Incoprea research · constraint tests · 3 November 2026

Three constraint tests, one of which failed

Computation only, except where explicitly labelled synthetic. A response-time law is refuted; an enzyme test is shown to have power on synthetic data (not evidence about enzymes); a fog model failed to discriminate and is reported as a construction failure rather than a refutation.

Three further constraint tests. Each asks the same question in a different field: does the claimed effect survive when the parameters are tied to something measurable rather than left free?

1. Response-time collapse

If tracking error depends only on R = τ/T, one curve should describe every system. At R = 1 the error spans 0.702 to 1.919 — a factor 2.74. Against the first-order curve the worst departure is 1.26 in absolute error amplitude.

System	error at R = 1
1st order	0.7016
2nd zeta=0.3	1.9193
2nd zeta=0.7	1.2223
2nd zeta=1.0	1.1145
2nd zeta=2.0	1.0298

Verdict: not supported. R is the dimensionless frequency; it was never going to carry order and damping. The hypothesis was a restatement, and the restatement does not extend.

2. Enzyme competition, constrained

The competitive-kinetics crossover is already fixed by k_cat/K_m; the only testable content is a departure. A synthetic two-site system was generated (clearly labelled synthetic), then fitted with a one-site model in two ways:

Fit	RMSE	Meaning
one-site, K_m free	0.020	the model absorbs the second site into a shifted K_m
one-site, K_m constrained to measured value	0.180	the departure becomes visible

The constrained model is 9.0× worse than the free one. That is the test working as designed: when the free fit is much better than the constrained fit, the system is not described by single-site competitive kinetics, and the size of the gap measures how far off it is.

But this is a synthetic demonstration, not evidence about any real enzyme. Its only content is that the test has power when the departure exists. Whether real enzymes show it is untested and requires the measured constants.

3. Fog collection: a model that failed to discriminate

The claim was that the optimum spacing-to-droplet ratio moves when collection and drainage are both scored. In the illustrative model used here, interception rises with spacing while the drainage fraction falls, and the product is maximised at the same spacing as interception alone — a shift factor of 1.00, i.e. no shift.

This test is uninformative as constructed, and the failure is mine, not the hypothesis's. The drainage term was monotone in the wrong direction, so no interior optimum could form. A useful version needs a drainage model with a genuine maximum (shedding that improves with spacing up to a point, then degrades), which was not built. The hypothesis remains planned, not refuted.

4. The constraint protocol, stated once

Write the model with every parameter either measured or explicitly free.
Count both.
Run the free version — expect the effect (that is the base rate, not evidence).
Run the constrained version.
Report both. The gap between them is the finding.

Applied three times on this page, the protocol produced one clean refutation, one synthetic demonstration of test power, and one honest construction failure. That mix is what a working protocol looks like.

5. New hypotheses

#	Hypothesis	Test
P3-H1	Two-parameter collapse (R, ζ) is minimal for second-order systems; no one-parameter version exists for order ≥ 2.	Fit surfaces; held-out prediction across unseen ζ and order.
P3-H2	Measured enzyme pairs will show constrained-fit residuals systematically larger than free-fit residuals, and the gap will track known allostery.	Use BRENDA/SABIO-RK constants; compare to allostery annotations.
P3-H3	A drainage model with a non-monotone shedding term produces an interior fog optimum that monotone drainage cannot.	Build the non-monotone term; map the optimum.

6. Limits

The enzyme section is synthetic and must not be cited as evidence about real kinetics.
The fog section failed as constructed; it is reported as a construction failure, not a refutation.
The response-time test uses eight chosen systems; the spread would change with a different set, though not the mechanism.

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.