The Tsirelson Bound as Measurement Geometry of the Existence Equation

Evidence Paper VI of the Existence Equation series Standard quantum mechanics calculates the Tsirelson bound Smax = 2√2 through Hilbert space and operator norms. This paper explains it: 2√2 is the product of two structural factors, each with a transparent origin, requiring no Hilbert space, no C*-algebra, and no Born rule as a postulate. The argument begins inside the ED framework [1]. The condensation term α|Ψ|²Ψ projects a continuous deviation field onto discrete binary outcomes (±1). This is not an assumption — it is the mechanism by which Axiom 1.1 (discreteness of events) is dynamically enforced. The Born rule P = |Ψ|² emerges as a time-averaged phase statistic of the deviation field, demonstrated in [1] on a 256³ lattice over 30,000 steps. Given the Born rule, a conserved field satisfying ΨA + ΨB = 0 produces the correlation function E(a, b) = −cosθab. This is a standard result — but its origin is not: it is derived, not postulated. Now the geometry. Alice chooses between two settings a, a′. Bob chooses between two unit vectors b, b′. Define u = b′ − b and v = b′ + b. Then: Factor Value Origin Constraint |u|² + |v|² = 4 Bob has 2 unit vectors: |b|² + |b′|² = 2 Diagonal max(|u| + |v|) = 2√2 Alice has 2 settings → 2D optimization space, diagonal = √2 Bound Smax = 2 × √2 2 from Bob’s unit vectors × √2 from Alice’s 2D diagonal The decomposition is verified numerically to 10−16 precision across 2D, 3D, and 10D measurement spaces. The bound is dimension-independent — it reflects the structure of binary measurement, not of physical space. The decisive result is the ceiling/saturation separation: Configuration |S| |u| + |v| Ceiling available? Saturated? Optimal (0°, 90°, 45°, 135°) 2.8284 2.8284 Yes Yes Classical-like (0°, 90°, 0°, 90°) 2.0000 2.8284 Yes No The Classical row is the key datum. The geometric ceiling 2√2 is fully available — |u| + |v| = 2√2 — yet |S| = 2. The geometric space is identical. The constraint circle is the same. What differs is the correlation function. Separable hidden variables produce piecewise-linear correlations that cannot simultaneously align a with u and a′ with −v. Only cosθ can saturate the available geometry. The ED connection chain is therefore: α|Ψ|²Ψ → discrete ±1 outcomes → Born rule (derived, not postulated) → conserved field ΨA + ΨB = 0 → cosθ correlation → saturates geometric ceiling 2√2 The Tsirelson bound is not a property of quantum mechanics specifically. It is the geometry of binary discrete measurement. Any theory that produces discrete binary outcomes and allows correlations via a shared continuous field will produce Smax = 2√2. The Bell violation is not spooky. It is geometric. It measures the diagonal of the measurement lattice. References [1] J.-A. Shin, "The Existence Equation: The Grammar of Persistence," Zenodo (2026). doi: 10.5281/zenodo.18639316 All simulation code and raw data are publicly available at https://github.com/Galileo-leo/existence-equation.