Information, Predictability, and the Special Status of FourDimensional Spacetime

Abstract. This note replaces a non-universal count of fields and counterterms by an operational, regulatorcontrolled test built from quantum Fisher information (QFI) and mutual information (MI). A matched family of freemassive scalar theories is constructed in spacetime dimensions d=3,4,5,6,7 using the same field content, periodichypercubic regulator, physical box size, dimensionless mass parameter, and separated-block partition protocol. Thevacuum QFI with respect to the dimensionless mass parameter is derived exactly on the lattice. Its continuumdensity has an ultraviolet threshold at spacetime dimension d=5: it is finite for d<5 and logarithmically divergent atd=5. This makes d=4 the highest dimension with finite mass-QFI density in this benchmark, but not a uniquemaximizer of the QFI itself. The mutual-information calculation is performed from Gaussian covariance matricesfor separated blocks; it is finite and regulator-stable in the numerical refinement shown, and it decreases withdimension for the chosen geometry. A fixed equal-weight composite score built from QFI and separated MItherefore does not select d=4. This negative result is retained as a falsification test rather than repaired by addingnew functionals. Relative entropy and modular data are consequently not introduced into the scalar score. Finally,an information-distance proxy constructed from two-site MI is shown to be monotone with lattice separation, whileWang et al.’s holographic distance bound is explicitly treated as an independent target criterion rather than appliedto the non-holographic free-scalar benchmark. The result is a reproducible research program: information geometrysupplies a controlled notion of distinguishability and relational structure, while any claim that it fundamentallyselects d=4 remains open.