Tsirelson Bound from sPNP Architecture

On the admissible kinematic/projected sector, Gaussoherence Π_Q₀ = e^{−Q₀²Δ_G} is a positivity- and trace-preserving heat-kernel coarse-graining, generated by the essentially self-adjoint Fisher–Laplace–Beltrami operator on the complete relational manifold Q_rel. Trace preservation is not a separate postulate, it follows from the equivariance continuity equation ∂_τρ + ∇^I(ρ G^IJ ∂_J S) = 0, the same condition that yields ∇^μT_μν = 0 in emergent spacetime. Every Gaussoherence-projected state is therefore a valid quantum density matrix. The causal kernel K(X,x;Q₀) enforces effective microcausality: projected observables at spacelike separation satisfy [O_A, O_B] = O(e^{−|x_A − x_B|/ℓ_F}), with ℓ_F the Fisher correlation length set by Q₀. For macroscopic separations this is indistinguishable from exact commutativity. Together these give the hypotheses of Tsirelson's commuting-operator theorem, from which CHSH ≤ 2√2 + O(e^{−|x|/ℓ_F}) follows without tensor-product assumptions. PR-box correlations (S = 4) lie outside this emergent commuting algebra projection structure, not by a separate axiom. Two scope restrictions apply. First, this argument holds in the kinematic regime; in the reflexive regime, where G_IJ[ρ] back-reacts on the generator, equivariance and trace-preservation remain open verification obligations. Second, the exponentially small correction to 2√2 is a distinctive sPNP prediction: at sub-ℓ F separations, the theory anticipates deviations from the exact Tsirelson ceiling that standard quantum mechanics does not predict. The Information Causality principle is the information-theoretic face of the same Q₀-bounded channel capacity that Step 2 encodes.

The Geometric Ceiling of Quantum Correlations

sPNP does not predict an experimental violation of quantum mechanics; rather, it derives the Tsirelson ceiling (S ≤ 2√2) as the ideal projected limit of a finite-resolution geometry. In standard quantum mechanics, this bound relies on the algebraic axiom of exact commutativity ([O_A, O_B] = 0) at spacelike separation. In sPNP, this commutativity is not assumed, it is delivered via the causal support of the projection kernel. Because physical projection occurs at a finite coarse-graining scale Q₀, the universe realizes these commuting-observable hypotheses through finite-resolution effective commutativity. Any deviation from 2√2 is not a prediction of super-quantum signaling, but a structural measure of the gap between exact and effective commutativity under this finite-resolution projection. Q₀ is not an ad hoc parameter; it is the same coarse-graining scale that already controls Gaussoherence, the projection kernel, and the kinematic limit. This derivation holds cleanly in the kinematic sector, which serves as the safe linear limit. In the state-dependent reflexive regime, where the Fisher metric back-reacts on the generator, full equivariance and trace preservation remain an explicitly open verification obligation. Thus, exact Tsirelson correlations belong strictly to the ideal, projected kinematic sector of the theory.