The Geometric Ceiling of Quantum Correlations
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 an extra postulate: it follows from the equivariance continuity equation
∂_τρ + ∇^I(ρ G^IJ ∂_J S) = 0,
the same conservation law that yields ∇^μ T_μν = 0 in the emergent spacetime description. 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}),
where ℓ_F is the Fisher correlation length set by Q₀. For macroscopic separations this is indistinguishable from exact commutativity. Together, these conditions give the hypotheses of Tsirelson’s commuting-operator theorem, from which
CHSH ≤ 2√2 + O(e^{−|x|/ℓ_F})
follows without any tensor-product assumption. PR-box correlations (S = 4) therefore lie outside the emergent commuting algebra, not because of a separate axiom, but because they are not realized by the projected operator structure of the theory.
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 not a claim of super-quantum signaling. It is a structural measure of the residual gap between exact and effective commutativity at finite resolution. In standard quantum mechanics, exact commutativity at spacelike separation is an algebraic axiom. In sPNP, it is delivered by the causal support of the projection kernel at the same finite coarse-graining scale Q₀ that also fixes Gaussoherence and the kinematic limit.
This is also the right place to read Information Causality. The principle is not added as a separate axiom; it is the information-theoretic shadow of the same geometry. The Q₀-bounded projection channel has finite Fisher-information capacity, so it limits how much of the upstairs relational nonlocality can survive coarse-graining into the emergent spacetime algebra. Exact Tsirelson correlations belong strictly to the ideal projected kinematic sector of the theory.
