Dark Energy from Gaussian Integer Degeneracy: Exact Spectral Quantisation, Lattice Phase Structure, and an Arithmetic Bridge to Wave Control
Description
Abstract We introduce a phenomenological framework in which dynamical dark energy emerges from geometric strain between a continuously expanding Friedmann–Lemaître–Robertson–Walker (FLRW) metric and a discrete arithmetic vacuum defined by the Gaussian integer lattice. The admissible vacuum states are integers s satisfying r₂(s) > 0, where r₂(s) = #{(x,y)∈ℤ² : x²+y²=s} is the Jacobi two-square representation count. The exact mathematical core is the coefficient sequence r₂(s) and its theta-series generating function ∑r₂(s)qˢ = θ₃(q)²−1; the cosmological equation of state and lattice phase behaviour are model-dependent consequences of coupling this arithmetic structure to an expanding background. Within the dominant subsector {r₂=4, 8, 16, 32}, the energy levels are logarithmically quantised: ln(x_k) = ln(16) − 2k•ln(2), verified to machine precision. The state r₂=16 sits at the critical point x=1, giving the conditional identity w = −1 + ε•ln(2). The hinge-law exponent satisfies β≊3ε empirically over 108 parameter combinations. A 4D Euclidean lattice realisation exhibits a robust defect-forming phase and an exact Diophantine vacuum lock at Φ=58 with strain 2/7. The generating function unifies dark energy, wave-control capacity, and the flat-torus Selberg trace as analytic operations on the same sequence. The framework predicts thawing dark energy wₐ > 0, in structural tension with current DESI dataset combinations.