Projective Data-Processing in Minimum-Production Renormalization

Finite networks with nonnegative quadratic production laws admit two exact and independent irreversible losses under constrained minimum-production (series) reduction: the physical mismatch loss and the target-current loss D_J, a resistance-weighted variance. This paper introduces the projective defect delta = kappa/Y and proves the exact coarse-fine identity delta_c − delta_f = K·D_J/(Y_c·Y_f) ≥ 0. Along nested charged reductions delta converges; the activity-weighted transition law is uniquely forced, and under it delta is a bounded supermartingale with explicit Doob compensator and a projective packing budget of expected total mass at most one. The full matrix analogue is established (range-compatible semidefinite formulation, fixed-range stability), together with exact relative-entropy chain rules for additive residual charges and scalar resistances, a dual parallel-channel frustration identity, quantitative near-equality estimates, and a classification of simultaneous equality cases. The theory is finite-dimensional and deliberately equation-agnostic: no Navier–Stokes regularity claim is made. This is the first paper of the M-series (M-I…M-V). The attached package contains the full LaTeX source, a deterministic verification suite with SHA-256 certificate and report, and provenance notes. The historical origin of the broader programme (Zenodo preprints v14.8 and v7.4) is cited as prior public disclosure only, never as a theorem source.