Time-Causal Scale Space from Hemigroup Axioms: Characterization of the Kernels

We characterize the time-causal scale spaces: the families of smoothing operators through which an uncommitted observer can measure a temporal signal at all scales at once. One classical axiom is weakened: the cascade property is stated as a two-parameter hemigroup, Φ⁠{y,z}Φ⁠{x,y} = Φ⁠x,z, retaining composition of measurements while dropping the tacit assumption that measurement stages are interchangeable. The admissible kernel families are then exactly the Laplace exponents of the self-decomposable laws, the marginals of self-similar additive (Sato) subordinators; reimposing interchangeability recovers the extremal stable densities together with a pure delay, exposing their heavy tails as the price of that assumption rather than of causality or covariance. The enlarged class contains members with all moments finite — foremost the Gamma family, implemented exactly, at integer shape, by cascades of first-order filter sections. Over the memory half-line, under a mild standing hypothesis, the scale evolution inverts into an evolution in present time, well posed in an explicit class: the memory line stores the Riemann–Liouville integrals of the past signal, the temporal N-jet reads that state instantaneously, and locality with the positive maximum principle singles out a degenerate transport member and a one-parameter Bessel family of media. Non-creation of temporal structure is settled in both directions: no nondegenerate member admits the scale-monotone form, while the endpoint form — no scale showing more sign changes than the signal — is attained exactly by the classified variation-diminishing members. The characterization and the signaling form are machine-checked in Lean 4, resting on two cited analytic facts. This record archives the v1.0.0 release: the monograph (paper.pdf), the machine-checked blueprint (blueprint.pdf), and the source repository including the Lean 4 formalization. Prose, mathematics, and figures are CC BY 4.0; the code directories (Formalization/, scripts/) are Apache 2.0, as described in the archive's LICENSE.md. Verification instructions are in the README.