Spectral Quantum Gravity: Emergent Three-Dimensionality and the Dark-Energy Scale from a Scale-Invariant Spectral Substrate

This work treats two numbers usually inserted by hand — the number of spatial dimensions and the magnitude of the vacuum energy — as quantities to be derived, and shows how far a single principle can take you. From one assumption (exact scale invariance, argued to be the output of a correctly applied maximum-entropy principle) applied to a single complex field — a spectral substrate — the framework derives, jointly and with an explicit standard of proof: — Three spatial dimensions. Topological stability of matter forces a complex field and exactly two base axes (U (1) defects have codimension two) ; the spectral axis supplies a third, and the spectral dimension is three in the ultraviolet and constant across the entire scale-invariant window — established in closed form via a Bessel heat-kernel theorem, not merely numerically. — The dark-energy scale. The coherence scale sits at the logarithmic midpoint of the Planck and de Sitter cutoffs, ℓₛ = (ℏG/c³Λ) ^1/4 ≈ 0. 04 mm — a parameter-free location, not a fit. Read in reverse, the 122-order smallness of the cosmological constant becomes a dimensional-transmutation exponent rather than a fine-tuning. — A conditional resolution of the cosmological-constant problem. Scale invariance together with the inversion symmetry of the cutoff window excludes the catastrophic vacuum-energy functional and selects the observed value. — The origin of time. Reflection positivity is proved (free sector, and at leading order), so the Euclidean theory continues to a Lorentzian one; the continued direction is the field’s own phase, activated by the condensate — so matter, masses, and time switch on together in a single symmetry breaking. The framework makes one sharp, presently testable prediction: a true cosmological constant, w = −1, with no time evolution — directly in the path of DESI and supernova programmes. What distinguishes the paper is its discipline as much as its results: every statement is tagged proved, computed, or assumed, and a single table separates what the principle outputs from what it hosts. The exploratory sectors — a ghost-free emergent spin-2 mode with all substrate corrections to the gravitational law Planck-suppressed (αg ∼ 10⁻⁶¹, predicting its own laboratory invisibility), and a forced non-abelian matter structure — are presented openly as lower-grade. The economy is the point: one field, one principle, the dimension and the dark-energy scale tied together.