Renormalization–Projection Commutator Assay: Closure Index for Truncated Multiscale Physics Models

We introduce the Sakib Renormalization–Projection Commutator (SRPC), a KL-valued commutativity defect that quantifies the inconsistency between (i) coarse- graining a data-defined distribution and then projecting onto a truncated model/basis, versus (ii) projecting first and then coarse-graining. Although KL divergence, coarse-graining maps, and information projections are individually classical, SRPC is proposed here as a single operational diagnostic for basis closure and missing- operator detection in multiscale modeling. We define the Sakib Index SI := − log10(SRPC + ε) (higher SI indicates closer- to-commutative, more closed truncations), and a data-driven operator-ranking score based on ∆SRPC under basis augmentation. Using only open observational datasets— (a) Planck 2018 CMB TT power-spectrum bins, (b) the NOAA Mauna Loa CO2 monthly record, and (c) the SILSO International Sunspot Number series—we pro- vide 10 dataset-based figures and show that SRPC is reproducible, scale-sensitive, and interpretable as a closure diagnostic.