A Dimensionless Multivariate Residual-State Representation for Selected Earth-System Observations

This preprint develops an auditable, dimensionless multivariate residual-state representation for selected Earth-system observations within the Planetary Common State (PCS) research program. Four monthly residual coordinates are constructed from distinct observational domains: global surface temperature anomaly (R_T), Mauna Loa atmospheric CO2 (R_C), Arctic sea-ice extent (R_S), and Atlantic Meridional Overturning Circulation observations from the RAPID-MOCHA-WBTS array (R_F). The coordinates are defined as standardized departures from declared observational references and statistical or climatological scales. They are not interpreted as universal tipping-point distances, probabilities of collapse, percentages of degradation, or bounded PCS L_i variables. The four-coordinate common validation window spans April 2004 through March 2024 (240 months). Validation includes integrity checks, held-out one-step persistence and AR(1) baselines, leave-one-domain-out sensitivity, raw, detrended and first-difference correlation analysis, principal-component analysis, calibration-sensitivity diagnostics, and an exploratory scalar-compression decision gate. PCA explains 50.87% of standardized variance with PC1, followed by 24.69%, 18.15%, and 6.29% for PC2 through PC4. The scalar-compression decision gate fails four of five declared governance criteria. The empirical representation retained by the present analysis is: R(t) = (R_T, R_C, R_S, R_F) A scalar PCS L(t) remains HOLD / disabled and is not presented as a validated physical state variable. The principal methodological result is that a common macroscopic representation does not require premature scalar aggregation. The accompanying archival package contains the audited manuscript, supplementary evidence, figures, frozen validation evidence, source files, bibliographic metadata, and integrity manifests supporting reproducibility. Scientific status: independent research preprint; not peer reviewed.