Exogenous cascades versus self-excitation: distinguishing Markov-Switching-Multifractal Cox processes from Hawkes processes, with an application to musical onsets

Bursty, hierarchically clustered event streams are almost universally modeled as self-exciting Hawkes processes. We study an alternative mechanism — exogenous hierarchical regime switching — realized as a doubly-stochastic (Cox) process whose intensity is a discrete-time Markov-Switching Multifractal (MSM): N_t ~ Poisson(λ0 · G_t^κ) with G_t the product of k mirror-balanced two-state multipliers on a geometric rate ladder. We derive the closed-form stationary moments, overdispersion, and autocovariance of the MSM-Cox counts, and show the autocovariance is an exact sum of geometrically-spaced exponentials — the 1/f mechanism. We then prove a mimicry result: because a Hawkes count autocovariance is itself an exponential sum, a Hawkes kernel can be fit to match the MSM-Cox second-order structure to R² > 0.999. The processes nonetheless separate at higher order: MSM-Cox is genuinely multifractal (MFDFA width h(-2)-h(2) > 0), has heavier inter-burst quiet-time tails, and — decisively — has an intensity that is autonomous of realized events, which the time-rescaling residual test detects. A likelihood cross-fitting study yields a strongly diagonal model-selection matrix at 10^4+ events, including for near-critical Hawkes (branching ratio 0.9), the regime where second-order mimicry is strongest. Applied to onset streams from the Fenophone generative instrument (an MSM-Cox by construction), the tools select MSM-Cox and attribute the decision to the discriminating statistics. Applied to real music (first corpus run 2026-07-12; seeded 120-piece samples per corpus on a 12-ticks/beat grid), min-BIC selection chooses MSM-Cox for 84% of MAESTRO piano performances — real piano onset streams prefer exogenous regime-switching over self-excitation — while Groove drum-kit streams on the beat grid select Poisson/Hawkes (58%/37%), consistent with the companion finding that the drum idiom's cascade lives in velocity, not count density. The earthquake positive control (run 2026-07-13, SCEDC M≥2.5 2010–2020): handed a Hawkes-true catalog, min-BIC selects Hawkes — decisively with a two-exponential (Omori-approximating) kernel — with the instructive twist that MSM-Cox out-fits a single-exponential Hawkes on those same aftershocks, quantifying on real data that likelihood selection between mechanism classes is fair only when each class is given comparable timescale flexibility (mimicry cuts both ways; §7a). Data provenance and attribution This record uses the MAESTRO v3.0.0 solo-piano corpus (Hawthorne, Stasyuk, Roberts, Simon, Huang, Dieleman, Elsen, Engel and Eck, “Enabling Factorized Piano Music Modeling and Generation with the MAESTRO Dataset”, ICLR 2019; hosted by Google Magenta) and the Groove MIDI Dataset v1.0.0 (Gillick, Roberts, Engel, Eck and Bamman, “Learning to Groove with Inverse Sequence Transformations”, ICML 2019). The raw corpora are not redistributed here. They are fetched at run time from their public sources by pinned download scripts that verify each archive against a recorded sha256 (MAESTRO: maestro-v3.0.0-midi.zip, sha256 70470ee253295c8d2c71e6d9d4a815189e35c89624b76d22fce5a019d5dde12c). Only derived quantities — fitted per-piece parameters, DFA/MFDFA statistics, model-selection verdicts, and the figures built from them — are deposited. MAESTRO is licensed CC BY-NC-SA 4.0 (non-commercial, share-alike), which controls over this record’s CC BY 4.0 grant wherever the two conflict, for any MAESTRO-derived material. The Groove MIDI Dataset is CC BY 4.0, so Groove-derived outputs carry no additional restriction. The null-floor calibration was re-cut (v2, 2026-08-30) The crosswalk’s null calibration is deposited at v2. The v1 record generated its iid “Fano-matched” null streams at the fitted latent Cox rate where the generators take a marginal mean, so its floor curve was measured at 0.577× the corpus’s own operating point. v2 re-runs the calibration mean-parameterized, adds the paired per-piece decorrelation tests, the unclamped surviving fraction, the off-crosswalk-domain counts and a tempo-grid sensitivity arm. v1 is retained unedited, with a dated pointer, because sealed records in this program are append-only. Part of the Feno program. The source repository is private; the reproduction materials for this record are contained in the record itself.