Theory Z VII: Vacuum Correlations, Constitutive Coherence, and Bell Correlations From the Microscopic Constitutive Vacuum to Dichotomic Correlations and CHSH

Abstract

Theory Z describes the physical vacuum as a microscopic dynamical substrate governedby a constitutive functional F[Φ]. The preceding stages of the program developed the vac-uum configuration space, constitutive fluctuations, localized excitations, and a framework inwhich effective internal modes may arise from localized solutions and associated Dirac-typeoperators. Theory Z VII extends this construction to the problem of spatially separatedcorrelations and asks whether Bell-type correlations can be formulated as a consequence ofthe constitutive vacuum rather than introduced as an independent postulate.The central distinction of the present work is between ordinary constitutive field corre-lations and correlations of an internal localized-mode sector. A scalar two-point functionG2(x, y) can generate long-range correlations, but by itself does not provide the dichotomicobservables required for a CHSH test. Following Bell’s locality criterion [1] and the experi-mentally realizable CHSH formulation [? ], the Bell sector must instead contain well-definedbinary observables A, B ∈ {−1, +1} together with a joint probability distribution.We therefore introduce a tensorial internal correlation sector associated with localizedconstitutive modes. For a rotationally invariant two-mode sector, the leading effective inter-action isHeffAB = E0 + J(r)σA ·σB.For J(r) > 0, the lowest-energy internal state is the antisymmetric two-mode sector. Itscorrelation tensor isDσiAσjBE= −δij ,up to a finite-distance constitutive coherence factor C(r). For linear polarization analyz-ers represented byn(α) = (cos 2α,sin 2α, 0),the resulting correlation isE(α, β) = −C(r) cos[2(α − β)] .The associated dichotomic probability distribution is1 P(A, B|α, β) = 14[1 − AB C(r) cos(2(α − β))] , A, B = ±1.The one-detector marginals are independent of the remote analyzer, so that the con-struction satisfies the operational no-signalling condition. For the standard optimal CHSHsettings,|SCHSH| = 2√2 C(r),and a Bell violation requiresC(r) >1√2.In the coherent limit C(r) → 1, the maximal quantum value|SCHSH| = 2√2is recovered, consistent with the quantum upper bound for dichotomic correlations.The principal microscopic question is whether the effective interaction J(r) and its tensorstructure can be obtained from the constitutive functional itself. We formulate this require-ment through the projection of the constitutive Hessian evaluated on a joint two-excitationconfiguration,Jij (r) = Zd3x d3y ψ†A(x)δ2FδΦ(x)δΦ(y)ΦABψB(y).The required isotropic Bell sector is then characterized byJij (r) = J(r)δij , J(r) > 0.This formulation separates the effective derivation from the remaining microscopic closureproblem: the existence and detailed structure of the localized internal doublet, as well asthe derivation of J(r) from a fully specified microscopic F[Φ], must be established ratherthan assumed. The resulting framework therefore turns Bell correlations into a quantitativespectral, variational, and statistical test of the constitutive vacuum.