A Scale-Invariant Spectral Substrate: Emergent Three-Dimensionality and the Dark-Energy Scale from a Single Principle

This work investigates whether the number of spatial dimensions and the magnitude of the vacuum (dark-energy) density can be derived rather than inserted, within a single framework: a complex spectral field defined on an abstract frequency axis together with emergent space. Throughout, results are separated explicitly into what is proved, what is computed (reliable numerics with analytic support), and what is assumed. The unconditional results are: (i) two emergent spatial axes follow from the substrate’s two necessary conserved labels (frequency and U (1) charge) ; (ii) without dynamics along the spectral axis the emergent dimension is exactly two, so a third axis requires a spectral kinetic term; (iii) the resulting operator is proved three-dimensional in the ultraviolet via a heat-kernel / Seeley–DeWitt argument, making the theory a dimensional crossover (three-dimensional at short distance, two-dimensional at long distance) ; (iv) the maximum-entropy spectral density survives the kinetic term; and (v) the Euclidean-to-Lorentzian signature flip is established at the Gaussian level via Osterwalder–Schrader reflection positivity, which follows from the same positive spectral density that ensures the absence of ghosts. A single further assumption — exact scale invariance of the substrate, argued to be the output of a maximum-entropy principle applied with the unit-invariant (Jeffreys) measure appropriate to a scale-like quantity rather than an independent postulate — then yields several consequences from one root: the spectral profile is fixed; the coherence scale emerges as the geometric mean ℓₛ = √ (LPl · LH) (the cosmic see-saw, agreeing with the dark-energy length scale to a factor of order two) rather than being an input; three-dimensionality extends across the full Planck-to-Hubble range, so that the observable universe is three-dimensional; and the vacuum-energy functional is selected uniquely. The last point addresses the cosmological-constant problem: for a scale-invariant spectrum the catastrophic linear moment is excluded as non-scale-equivariant, and the unique scale-equivariant estimator (the logarithmic centre of mass) lands on the see-saw value ρDE ~ 1/ℓₛ⁴ rather than the Planck value, so the 10¹²⁰ discrepancy does not arise. The work is explicit about its limitations. The central results beyond the dimensional crossover are conditional on the single scale-invariance assumption; the dark-energy treatment is strong as a mechanism (why the Planck catastrophe is avoided) but matches the observed value only at the order-of-magnitude level. The principal open problem is stated precisely: exact scale invariance must break to give the world its masses, and the breaking generates a scale near ℓₛ; whether this breaking can be confined to the matter sector without renormalizing the spectral profile — a sharp decoupling condition — determines whether the three-dimensional regime survives. Further open problems include reflection positivity of the interacting theory, the uniqueness of the emergent time direction, radiative Lorentz universality (constrained by gamma-ray-burst timing), and a non-abelian completion indicated from several directions. This is an exploratory theoretical framework, not a completed theory. Its intended contribution is not priority on any single ingredient — emergent dimensionality, the ~85 μm dark-energy scale, and Planck–Hubble two-scale pictures all have precedents (e. g. Cao–Carroll–Michalakis; Dupays et al. ; the Swampland dark-dimension program; Padmanabhan) — but the joint derivation of the dimensional and dark-energy results from a single scale-invariance principle, without a compact extra dimension or a Casimir mechanism, and with ℓₛ as an output rather than an input. Keywords: emergent spacetime; spectral dimension; dark energy; cosmological constant problem; scale invariance; maximum entropy; reflection positivity; dimensional crossover