From Ledger to Born: The Square is Forced

Born's rule is the gate through which Hilbert-space amplitude becomes laboratory frequency. In standard quantum mechanics it is empirically secure but introduced as an operational probability postulate. Gleason's theorem, Busch/POVM extensions, decision-theoretic derivations, and envariance each identify important parts of the structure. What they do not by themselves supply is a single substrate object that is being counted before probability is introduced. This paper gives the QTT answer: the counted object is a finite-capacity, address-indexed, real-J ledger. Conditional on QTT axioms A1–A7 and the laboratory condition of a calibrated repeated preparation, the Born rule is derived as a theorem. The proof is deliberately modular and triangulates three independent routes. Route I (exchangeability): The de Finetti / Hewitt–Savage theorem gives a frequency limit, with a fixed Born number only after conditioning on a single ergodic preparation sector. The directing-measure structure is illustrated with explicit examples of the calibrated and uncontrolled-mixture cases. Route II (capacity symmetry): The per-tick capacity functional on real-J address amplitudes is shown to be additive, gauge-invariant, J-basis-invariant, and regular. These properties uniquely force the address weight μ(w) = ||ρ(w)||²_J. Any non-quadratic power q ≠ 2 violates J-basis invariance; this is demonstrated by an explicit two-address calculation. Route III (Gleason/Busch at fixed address): A5/A6 instrument equivalence physically discharges the non-contextuality assumption of Gleason's theorem. The effect-Gleason theorem of Busch and Caves–Fuchs–Manne–Renes then fixes the conditional outcome law at each address. Combined, the three routes yield the resolved Born formula p(α | M) = Σ_w ||ρ(w)||²_J ⟨ψ_w | Π_α ψ_w⟩_J = Tr_J(ρ_eff Π_α), with ρ_eff = Σ_w ||ρ(w)||²_J |ψ_w⟩⟨ψ_w|_J. In a single laboratory-pure projection sector this reduces to the familiar position rule P(X) = ∫_X ||Ψ(x,τ)||²_J d³x. No fitted constant, collapse rate, rational-agent axiom, environment cut, or two-clock cos(π/8) input appears in the proof. The shocking point is also the narrow one: grant finite additive real-J address capacity and fixed-address instrument equivalence, and the square is no longer optional. Version 1.2 adds an explicit illustration of the directing measure (Remark 4) immediately after Lemma 1, making the role of the calibration condition concrete with worked single-preparation and uncontrolled-mixture cases. All substantive content from v1.1 is preserved. Status: conditional theorem inside a QTT-axiomatic framework, not a claim that the axioms are established physics. The paper claims that, conditional on A1–A7 and a calibrated repeated-preparation ensemble, Born's rule is forced rather than fitted. Falsifiers are listed in Section 10: calibration failure, frequency failure, non-quadratic capacity, fixed-address contextuality, or any reproducible Born-rule violation under controlled conditions.