Tauberian Decay Criteria for Sigma-Time Transform Methods

This paper rebuilds the theorem spine for the Tauberian layer of Sigma Transform Calculus. A bounded-variation clock is first resolved into mass coordinates, where the sigma transform becomes an ordinary Laplace-type transform of the selected signal. The paper then records the decay criteria that can be transported through that representation: transform naturality, Abelian estimates for exponential and polynomial decay, a conditional Tauberian reduction principle, monotone window-to-pointwise decay, a resolvent corridor for imported Hilbert-space stabilization theorems, mixed continuous/atomic kernel-clock stabilization, and explicit rate transfer from sigma time to physical time. The spine is intentionally conditional. It identifies exactly which transform-domain, monotonicity, boundaryregularity, resolvent, atomic-product, and clock-growth hypotheses are required before a decay conclusion can be claimed.