Exploratory Sweep Report on Jₑff: Exclusion of Spectral-Geometric Candidate Paths for the Translation Factor in the Lepton Mass Formula and a Test of the Primitivity Hypothesis — A Stage Report on the Origin of Jₑff within the 𝒯⁶⁴ Framework of Yuanxian Theory

Abstract

Yuanxian Theory Core Paper No. 7, “Reading Paths of the 24 Fundamental Physical Constants” (10.5281/zenodo.22239740), has completed the structural writing of the lepton mass formula\[m_\ell = m_0 \cdot F(k) \cdot J_{\mathrm{eff}},\]where \(J_{\mathrm{eff}}\) is the translation factor that maps the spectral-geometric information density of \(\mathcal{T}^{64}\) onto the four-dimensional lepton-mass cross-section. That paper explicitly states that the origin of \(J_{\mathrm{eff}}\) remains to be proved: its numerical value has been locked (\(\approx 1.342\)), yet its topological provenance has not been established. The present note records, as a matter of fact, five candidate derived paths that have been examined and excluded—the 64th-root of a ratio of spectral determinants, a normalisation-factor correction, a fully regularised determinant, a ratio of curvature-spectrum eigenvalues, and a ratio of two spectral-geometric readings. None of these paths reproduces the locked value 1.342. **Honest statements of this note**:1. The search space consisting of the five candidate paths is an artificial restriction; it does not exhaust all possible derived paths;2. The numerical value 1.342 of \(J_{\mathrm{eff}}\) originates from a backward calibration of the lepton mass formula against experimental data and is therefore a posterior locked value;3. Consequently, the sweep method at the present stage can neither prove that “no derived path exists” nor prove that “\(J_{\mathrm{eff}}\) is a primitive term”. **Current status**: In the five spectral-geometric derived paths that have been tried, \(J_{\mathrm{eff}}\) has not been recovered. It may be a primitive term, or it may be the result of a still-undiscovered correct derived path. Primitivity has **neither been proved nor been disproved**. The contribution of this note is to exclude several derived candidates and thereby narrow the search space for subsequent work. --- 元宪理论第七号论文《24个基本物理常数的读出路径》(10.5281/zenodo.22239740)已完成了轻子质量公式的结构写出:\[m_\ell = m_0 \cdot F(k) \cdot J_{\mathrm{eff}},\]其中 \(J_{\mathrm{eff}}\) 是连接 \(\mathcal{T}^{64}\) 谱几何信息密度与四维轻子质量截面的翻译因子。该论文明确指出:\(J_{\mathrm{eff}}\) 来源待证——其数值已被锁定(\(\approx 1.342\)),但它的拓扑起源尚未被证明。 本文如实记录了已被排除的五条候选派生路径——谱行列式比值开方、归一化因子修正、完整正则化、曲率谱特征值比值、谱几何读数之比。所有路径的计算结果均无法复现 \(J_{\mathrm{eff}}\) 的锁定值 1.342。 **本文的诚实声明**:1. 五条候选路径的搜索空间是人为设定的,并未穷尽所有可能的派生路径;2. \(J_{\mathrm{eff}}\) 的数值 1.342 来自轻子质量公式对实验值的反向校准,属于后验锁定值;3. 因此,清扫法在当前阶段不能证明“不存在任何派生路径”,也不能证明“\(J_{\mathrm{eff}}\) 是原生项”。 **当前状态**:在已尝试的五条谱几何派生路径中,\(J_{\mathrm{eff}}\) 均未能被复现。它可能是原生项,也可能是尚未找到的正确派生路径的结果。原生性**尚未被证明**,也**未被证伪**。本文的贡献在于明确排除了若干派生候选,为后续研究缩小了搜索范围。