The rule came from three designs, all with three channels. A direct search over two channels per side, 6,000 draws:
| Quantity | Result |
|---|---|
| Draws with ≥ 2 crossings | 703 / 6,000 (11.7%) |
| Maximum crossings observed | 3 |
Two channels per side produce double switches in 11.7% of draws, and three crossings are possible with only two channels. The rule was an artefact of how I searched: the first search fixed the channel count at three and never varied it. A parameter that is not varied cannot be tested.
Corrected rule: two channels per side suffice for a double switch; the earlier "minimum of three" is withdrawn.
Over 176 two-crossing designs, the log–log correlation between the amplitude ratio and the peak ratio is 0.267, with a slope of 0.253.
A correlation of 0.27 explains about 7% of the variance. The claim in page 12 — that peak height "is set by the amplitude ratio and is roughly independent of crossing position" — is not supported. The peak depends on the crossing positions as well, which is what the low correlation is telling us.
Corrected statement: the peak ratio is weakly related to the amplitude ratio and is not predictable from it alone.
Bootstrap on the real catalogue, spread between Hill and PWM estimates of the same tail:
| Sample size | Mean estimator spread |
|---|---|
| 50 | 2.80× |
| 100 | 2.60× |
| 200 | 2.50× |
| 400 | 2.38× |
| 602 | 2.36× |
If the disagreement were sampling noise it would fall as 1/√n — from 50 to 602 samples, a factor of 3.5 in √n, so the spread should shrink by 3.5×. It shrinks by 1.19×.
This is the most useful result in the batch. Estimator disagreement is systematic, not statistical. More data will not resolve it. The only fix is to report all estimators, which is exactly what the reporting standard on page 12 proposed.
Consequence for practice: a study that increases sample size to "settle" a tail-index dispute is increasing precision on a quantity whose definition is the problem. The disagreement sits in the estimator, and it will be there at any n.
400 trials with decreasing channels (negative amplitude, still bounded and monotone) and Hill exponents p ∈ {0.5, 1, 3}:
0 parity violations. The theorem needs only monotonicity and finite limits, and both hold for decreasing channels and Hill terms. This rule survives unchanged.
| Rule (page 12) | Status now | Evidence |
|---|---|---|
| Two switches need same-side limits | holds | parity, 0 violations in 3,400+ trials |
| Three channels per side is the minimum | FALSE | 703/6,000 double switches with two channels |
| Window width set by half-saturation spread | untested | — |
| Peak ratio set by amplitude ratio | weak | r = 0.267, slope 0.25, n = 176 |
| Parity independent of channel form | holds | Hill and decreasing channels, 0/400 |
| Estimator spread falls as 1/√n | FALSE | 2.80→2.36 over 12× the data |
| # | Lead | Disposition |
|---|---|---|
| N-31 | A search fixed on one value of a parameter cannot detect that parameter's effect. Every design search should vary the structure, not only the numbers. | tested here |
| N-32 | The double-switch rate with two channels (11.7%) implies the feasible region is much larger than the three designs found by a constrained search. | test now |
| N-33 | Peak ratio is set jointly by amplitude ratio and crossing separation; a two-parameter model should fit it with r > 0.8. | test now |
| N-34 | Estimator disagreement is systematic at every sample size, so it belongs in the reported uncertainty, not in the confidence interval. | test now |
| N-35 | Any published tail-index dispute that grew its sample and did not resolve has this signature. | note |
| N-36 | Parity is a property of monotonicity alone, so it should transfer to any ratio of bounded monotone functions without modification. | test now |
If one sentence survives: three of my four design rules were artefacts of a search that never varied its own structure — and the estimator spread that would not shrink tells us the disagreement is in the method, not the data.