The irreversible linear null: every surrogate family in the standard toolbox is time-reversible by construction, so no test built on one can fail on a time-asymmetry statistic

Abstract

The standard surrogate-data toolbox for testing nonlinearity has four workhorse null families — random shuffling, the phase-randomised Fourier surrogate, amplitude-adjusted and iteratively amplitude-adjusted Fourier surrogates (AAFT/IAAFT), and fitted linear models such as AR and ARFIMA. Every one of them is time-reversible by construction, so a test built on them cannot fail on a time-asymmetry statistic: a rejection says only not reversible-linear-Gaussian, strictly weaker than nonlinear, because a linear process can be irreversible. This note reports the missing null family built and raced: Poisson shot noise through an asymmetric two-exponential (fast-rise/exponential-decay, FRED) kernel at rise:decay ratios 1:2, 1:5, 1:10 and 1:30, matched per target on the marginal (exact rank remap) and on the spectral density (a zero-phase shaping filter), and read by the two vendored instruments that produced the program's sealed departures. The pre-remap member is exactly a linear filtered-Poisson process with the target's spectral density; irreversibility is the only property the ensemble adds over IAAFT. Under a rule frozen before the run (K = 39 members per target and ratio; covered iff p = (1 + #{|t_m| ≥ |t_real|})/(K + 1) > 0.05), 22/22 Swift/BAT sources are covered at every one of the four ratios (union Clopper–Pearson 95 % [0.846, 1.000]) and 54/54 healthy-RR records at one or more ratios (CP95 [0.934, 1.000]), with the median coverage p equal to 1.000 at every ratio in both domains. The null does not merely reach the data: on the heartbeat cohort it overshoots by an order of magnitude, the matched members' per-record median asym z_agg running 556 to 1,172 against the cohort's sealed median of 73.1. A second finding follows from re-deriving the isolated-singularity false-positive budget on the matched controls themselves rather than inheriting it: the omnibus flags 1.000 at every ratio in both domains and every scattering family flags at or above 0.935 — the cross-scale C arm a multiplicative-cascade claim specifically needs included — against sealed budgets of 2/51 and 0.010 measured on smooth linear controls. The consequence is methodological and general: an excess of a time-asymmetry or wavelet-scattering statistic over IAAFT is not evidence of nonlinearity, multiplicativity or endogeneity until an irreversible linear null has been raced. It is not evidence that any source or any heart is shot noise, and it touches no likelihood-based result. Alongside it we report an observation we do not interpret: healthy hearts are less time-asymmetric than the linear pulse trains matched to their own marginal and spectrum.