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

This work treats two quantities usually taken as inputs — the number of spatial dimensions and the scale of the vacuum energy — as things to be derived from a single complex field, a “spectral substrate, ” under one assumption: exact scale invariance. Every result is tagged by its epistemic status (proved, computed, argued, or exploratory), and the central claims are kept strictly separate from the lower-grade material, so the reader always knows what rests on a theorem and what on a model. The proved/computed core: topological stability of matter forces a complex field with at least two spatial axes; a third appears once the field propagates along its spectral direction, fixing the spectral dimension at three in the ultraviolet and — via a closed-form heat-kernel theorem — across the entire scale-invariant window. The same scale invariance fixes the coherence scale as the geometric mean of the Planck and de Sitter cutoffs, ℓₛ = (ℏG/c³Λ) ^ (1/4) ≈ 0. 04 mm, and excludes the catastrophic vacuum-energy functional — a conditional resolution of the cosmological-constant problem. Reflection positivity is proved, and the continued Euclidean direction is the condensate-activated phase fiber, so matter, masses, and time switch on together in one symmetry breaking. Explicitly exploratory, lower-grade sectors then ask what such a substrate could host: a ghost-free emergent graviton with all corrections Planck-suppressed (the framework predicts its own laboratory invisibility), a non-abelian matter sector forced once the matter scale is generated dynamically, a gapped vector-like emergent fermion, and meV-scale Majorana-neutrino signatures tied to the dark-energy scale. The chirality of Standard-Model matter is located as a single sharp open question — not claimed solved. Falsifiable handle: the dark-energy equation of state obeys w (z) ≥ −1 with no phantom crossing; a confirmed phantom branch would falsify the dark-energy account. Two further meV-scale tests — neutrinoless double-beta decay and the neutrino-mass sum — are tied to the same scale. This is a self-archived preprint and an ongoing research program, not a peer-reviewed result. The dimensional and dark-energy results rest on proofs and validated numerical checks; the gravitational and matter sectors are consistency arguments and model computations, flagged as such throughout.