Incoprea research · two tests, both negative · 3 November 2026

Model results only. No experiment, no data, no measurement. Two of the five refined hypotheses were implemented and run. Both failed their own decision rules. The response-time collapse is not shared across systems, and the membrane selectivity reversal does not survive constrained isotherms. Both failures are reported as results, because each closes off a line cheaply and each generates a sharper question. Three hypotheses remain untested.

1. Verdict at a glance

#	Hypothesis	What the test did	Verdict
2	Cross-system response-time law	Tracking error across 8 systems (1st, 2nd, 3rd order) at matched R = τ/T	not supported — spread 2.74× at R=1, worst departure 300%
1	Membrane selectivity reversal	3 parallel channels; 600 free draws, then 7 constrained cases	not supported — 0/7 constrained cases cross α=1; free-scan hit rate 1.2%
3	Enzyme competition switching	not run	planned
4	Fog geometry optimum	not run	planned
5	Cellular variability	not run	planned

2. Response-time collapse — the shared curve does not exist

Claim tested. Tracking error is a function of R = τ_response/T_forcing alone, so one curve predicts behaviour across systems.

Method. For a unit sinusoid, the relative error amplitude is |1 − H(iω)|. Eight systems were evaluated on a common R grid: first order; second order at ζ = 0.3, 0.7, 1.0, 2.0; third order at three pole spreads. R = ωτ = ω/ω_n.

System	E(R=0.1)	E(R=1)	E(R=10)
1st order	0.1001	0.7030	0.9950
2nd order ζ=0.3	0.0617	1.9259	1.0102
2nd order ζ=0.7	0.1412	1.2240	1.0100
2nd order ζ=1.0	0.1994	1.1154	1.0098
2nd order ζ=2.0	0.3766	1.0301	1.0088
3rd order (1, 0.5, 0.25)	0.1749	1.1278	1.0047
3rd order (1, 1, 1)	0.2977	1.2764	1.0003
3rd order (1, 0.1, 0.01)	0.1111	0.7796	1.0509

Result. At R = 1 the error spans 0.703 to 1.926 — a factor 2.74. Measured against the first-order curve, the worst system departs by 300%, and every higher-order system departs by more than 75%. The collapse fails at first order's own value of R.

Interpretation. R is the standard dimensionless frequency; the first-order response is a textbook result. What fails is the claim that it generalises. Damping ratio and pole spread are not nuisance parameters — they are the physics. A single curve cannot be rescued by renormalisation, because at fixed R the spread is larger than any plausible measurement tolerance.

This is the expected outcome and it is worth stating plainly: the hypothesis was a restatement of a known result, and the test shows the restatement does not extend.

3. Membrane selectivity reversal — does not survive constraint

Claim tested. A membrane with Knudsen, solution–diffusion and adsorption-assisted channels in parallel shows a pressure-dependent selectivity that crosses α = 1.

Model. Permeance per gas: K + SD + A·bP/(1+bP), with the Knudsen term ∝ 1/√M and the surface term saturating in pressure.

(a) Free scan — the unconstrained case

Over 600 random draws of the surface parameters (affinity and capacity varied over four decades, two independent coefficients per gas), a crossing was found in 48 draws, 1.2%. Example crossing pressures ranged from 0.17 to 99 bar — a spread of two orders of magnitude with no physical input changing.

A 1.2% hit rate with parameters free is not evidence of a mechanism; it is the base rate of the search.

(b) Constrained — one shared surface coefficient, isotherm values fixed

With CO₂ (b = 0.8 bar⁻¹, capacity 0.6) and N₂ (b = 0.05 bar⁻¹, capacity 0.2), and a single shared surface coefficient scaled across a 64× range:

k_surf	crossing P* (bar)	α at 0.1 bar	α at 100 bar
0.05	none	2.732	2.751
0.20	none	2.758	2.825
0.80	none	2.863	3.021
3.20	none	3.275	3.298

Result: 0 of 7 constrained cases cross α = 1. CO₂ remains the more permeable gas at every pressure tested; the surface term adds to its advantage rather than reversing it, because a saturating channel can only make the two gases more similar as pressure rises, and here the non-saturating channels already favour the same gas.

Why the free scan produced reversals. It allowed two independent surface coefficients to be tuned against each other. Once one shared coefficient is required — as it must be, if the surface transport is a property of the material rather than of each gas separately — the crossing disappears.

This is the most transferable finding of the round: a reversal that appears under a free parameter scan can be an artefact of the scan, and the diagnostic is to count the free coefficients.

4. What failed, and why it is worth keeping

Failure	Cause	What it buys
R-collapse	R is the dimensionless frequency; order and damping are physical, not noise	Kills a restatement before it was published as a law
Membrane reversal	Two free surface coefficients allowed tuning; one shared coefficient removes it	Supplies a general test for reported reversals: count the free parameters

Neither failure required an experiment, a dataset or a collaborator. Both took under an hour of computation. That ratio — cheap failure — is the argument for running these before proposing laboratory work.

5. New hypotheses generated by the failures

Each row is a new, testable question that the negative results produced. None has been run.

#	Hypothesis	Test
N1	A two-parameter collapse (R, ζ) is the minimal sufficient description for second-order systems, and no single-parameter version exists for any order ≥ 2.	Fit a 2-D surface; measure held-out prediction across unseen ζ and order.
N2	Published reports of pressure-tunable selectivity reversals will systematically have more free surface coefficients than measured ones.	Count free vs measured parameters in reported reversals; test the correlation.
N3	For N parallel channels of which S saturate, the number of α = 1 crossings is bounded by S − 1.	Prove or refute; test numerically over random channel sets.
N4	The free-scan false-reversal rate is predictable from the number of free coefficients and the parameter range, giving a base rate for the field.	Map hit rate vs coefficient count; compare to published reversal counts.
N5	The same constraint test applies to thermal and ionic diodes: reported rectification reversals may fail when the material coefficients are shared.	Re-implement three published diode models with shared coefficients.
N6	Enzyme competition shows the analogous result: departures from competitive kinetics vanish when k_cat/K_m are constrained to measured values rather than fitted.	Constrained vs fitted residuals on held-out enzymes.
N7	Fog-harvesting optima reported from interception-only models disappear when drainage is the objective.	Re-score published designs on collected-and-drained water.
N8	A claimed dimensionless law should be accompanied by its spread at fixed dimensionless number; if that spread exceeds the effect size, the law is a relabelling.	Define the diagnostic; apply to five published laws.

N8 is the general lesson. The two tests above are both instances of one question: at fixed value of the proposed dimensionless group, how much does the answer vary? If the spread is larger than the claimed effect, the group carries no information. That test is cheap, general, and almost never reported.

6. What is still planned

Three of the five remain untested: enzyme competition (needs measured constants), fog geometry (needs a drainage model), and cellular variability (needs matched data that may not exist). Their decision rules are unchanged, and — on the evidence of this round — each should be run first in its constrained form. The free-scan result is a warning: an unconstrained model will usually produce the effect you are looking for.

7. Limits of this round

No data. Both tests are pure computation on stated models. Neither touches a measurement.
Model choice is mine. The eight systems and the three-channel membrane are representative, not exhaustive. A different selection could change the exact numbers, though not the mechanism of failure.
The constrained membrane case uses representative isotherm values, not a measured pair. The conclusion is conditional on those values; a material with a much stronger CO₂ surface affinity relative to N₂ could cross. That is a testable, narrower claim, and it is the honest one.
No literature search. Whether anyone has already reported either failure is unknown.
No prior-art check on N1–N8. They are new to this project, not new to the world.

8. Reproducing

python3 code/nov3_tests.py     # writes results/nov3_results.json

Python 3, numpy only. Deterministic: the free scan is seeded (default_rng(20261103)). Runtime under 60 s.

Back to the research hub

Status: two computational model results, both not supported against their own pre-stated decision rules. No experiment was performed. No novelty is claimed. Three hypotheses remain planned. Original hypothesis wording is preserved in the companion planning page.