Operator-Algebraic Quantum Uncertainty and Bell Sharpness

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The geometric R1–R34 programme combines Gaussian heat control, three-torus Fourier geometry, a solid-torus Dirichlet disk and Möbius holonomy. It constructs exact zero-variance commuting relational observables alongside nonzero local noncommuting uncertainties, and studies their Gaussian spectral dynamics, energy constraints, and Bell protection. Subsequent geometry-free R35–R36 theorems establish the general continuous-spectrum Laplace fingerprint, optimal fidelity bounds from stabilizer means, sharp local–relational variance transfer, and exact nonselection and statewise leakage counterexamples. Earlier historical propositions are retained with their original hypotheses; a structural source audit is not represented as an independent verification of every historical proof. The R37 continuation addresses three outstanding operational interpretations directly. First, the natural half-density transport of a fully coupled configuration-space metric factorizes into local A/B maps precisely when the mixed second derivatives of its logarithmic volume density vanish. For a prescribed operational tensor product, a bounded nonproduct density distortion nevertheless retains Bell fidelity at least sech2 δ, giving explicit quantitative entanglement and one-way distillation thresholds. A smooth conformally mixed ten-dimensional four-structure metric realizes this bound with genuinely nonfactorizing volume. Second, the permanent Gaussian generator Λ(I −e−A/Λ) is proved not to be a local finite-order differential Hamiltonian on an infinite-dimensional compact elliptic model, despite its exact heat-average representation. Third, an ancilla-based six-syndrome nondemolition instrument is constructed explicitly, and a product-input entanglement witness proves that coherent Bell parity … [excerpt; full original in PDF]

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