Pair Observable Spectral Dataset (σ_pair, δ_pair) across Heis₃(ℤ/qℤ) — Cosmochrony Programme

This dataset provides numerical results for the spectral pair observable σ_{pair}(n) and the associated statistics $\delta_{pair}$, computed on finite Heisenberg groups Heis₃(ℤ/qℤ) for a range of prime moduli q. The data are generated from a reproducible pipeline based on breadth-first shell decompositions and block-capacity measurements. For each conjugate pair (c, q−c), the dataset records depth-resolved estimates of $\sigma_{pair}(n)$, empirical distributions of $\sigma_{pair}$, and fitting-window parameters (n₀, n₁) used for asymptotic analysis. The dataset originates from the pair-level spectral campaign introduced in O25 and has been extended to support subsequent developments, including refined window selection, scaling analyses in n₁(q), and representation-theoretic interpretations of $\sigma_{pair}$. It is intended to provide a stable empirical basis for testing structural properties such as monotonicity, inter-pair concentration, and finite-size effects. All files are provided in NumPy .npz format and include metadata describing generation parameters (q, sampling size, BFS coverage, block structure) to ensure full reproducibility.本数据集提供了谱对可观测量（spectral pair observable）$sigma_{pair}(n)$与关联统计量$delta_{pair}$的数值结果，该结果针对一系列素数模$q$，在有限海森堡群（finite Heisenberg group）$ ext{Heis}_3(mathbb{Z}/qmathbb{Z})$上计算所得。 该数据集由基于广度优先壳层分解与块容量测量的可复现计算流程生成。 针对每一组共轭对$(c, q-c)$，本数据集记录了$sigma_{pair}(n)$的深度解析估计值、$sigma_{pair}$的经验分布，以及用于渐近分析的拟合窗口参数$(n_0, n_1)$。 本数据集源自O25提出的谱对级研究项目（pair-level spectral campaign），并已针对后续研究进展完成扩展，涵盖精细化窗口选择、$n_1(q)$的尺度分析，以及$sigma_{pair}$的表示论阐释。其旨在为检验单调性、对间集中度与有限尺寸效应等结构性质提供稳定的经验基础。 所有文件均以NumPy .npz格式提供，内含描述生成参数（$q$、采样规模、广度优先覆盖度、块结构）的元数据，可确保结果完全可复现。