The Born Rule from S² Geometry: Three Independent Routes to P = |⟨φ|ψ⟩|², with Geometric Resolution of Schrödinger's Cat, the Three-Polariser Paradox, the Tsirelson Bound, and the Cosmological Decoherence Floor

The Born rule — that the probability of obtaining outcome |phi> in a measurement of a quantum system in state |psi> is P = ||² — is one of the most empirically well-confirmed and conceptually contested statements in physics. Existing derivation attempts (Gleason’s theorem; Deutsch–Wallace decision theory; Zurek envariance; many-worlds branch counting) each establish it under their own assumptions, but none has achieved universal acceptance. We show that in the Three Time Dimensions (3+3) spacetime framework (de Haan 2026, book DOI 10. 5281/zenodo. 19633127), in which the third time dimension t₃ is compactified as a discrete two-sphere S² with 2¹52 Planck-area cells, the Born rule emerges through three independent geometric routes that all converge on the same result P = ||². Route 1 (Malus’s Law on S²): the wavefunction is literally a direction on the t₃ S²; the Bloch sphere IS this S²; Malus’s 1809 law P = cos² (theta) is identical to the Born rule for bosons, with SU (2) double cover giving cos² (theta/2) for fermions. Route 2 (Gleason + S² non-contextuality): Gleason’s 1957 theorem uniquely selects ||² given non-contextuality, which in (3+3) is a consequence of the S² Riemannian metric rather than an independent postulate. Route 3 (Lévy measure + t₂ isotropy): the t₂ expansion perturbs all t₃ directions isotropically; the unique continuous binary-outcome probability measure is the Lévy / Haar-invariant spherical measure, which gives P (+nA) = cos² (theta/2). The three routes use distinct mathematical tools (projection, Hilbert-space theorem, stochastic analysis) and distinct physical inputs (geometry, Riemannian metric, dynamical isotropy). They must converge if the geometric identification is correct; they would diverge if it were wrong. Malus’s 1809 law is the Born rule avant la lettre. The empirical optical law taught to undergraduates as a fact is, under this reading, a direct geometric consequence of photon polarisations being directions on S² — a statement about projection geometry that predates quantum mechanics by a century. The same geometric identification simultaneously resolves four foundational puzzles: (1) Schrödinger’s cat is never in superposition (the cat’s ~10²9 t₃ modes are entrained by environmental t₂ perturbation within femtoseconds of the first atomic interaction) ; (2) the three-polariser paradox dissolves (non-commuting measurements are non-commuting rotations on S²) ; (3) the Bell/Tsirelson bound 2sqrt (2) is a geometric consequence of S² having dimension exactly 2 — empirical confirmation by loophole-free Bell tests (Hensen, Giustina, Shalm 2015) confirms the S² identification; (4) the cosmological decoherence floor taucosm = 1/H₀ ~ 14. 5 Gyr is a structural upper bound on quantum coherence set by the t₂ cosmic expansion — distinguishing the framework from many-worlds interpretations. Three supporting elegance claims establish the framework’s quantum-foundational coherence: (A) the complex-valuedness of psi in C follows from t₂ being a rotation (the imaginary unit i is the generator of the t₂ precession rotation) ; (B) the Schrödinger equation i hbar dpsi/dt = H psi is the infinitesimal form of t₂ precession rotating the phase of the t₃ configuration; (C) the canonical commutation x, p = i hbar follows from the foam lattice structure. The paper is complementary to the companion Quantum Computing preprint DOI 10. 5281/zenodo. 19651560 which uses the Bloch-sphere-as-S² identification for applied hardware-level proposals. The Born Rule paper provides the foundational underpinning for why that identification is correct at the level of quantum-mechanical first principles. The paper is honest about what remains open. Four specific items are explicitly acknowledged (§11): (1) the framework does not yet have a 6D Lagrangian formulation from which the Born rule would fall out of path-integral quantisation; (2) the uniqueness of the S² identification is only partially addressed — higher-dimensional spheres are ruled out by the Tsirelson bound, but tori and Calabi–Yau alternatives are not examined; (3) the path-integral reformulation of quantum mechanics within (3+3) has not been constructed; (4) detailed comparison with QBism, relational quantum mechanics, and consistent histories is outside the paper’s scope. Falsifiability. Six routes: Tsirelson-bound overshoot/undershoot beyond 2sqrt (2) + measurement precision; indefinite quantum coherence beyond cosmological timescales; real-valued QM experimental confirmation (already ruled out by Renou et al. 2021, reconfirmed by independent tests) ; detection of genuine contextuality; detection of discrete S² cell structure at near-Planck scales. The Born rule, in (3+3), is not a mysterious postulate; it is Malus’s Law on a compact sphere that nobody knew was there.