates indefinitely in a non-equilibrium steady state . That persistent
current diagnoses broken detailed balance in the stated diffusion model. The gradient part illustrates the Free
Energy Principle’s descent; the solenoidal part supplies conservative circulation in this example [ Friston et al. ,
2023].
Read the two panels together and the headline of Active Inference is visible at a glance: the same imperative —
minimize free energy — produces relaxation in one regime and persistent stationary current in the other. The
rest of this overview unpacks that picture in the spirit of show, not tell : each section adds one runnable, minimal
demonstration rather than another layer of formalism. sec. 3 makes the inference concrete — that minimizing
variational free energy over a generative model is Bayesian updating, and that free energy is a tight upper bound
on surprise. sec. 5 makes the mechanics concrete — integrating the flow above into worked equilibrium and non-
equilibrium trajectories and quantifying the persistent current that separates these two model regimes. sec. 6
turns to action — evaluating expected free energy over policies, whose pragmatic and epistemic terms produce
the explore/exploit balance from one rule. sec. 7 exhibits conditional screening in a specified Gaussian model;
a representational interpretation requires further hypotheses. sec. 8 quantifies the far-from-equilibrium regime
outright — the analytic probability current and the entropy-production rate that is zero exactly at detailed balance
and positive out of it. Finally sec. 9 closes the loop on rigour: the load-bearing theorems — Gibbs’ inequality, the
free-energy bound, the conservative solenoidal current, the sign of entropy production — are machine-checked in
Lean 4 , proved from the kernel’s axioms with no sorry: the real-valued exposition imports the FepSketches library
maintained in fep_formal.
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Figure 1: One landscape, two regimes. Both panels place the system on the same Gaussian free-energy bowl
(grey contours; the black dot marks the mode, where free energy is least). (a) Equilibrium: the flow is pure gradient
descent on free energy, 𝑓 = −∇𝐹 (blue arrows). Detailed balance holds, so trajectories fall to the mode and come to
rest with no net probability current. (b) F ar-from-equilibrium (NESS): an antisymmetric solenoidal operator
𝑄 adds a conservative current, 𝑓 = (𝑄 − Γ)∇𝐹 (orange streamlines), that flows along the level sets of 𝐹. Free
energy is still minimized, but probability mass circulates without end — a non-equilibrium diagnostic in this model.
The Free Energy Principle is the descent term; Bayesian mechanics adds the circulation. Internal states 𝜇 infer
external states 𝜂 across a Markov blanket (sensory 𝑠, active 𝑎). Figure rendered deterministically by scripts/gene
rate_overview_figure.py over the tested flow in src/free_energy_flow.py.
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3 Perception as Inference
The extended abstract makes a strong claim in one sentence: minimizing variational free energy is approximate
Bayesian inference, carried out by the physical dynamics of the thing itself . This section makes that claim runnable.
It is the smallest generative model that still says something true, and everything below is checked against closed-form
Bayes in src/generative_model.py.
A system holds a generative model of a hidden cause 𝑥 and a sensation 𝑠: a prior 𝑝(𝑥) over states and a likelihood
𝑝(𝑠 ∣ 𝑥) saying which sensations each state tends to produce. Together they fix the joint 𝑝(𝑠, 𝑥) = 𝑝(𝑠 ∣ 𝑥) 𝑝(𝑥), the
evidence 𝑝(𝑠) = ∑𝑥 𝑝(𝑠, 𝑥), and — by Bayes’ rule — the posterior 𝑝(𝑥 ∣ 𝑠) = 𝑝(𝑠, 𝑥)/𝑝(𝑠). The system never sees 𝑥;
it only ever has 𝑠 and its own model. Inference is the problem of recovering 𝑝(𝑥 ∣ 𝑠) from those.
Active Inference solves it without ever computing 𝑝(𝑠) directly, through a belief 𝑞(𝑥) and a single scalar, the
variational free energy :
𝐹 [𝑞] = KL[𝑞(𝑥) ‖ 𝑝(𝑥)]⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟
complexity
− 𝔼 𝑞[ln 𝑝(𝑠 ∣ 𝑥)]⏟⏟⏟⏟⏟
accuracy
. (1)
The decomposition in eq. 1 separates complexity from accuracy. Two facts, both asserted to machine precision in
the tests, make 𝐹 the whole game. First, 𝐹 is an upper bound on surprise : 𝐹 [𝑞] = −ln 𝑝(𝑠) +KL[𝑞‖𝑝(𝑥 ∣ 𝑠)] ≥
− ln 𝑝(𝑠), because a KL divergence is never negative. Surprise − ln 𝑝(𝑠) is exactly what an organism “wants” to keep
low — the improbability of its own sensory state — but it cannot compute it. It can compute 𝐹, from quantities
it already has. Second, the bound is tight at the posterior : 𝐹 is least, and equals the surprise, precisely when
𝑞 = 𝑝(𝑥 ∣ 𝑠) . So the belief that minimizes free energy is the Bayesian posterior, and the minimized value is the
surprise it bounds. Inference falls out of minimization.
fig. 2 shows this happening. Take a two-state model with prior 𝑝(𝑥) = (0.60, 0.40)and a likelihood under which state
𝑥0 makes the observed sensation 𝑠 far more probable than 𝑥1 does. Panel (a) starts from a maximally uncommitted
uniform belief and runs gradient descent on 𝐹 over the belief simplex; the belief mass migrates toward the exact
posterior 𝑝(𝑥0 ∣ 𝑠) = 0.54/0.62 ≈ 0.87 . Panel (b) tracks 𝐹 across the same descent: it falls monotonically to the
surprise − ln 𝑝(𝑠) (dashed line), and the shrinking shaded gap is exactly KL [𝑞𝑡‖𝑝(𝑥 ∣ 𝑠)] — the belief’s remaining
error. Perception, here, is nothing but this descent: the system lowers free energy by updating beliefs, and in doing
so recovers the posterior it could not compute directly.
The complexity–accuracy split names the trade-off. Accuracy rewards beliefs that explain the sensation; complexity
penalizes moving away from the prior. Perception moves beliefs only as far as the evidence warrants — the same
regularized fit that makes the free-energy principle a principle rather than a slogan. Action, treated in the next
demonstrations, minimizes the same 𝐹 in the other direction: by changing 𝑠 rather than 𝑞.
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Figure 2: Perception as free-energy minimization. A two-state generative model (prior 𝑝(𝑥) = (0.60, 0.40);
likelihood favouring 𝑥0 for the observed 𝑠). (a) From a uniform starting belief, gradient descent on variational
free energy drives the belief over hidden states to the exact Bayesian posterior 𝑝(𝑥 ∣ 𝑠) (orange), passing through
intermediate beliefs (blue) on the way. (b) Along the same descent the free energy 𝐹 [𝑞𝑡] falls monotonically to the
surprise − ln 𝑝(𝑠) (dashed) that it bounds; the shaded gap is exactly KL [𝑞𝑡‖𝑝(𝑥 ∣ 𝑠)], the belief’s error, which closes
as inference completes. Minimizing free energy is Bayesian updating. Rendered deterministically by scripts/gener
ate_belief_update_figure.py over the tested core in src/generative_model.py.
4 Methods and Reproducibility
The examples are deliberately small models with declared inputs, rather than empirical estimates of biological or
artificial-system performance. Numerical parameters are shared between figure builders and tests in src/demos.py.
Manuscript tokens are computed by the same numerical cores and hydrated before the template renders the paper.
Each figure caption identifies the corresponding model or script; the figure registry binds labels, filenames and
builders.
Finite probability channels store outcomes in rows and hidden states in columns. Their columns are normalized;
beliefs are probability vectors. Conditioning enumerates reachable observations or observation histories. The
decision-value calculation compares the best prior commitment with optimal observation-contingent commitments
under the same state, actions, utilities and acquisition cost. Impossible outcomes do not acquire invented posteriors.
These finite enumerations cover their specified branch spaces, while their Python arithmetic remains floating point
with explicit validation and test tolerances. They do not establish an optimum outside the declared model class.
Independent references are chosen for each model: direct Bayes conditioning for belief updates, alternative free-
energy decompositions, exhaustive deterministic feedback policies for decision value, the linear closed form for RK4
paths, and Gaussian covariance and precision identities for blanket screening. The OU stationary calculations
additionally require matched diffusion and positive dissipation. A deterministic path is not substituted for sta-
tionary current. Information is reported in nats; utility and observation cost share declared utility units; physical
thermodynamic interpretations require extra scaling.
Counterexamples exercise interpretation boundaries. Independent cues provide no context information. Informative
cues with state-independent utilities give no instrumental gain. Excessive observation cost can make net gain nega-
tive. Partial blanket conditioning and altered precision couplings need not preserve screening. Invalid probabilities,
support mismatches and nonfinite quantities are rejected. The formal tests also check that false Lean statements
fail, rather than relying only on successful proof builds.
The package’s installed namespace is active_inference_synthetic. Reproduction from the standalone project uses:
uv sync --locked --extra dev
uv run pytest tests/ --cov=src --cov-fail-under=90
uv run python scripts/generate_all_figures.py
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uv run python scripts/run_teaching_experiments.py
The last command records inputs, computed decision values and source hashes for the informative, uninformative
and irrelevant-payoff conditions. A changed or missing source changes or blocks that receipt. Such a receipt
establishes the inputs of a particular execution; it is not certification of a scientific interpretation.
The independent formal layer builds with lake build from formal/ and pins FEP_Formal through Lake. The
publication renderer belongs to the template repository. Python execution, Lean checking, rendered-document in-
spection and publication-file integrity are separate verification steps. The accompanying source archive supplies the
authored numerical, test, formal and manuscript files; it excludes local environments, credentials and intermediate
runtime state. The resource guide identifies further mathematical and educational materials sec. 15.
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5 Worked Regimes: Equilibrium and Non-Equilibrium
The overview figure (fig. 1) contrasts two flows — one that comes to rest, one that circulates — as arrow fields.
A flow field says where a system would go next; it does not, on its own, show where a system ends up . This
section integrates the flow into actual trajectories, so the equilibrium ↔non-equilibrium distinction becomes a
path you can watch, and the rotation diagnostic becomes a number rather than an adjective. The integrator and
its diagnostics live in src/trajectories.py, built directly on the tested flow of src/free_energy_flow.py.
Both regimes obey the Bayesian-mechanics flow 𝑓 = (𝑄 − Γ)∇𝐹 on the same isotropic Gaussian free-energy
landscape. The only difference is the antisymmetric solenoidal operator 𝑄: zero at equilibrium, nonzero in a
non-equilibrium steady state (NESS). A fixed-step fourth-order Runge–Kutta scheme (no adaptive stepping, no
randomness) advances a starting point under this flow. That the integrator can be trusted is not assumed: at
equilibrium the flow reduces to the linear system ̇ 𝑥 = −𝛾 Π 𝑥(with Π the precision), and the tests confirm the RK4
path matches the closed form 𝑥(𝑡) = 𝑥0 𝑒−𝛾 Π 𝑡 — i.e. 𝑥0𝑒−𝛾𝑡 at unit precision — to better than 10−6.
fig. 3 shows the two fates. In panel (a), the equilibrium trajectory ( 𝑄 = 0 ) falls straight down the free-energy
gradient to the mode and stops. Free energy decreases monotonically along the path, the radius collapses to under
1% of its starting value, and the net angle swept about the mode is zero to numerical precision — detailed balance,
no current. This is a rock settling, a battery draining, a belief that has finished inferring.
Panel (b) is the NESS (𝑄 ≠ 0). The dissipative term still pulls the system down 𝐹 — free energy again decreases
monotonically, so the system is still minimizing surprise — yet the solenoidal term drives it around the level sets
of 𝐹 at the same time. The result is a spiral that sweeps many full turns before relaxing inward.
How much it winds is worth stating carefully. The total net rotation — the figure’s ≈ −4.58 turns over the
demonstration window 𝑇 = 12 — is not an intrinsic property of the NESS; it grows with 𝑇 and would diverge as
𝑇 → ∞ . The intrinsic, initial-condition-free quantity is the rotation rate . For this isotropic-Gaussian flow the
angular velocity is analytically constant, ̇𝜃 = −𝜔 Π radians per unit time (with Π the precision), so the rate is
≈ −0.38 turns per unit time ( −𝜔 Π/(2𝜋)at 𝜔 = 2.4, Π = 1) for any starting point off the mode — and it is exactly
0 at equilibrium for every window. That zero-versus-nonzero rate is the honest, quantitative witness that detailed
balance is broken. The tests pin it directly: net rotation equals −𝜔 Π 𝑇 across several starting points to a tenth of
a percent.
One more distinction keeps the claim precise. Taken on its own, the deterministic trajectory here relaxes — it
spirals into the mode and would sit there. A genuine non-equilibrium steady state is a property of the stochastic
system: a stationary density 𝜌 ∝ 𝑒 −𝐹 carrying a divergence-free probability current 𝑗 = 𝑄∇𝐹 𝜌 that never dies.
What the figure shows is that system’s deterministic drift , and the spiral is the mechanistic origin of that current,
not a measurement of it. So the honest reading of “NESS” here is precise: the nonzero winding witnesses the
solenoidal 𝑄 ≠ 0 that breaks detailed balance and thereby renders the corresponding stochastic stationary state a
non-equilibrium one. Reverse the sign of 𝑄 and the circulation reverses with it, as the tests confirm. (sec. 8 supplies
the measurement this section withholds — the actual probability current and its entropy-production rate.)
Read the two panels against the paper’s teaching aim. Descent is common to both: the free-energy landscape
decreases along each deterministic trajectory. Nonzero solenoidal drive additionally produces rotation. Persistent
stationary probability current belongs to the corresponding stochastic model and is quantified in the next steady-
state section. These are distinct demonstrations of relaxation and non-equilibrium circulation; they do not establish
life, intelligence, or perpetual motion of an individual deterministic trajectory.
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Figure 3: W orked trajectories over one landscape. Both panels integrate the Bayesian-mechanics flow 𝑓 =
(𝑄 − Γ)∇𝐹 from the same starting point (coloured dot) over the same Gaussian free-energy bowl (grey contours;
black dot marks the mode). (a) Equilibrium (𝑄 = 0 ): pure descent falls radially to the mode and rests — free
energy monotonically decreasing, rotation rate ≈ 0 turns/time. (b) NESS (𝑄 ≠ 0 ): the same descent plus a
solenoidal term spirals inward at a constant rate ≈ −0.38 turns/time ( ≈ −4.58 turns over the window 𝑇 = 12 )
while free energy still decreases monotonically. The nonzero rotation rate witnesses the broken detailed balance in
the corresponding stochastic model; sec. 8 computes its stationary probability current. Rendered deterministically
by scripts/generate_trajectory_figure.py over the tested cores in src/trajectories.py and src/free_energy_flow
.py.
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6 Action: Variational vs Expected Free Energy
sec. 3 showed perception: minimizing variational free energy 𝐹 over beliefs about what is happening now. But a
thing that only perceived would be a spectator. Active Inference closes the loop by minimizing the same currency in
the other direction — over actions, scored by the free energy they are expected to produce. That forward-looking
quantity is expected free energy 𝐺, and it is what turns inference into behavior. The core is src/expected_free
_energy.py; every value below is checked against a hand-worked reference.
The two free energies differ only in tense. 𝐹 [𝑞]grades a belief against a sensation already in hand — it is retrospective,
and its minimum is the posterior. 𝐺(𝑎) grades a candidate action against the sensations it would generate, averaged
over what the agent currently believes. For an action 𝑎, belief 𝑞(𝑠), outcome model 𝑝(𝑜 ∣ 𝑠, 𝑎) , and preferences
𝑝pref(𝑜) — a distribution over the outcomes the agent wants to encounter — 𝐺 splits into two terms the agent seeks
to make large (so it minimizes their negatives):
𝐺(𝑎) = − 𝔼 𝑞(𝑜∣𝑎)[ ln 𝑝pref(𝑜)]⏟⏟⏟⏟⏟⏟⏟
pragmatic value
− 𝔼 𝑞(𝑜∣𝑎)[KL[ 𝑞(𝑠 ∣ 𝑜, 𝑎) ‖ 𝑞(𝑠) ]]⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟
epistemic value
. (2)
Pragmatic value is expected utility: how well an action’s likely outcomes match what the agent prefers. Epis-
temic value is expected information gain — how much the action’s outcomes would sharpen the belief about the
hidden state, which is exactly the mutual information 𝐼(𝑠; 𝑜 ∣ 𝑎) and is always non-negative. One is the drive to get
what you want ; the other is the drive to find out . Both live inside a single number.
fig. 4 shows why that unification matters. An agent is unsure which of two contexts holds ( 𝑞(𝑠) = (0.50, 0.50)) and
can take one of two actions: a cue that would reveal the context (highly informative, epistemic value +0.37 nats,
but preference-neutral) or a bet that pays off regardless of context (uninformative, epistemic value 0, but potentially
rewarding). The only thing that changes between the two panels is the preferences. Under flat preferences (panel
a) the agent has nothing to chase, so the epistemic term decides: it takes the cue and explores (𝐺 = 0.33 vs
0.69). Under sharp preferences (panel b) the pull of the reward dominates, and the same minimization takes the
bet and exploits (𝐺 = 0.43 vs 0.84). This is a deliberately constructed two-action example, chosen to isolate the
mechanism; what it demonstrates is structural, not incidental to these numbers. The explore/exploit trade-off is
not a separate rule bolted onto the agent — there is one rule, arg min 𝑎 𝐺. Because 𝐺 carries both a pragmatic
and an epistemic term, exploration and exploitation are the two limits of a single balance, and which one wins is
decided by the preferences alone.
This is the whole architecture in miniature. Perception ( 𝐹) and action ( 𝐺) are one imperative — minimize free
energy — read in two tenses. Inference makes beliefs fit the world; action makes the world fit beliefs and preferences,
gathering information where it is uncertain and reward where it is confident.
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Figure 4: One rule yields explore and exploit. An agent uncertain between two contexts ( 𝑞(𝑠) = (0.50, 0.50))
chooses between an informative “cue” (epistemic value +0.37, preference-neutral) and a rewarding but uninformative
“bet” (epistemic value 0). Bars are expected free energy 𝐺 per action (lower is chosen); each is annotated with
its pragmatic/epistemic split. (a) Flat preferences: the epistemic term dominates, so the agent explores (cue,
𝐺 = 0.33 ). (b) Sharp preferences: the pragmatic term dominates, so the same arg min 𝐺 rule exploits (bet,
𝐺 = 0.43 ). A single rule; the balance of 𝐺’s two terms — not a hand-coded switch — is what turns over between
the panels. Rendered deterministically by scripts/generate_efe_figure.py over the tested core in src/expected_fr
ee_energy.py.
7 The Markov Blanket
Everything so far assumed a clean split between a system and its world — a belief 𝑞(𝑠) about hidden states,
sensations 𝑜 that are not those states, actions that reach out. That split is not a modeling convenience; it is a
structural claim, and it has a name: the Markov blanket . This section shows what the blanket actually asserts,
using the one case where it is exactly computable — a jointly Gaussian world. The core is src/markov_blanket.py.
Partition the variables of a system-plus-environment into four roles: external states 𝜂 (the world), internal states
𝜇 (the system’s own states, which parameterize its beliefs), and the blanket that lies between them — sensory
states 𝑠 (world → system) and active states 𝑎 (system → world). The blanket’s defining property is a conditional
independence:
𝑝(𝜂, 𝜇 ∣ 𝑠, 𝑎) = 𝑝(𝜂 ∣ 𝑠, 𝑎) 𝑝(𝜇 ∣ 𝑠, 𝑎). (3)
In this example the endpoints are marginally dependent, but their coupling is screened by the blanket. Given sensory
and active coordinates, knowing the external coordinate supplies no additional information about the internal
coordinate under the stated Gaussian law. A causal interpretation would need a dynamical or interventional model.
Conditional independence alone does not establish a posterior encoding or a cognitive interpretation.
fig. 5 makes both halves of the claim visible on a Gaussian built so that the 𝜂–𝜇 block of its precision matrix
(the inverse covariance) is exactly zero. Panel (a) is the ordinary correlation matrix: 𝜂 and 𝜇 are plainly correlated
(corr = 0.19): they are statistically coupled through the blanket. Panel (b) is the precision matrix, and the payoff
is the highlighted cell: Π𝜂𝜇 = 0. In a Gaussian a zero precision entry is exactly a conditional independence given
the remaining variables — so the zero at (𝜂, 𝜇) says 𝜂 ⟂ 𝜇 ∣ {𝑠, 𝑎} . Conditioning on the blanket collapses the
partial correlation between internal and external states to zero (the tests confirm it vanishes to 10−10 via the Schur
complement), even though conditioning on only part of the blanket does not — both the sensory and active states
are needed to close the channel.
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7.1 A matched native Gaussian example
A second fixture matches the pinned upstream canonical four-axis Gaussian carrier. Its ordered variables are
external, sensory, active, and internal. The exact precision and covariance arrays live in src/demos.py; tests verify
their rational inverse relation. External and internal marginal covariance is 0.0417, while conditioning on sensory
and active yields a diagonal endpoint covariance. The native Lean adapter binds the canonical precision to a
positive-definite Gaussian law and uses upstream conditional independence for every supplied center, including its
almost-everywhere conditional-kernel interpretation.
The corresponding perturbed endpoint precision supplies a genuine boundary: adding a nonzero external-internal
coupling produces conditional covariance −1/15 and native non-independence. The original numerical fixture and
this canonical native fixture are distinct examples. Formal coverage is claimed for the matched canonical carrier,
not inferred for arbitrary precision matrices from the appearance of a zero cell.
Conditional independence alone does not imply that internal states encode a posterior, that the partition is causal,
or that an organism is intelligent. Those interpretations require a generative or dynamical model and additional
assumptions. The finite perception-action example makes such a model explicit rather than inferring it from the
Gaussian partition.
Figure 5: Marginal dependence, conditional independence. A jointly Gaussian world over external 𝜂, sensory
𝑠, active 𝑎, internal 𝜇, built with a zero 𝜂–𝜇 precision block. (a) Marginal correlation: 𝜂 and 𝜇 are correlated
(0.19, highlighted) — they are statistically coupled through the blanket. (b) Precision (inverse covariance): the
𝜂–𝜇 entry is exactly 0, which in a Gaussian is conditional independence given the other variables — so 𝜂 ⟂ 𝜇 ∣ {𝑠, 𝑎}
and the partial correlation given the blanket is 0. (c) The matched upstream canonical fixture has a diagonal
conditioned endpoint covariance despite nonzero marginal coupling. The sensory and active states screen the
internal states from the external world. Rendered deterministically by scripts/generate_markov_blanket_figure.p
y over the tested core in src/markov_blanket.py.
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## Page 14

8 The Steady State, Quantified
sec. 5 was deliberately careful about one thing. The deterministic spiral it drew is the drift of a non-equilibrium
steady state, not the state itself — a single trajectory relaxes to the mode and stops, while a genuine steady state is
a property of a whole stochastic ensemble, with a current that never dies. This closing section supplies that missing
object, still analytically and with no randomness, in src/steady_state.py. It is where “far from equilibrium” stops
being a picture and becomes two numbers: a probability current and an entropy-production rate .
Give the Bayesian-mechanics flow a matched noise term — the Ornstein–Uhlenbeck diffusion 𝑑𝑥 = −𝑀 𝑥 𝑑𝑡 +√
2𝐷 𝑑𝑊 with drift 𝑀 = (Γ − 𝑄)𝑃 and, by the fluctuation–dissipation relation, diffusion 𝐷 = Γ . Three exact
results follow, each checked against a closed form in the tests.
First, the ensemble settles into a stationary density . Its covariance solves the Lyapunov equation 𝑀 Σ + Σ𝑀T =
2𝐷, and with 𝐷 = Γ the solution is exactly Σ = 𝑃 −1 — the system’s own belief covariance. The distribution the
noisy system lives in is precisely 𝜌 ∝ 𝑒 −𝐹 : free-energy minimization and the steady state are the same fixed point.
Second, that density carries a probability current 𝑗(𝑥) = 𝑏(𝑥)𝜌 − 𝐷∇𝜌 = (𝑄𝑃 𝑥) 𝜌(𝑥). When 𝑄 = 0 the current
is zero everywhere (detailed balance — the density just sits), and when 𝑄 ≠ 0 it is a divergence-free field that
circulates around the mode without end. This is the sustained current the deterministic trajectory could only
gesture at; here it is an explicit vector field, orthogonal to ∇𝐹 (the tests confirm both ∇⋅ 𝑗 = 0 and 𝑗 ⟂ ∇𝐹 to
machine precision).
Third, the current has a price. The entropy-production rate 𝜎 = tr(𝐷−1𝑄𝑃 Σ 𝑃 T𝑄T) measures the system’s
irreversibility, and for the isotropic model it is exactly 𝜎 = 2 𝜔 2 Π/𝛾 (with Π the precision). It is strictly positive
out of equilibrium and exactly zero if and only if 𝜔 = 0 . Entropy production is the thermodynamic cost of
staying far from equilibrium — of keeping the current running — and it is the sharpest available statement of the
difference between the two regimes: not the shape of a path, but a non-negative number that vanishes precisely at
detailed balance. (Two conventions, stated plainly: the choice 𝐷 = Γ is the fluctuation– dissipation relation with
𝑘B𝑇 ≡ 1 absorbed, so 𝜎 is a pure rate — the dimensional entropy-production rate is 𝑘B𝜎 and the dissipated power
is 𝑘B𝑇 𝜎; and crucially the nonequilibrium character is certified by the current and 𝜎, not by the density — the
Gaussian 𝜌 on its own looks equilibrium-like, so it is the circulation, not the shape of the steady state, that marks
the regime as nonequilibrium.)
fig. 6 shows both panels over the same stationary density. At equilibrium (a) the current field is identically zero and
𝜎 = 0 : nothing circulates and nothing is dissipated. In the NESS (b) the current curls steadily around the mode and
𝜎 > 0 : the same density, now maintained by a perpetual, entropy-producing flow. Read against the whole overview,
this is the final turn of the one thesis — and the point at which the mathematics reconnects to the motivation.
A system that minimizes free energy need not merely come to rest at its expected states; a nonequilibrium steady
state holds those states while circulating through them, at a strictly positive entropy cost. The model makes the
sustained current and its irreversible cost precise and measurable under the stated diffusion assumptions. This
motivates connections to living and synthetic systems without proving biological adequacy or intelligence.
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## Page 15

Figure 6: Probability current and entropy production. Both panels place the same stationary Gaussian
density (grey contours) under the Bayesian-mechanics diffusion. (a) Equilibrium (𝑄 = 0 ): the steady-state
probability current is zero everywhere and the entropy-production rate is 𝜎 = 0 — detailed balance. (b) NESS
(𝑄 ≠ 0): a divergence-free current 𝑗 = (𝑄𝑃 𝑥)𝜌 circulates around the mode (orange arrows), sustained at an entropy-
production rate 𝜎 = 2𝜔 2Π/𝛾 > 0 (here 𝜎 ≈ 11.5 nats/time). The current vanishes at the mode (where 𝑄𝑃 𝑥 = 0)
and in the tails (where 𝜌 → 0), peaking on an annulus between. Rendered deterministically by scripts/generate_s
teady_state_figure.py over the tested core in src/steady_state.py.
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9 Formal Reference and Verification
This paper supplies worked models and exposition. Its formal layer selects real-valued results from the Institute’s
fep_formal repository [Friedman, 2026a,b]. Lake imports its FepSketches Lean modules from the pinned revision
in formal/lakefile.lean. Generic proofs remain upstream; Synthetic’s Python cores do not import fep_lean.
9.1 Selected results and hypotheses
ActiveInferenceFormal.Geometry selects the flow and steady-state results; ActiveInferenceFormal.Information
selects finite information, VFE, and EFE results. Their ℝ_ theorem names correspond to the following claims:
• ℝ_solenoidal_orthogonal and ℝ_flow_work_eq_dissipation: the antisymmetric flow is orthogonal to the gra-
dient, and only dissipation contributes work. Non-increasing work requires nonnegative dissipation.
• ℝ_detailed_balance_iff: zero solenoidal flow is equivalent to zero drive at a nonzero gradient. The gradient
condition cannot be dropped.
• ℝ_entropyProduction_nonneg and ℝ_entropyProduction_eq_zero_iff: the sign and zero condition for the se-
lected isotropic OU expression; the equivalence requires strictly positive precision and dissipation.
• ℝ_finiteKL_nonneg, ℝ_finiteKL_self, and ℝ_finiteKL_eq_zero_iff: finite information laws for the upstream
normalized carriers and totalized functional.
• ℝ_surprisal_le_VFE, ℝ_VFE_at_posterior, and ℝ_negative_VFE_le_logEvidence: posterior-form VFE bounds
and attainment for a finite generative model with positive predictive evidence.
• ℝ_EFE_decomposition, ℝ_EFE_minimizer_exists, and ℝ_epistemicValue_nonneg: the pragmatic/epistemic iden-
tity, finite-policy minimum, and information-gain sign for the stated upstream model.
The source signatures, rather than these summaries, own the full hypotheses. The upstream methods and formal-
kernel guides describe the carriers, assumptions, and reviewed scope. The repository’s docs/positioning.md provides
their section-level navigation map.
9.2 Numerical and formal evidence
A Lean theorem checks its exact proposition under its hypotheses. A numerical test checks implementation behavior
against its references and tolerances. Neither replaces the other. In particular, the Python KL helper rejects zero
reference mass on positive belief support, while upstream finite KL has a real-valued totalization with different
boundary behavior. Comparison requires a common support domain and a justified carrier match.
Witnesses.lean retains concrete controls: a permitted zero-work case, positive work when the nonnegative-
dissipation hypothesis is violated, and the upstream disjoint-support totalized-KL boundary. Audit.lean prints the
axiom footprint of every selected result and witness. The Python verification suite invokes the real build, checks
the complete audit for sorryAx, and asks Lean to reject an explicit false real inequality.
9.3 Qualified blanket and planning adapters
GaussianBlanket.lean matches an additional canonical four-axis precision to the upstream native Gaussian carrier.
It reuses positive definiteness, marginal endpoint covariance, conditional product rows, and endpoint conditional
independence at every center. A supplied coupled endpoint perturbation has native non-independence. This covers
the matched canonical example in sec. 7, not the separate original precision fixture or arbitrary Gaussian models.
Planning.lean uses Boolean observation histories as the outcome carrier. Its normalized terminal-history preference
adds the explicit constant ln 2 to the terminal-preference/history-information score. It proves predictive and pos-
terior mass identities, a fair-coin terminal/cumulative mismatch, and a repeated-revealing-observation boundary.
The Python implementation is compared numerically to that mathematical carrier in src/planning_bridge.py. Nei-
ther the identity nor the source-name correspondence certifies floating-point code or general observation-contingent
policy-tree optimization.
Together these adapters make the paper’s transfer boundaries constructive: state the carrier, bind an actual example,
and test a condition that should break the conclusion. The closed-loop agent and hierarchical model remain
independently tested numerical demonstrations. The paper does not establish empirical intelligence or universal
formal coverage from these examples.
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## Page 17

10 Planning: Policies as Action Sequences
The one-step EFE example can be extended to a finite sequence of actions on a static hidden state. The local
algorithm enumerates every observation branch, updates beliefs along that branch, and scores a declared terminal-
preference objective. It does not optimize a general feedback policy. With 𝜋 = (𝑎1, … , 𝑎𝑇 ), its branch law is
𝑞(𝑜1∶𝑇 ∣ 𝜋) = ∑
𝑥
𝑞(𝑥) ∏
𝑡
𝑝(𝑜𝑡 ∣ 𝑥, 𝑎𝑡). (4)
The score implemented in src/policy_planning.py is
𝐺terminal(𝜋) = −𝔼[ln 𝐶(𝑜𝑇 )] − 𝐼(𝑥; 𝑜1∶𝑇 ∣ 𝜋). (5)
The score in eq. 5 uses the predictive history in eq. 4. Terminal pragmatic value is collected only at the last step.
Epistemic value is the expected sequential posterior KL gain, checked independently against mutual information
of the complete observation history. The two-step result selects cue-then-bet at 0.07, compared with 0.43 for
bet-then-bet. Its information contribution is 0.37.
The bet channel here is state-independent. Observing the cue does not improve that fixed bet’s reward distribution.
Cue-then-bet wins through the objective’s information term while retaining the preferred terminal outcome law.
Genuine observation-adaptive commitment is demonstrated separately in sec. 11. Reading this open-loop score as
an instrumental advantage or a closed-loop control theorem would exceed what the example computes.
10.1 An explicit upstream carrier bridge
For Boolean observations, define one history outcome (𝑜1, 𝑜2) with product likelihood and normalized preference
𝐶𝐻(𝑜1, 𝑜2) = 𝐶(𝑜 2)/2. Upstream one-step EFE on that finite history carrier equals the local terminal/history
objective plus the constant ln 2. The computed normalization offset is 0.6931, and the cue-then-bet history-channel
score is 0.7600. The offset does not change the ranking of the same policy set. src/planning_bridge.py tests the
correspondence numerically; formal/ActiveInferenceFormal/Planning.lean proves the exact finite-carrier identity
and predictive mass law.
This differs from repeatedly summing one-step EFE at every stage. A fair uninformative two-step model has
terminal preference cost ln 2 but cumulative preference cost 2ln 2. A repeated perfect observation resolves the
static state only once: history mutual information is ln 2, not twice that value. The native controls preserve these
distinctions. The Lean identity is about its explicit normalized carrier; it does not certify the floating-point branch
implementation or general policy-tree optimization.
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## Page 18

Figure 7: Finite sequence scoring. The four two-step sequences share a static hidden state. Cue-then-bet
minimizes the declared terminal-preference minus history-information objective. The bet remains state-independent;
the advantage is information scoring, not an observation-conditioned improvement in its payout.
11 A Closed-Loop Synthetic Agent
The previous examples separate inference and policy scoring. Here one finite agent connects them: predict obser-
vations, select an action by one-step EFE, receive a cue, update its belief by Bayes’ rule, and choose a subsequent
commitment. The hidden context is left or right and remains fixed during the episode. The outcome alphabet
distinguishes cue-left, cue-right, reward, and loss. Action-conditioned likelihoods and positive outcome preferences
are explicit in src/synthetic_agent.py.
The initial action set is cue, left, and right. The cue minimizes the initial EFE, with value 0.9163. A cue-right
observation has predictive probability 0.5000 and gives posterior right-context mass 1.0000. The second-stage
action set is explicitly restricted to left and right: information gathering has finished, and the agent must commit.
Its right-action EFE is 0.5276, compared with 4.6052 for left. Cue-left gives the opposite decision. These are
observation-contingent commitments, not a fixed action sequence.
The matched counterexample replaces the cue with a context-independent coin. An observed cue-right label then
leaves right-context mass at 0.5000. The commitment scores tie, and the declared deterministic first-index tie rule
selects left for either cue label. The label alone does not convey information: its state-dependent likelihood does.
The tests check both real informative branches, the uninformative branch, impossible evidence, invalid indices, and
normalization.
This is a deliberately small synthetic agent, with a supplied observation whose likelihood is checked against its
environment model. The episode includes inference and an observation-sensitive next action. It does not learn
its likelihood, discover its preferences, solve an arbitrary partially observed control problem, or establish empirical
intelligence. EFE and policy support still depend on the stated model [ Parr et al. , 2022].
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## Page 19

Figure 8: Observation changes the next decision. Informative cue-left and cue-right produce different posterior
beliefs and opposite commitments. Replacing the cue with a state-independent coin leaves the prior unchanged and
fails to produce an observation-sensitive commitment. Every panel comes from the same exact finite inference and
action-selection core.
12 When Information Improves a Decision
Information gain and instrumental value answer different questions. Mutual information measures expected belief
change in nats. Instrumental value asks whether an observation improves a subsequent choice under a specified
utility. The open-loop bet in sec. 10 does not use hidden context. The closed-loop agent in sec. 11 does. We
can isolate this difference with exact finite Bayesian decision analysis, without asserting that utility is EFE or
thermodynamic free energy.
Let the static hidden state have prior 𝑞(𝑥), the cue channel be 𝐴(𝑜 ∣ 𝑥), and the available commitments have finite
utilities 𝑢(𝑎, 𝑥). With no cue, the best expected utility is 𝑉0. With a cue, a feedback policy can choose a different
commitment for each reachable observation:
𝑉0 = max
𝑎
∑
𝑥
𝑞(𝑥)𝑢(𝑎, 𝑥), 𝑉 obs = ∑
𝑜
max
𝑎
∑
𝑥
𝑞(𝑥)𝐴(𝑜 ∣ 𝑥)𝑢(𝑎, 𝑥). (6)
The joint-mass expression avoids defining a posterior on an impossible outcome. On reachable outcomes it equals
expected optimal posterior utility. The policy that ignores the observation and always selects a prior-optimal action
is among the admissible feedback policies. Therefore 𝑉obs ≥ 𝑉 0: for each outcome the maximum is at least the
value of that constant action, and summing over outcomes recovers 𝑉0 by channel normalization. This elementary
finite argument is an exposition here, not a newly imported Lean theorem.
For a fixed acquisition cost 𝑐 in the same utility units, define
VOI = 𝑉 obs − 𝑉0, VOInet = 𝑉 obs − 𝑉0 − 𝑐. (7)
The gross value is nonnegative under these assumptions; the net value can be negative. An optional acquisition is
worthwhile under this utility objective only when its net value is positive. Unlike EFE’s explicit epistemic reward,
this calculation rewards information solely through its later consequences. Utility units, log-preference scores, and
nats must not be silently substituted for one another. If utility is rescaled, its acquisition cost must be rescaled
consistently.
The worked channel has known accuracy 0.9000 and a uniform prior. Correct commitment receives unit utility and
incorrect commitment receives zero. Exact enumeration gives baseline utility 0.5000, informed utility 0.9000, and
gross value 0.4000. With cost 0.1500, net value is 0.2500. The same cue carries 0.3681 nats of information when
the utility matrix is replaced by state-independent outcomes, yet its gross instrumental value then vanishes. An
independent cue gives both zero information and zero gross value.
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Figure 9: Decision value and information have distinct units. The left panel sweeps known cue accuracy for
an observation-contingent commitment, its acquisition cost, and a state-independent utility counterexample. The
right panel shows the same channel’s mutual information. Positive information does not imply positive instrumental
value; a beneficial cue can also be too costly. Both panels use exact finite computation, not sampled performance
estimates.
src/value_of_information.py enumerates every reachable observation; tests independently enumerate every deter-
ministic feedback policy. Unreachable outcomes have no invented posterior or action. Ties choose the lowest action
index. The sweep holds the prior and utilities fixed and treats the channel as known. Miscalibrated models, chang-
ing hidden states, action-dependent sensing costs, limited computation, and learned preferences require additional
models. This is a controlled bridge from learning to acting, not a claim of general adaptive intelligence.
Enumeration covers the complete finite policy space, but the Python arithmetic is floating point. Tests use in-
dependently calculated references and explicit tolerances; nonfinite inputs and unrepresentable utility results are
rejected.
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## Page 21

13 A Two-Level Generative Hierarchy
A hierarchy gives the model another hidden level without changing the logic of conditioning. Let context 𝑐 generate
state 𝑥, which generates observation 𝑜. The factorization is
𝑝(𝑐, 𝑥, 𝑜) = 𝑝(𝑐) 𝑝(𝑥 ∣ 𝑐) 𝑝(𝑜 ∣ 𝑥). (8)
Top-down prediction marginalizes the context law through the state channel. Bottom-up inference conditions the
joint on the observed outcome, then marginalizes the posterior to each hidden level. This preserves the joint rather
than updating the levels independently. The exact categorical core is src/hierarchical_model.py; matrix columns
are conditional distributions.
For the worked observation, predictive state-zero mass is 0.5600 and observation evidence is 0.6360. The posterior
state-zero mass becomes 0.7925 and the posterior context-zero mass becomes 0.7358. The observation therefore
updates both levels while their posterior dependence remains represented in the joint. fig. 10 shows the downward
prediction and upward update.
A real control makes the observation channel independent of state. Conditioning then leaves both hidden marginals
unchanged. An impossible observation raises an error instead of manufacturing a posterior. Tests compare the full
joint and each posterior with independently enumerated products, and reject malformed channels and out-of-range
observations.
This is exact inference in a finite Bayesian hierarchy, not a neural network, a variational message-passing algorithm,
or a learned deep model. Its purpose is to show how a higher-level model supplies a lower-level prior and how
observations update that higher level through the shared generative structure [ Parr et al. , 2022].
Figure 10: Prediction downward, inference upward. The context law induces a state prediction and an
observation prediction. Conditioning on one observation updates both context and state using the same posterior
joint, without discarding their dependence.
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## Page 22

14 Discussion: Informational and Thermodynamic Interpretation
The demonstrations connect several mathematical settings without identifying their free energies merely by name.
Variational free energy is a functional of a recognition distribution and a generative model; with natural logarithms
it is measured in nats. Expected free energy scores predicted outcomes and states under a policy, with pragmatic
and epistemic terms in the same informational units. Thermodynamic free energy is an energy quantity, measured
in joules. A relation between these objects requires a stated physical model and scaling.
14.1 Equilibrium and stationary circulation
The OU example fixes a Gaussian stationary density and a matched-noise fluctuation-dissipation convention. Zero
solenoidal drive gives detailed balance. Nonzero drive gives a divergence-free stationary probability current and
positive entropy production for positive precision and dissipation. The deterministic drift trajectory still relaxes;
it is not a perpetual orbit. Equilibrium’s zero probability current also does not imply zero microscopic stochastic
motion [ Da Costa et al. , 2021, Friston et al. , 2023].
The Gaussian landscape used in the drift is dimensionless. Under an explicit Gibbs model it can be related to an
energy landscape through 𝐹 (𝑥) = 𝑈 (𝑥)/(𝑘B𝑇 ) +constant, with temperature in kelvin. For an isothermal physical
interpretation using the corresponding local thermodynamic assumptions, an informational entropy-production rate
𝜎 can be scaled to 𝑘B𝜎 and dissipated power to 𝑘B𝑇 𝜎. The code does not estimate temperature, friction, physical
work, or metabolic expenditure from data. The conversion is conditional on a physical interpretation, not an
empirical result of the numerical figure.
14.2 Statistical boundaries and synthetic agents
A Gaussian precision zero is a conditional-independence statement under the Gaussian law. It does not by itself
specify causal channels, action effects, model learning, or agency. The finite closed-loop example supplies a separate
explicit perception-action carrier, while the hierarchy supplies a separate multi-level inference carrier. A shared
vocabulary is a guide to comparison; transporting a theorem requires compatible carriers and hypotheses.
Bayesian mechanics provides a framework for relating probabilistic dynamics, statistical boundaries, and belief
interpretation [ Ramstead et al. , 2023]. The Free Energy Principle supplies the broader organizing perspective
[Friston, 2010]. These worked examples introduce that perspective while keeping its assumptions visible. They do
not establish that every persistent current is intelligent, every conditional-independence partition is an agent, or
every informational objective is a measured physical free energy.
The intended next use is reproducible exploration: change a channel, a preference, a precision entry, or a drive
parameter; rerun the tests and figures; and inspect which conclusion survives. For deeper mathematics, the pinned
fep_formal methods and formal-kernel documentation provide the declarations, support conventions, and scope
reviews used by this paper’s exposition.
14.3 Reading and reusing the Institute repositories
Synthetic supplies the worked teaching models and paper. The Institute’s fep_formal provides the pinned mathemat-
ical reference used in sec. 9 [Friedman, 2026a,b]. Its FepSketches Lean interface is the actual formal dependency;
fep_lean and fep-lean are upstream Python and command-line interfaces for readers who want to inspect the
broader catalogue.
Generalized Notation Notation (GNN) is a complementary language and toolchain for specifying generative models
[Smékal and Friedman , 2023, Friedman, 2026c]. It is relevant when moving these explicit priors, likelihoods, prefer-
ences, and state variables into a shareable model specification. The software citation identifies the reviewed GNN
source snapshot . Synthetic currently constructs its arrays directly; it neither imports gnn nor parses or executes a
GNN specification. An extension would need an explicit mapping of dimensions, channel orientation, observations,
preferences, and control assumptions, followed by numerical equivalence and boundary tests. A GNN representation
would not itself certify an upstream Lean theorem or a thermodynamic interpretation.
The numerical distribution installs as active_inference_synthetic; checkout scripts retain their local src imports.
Readers can reproduce the finite models independently of either upstream Python package. The repository’s
positioning guide separates mathematical reuse, model specification, and template-based manuscript rendering,
with revision-bound links and citation metadata for each upstream project.
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## Page 23

15 A Reader’s Route Beyond Synthetic
Reproduce a worked model first, then choose the resource that answers the next question. Mathematical proof,
shared vocabulary, model specification, software implementation, and educational discussion are different forms
of support. Links below are navigation resources; they do not add runtime dependencies or extend the formal
guarantees of the selected Synthetic carriers.
Question Resource How to use it
Which mathematical hypotheses are
required?
FEP_Formal Inspect exact declarations and
assumption reviews; Synthetic uses
its pinned FepSketches Lean
interface.
How can a model be written as a
shareable specification?
Generalized Notation Notation Compare state dimensions, channel
orientation, preferences and control
assumptions before implementing a
mapping.
What does a term mean in the
shared vocabulary?
Active Inference Ontology Read definitions and typed
relations; a vocabulary entry is not
a theorem or empirical validation.
Where can I study the foundations
systematically?
fundamentals and courses Use reading-group companion
materials and structured curricula;
the fundamentals repository
describes itself as unofficial
textbook support.
Where can I explore a broader
executable agent implementation?
cognitive Inspect the discrete/continuous
model interfaces and their
validation commands; compare
conventions before reusing code.
Where can I find talks and their
transcripts?
ActiveInferenceJournal Navigate the structured
video/transcript archive and check
the original recording and cited
papers.
The bibliography distinguishes FEP_Formal and GNN software snapshots from their scholarly papers [ Friedman,
2026a,b,c, Smékal and Friedman, 2023]. The additional learning resources are maintained independently. Repository
availability or a passing software test is not evidence that a particular explanation is scientifically correct.
For a constructive next experiment, change one cue channel, utility, preference, precision entry or drive parameter.
Predict the result before computing it, include a control that should falsify the intended conclusion, and compare the
new result with the analytical reference. If the modification crosses into a different mathematical carrier, document
the transfer assumptions before citing an upstream theorem. The repository’s resource and methods guides provide
direct links, runnable examples, and the limits of each execution surface.
The ontology and transcript archive have their own software/dataset citation records [ Active Inference Institute ,
2023, Active Inference Institute ]. Their citation metadata is independent of the mathematical and scholarly sources
used for the worked models.
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## Page 24

References
Active Inference Institute. ActiveInferenceJournal. Video-library transcript and metadata dataset. URL https:
//github.com/ActiveInferenceInstitute/ActiveInferenceJournal. Dataset DOI: https://doi.org/10.5281/zenodo.2
1074926. Dataset identity from CITATION.cff; accessed 2026-10-11.
Active Inference Institute. Active inference ontology. Controlled vocabulary repository, 2023. URL https://github
.com/ActiveInferenceInstitute/Active_Inference_Ontology . Release DOI: https://doi.org/10.5281/zenodo.797
2289. Citation metadata from CITATION.cff, version v5, released 2023-05-25; accessed 2026-10-11.
Lancelot Da Costa, Karl Friston, Conor Heins, and Grigorios A. Pavliotis. Bayesian mechanics for stationary
processes. Proceedings of the Royal Society A , 477(2256):20210518, 2021. doi: 10.1098/rspa.2021.0518.
Daniel Ari Friedman. fep_formal: Towards Lean 4 formalization of the Free Energy Principle. Active Inference
Institute software repository, 2026a. URL https://github.com/ActiveInferenceInstitute/fep_formal/tree/
99276926da1df4287a2e13e9aa7f6d1822cfdca3 . Source revision 99276926; citation metadata from the pinned
CITATION.cff, accessed 2026-10-11.
Daniel Ari Friedman. Towards Lean 4 formalization of the Free Energy Principle: AI-driven theorem sketching and
verification for Active Inference and Bayesian Mechanics. Scholarly paper, Active Inference Journal, 2026b. URL
https://doi.org/10.5281/zenodo.23220027.
Daniel Ari Friedman. Generalized Notation Notation (GNN). Active Inference Institute software repository, 2026c.
URL https://github.com/ActiveInferenceInstitute/Generalized_Notation_Notation/tree/9255e19f951d4c9c146
1514464d420bd3fc8b912 . Source revision 9255e19f; project concept DOI, not a version-specific archive; citation
metadata accessed 2026-10-11.
Karl Friston. The free-energy principle: a unified brain theory? Nature Reviews Neuroscience, 11(2):127–138, 2010.
doi: 10.1038/nrn2787.
Karl Friston, Lancelot Da Costa, Noor Sajid, Conor Heins, Kai Ueltzhöffer, Grigorios A. Pavliotis, and Thomas
Parr. The free energy principle made simpler but not too simple. Physics Reports , 1024:1–29, 2023. doi:
10.1016/j.physrep.2023.07.001.
Thomas Parr, Giovanni Pezzulo, and Karl J. Friston. Active Inference: The Free Energy Principle in Mind, Brain,
and Behavior . MIT Press, Cambridge, MA, USA, 2022. ISBN 978-0-262-04535-3. doi: 10.7551/mitpress/12441
.001.0001.
Maxwell J. D. Ramstead, Dalton A. R. Sakthivadivel, Conor Heins, Magnus Koudahl, Beren Millidge, Lancelot
Da Costa, Brennan Klein, and Karl J. Friston. On bayesian mechanics: a physics of and by beliefs. Interface
Focus, 13(3):20220029, 2023. doi: 10.1098/rsfs.2022.0029.
Jakub Smékal and Daniel Ari Friedman. Generalized notation notation for Active Inference models. Scholarly
paper, Active Inference Journal, 2023. URL https://zenodo.org/records/7803328.
23


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*Extraction method: pypdf*