Incoprea research · free-parameter base rates · 3 November 2026

Reversals that are artefacts of freedom

Computation only. No data, no experiment. A selectivity reversal was manufactured from random coefficients at four levels of model freedom. The base rate rises from 0% at one or two free coefficients to 24% at four. A screening bound on the number of possible crossings is tested and survives 600 trials.

A "reversal" is any claim that a ranking flips: gas A separates better than B at low pressure and worse at high; membrane X is selective one way and then the other; a diode rectifies and then inverts. These claims are common and they are easy to produce. This page asks how easy.

The answer: the probability of manufacturing a reversal rises steeply with the number of free coefficients in the model, and a reversal that survives when the coefficients are constrained is a different and much stronger thing than one that does not.

1. Base rate versus freedom

A three-channel permeance model was run 400 times at each level of freedom. At level k, exactly k of the four surface coefficients (affinity and capacity for each of two gases) were drawn from decade-wide ranges; the others were fixed.

Free coefficients	Draws showing an α = 1 crossing
1	0.0%
2	0.0%
3	13.2%
4	24.0%

Frequency of an apparent selectivity reversal against the number of free coefficients. With one or two free coefficients the reversal does not appear; by four it appears in roughly one draw in four.

With one or two free coefficients the reversal essentially never appears. With three it appears in 13.2% of draws; with four, in 24.0%. Nothing physical changed between those rows. The only difference is how much freedom the modeller allowed.

Consequence: a published reversal should be accompanied by a count of free versus measured coefficients. Without that count, the reader cannot distinguish a mechanism from a fitting freedom.

2. Is there a bound on how many crossings are possible?

If each saturating channel can contribute at most one sign change to the difference of two sums, then with S saturating channels the number of crossings should be at most S − 1. The test: 600 random channel sets with 2–5 channels and 0–5 saturating channels.

Result: 0 violations in 600 trials. With zero saturating channels the maximum number of crossings observed was 0 — as the bound requires, since a sum of monotone pressure-independent terms cannot cross.

The bound is not a deep theorem, but it is a useful screening rule: if a reported reversal has no saturating channel in the model, the reversal is not in the model's physics — it is in its parameters.

3. The membrane case, constrained

The same model with one shared surface coefficient and fixed isotherm values:

k_surf	α at 0.1 bar	α at 100 bar
0.05	2.732	2.751
0.2	2.758	2.825
0.8	2.862	3.021
3.2	3.275	3.298

0 crossings in 4 constrained cases. The selectivity stays between 2.73 and 3.30 across a 64-fold change in the shared coefficient. The surface term increases the same gas's advantage at low pressure and lets it decay at high pressure, which narrows the gap but does not close it.

4. The protocol this suggests

Count the free coefficients in the model that produces the reversal.
Count the ones that were independently measured.
If free > measured, re-run with one shared coefficient and measured isotherms.
If the reversal survives, report the pressure at which it occurs and the coefficient that sets it.
If it does not, report that the reversal was an artefact of freedom. That is a publishable negative.

5. New hypotheses

#	Hypothesis	Test
P2-H1	Published reversals in membrane, thermal-diode and ionic-diode literature will correlate with free-coefficient count.	Code 40 papers on both axes; test the association.
P2-H2	Reversals in systems with no saturating channel are always parameter artefacts.	Screen reported reversals for a saturating term.
P2-H3	A "constrained replication" requirement would remove a large fraction of published reversals.	Re-run three published models with shared coefficients.

6. Limits

The base rates are properties of my parameter ranges, not of nature. A different range moves the percentages.
The model is three parallel channels with Langmuir saturation. Real membranes have concentration polarisation, non-ideal sorption and support layers, none of which are here.
The isotherm values are representative, not measured. The conclusion is conditional on them.

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.