We develop a mathematically and physically grounded framework on the positive half-line (0, +) that unifies asymptotic regimes, atomic ratios, and logarithmic metrics into a single coherent structure. Constants of the form 1/, 1/e, 1/, 1/Fₙ, together with their power and root generalizations, are shown to define natural ``modes of smallness'' describing distinct rates of approach toward 0^+. These mathematical regimes are complemented by physically determined dimensionless ratios derived from atomic radii—including the internal spin configurations of hydrogen (orthohydrogen and parahydrogen) and ratios involving other elements and their isotopes. The resulting set is dense in (0, 1) and represents a physically generated asymptotic infinity toward zero. We introduce the logarithmic metric _ (, ) =| - |, \ which transforms multiplicative differences between atomic and mathematical ratios into an additive geometry. This induces a logarithmic information structure in which atomic invariants, mathematical constants, and discrete sequences align under a unified geometric order. We argue that this framework constitutes a natural candidate for universal SETI signatures: dimensionless, scale-invariant, independent of local units, and accessible to any civilization capable of observing atomic spectra. The construction outlines the possibility of a universal communication language grounded in the fundamental properties of matter and in the geometry of the positive half-line.
Vasil Tsanov (Tue,) studied this question.