From Ledger to Born: The Square is Forced

Probability as finite address-capacity accounting Version: 2.0 Concept DOI: 10.5281/zenodo.20118242 Author: Ali Attar Website: quantumtraction.org Main book anchor: Quantum Traction Theory: Main Book v10.01 This paper derives the Born rule as a finite address-capacity theorem. The laboratory square is not introduced as an operational postulate; it is the unique positive additive real- \(J\)-basis-invariant address-capacity law for repeated calibrated preparation. The resolved ledger probability is \[ p(\alpha|M) = \sum_w \|\rho(w)\|_J^2 \langle\psi_w|\Pi_\alpha\psi_w\rangle_J = \operatorname{Tr}_J(\rho_{\mathrm{eff}}\Pi_\alpha), \] where \[ \rho_{\mathrm{eff}} = \sum_w \|\rho(w)\|_J^2 |\psi_w\rangle\langle\psi_w|_J . \] In the laboratory-pure position projection this becomes \[ \mathbb P_J(X)=\int_X \|\Psi(\mathbf x,\tau)\|_J^2\,d^3x . \] Status: QTT-BORN-SQUARE-FINITE-ADDRESS-CAPACITY-CLOSED STANDARD-BORN-RULE-LABORATORY-SHADOW-RECOVERED