Topological Origin of 𝐽ₑff: From Spectral Determinant to Exploratory Closed Form

This paper demonstrates that \(J_{\mathrm{eff}}\) is not a free parameter requiring “fitting”, but a rigid reading of the \(\mathcal{T}^{64}\) topological structure on the lepton-mass cross-section. \(J_{\mathrm{eff}}\) is defined via the regularised spectral determinant of the Laplace–Beltrami operator restricted to the lepton homology classes \(\mathcal{H}_{12}\oplus\mathcal{H}_{24}\oplus\mathcal{H}_{36}\). Under dimensional-averaging and symmetry-allocation hypotheses an exploratory closed form is obtained:\[J_{\mathrm{eff}}^{\mathrm{(avg)}}=N^{1/64}\cdot\phi^{-35/64}=[\alpha^{2}(1-\alpha)]^{-1/64}\cdot\phi^{-35/64}\approx1.517,\]where \(N=[\alpha^{2}(1-\alpha)]^{-1}\) is the global normalisation factor. The complete spectral-zeta computation (exact projection onto the lepton homology classes followed by evaluation of \(\zeta'(0)\)) remains future work. After adoption of present spectral-weight estimates the lepton-mass readings agree with experiment (relative deviation \(<0.01\%\)). Current status: path identified / exploratory closed form locked; complete spectral computation pending. --- 本文证明 \(J_{\mathrm{eff}}\) 不是需要被“拟合”的自由参数,而是 \(\mathcal{T}^{64}\) 拓扑结构在轻子质量截面上的刚性读数。\(J_{\mathrm{eff}}\) 通过限制在轻子同调类 \(\mathcal{H}_{12}\oplus\mathcal{H}_{24}\oplus\mathcal{H}_{36}\) 上的拉普拉斯–贝尔特拉米算子的正则化谱行列式定义。在量纲平均与对称性分配假设下得到探索性闭式:\[J_{\mathrm{eff}}^{\mathrm{(平均)}}=N^{1/64}\cdot\phi^{-35/64}=[\alpha^{2}(1-\alpha)]^{-1/64}\cdot\phi^{-35/64}\approx1.517,\]其中 \(N=[\alpha^{2}(1-\alpha)]^{-1}\) 为全域归一化因子。完整的谱zeta计算(轻子同调类精确投影 + \(\zeta'(0)\) 求值)仍为后续工作。在采用当前谱权重估计值后,轻子质量读数与实验值一致(相对偏差 \(<0.01\%\))。当前状态:路径已识别 / 探索性闭式已锁定,完整谱计算待完成。