2.2. Operation gating (R4, R5, R6)🔗

R4 — Operations are gated by kind (the additive law). Arithmetic that keeps a quantity within its kind is well-typed; arithmetic across incompatible kinds is a compile-time type error, not a runtime check. Width + Width = Width type-checks; Width + Height does not — and, decisively, neither does Torque + Energy, even though torque and energy share a dimension. Matching dimension is not licence to add.

R5 — The interaction algebra is a partial, typed, ternary product. Some kinds combine across kinds to yield a third: torque times plane angle is energy, force times length is work. The combination is partial and curated — most pairs combine to nothing (fuel-consumption times rainfall is a category error, not a number) — and it carries a division dual, with multiplication and division proved inverse. This multiplicative combination is distinct from R4's additive, same-kind gate.

R6 — Operator availability is gated by the measurement scale, monotonically. Which operations even exist between two properties is fixed by their scale type, \mathrm{nominal} \sqsubset \mathrm{ordinal} \sqsubset \mathrm{interval} \sqsubset \mathrm{ratio}: a nominal property admits only =, a ratio property admits \times and \div, and a richer scale licenses every operation a poorer one does. This gate is orthogonal to kind — kind says which properties may combine, scale says which operators are defined at all. This requirement is now realized on the real standard: ISO 80000-5 lists thermodynamic temperature (ratio-scale) and Celsius temperature (interval-scale) at the same dimension Θ, so a ratio of Celsius temperatures is undefined where a ratio of thermodynamic temperatures is not — the scale type, not the dimension, separates them (see the ISO 80000-5 chapter). IEC 80000-6 repeats the pattern on electromagnetism: electric potential is gauge-dependent (fixed only up to an additive reference), hence interval-scale, while electric potential difference, of the same dimension V, is ratio-scale (see the IEC 80000-6 chapter).