Quantum Dust from the Curse of Dimensionality

Abstract

Why do unrelated approaches to quantum gravity nearly all find spacetime two-dimensional at the shortest scales? Each theory answers only within its own dynamics; we highlight a single kinematic route to the same value, one assuming no field equation and living in the geometry of the space of states alone. That route is concentration of measure on the Fubini-Study geometry of pure states, which forces the pairwise distances of a random sample to equalize as the dimension grows, so any finite sample collapses to an equidistant dust whose thresholded metric graph is the complete graph. Handed this dust, a diffusion probe reads it as two-dimensional in the large-sample limit, the value the running spectral dimension takes at the dust's single relaxation scale, a property of the measurement rather than the structure; this convergence on two is not, by itself, evidence that spacetime is two-dimensional. Whether a given two is such an artifact is governed by the Laplacian spectrum near zero, and whether that reading carries across an emergence map is the condition we call spectral faithfulness; a single relaxation scale encodes no spectral dimension that tells one structure from another. The collapse, the probe value, and the eigenvalue-density criterion are machine-checked in Lean 4 against Mathlib, resting on the standard Beta law of overlaps; a power-law tail of small eigenvalues reads a genuine dimension, a single scale above a gap reads two at its own clock, and a gapped two-scale band reads off the universal line. These classes are run on graph-Laplacian proxies, and whether a link-graph reading carries to the physical nonlocal operator is left open. The spectral test reads the eigenvalue density near zero and separates, on a given structure, a measurement artifact from a dimension the structure genuinely expresses.