Closing the Tsirelson Loophole in Relativistic Quantum Field Theory: Split Inclusions, Hyperfinite Local Algebras, and the Effective Dimension of the Vacuum

MIP*=RE implies that correlations of commuting observables on an infinite-dimensional Hilbert space (C_qc) can lie at a finite distance from every limit of finite-dimensional tensor-product correlations (C_qa). Such correlations would let a finite experiment detect an actual infinity with a modest number of rounds. In earlier work this was identified as the only route that escapes the resource bounds on certifying Hilbert-space dimension. Here we ask whether relativistic quantum field theory allows Bell correlations in this gap. Under standard structural assumptions, it does not. Suppose the theory has the split property, which follows from the Buchholz–Wichmann nuclearity condition and holds for free fields. Then the correlations of local measurements in strictly spacelike separated regions are tensor-product correlations and lie in C_qa. If the local algebras are hyperfinite, the same holds for regions that are merely spacelike, with no gap between them. Buchholz, D'Antoni and Fredenhagen showed this hyperfiniteness under a scaling-limit assumption. We give a self-contained proof based on semidiscreteness. Alice's measurements are compressed through matrix algebras by unital completely positive maps, and the resulting correlations converge to the original ones. The same applies to the type II₁ observer algebra of a de Sitter static patch, constructed by Chandrasekaran, Longo, Penington and Witten. This algebra is a corner of a crossed product of such an algebra by ℝ, so it is again injective. Within these frameworks, local measurements in spacelike separated regions cannot realise the gap found by MIP*=RE. This excludes the fourth of the logical possibilities listed by Cabello, Quintino and Kleinmann. We then bound, from the state side, how much dimension a Bell test can certify. Compressing each party to the support of its ε-smoothed reduced state reproduces all correlations to within 2(√ε_A + √ε_B) + ε_A + ε_B. Since N rounds resolve correlations only to N^(-1/2), an experiment with N rounds certifies at most the smooth max-entropy of the local state at smoothing ε ~ 1/N. We compute this effective dimension exactly for the vacuum of a lattice free scalar field. The single-particle entanglement energies form two towers, one per block endpoint, with a spacing set by the mass, so only a handful of modes are active. The effective dimension follows an area law: it is at most 9 bits for a 128-site block at ε = 10⁻⁶, against 553 bits from naive counting. As ε decreases from 10⁻¹ to 10⁻¹², it grows only from 2.3 to 13.1 bits. A product bound over thermal modes shows that the growth is polylogarithmic in 1/ε for any regularised block. In the continuum, the corresponding object is the canonical splitting type I factor of Doplicher and Longo, whose vacuum entropy coincides with the reflected entropy. Evaluating its smooth max-entropy as a function of the splitting distance is left for future work. Together with the certification budget of the companion paper, the result gives a coherent picture. Physics uses infinite-dimensional Hilbert spaces and type III algebras. Yet, within these frameworks, the correlations that local measurements can produce are reproduced to any finite precision by finite-dimensional models. The size of these models is controlled by entanglement entropy. Code, data and figures: https://github.com/Ruqing1963/qft-tsirelson-loophole