Why Amplitudes Add: A finite boundary derivation of the addition rule

A detector can record an arrival without recording its route. Suppose its pairing rule sees only the retained record and admits every self-pair. It must then admit every pair of alternatives with that record. A product weight and a common unit complex clock turn this pair sum into the squared magnitude of an amplitude sum. Amplitudes add within a record; weights add between orthogonal records. A conserving pair separated by one tick selects a quarter-turn clock. For partial records, positive pair laws are exactly the overlaps of unit record vectors. Four phase settings recover these overlaps, and a three-record compatibility condition constrains their simultaneous realization. Complete linear output maps preserve total weight exactly when they are isometries. A two-output apparatus connects record overlap to normalized fringes and shows why ignoring an existing route record leaves its interference suppressed. Physical applications require the stated record states, pairing law, and apparatus transformation.