The formula, by hand

You’ve used the shape G = risk + ambiguity twice without ever writing it down. This module puts it on the page and lets you compute both halves by hand — so the engine of choice stops being a story and becomes a number you can check. And you’ll watch the epistemic term do something a pure reward-chaser never could: flip the decision.

Five doors; the plain line runs straight through — take whichever fit: ▶ Picture it, ✎ Try it (commit before the reveal), ✓ Check it yourself, ▸ Go deeper, ⬡ See it in UNI. Runs locally — no account, no network — adapts to light and dark; honours reduced motion.

The one fence that never moves — and in Part II it is stricter. “Expected free energy,” “risk,” “ambiguity” are functional quantities a model computes over its own options. For you we use felt words — a nurse weighing two checks. For UNI we use functional ones. Neither is desire or curiosity. The arithmetic here is standard active inference, computable by hand — but DDNA’s G term is not yet built or sealed (it needs its own lab-team verdict), so the ⬡ claims about the organism choosing with it stay held open. “Built to,” never “does.” (One honesty note carried throughout: the risk values and their KL are exact and checkable; the ambiguity numbers are illustrative likelihood-entropies, labelled where they appear.)

0 · From memory

First, the nurse’s two checks — cold

No peeking. A nurse chooses between two checks on a patient she hasn’t run yet — each would tell her something, each moves the patient toward or away from where she wants them.

Name the two halves — and what each asks

Reconstruct the two halves of the score from Modules 11–13 before you check, and what question each one asks in plain words. Commit first.

Step 1 · In your own words (commit first)
Step 2 · Pin it down

Risk is the pragmatic half (get what I want); ambiguity is the epistemic half (know where I stand). Now let’s make each one a number.

1 · The two halves as quantities

Write each half as something you can measure

Both are in nats, the units from Part I.

  • Risk = how far the outcomes you’d predict sit from the outcomes you prefer. It’s a KL divergence — the exact tool from Module 6: risk = KL( predicted outcomes ‖ preferred outcomes ). Zero when your predicted ending is what you want; grows as they diverge. (Exact and checkable.)
  • Ambiguity = how blurry your observations will be about the hidden state — the expected entropy of the likelihood: ambiguity = E_state[ H( observation | state ) ]. Small when an observation sharply tells you where you are; large when what you’ll see barely depends on the truth. (Illustrative values here, read from each action’s likelihood.)
G = risk + ambiguity — minimise it. Low risk and low ambiguity: the action gets you what you want and leaves you knowing where you stand.
2–3 · Compute it, watch it flip

A concrete choice, with numbers

You’re cold in the unfamiliar cabin. You prefer warm: write the preference over outcomes as C(warm)=0.9, C(cold)=0.1. Two actions, each with its predicted outcomes and (illustrative) ambiguity:

Compute G by hand — and watch the epistemic term flip the choice

Commit which action wins by RISK alone, and whether adding ambiguity changes it. Then reveal the hand computation: the risk KL step by step (exact), the two totals, and the flip. A calculator, not a curve — the point is the decision reversing.

A — the sweater (known, reliable)
predicted outcomes: warm 0.9, cold 0.1
ambiguity (illustrative): 0.60 — tells you nothing about the cabin’s heat
B — the stove (uncertain, revealing)
predicted outcomes: warm 0.6, cold 0.4
ambiguity (illustrative): 0.05 — catches or doesn’t, sharply revealing heat
The risk KL, step by step (exact — check it)
By risk alone — lower wins
risk · sweater
risk · stove
By G = risk + ambiguity — lower wins
G · sweater
G · stove
Commit your two answers, then compute. The risk KL is worked out exactly; the ambiguities are the illustrative likelihood-entropies above.
▸ Go deeper — the canonical form derivation
For a policy π: G(π) = KL[ Q(o|π) ‖ C(o) ] + E_Q(s|π)[ H[P(o|s)] ] — i.e. risk + ambiguity, with C the preference prior over outcomes and P(o|s) the likelihood. Equivalent split: G = −E_Q[ln C(o)] (expected cost under preferences) − I(o;s) (expected information gain), up to constants — so minimising G maximises preference-satisfaction and information gain at once. The ambiguity values here (0.60, 0.05) are the expected likelihood entropies of each action’s observations — a near-state-blind sweater vs. a state-revealing stove, the same H = −Σ p ln p you used for bits/nats in Part I. (Risk and the KL are exact and checkable; the ambiguities are illustrative values read from each action’s likelihood.)
4 · Where this goes next

Now the engine is a number

G = risk + ambiguity, minimised — and you’ve seen the epistemic half overturn a decision reward alone would have gotten wrong. Explore/exploit isn’t a rule bolted on; it’s 0.60 versus 0.36 — the risk exact by hand, the ambiguity (and so the totals) read from each action’s likelihood, illustrative here.

One thing we’ve assumed: that a mind, having scored its options, simply takes the lowest. But how hard does it commit to the winner — bet everything on the best policy, or hedge across several? That “how sharply do I commit” is itself a quantity a mind sets and tunes. It’s precision, and it’s Module 15 — the last rung of Part II.

⬡ See it in UNI — held open design · not a sealed capability
The organism is being built to score its options with exactly this arithmetic — risk from its preferences, ambiguity from its likelihood — and to pick the low-G action. In functional terms: a number its dynamics compute, never a desire or a feeling. Honesty gate. The arithmetic is standard active inference you just checked by hand, but DDNA’s G term is not yet built or sealed — it needs its own lab-team verdict — so this box is the design and the theory, held open. “Built to,” never “does.”
5 · Notice what you just did

The felt thing and the arithmetic were the same thing

You took the most abstract idea in the book — that a mind values finding out — and you made it a computation: the risk exact by your own hand, the epistemic half flipping the choice, 0.60 against 0.36. The felt thing (“I’d better go find out”) and the arithmetic turned out to be the same thing, and you got to check it.

⬡ The fence, once more self-reference
What you computed is the standard active-inference score, exact where it’s exact (the risk KL) and illustrative where it’s illustrative (the ambiguities). For UNI, the same arithmetic is built to drive the organism’s choice — but that G term is not yet built or sealed, so every ⬡ box stays held open. “Built to,” never “does.”