What the paper does. SQG asks whether two numbers usually put in by hand — the number of spatial dimensions and the vacuum-energy scale — can instead be derived from one structure, a single complex field on a spectral axis. Throughout, results are tagged proved, computed, or assumed, keeping the load-bearing core separate from exploratory extensions. The proved core. Topological stability of matter forces the field to be complex and the base to have at least two dimensions (U (1) defects have codimension exactly two) ; minimality fixes two. A spectral kinetic term activates a third axis, and the heat-kernel spectral dimension is three in the ultraviolet. One assumption — exact scale invariance, argued to be corrected maximum entropy rather than an independent postulate — then fixes the spectral profile (uniqueness lemma), keeps the dimension at three across the scale-invariant window (closed-form Bessel heat-kernel theorem, confirmed numerically), and fixes the coherence scale as the geometric mean of the Planck and de Sitter cutoffs, ℓₛ = (ħG/c³Λ) ^ (1/4) ≈ 0. 04 mm, at their logarithmic midpoint. The same invariance excludes the catastrophic (Planck-valued) vacuum-energy functional — a conditional resolution of the cosmological-constant problem. Reflection positivity is proved for the free sector and at leading order in 1/N, giving the Euclidean-to-Lorentzian signature flip; the continued time direction is the field’s own phase, the unique direction whose kinetic weight the condensate creates, so that matter, masses, and time emerge together from one symmetry breaking. The exploratory part (lower grade). Within the bulk-entanglement (entanglement-equilibrium) framework the substrate supplies several inputs that programme assumes and yields a ghost-free emergent spin-2 mode — a collective oscillation of the emergent geometry, not a fundamental quantum — compatible with unimodular gravity. All substrate corrections to the gravitational law are Planck-suppressed (α ∼ (ℓPl/ℓₛ) ² ∼ 10⁻⁶¹), so the framework predicts its own laboratory invisibility, its phenomenology residing in the dark sector. A non-abelian internal structure is forced once the matter scale is generated dynamically (Coleman–Gross), and a structural obstruction is identified to obtaining emergent fermions from the parity-even, short-range-entangled bosonic substrate. Status and intent. This is independent, single-author theoretical work. The explicit standard is internal consistency and honest grading: the dimensional and dark-energy results rest on proved statements and validated computations; the gravitational and matter results are conditional and exploratory, as flagged throughout. The work is shared for scrutiny and critique, not presented as established physics.
Karol Frank (2026) studied this question.