Theory notes arising from the batch. These are arguments, not measurements, and none has been checked against the literature.
Let the selectivity be α(P) = N(P)/D(P), where each of N and D is a sum of channels. Write each channel as K + A·bP/(1+bP). A pressure-independent term contributes no sign change. A saturating term contributes exactly one inflection: it rises from 0 to A, monotonically, with a single curvature change.
The difference α(P) − 1 has the sign of N(P) − D(P), which is a sum of terms each monotone with at most one inflection. Such a sum can change sign at most once per saturating term, so with S saturating channels the number of crossings is at most S. The observed test is consistent with a bound of S − 1 in the sampled cases, but the proof of S − 1 would require an argument about the cancellation of the leading terms, which has not been made.
Status: the S bound is argued informally; the S − 1 bound is observed but not proved. The 600-trial test found no violation of "zero saturating channels implies zero crossings", which is the part the argument actually secures.
Let a claimed dimensionless law be f(Π) for some group Π. Define the spread
S(Π) = max over admissible systems of f(Π) / min over admissible systems of f(Π)The law is informative only if the effect it claims to explain exceeds S(Π). For the eight laws audited here, S ranges from 1.13 to 125.89. In every case the effect being claimed — a 20% improvement, a factor of two, a sign change — is smaller than the spread.
It would be trivial if the spread were measurement noise. It is not: the spread is between correct models in different regimes, evaluated at the same group value. The group is doing its job — collapsing a family — but the family is parameterised by the omitted variables, and that parameterisation is what a law has to specify.
The base-rate result can be read as a statement about model capacity. A reversal requires the sign of N(P) − D(P) to change, which requires at least one saturating channel and at least one parameter free enough to place the crossing inside the observation window.
With four free coefficients, the modeller can place the crossing almost anywhere; with one, the crossing is either fixed or absent. The measured progression — 0%, 0%, 13.2%, 24.0% at 1, 2, 3, 4 free coefficients — is consistent with a simple counting argument: the number of ways to place a crossing grows combinatorially with the freedom, so the probability of landing in the window grows with it.
| # | Question | Why it matters |
|---|---|---|
| Q1 | Is there a general bound on crossings for a sum of monotone channels with S saturating members? | Would turn a screening rule into a theorem. |
| Q2 | Does the spread criterion predict which published laws replicate? | Would make the criterion prospective rather than retrospective. |
| Q3 | Can free-coefficient count be extracted automatically from papers? | Would allow screening at scale. |
| Q4 | Do the same base rates hold in thermal and ionic diode models? | Would generalise the reversal result beyond membranes. |
Status: computational model results, 3 November 2026. No experiment, no data, no measurement. No novelty claimed; no literature search performed. Code and raw JSON accompany this page set.