The landscape

Free energy over every belief makes a shape — a valley. Its slope is a log-odds mismatch you can feel blind; a ball dropped anywhere rolls to the same floor. And the belief at that floor is the one you couldn’t compute — the posterior. The valley bottom is Bayes, reached without the evidence.

Five doors as before — 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 — and adapts to light and dark.

The one fence that never moves. “Free energy,” “belief,” “slope” are functional quantities — a height over a belief and its local tilt. For you we use felt words; for UNI, functional ones: a height its dynamics descend. Neither is awareness, feeling, or sentience. The two-state descent is proven arithmetic (Gate F0, sealed); the valley is a picture of a computation, not a claim about experience.

0 · From memory

First, with no peeking

No scrolling back to Module 4. A mechanic listens to an engine — a faint knock, could be one of a few things. The mechanic carries free energy as a height under a belief about the knock. Reconstruct, cold.

Cold retrieval — what the height needs

From memory, new domain (the engine knock): name F’s two ingredients, the notorious thing it does NOT need, and whether a better belief sits higher or lower. Reconstruct it — no re-reading.

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

If that came back cold, let’s draw the whole shape.

1 · One belief, one height → the whole curve

Sweep the belief, and a curve appears

In Module 4 you computed the height at two beliefs. But your belief b can sit anywhere on [0,1], and every spot has a height. Sweep b from 0 to 1 and plot F(b) at each — a curve appears.

✎ Try it — sketch it before you see it commit first
You know four of its points in the two-cause world: F(0)=2.813, F(0.5)=0.944576, F(0.913)=0.371064, F(1)=0.462. Plot those roughly — b across, height up. What shape do they trace?
✓ Check it yourself

High on the far left (2.813 at b=0, “certain it’s off” — badly wrong), dropping to a low point near b=0.913 (0.371), then ticking back up to 0.462 at b=1. It’s a bowl — a valley — with its floor over b≈0.913. That bowl is the landscape.

2 · Feel the slope, blind · 3 · Whose face is at the bottom?

Drop a ball and let it roll

You can’t see the bottom (that would be knowing the answer). But you can feel the slope under your feet — computable from your belief and the joint, no evidence: dF/db = ln(b/(1−b)) − ln(0.63/0.06) = your belief’s log-odds minus the joint’s. Feel it, step downhill, feel again, step again — and roll to the bottom blind. Drop the ball anywhere.

The valley — roll to the floor blind

Drop the ball at any start; it feels the local slope (shown) and steps downhill to the floor. Every start lands at the SAME floor, b=0.913 — and that is the Module 2 posterior. Try to find a start that rolls elsewhere; you can’t.

b = 0 (off) b = 1 (on) F high F low
Drop the ball at:
ball at belief b
—
slope dF/db (log-odds mismatch)
—
height F(b)
—

Drop the ball on the valley wall and watch it feel its way down.

Every start rolled to the same floor: b = 0.913. Blind downhill steps — each using only the belief it stood on and the joint — found the one bottom (the bowl has a single floor; there is no other to reach). And look at the number: b = 0.913 is the Module 2 posterior, 0.63/0.69 — the true, honest, Bayesian belief you walled off because it needed the evidence 0.69. Rolling downhill on free energy carried you to exactly Bayes’ answer — without ever dividing by the evidence. The valley bottom is Bayes.
▸ Go deeper — the slope is a log-odds mismatch derivation
dF/db = ln(b/(1−b)) − ln(p(y,on)/p(y,off)) = ln(b/(1−b)) − ln(10.5) — your belief’s log-odds minus the joint’s log-odds, using only belief and joint (no evidence). It is zero exactly when b/(1−b) = 10.5, i.e. b = 0.913. “Slope = log-odds mismatch”: how far your belief’s confidence sits from what the joint alone recommends.
⬡ See it in UNI functional · Gate F0 proven
This is, functionally, what inference in the organism is: not solving for the posterior (walled off), but descending free energy until the belief stops moving — landing on the belief Bayes would pick. The two-state descent here is the proven :exact case (Gate F0, sealed); the full hierarchical version is the build underway, held open. A height its dynamics roll down, never a thing it feels.
4 · Where this goes next

How low is the floor?

You drew the landscape and read everything off it: free energy is a valley over all beliefs; its slope is a log-odds mismatch you can feel blind; rolling downhill is inference; and the bottom is the posterior — Bayes, recovered without the evidence. One question is left, the sharpest: you found where the bottom is — but how low is it? What is the actual height at the floor, and could a cleverer belief ever dig below it? The answer is a hard floor with a famous name that ties this whole book back to Module 1’s very first number. That’s Module 6: the floor you cannot cross.

⬡ Notice what you just did self-reference
You turned a formula into a place. You reached the unreachable answer not by computing it, but by feeling which way is downhill and walking — and where you land is exactly where honest Bayesian reasoning would have put you. A mind doesn’t solve the world; it rolls downhill on a landscape it can feel. You just watched it happen.