The width and the null

Abstract

The standard test for multifractality in a time series compares the width of an MFDFA singularity spectrum against an IAAFT surrogate ensemble, and a null result is routinely read as evidence that the series is linear. We measure that test's power on its own canonical positive control and find it near zero: on a deterministic binomial multiplicative cascade the MFDFA width flags 1 of 24 realisations — statistically indistinguishable from the rank test's own false-positive floor of 2/51 ≈ 0.0392 — where a wavelet-scattering omnibus at matched specificity flags all 24. The mechanism is measured rather than asserted: on the binomial the IAAFT null hands back 0.939 of the original's own width, against about half on every stochastic cell with a non-trivial width, because the binomial's multifractality is maximally marginal-encoded and IAAFT returns the marginal exactly. A pre-registered σ-axis battery then falsifies the paper's own broader hypothesis: the blindness is not generic to multiplicative cascades but specific to the deterministic multiplier — the width's detection rate climbs 0.042 / 0.375 / 0.750 / 0.875 across σ = 0.30 / 0.50 / 0.70 / 1.10 while the omnibus holds 0.958–1.000, and at the binomial's own σ-equivalent the stochastic twin is flagged 18/24 where the deterministic original is flagged 1/24. The paper absorbs the falsification as the stronger argument: the converse inference "null ⇒ linear" dies not because power is uniformly zero but because it is truth-dependent by a factor of twenty in a direction no analyst can observe. A reporting standard is proposed, and the omnibus's own high side (isolated singularities; from-scratch specificity) is priced rather than hidden.