This work studies the spatial dimensionality and the vacuum-energy scale of a single complex field — a spectral substrate — treating both as quantities to be derived rather than inserted. From a single assumption, exact scale invariance (argued to be the output of a correctly applied maximum-entropy principle rather than an independent postulate), the construction obtains: a third spatial axis as an ultraviolet spectral-dimension property, constant across the scale-invariant window; the coherence/dark-energy scale as the geometric mean of the Planck and de Sitter cutoffs, lₛ = (ħG/c³Λ) ^ (1/4), the non-trivial content being its location at the logarithmic midpoint; and the exclusion of the catastrophic vacuum-energy functional by a scale-equivariance argument — a conditional resolution of the cosmological-constant problem. Reflection positivity of the free sector gives the Euclidean-to-Lorentzian signature flip. A separate, explicitly exploratory part examines the gravitational sector within the bulk-entanglement (Cao–Carroll) framework, where the substrate supplies as derived or measured properties several inputs that programme assumes, yielding a massless, ghost-free emergent spin-2 mode compatible with unimodular gravity. A falsifiable prediction follows: a Yukawa-type correction to the Newtonian potential at range λ ≈ 44 μm, fixed parameter-free by the dark-energy scale, with a bounded amplitude testable by short-range torsion-balance experiments. Results are tagged throughout as proved, computed, assumed, or open. The gravitational and matter sectors are of explicitly lower evidential grade than the dimensional and dark-energy results. The fermion mass hierarchy and the emergence of a non-abelian gauge structure are stated as open problems; the substrate explains that fermions are massive (soliton self-energy) but not the mass values, a limitation it shares with the Standard Model.
Karol Frank (2026) studied this question.