2.9. Unit classification (R13)🔗

R13 — Unit classification needs a third category beyond base and derived: scale-spanning units. The SI sorts units into base (kilogram, metre, second, ampere, kelvin, candela, mole) and derived (everything built from them). Finkelstein and Whitehead (Eur. J. Phys. 46 (2025) 035701) argue this dichotomy is insufficient: of the seven base units only four — the kilogram, metre, second, and ampere — are genuinely dimensionally independent, while the kelvin (the dimension of energy, via the Boltzmann constant), the candela (the dimension of power, via the luminous efficacy), and the mole (a dimensionless number, via the Avogadro constant) are not — yet each is kept a distinct coherent unit because it carries a human-selected coefficient sized to span vast scales. They are scale-spanning units, a third category. R13 specifies that category, and its point is the unit-layer twin of R1: whether a unit is scale-spanning is not a function of its dimension alone. The dimension layer can detect that the candela and the mole are mechanically reducible (to power and to one), but it cannot detect the kelvin — this work keeps thermodynamic temperature an independent dimension (ISO 80000-5 needs it), so the kelvin's scale-spanning character is visible only through its coefficient, never through dimensional analysis. The ampere, by contrast, is dimensionally independent (it carries the electromagnetic generator), a genuine base unit — so the third category is non-empty and distinct from the first. In plain engineering terms: just as a kind carries more than its dimension, a unit's category carries more than its dimension; the candela and mole (ISO 80000-7) and the kelvin (ISO 80000-5) are where this lands on the standard, the ampere (IEC 80000-6) the base-unit foil. The category's nonlinear extreme is the level — the decibel of ISO 80000-8 and of EM practice (dBm, field levels) — and it is now construction rather than aspiration: LevelKind presents a ratio root kind through a logarithmic chart against a named reference, with the reference and the ISO 80000-1 power/root-power role carried as kind identity (dBm and dBW are distinct kinds), the level kind ordinal-as-a-kind so the certified addition is unprovable at it (L₁ + L₂, the classic domain error, is a type error), differences landing in a provably reference-free gain kind that composes by ordinary certified addition, and the role-independent energetic combination as the licensed way two sources meet. The kelvin is the linear member of R13's category; the decibel is its chart-bearing one, and it lives beside the kind — deliberately not in a unit system of pure rescalings, which cannot express even the affine °C, let alone a logarithm with a reference.