Dark Energy from Gaussian Integer Degeneracy: Exact Spectral Quantisation, Lattice Phase Structure, and an Arithmetic Bridge to Wave Control

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.摘要 我们提出了一个现象学框架，其中动力学暗能量源于连续膨胀的弗里德曼-勒梅特-罗伯逊-沃尔克（Friedmann–Lemaître–Robertson–Walker, FLRW）度规与由高斯整数格定义的离散算术真空之间的几何应变。可允许的真空态为满足$r_2(s) > 0$的整数$s$，其中$r_2(s) = #{(x,y)inmathbb{Z}^2 : x^2+y^2=s}$为雅可比二平方表示数（Jacobi two-square representation count）。该框架的精确数学核心为系数序列$r_2(s)$及其θ级数生成函数$sum r_2(s)q^s = heta_3(q)^2 - 1$；宇宙学态方程与格点相变行为是将该算术结构耦合至膨胀背景后的模型依赖结果。在主导子扇区${r_2=4, 8, 16, 32}$中，能级呈对数量子化形式：$ln(x_k) = ln(16) - 2kcdotln(2)$，该结果已验证至机器精度。$r_2=16$的态位于临界点$x=1$处，由此得到条件恒等式$w = -1 + varepsiloncdotln(2)$。通过对108组参数组合的实证分析，铰链律指数满足$eta approx 3varepsilon$。一个四维欧几里得格点实现展现出稳定的缺陷形成相，以及在$Phi=58$、应变为$2/7$处的精确丢番图真空锁定（Diophantine vacuum lock）。该生成函数将暗能量、波控能力与平环面塞尔伯格迹（Selberg trace）统一为作用于同一序列的解析运算。该框架预言了解冻型暗能量$w_a > 0$，这与当前暗能量光谱仪（Dark Energy Spectroscopic Instrument, DESI）的数据集组合存在结构上的张力。