Blog
A detectability floor of n ≈ 3.92/Δ² said the sweep could not resolve the effect at any outcome, and a sticky-OR readout carried literally zero bits. Both knowable in advance.
Two results from a hardware bring-up, both provable on paper, both of which would have saved a week if applied before the runs instead of after.
For a difference between two proportions at alpha = 0.05 and power 0.8, the sample size needed is roughly:
n ≈ 3.92 / Δ² so Δ_min ≈ √( 3.92 / n )The sweeps used 4 to 6 samples per arm. At n = 6 that puts the smallest resolvable difference at 81 points — while the entire plausible effect being looked for was under 30.
The experiment could not have produced a valid answer regardless of outcome, and that was knowable before any data was taken.
This is the useful shape of the result. It does not say the answer was wrong — it says no answer was available, so whatever came out was noise wearing a conclusion. Two numbers, a square root, and it costs nothing to check while the rig is still on the bench.
A sticky-OR over a window — set a bit if any sample was high, read it at the end — is the natural way to catch a transient. If the window is long enough to contain at least one high sample under either hypothesis:
P(T=1 | H₁) = P(T=1 | H₂) = 1
Λ ≡ 1 I(T;H) = 0The likelihood ratio is identically one, so the mutual information between the readout and the hypothesis is exactly zero. Not small. Zero. And repetition does not help: averaging N independent readings of a variable that carries no information gives no information.
Report AND as well as OR over the same window:
| OR | AND | state |
|---|---|---|
| 0 | 0 | stuck low |
| 1 | 0 | toggling |
| 1 | 1 | stuck high |
| 0 | 1 | impossible — a check on the rig itself |
Initialise AND to 1 and OR to 0. The AND leaving 1 also proves the clock ran, which the OR cannot show. And the fourth row can never occur, so if it does the instrument is broken rather than the design.
One says the experiment cannot resolve the effect. The other says the readout cannot carry the answer even if it could. Both are properties of the setup rather than of the result, both are provable before any run, and both were discovered afterwards — which is the expensive way to learn either of them.
Every figure above is measured, and the limits are named with it.