The sin²θ_W Two-Loop Test — 3/13 Is Not the Limit: Two-loop overshoots 3/13. The perturbative series targets the measured value, not the integer ratio. sin²θ_W = 0.23133, miss 0.048%.

The sin²θ_W Two-Loop Test — 3/13 Is Not the Limit: Two-loop overshoots 3/13. The perturbative series targets the measured value, not the integer ratio. sin²θ_W = 0.23133, miss 0.048%. This paper is part of the HOWL research archive—a collection of physics papers exploring integer fraction derivations across multiple domains using exact arithmetic and automated comparison. Abstract The weak mixing angle sin²θ_W is one of the most precisely measured quantities in particle physics (0.23122 at the Z mass). At the GUT scale, the SU(5) tree-level value is 3/8 = 0.375. One-loop running with the Cabibbo Doublet betas predicts sin²θ_W = 0.22845, undershooting by 1.20% (PHYS-27). The value 3/13 = 0.23077 — the ratio of the number of fermion generations (3) to the absolute value of the modified SU(2) beta numerator (13) — sits between the one-loop prediction and measured, differing from measured by only 0.20%. This paper tests whether the two-loop running converges toward 3/13. It does not. The two-loop prediction with the full SM+VL b_ij matrix gives sin²θ_W = 0.23133, which OVERSHOOTS both 3/13 and the measured value. The ordering is: one-loop (0.22845) < 3/13 (0.23077) < measured (0.23122) < two-loop (0.23133). The perturbative series crosses 3/13 between one-loop and two-loop, then continues past measured. The two-loop miss from measured is 0.048% — the most precise sin²θ_W prediction in the series, but an overshoot, not an undershoot. 3/13 is not the perturbative limit. The convergence target is the measured value. Falsification Criteria All papers in this archive are subject to falsification through direct comparison to published experimental measurements. Each derived value is tested against independent data with explicit PASS/FAIL criteria. Any derived value that fails its comparison is documented and published alongside the successes. Research Context This archive documents an ongoing research program in integer fraction physics. The methodology is: derive values from gauge group integers using exact fraction arithmetic, compare to published measurements, and document all results including failures. The archive spans multiple physics domains connected through the soliton boundary framework described in the constituent papers. Package Contents manuscript.md: The complete derivation and supporting analysis. README.md: Navigation, dependencies, and citation (Registry: HOWL-PHYS-34-2026). Dependencies: HOWL-PHYS-1-2026, HOWL-PHYS-13-2026, HOWL-PHYS-27-2026, HOWL-PHYS-28-2026, HOWL-PHYS-30-2026 Motto: Derive. Compare. Publish.Status: Complete