Why Amplitudes Add Quantum probability as the bookkeeping of a finite boundary

Quantum mechanics assigns each alternative a complex amplitude, adds the amplitudes, and squares the result. Textbooks postulate this. Here it is built from one physical situation: a two-sided cut with finitely many states, holding paired one-sided records of alternatives it cannot fully distinguish, obliged to keep the ledger's books. The pairing is the purchase: block weights are bilinear in the shares, which is where the square enters, priced as a premise. The rest is forced. Indistinguishable alternatives must fuse. The cut's bits carry only a sign, so phase rides posting order; requiring the one fusion that constitutes recording to leave totals untouched has no real solution and forces the imaginary unit, while route fusions interfere freely and cancel exactly over the clock's cycle. The price is a seven-premise inventory, two entries carrying proofs of undecidability, the sharpest being that the golden ratio makes balanced books provably blind to which alternatives a boundary holds.