Non-Inertial Response of Correlations: From Scalar Bell Observables to an Extended Correlation Tensor

A standard Bell observable is a scalar correlation associated with a selected pair of local measurement directions. We formulate it as a projection of the correlation block of a complete two-particle tensor and distinguish two fundamentally different angular sectors. The central result is a reversal of the sign multiplying the angular cosine law: the photon sector has a positiveprefactor, whereas the fermionic singlet sector has a negative prefactor. For coincident calibrated settings, the Bell observable is positive for photons and negative for fermions. This sign difference can be used for experimental identification of the two types of objects. For photons, this result follows from averaging two projection amplitudes over the complete non-inertial phase interval; the fermionic sign follows from the negative exchange holonomy of the complete phase--momentum sector. Mapping the phase directions to the physical axes of linear polarizers produces the corresponding double-angle dependence. Accordingly, the two cases are distinguished by their correlation tensors rather than by different definitions of the Bell observable. The photon Stokes correlation tensor has positive linear-polarization components and a negative circular-olarization component, whereas the fermionic singlet is described by an isotropic negative correlation tensor. A motion-dependent extended tensor and its generally frequency-dependent non-inertial susceptibility are introduced. A phase-synchronous experiment with mechanical and equivalent optical modulation is proposed to separate calibrated basis rotations from a residual response of the correlation structure. The same phase construction yields binary joint probabilities and recovers the Tsirelson bound, with opposite signed optimal CHSH combinations for the photon and fermionic sectors.