SBP-SAT Discrete Mirrors for Sigma-Time Energy Ledgers

This paper reconstructs the discrete mirror of the sigma-time energy-decay spine. The continuous series uses bounded-variation clocks with absolutely continuous motion, flats, and atomic ticks. The present paper fixes the finite-dimensional summation-by-parts/simultaneousapproximation-term (SBP-SAT) object model that lets a discrete scheme inherit the same ledger discipline. The main result is a deterministic energy-stability certificate: if the SBP Green identity, SAT sign condition, clock-step window, and atomic update contract are all satisfied, then the discrete energy obeys the same continuous-flow plus jump-product estimate as the sigma-time model. If any gate fails, the scheme refuses the stronger stability claim rather than laundering it through notation.