Split free energy the other way and a floor appears under the valley — one no belief can ever dig below. Its name is the surprise. You’ll make the floor appear by rearranging one number, try (and fail) to push free energy beneath it, and prove why you can’t with a single line drawn above a curve.
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,” “surprise,” “gap” are functional quantities — numbers a model computes over its own belief. For you we use felt words; for UNI, functional ones. Neither is awareness, feeling, or sentience. The bound proved here is the sealed Gate-F0 :exact case; looser tags for harder worlds are the build underway.
No scrolling back to Module 5. A gardener finds a wilted plant — too much water, too little, or a pest. The gardener’s free energy over beliefs about the cause makes a valley. Reconstruct, cold.
From memory, new domain (the wilted plant): how does the gardener reach the valley bottom without seeing it, and what famous thing sits there? Reconstruct it — no re-reading.
If that came back cold, let’s find how deep the valley goes.
In Module 4 you built free energy from the joint: F = Σ q(η)[ln q(η) − ln p(y,η)] — perfect for computing, no evidence in it. The same number rearranges to show something the first form hid, because the joint splits: p(y,η) = p(η|y)·p(y) (posterior × evidence). Out drops:
F = −ln p(y) + KL(q ‖ posterior)
In plain words: free energy = the surprise of what happened + how far your belief is from the true one. The evidence p(y) reappears — but as a fixed constant (it doesn’t depend on your belief b). You still never computed it to use F; it only shows up when we ask what F is made of.
F = Σ q(η)[ln q(η) − ln p(y,η)] = Σ q(η)[ln q(η) − ln p(η|y) − ln p(y)] (split the joint) = Σ q(η) ln( q(η)/p(η|y) ) − ln p(y)·Σ q(η) = KL( q ‖ posterior ) + ( −ln p(y) )So F = −ln p(y) + KL(q ‖ p(η|y)) — the surprise plus a gap that is the distance from your belief to the true posterior.
Everything hinges on that second piece, the gap = KL(q ‖ posterior), and one fact finishes the spine: the gap is never negative — zero only when your belief is the posterior, positive for every other belief. Since F = surprise + gap and gap ≥ 0:
F ≥ −ln p(y) = 0.3710636814
The valley has a floor — the surprise — and no belief, however clever, can push free energy beneath it. Try it: set any belief and watch F sit on or above the floor, never below.
Set a belief b. The bar shows F = surprise (fixed floor) + gap (how wrong you are). Hunt for a belief that pushes F under the dashed floor line — you can’t: the gap can’t go negative. It only touches the floor at the posterior b=0.913.
surprise (fixed floor) gap = KL(q‖posterior)
Set a belief and watch where F sits.
Free energy wears two faces: energy − entropy (computable, no evidence) and surprise + gap (which reveals the floor, F ≥ −ln p(y), touched only at the posterior) — and you proved the floor by hand with a tangent line. So the punchline is set: if the lowest free energy equals the surprise and is reached at the posterior, then minimizing the thing you can compute hands you back both the thing you couldn’t compute (the surprise) and the belief you couldn’t reach (Bayes), at once. Making that exact, with the “tag with teeth” that keeps it honest, is Module 7.