Interference Conservation Forces the Quadratic Norm

Why does a conserved wave quantity depend on the square of amplitude? For a sum-and-difference mixer, conservation of an additive positive power of a norm forces the power to be two, the common gain to be 1/sqrt(2), and the norm to come from an inner product. An empty input and a dark output fix the constants; arbitrary input pairs fix the geometry. A third run allows the two output gains to be independently unknown: conservation forces them to agree. For imperfect conservation, both protocols give the same sharp exclusion bound on nonquadratic powers. A strictly increasing port contribution can replace the power premise; full conservation again forces a square. The triangle inequality also follows from the remaining size axioms and conservation. Register symmetries select equal coordinate weights. Applying these results to a device requires independently calibrated amplitudes and port contributions; interpreting normalized squared sizes as probabilities requires an outcome premise.