The physical DNS corpus from which Pilot A and Pilot B are drawn. The present study uses two preregistered, spatially and temporally non-overlapping cutouts of the same forced-isotropic realization. This upstream dataset DOI anchors the physical provenance independently of the UNNS analysis and Zenodo extraction archive.

Germano’s operatorial treatment organizes turbulent fields through filtered representations at different resolution levels. It is the closest conventional analogue to the scale side of the present construction, although D□ measures compatibility of structural routes rather than filtered stresses or a Germano identity.

Ghosal and Moin show explicitly that filtering and differentiation need not commute when filter width varies, making operation-order error a legitimate turbulence quantity in its own right. Their commutation problem is mathematically different from scale–time route commutation, but supplies an established precedent for treating noncommutation as physically meaningful.

Friedrich and Peinke describe the turbulent cascade through conditional transition probabilities across scale and a Fokker–Planck process. This provides the clearest conventional bridge to a route or transition grammar: their framework asks how states transition through scale; the present work asks whether scale transitions remain compatible with temporal evolution.