Dual-Path Topological Readout of J_eff — From the Exact Value at Step 10 to the Complete Structure at Step 15

This paper presents two topological readout paths for \\(J_{\\mathrm{eff}}\\): - **Path A**: phase point at step 10, \\(J_{\\mathrm{eff}}=\\tan(2\\pi\\times10/68)\\approx1.34189\\);- **Path B**: phase point at step 15, \\(J_{\\mathrm{eff}}=J_0\\cdot\\cos\\Delta\\phi/\\kappa\\approx1.34189\\). The two paths are complementary: Path A supplies the exact value via round-trip exchange (outbound 5 steps + return 5 steps) aligned with the minimal self-referential cycle \\(i^4=1\\); Path B supplies the topological structure at the three-generation fermion fixed point, including the explicit screening factor \\(\\kappa=64\\pi^2(1-1/612)\\). Both paths yield the identical numerical value with 0.007% deviation from the target 1.342, attributed to second-order projection blur. --- 本文给出 \\(J_{\\mathrm{eff}}\\) 的两条拓扑读出路径: - **路径 A**:第10步相位点,\\(J_{\\mathrm{eff}}=\\tan(2\\pi\\times10/68)\\approx1.34189\\);- **路径 B**:第15相位点,\\(J_{\\mathrm{eff}}=J_0\\cdot\\cos\\Delta\\phi/\\kappa\\approx1.34189\\)。 两条路径互补:路径 A 通过往返交换(去程5步 + 回程5步)并与最小自指循环 \\(i^4=1\\) 对齐给出精确值;路径 B 在三代费米子不动点给出拓扑结构,含显式屏蔽因子 \\(\\kappa=64\\pi^2(1-1/612)\\)。两条路径数值完全一致,与目标值1.342的偏差均为0.007%,归因于二阶投影模糊。