The Born Rule from Finite Address-Capacity Counting: A Conditional QTT Representation and Uniqueness Theorem

Why does quantum probability use the square rather than another power? This paper separates that question into a source theorem and two explicit laboratory bridges. For a finite address block, every nonnegative, regular, orthogonally additive capacity that is invariant under real-\(J\) gauge and basis rotations is forced into the quadratic form \[ c(w)=\lVert\rho(w)\rVert_J^2. \] Within that declared symmetry class, the exponent is not fitted and no higher-power alternative survives. The laboratory Born law follows when normalized source capacity is identified with long-run address occupancy, \(\mu(w)=c(w)\), and the fixed-address outcome functional is normalized, effect-additive, and instrument-equivalent. The resolved law is then \[ p(\alpha\mid M) =\sum_w \lVert\rho(w)\rVert_J^2 \langle\psi_w\vert\Pi_\alpha\psi_w\rangle_J =\operatorname{Tr}_J(\rho_{\rm eff}\Pi_\alpha), \] with \[ \rho_{\rm eff} =\sum_w \lVert\rho(w)\rVert_J^2 \vert\psi_w\rangle\!\langle\psi_w\vert_J. \] For a laboratory-pure position projection this becomes \[ \mathbb P_J(X) =\int_X\lVert\Psi(\mathbf x,\tau)\rVert_J^2\,d^3x. \] The logical ledger is deliberately split. The quadratic capacity theorem is closed in its declared finite-block class. The capacity-frequency identification and the Gleason-Busch trace representation are printed as conditional bridges with their premises exposed. Standard Born-rule data confirm the recovered laboratory law; experiments that vary address access or break the declared bridge conditions are needed to select the QTT ontology independently. Release: Version 2.1 Stable concept DOI: 10.5281/zenodo.20118242 Main book: Quantum Traction Theory: Main Book v10.01 Website: quantumtraction.org Public reader routes: Corpus Tree family Derivation Atlas: Born node Observatory: Born-rule row Lexicon: Born rule Two Universes Status ledger: QTT-BORN-QUADRATIC-CAPACITY-CLOSED-DECLARED-CLASS QTT-BORN-CAPACITY-FREQUENCY-BRIDGE-CONDITIONAL QTT-BORN-TRACE-REPRESENTATION-CONDITIONAL STANDARD-BORN-RULE-RECOVERED QTT-ONTOLOGY-INDEPENDENT-EMPIRICAL-SELECTION-NOT-CLAIMED