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) ||²J. 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) = Σw ||ρ (w) ||²J ⟨ψw | Π_α ψw⟩J = TrJ (ρₑff Π_α), with ρₑff = Σw ||ρ (w) ||²J |ψw⟩⟨ψw|J. In a single laboratory-pure projection sector this reduces to the familiar position rule P (X) = ∫X ||Ψ (x, τ) ||²J 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.