A work sample from a research proposal. The proposal is about whether one pore-size law explains why two zeolites, HZSM-5 and HMCM-22, give different products from ethanol. This page does not contain the proposal. It shows how the proposal’s test is defined, and runs it on invented numbers. It is not peer-reviewed, I do not hold a PhD, and no real data are shown here.
The claim, Theorem 1(iii) of my paper (preprint V6), is that the selectivity of a zeolite channel with pore radius r is σ = 1 − (r*/r)², where r* is the self-trapping width. The paper adopts the fixed-norm form r* = 4aJ/(λP) and keeps the fixed-amplitude form r* = a√(J/λ) as the alternative. The test below does not depend on which holds, because (1 − σ)r² = r*² in both. If r* is the same for both materials, that quantity must come out equal for both zeolites. So the test needs no fitted lattice spacing. It needs one ratio.
One scope limit from the paper: the smooth law is derived for one-dimensional channels. In cages there is no continuous r*, only a threshold, so a cage radius is not a valid input. Which pore dimension to use for each material is therefore a pre-registered choice, not a free one.
Call the ratio R = (1 − σZ) rZ² / [(1 − σM) rM²]. The law predicts R = 1. The rule passes if |R − 1| ≤ tol, where the tolerance is set from replicate-run repeatability before the confirmatory runs. A blank tolerance asks for one, and a σ outside 0 to 1, or a radius that is not positive, is invalid. The numbers below are invented for illustration.
The proposed confirmatory design is Prediction 1b in the V7 draft of the paper (a draft branch, not yet deposited; the live paper is V6). It is a full factorial of five factors at three levels, 35 = 243 runs, plus one control: 244 runs per material, 488 in total. The five factors are proposed, not confirmed. The single control gives only a weak repeatability estimate, so more centre repeats are needed. No run has been made, and cost and feasibility are not yet quoted.
The rule above was written the way everything in this series is: by hand first, then as code, then run on a damaged copy to confirm the check can fail. This chapter has its own verifier, book26/ch05-verify.py. It ran failing with this page absent, then passing, and it rejects five damaged copies, including one with the decision rule reversed.
The reason for that habit is a record of things that went wrong, all in the public audit log: formal theorems that compiled but proved something other than their names claimed, an inverted scaling law in a published prediction, a backwards sign in a selectivity claim, a sign error in a numerical stepper, and a prediction (a “helical phase”) that was cut because its premise was false. A tool that proves what it says is only useful if the failures are written down next to it.