Spatial scale space from hemigroup axioms: characterization of the kernels on the line

We characterize the spatial scale spaces on the line: the families of smoothing operators through which an uncommitted observer can measure a signal at every scale, under translation and reflection symmetry. The axioms are those of the extended class of Pauwels, Van Gool, Fiddelaers and Moons, with one weakened. Their recursivity, a one-parameter semigroup in scale, is stated as a two-parameter hemigroup Φ_{s,t}Φ_{r,s} = Φ_{r,t}, which retains the composition of measurements and drops the tacit assumption that an increment depends only on the scale difference, the stationarity of the increments in the scale parameter. Two things then change. Without positivity no classification of the form proved here exists: the remaining axioms admit an infinite-dimensional family containing the Gaussian. With positivity the admissible kernels from the origin are exactly the symmetric self-decomposable laws. Each family is determined by a Gaussian coefficient and a nonincreasing function that sets the rate of displacements at every size. The Gaussian families form the one ray with a Gaussian coefficient alone, an extreme ray of the cone rather than the whole of it. Reimposing stationarity, in whichever scale parametrization the family has it, selects the symmetric stable family, the positive part of the extended class, with the bound α <= 2 proved from the representation. The class also contains members with all moments finite, the variance-gamma family among them, whose kernels are the Mat\\'ern covariance functions for smoothness above zero. The kernels from the origin are absolutely continuous and unimodal. The contribution is the derivation of this class from measurement axioms on the line; locality, non-creation of local extrema, recursive implementation and higher dimension are outside the paper. The characterization is machine-checked in Lean 4, resting on two cited analytic facts, Bochner's theorem and the symmetric Lévy-Khintchine representation; the regularity of the kernels rests on two more. Version 0.1, the first release: sections 1–8 with an appendix. Thirty-three numbered statements are machine-checked in Lean 4 except the signed construction of one clause; the characterization rests on two cited analytic facts (Bochner's theorem, the symmetric Lévy–Khintchine representation). The accompanying archive is the verification export: the Lean development the statements rest on, buildable with lake build, whose axiom guard reproduces the block printed in section 1.1; the blueprint chapters behind sections 2–7; the paper sources; the axiom ledger; and both external review rounds verbatim with the response plan. Development repository: private until the two further modules it holds are released.