Sharp Logarithmic Retained-Score Laws for Local Decoders in Critical Planar Ising Models

We study the loss incurred when a raw observation that is sufficient for the global-spin-even quotient scores considered here is processed only by a spatially local additive decoder. The nearest-neighbor bond field determines the spin configuration modulo the global spin flip, so every even score used below is measurable from the raw observation; the loss instead comes from the restricted decoder architecture. In the critical nearest-neighbor square-lattice model, for every fixed decoder radius we prove a sharp inverse-log optimal retained-score law over the full fixed-range nonlinear local bond algebra. We also exhibit a mixed free-boundary/bulk protocol with a two-log law. Finally, for each fixed finite-range, translation-invariant, spin-flip-even perturbation potential in a constructive renormalization-group class, we prove a boundary-source/bulk-energy three-log law at sufficiently small coupling along the tuned critical family. The common mechanism is a separation between global score information and the marginal thermal or boundary channels accessible to a restricted local-additive decoder. The proof combines exact quotient structure, finite Pfaffian reductions, discrete $s$-holomorphic regularity, boundary primary-field convergence and fusion, correlation sign control, and source-weighted Gallavotti–Nicolò expansions. We do not claim that the exponents $1, 2, 3$ form a universal classification, nor that the associated amplitudes are universal.