VFE / ELBO / EFE Bound Discipline for the Agent Layer

Scope

This document specifies the variational free energy (VFE), evidence lower bound (ELBO), and expected free energy (EFE) identities that govern the simulation lab's agent layer — model comparison and policy scoring only. It is not a physics proof and not a clinical claim. Per lab/docs/SCIENTIFIC_LIMITS.md, the host system implements a live active-inference loop (SP.*: SP.Sim.step/1 runs Body.step → Sensor.transduce → agent_mod.decide → interpret_all, and SP.Brain.Agent.decide/3 commits the chosen action so the next tick's prior and transition-learning condition on it — a closed perceive→infer→act→learn loop where actions change future observations). This lab layer adds no separate agent loop; it only documents the bound discipline the host agent's VFE/ELBO/EFE math is held to, and fences the language.

Equations (preserved exactly)

VFE — upper bound on surprisal (Class A; docs/MATH_DERIVATIONS.md 8, 20–22):

$$F[q] ;=; \int q(\eta|r),\ln\frac{q(\eta|r)}{p(y,\eta|m)},d\eta ;=; D_{\mathrm{KL}}!\big(q(\eta|r),|,p(\eta|y,m)\big) ;-; \ln p(y|m)$$

ELBO (Class A):

$$\mathrm{ELBO}(q) ;=; -F[q] ;=; \mathbb{E}_q[\ln p(y,\eta|m)] + H[q] ;\le; \ln p(y|m)$$

EFE — policy scoring (Class B; Sajid, Ball, Parr & Friston 2019/2021, arXiv:1909.10863):

$$G(\pi) ;=; \mathbb{E}{q(o,s|\pi)}!\big[\ln q(s|\pi) - \ln p(o,s|C)\big] ;=; \underbrace{\mathbb{E}[D{\mathrm{KL}}(q(o|\pi)|p(o|C))]}{\text{risk}} + \underbrace{\mathbb{E}{q(s|\pi)}[H[p(o|s)]]}{\text{ambiguity}} ;-; \underbrace{\mathrm{IG}}{\text{info gain}}$$

Selection (canonical repo form, CLAUDE.md §2/§3): $Q(\pi) = \mathrm{softmax}\big(\ln E - \gamma, G(\pi)\big)$.

Variables and Units

• $F[q]$: variational free energy [nats]. Lower is better.
• $q(\eta|r)$: moving recognition density over hidden causes $\eta$, parameterized by recognition states $r$ [dimensionless density].
• $p(\eta|y,m)$: exact posterior under model $m$ [density] — model-internal, not the world/process.
• $p(y|m)$: model evidence [probability]; $-\ln p(y|m)$ = surprisal [nats].
• $H[q]=-\mathbb{E}_q[\ln q]$: entropy [nats]. $\gamma$: policy precision [dimensionless]; $E$: habit/prior. $C$: log-preferences. IG: expected information gain [nats, subtracted → epistemic drive].

Worked Example (numbers)

Binary hidden cause $\eta\in{0,1}$, model posterior $p(\eta{=}1|y,m)=0.8$. If $q(\eta{=}1|r)=0.5$:

$$\mathrm{KL} = 0.5\cdot\ln\tfrac{0.5}{0.8} + 0.5\cdot\ln\tfrac{0.5}{0.2} = 0.5(-0.470) + 0.5(+0.916) = \mathbf{0.223144\ \text{nats}}$$

So $F = -\ln p(y|m) + 0.223144$ — above the surprisal floor by exactly the KL gap. (Per the verdict correction: $-0.470$ and $+0.916$ are the unweighted logs; the 0.5-weighted contributions are $-0.235$ and $+0.458$.) If instead $q=0.8$ (matched to the posterior), KL $=0$, the bound is tight, and ELBO $=\ln p(y|m)$. Minimizing $F$ drives $q$ toward the posterior; it never drops below $-\ln p(y|m)$.

Dimensional Check

All terms in nats. VFE: KL [nats] $-\ln p(y|m)$ [nats] $\Rightarrow$ [nats]. ELBO: expected log-joint [nats] $+$ entropy [nats] $\le$ log-evidence [nats]. EFE: risk (KL) $+$ ambiguity (entropy) $-$ info-gain, all [nats] $\Rightarrow G$ [nats]; info-gain subtracted lowers $G$ (sign matches CLAUDE.md §2/§4). PASS (independently re-verified in verdict).

Evidence Class per Claim

• VFE is an upper bound on surprisal; $F\ge-\ln p(y|m)$; ELBO $\le\ln p(y|m)$ — Class A (derived in-repo).
• $q$ vs $p$, and $p(\eta|y,m)$ ≠ the world — Class A invariant.
• EFE decomposition (risk + ambiguity − info-gain), $Q(\pi)$ selection — Class B (peer-reviewed).
• KL-gap $=0$ iff $q=$ posterior; 1e-10 acceptance threshold — Class A (theorem on tree/junction-tree only; regression check on loopy graphs).

Falsification Conditions

• Observe $F < -\ln p(y|m)$ for any $q$ → the bound identity is contradicted-by-test (would indicate a code/derivation defect).
• ELBO exceeding $\ln p(y|m)$ → inequality inverted → contradicted.
• On a declared tree/junction-tree, $\max|q_{BP}-q_{\text{enumerated}}| \ge 10^{-10}$ → the exactness claim fails (downgrade to VARIATIONALLY-CONTROLLED).

Outside This Model (fence)

This domain proves no physical mechanism: not gravity ($g=GM/R^2$ stands on its own; $g=kP$ is X / contradicted-by-test), not ozone chemistry, not water formation, not the origin of life. Free energy here is information-theoretic (energy−entropy over densities), not literal Helmholtz/Gibbs (no $pV$, no temperature). ELBO/VFE rank models within an assumed hypothesis space — they are surrogates for intractable log-evidence, not truth certificates; "a simulation proves reality" is forbidden. The host runtime is a closed perceive→infer→act→learn loop (actions change future observations; SP.Sim/SP.Runtime.Agent/SP.Brain.Agent), but this lab domain proves no physical mechanism and registers no capability beyond that — the standing not-yet-shown fence (lab/docs/SCIENTIFIC_LIMITS.md) is that no novel, pre-registered, out-of-sample prediction a mainstream baseline cannot make has yet survived. "Trauma-locked agent" = a parameter-regime analogy (maladaptive priors / miscalibrated $\gamma$), never a clinical claim. Exactness is always relative to a declared factorization. Citation: Sajid et al. = 2019 (arXiv:1909.10863) / 2021 (Neural Computation), not 2020.