NSBH joining frequencies gate — 27 September 2026

The frozen specification was committed as f18bc77 before numerical native comparison. It transcribes Thompson et al. (2020), Table I literally. The scalar reference uses high-precision arithmetic and the independent scalar branch factors; no LAL source or binary output is used as an equation oracle. The native helper is called by the production amplitude path in both backends.

The frozen grid has 30 controlled and 180 composed inputs, covers every class and the exact \(f_{\rm tide}=f_{\rm RD}\) boundary, and supplies rather than derives \(f_{\rm RD}\). All paper-defined frequencies in all 210 cases passed the 70-versus-90-digit scalar stability limit of \(10^{-30}\) absolute. Torch and JAX CPU FP64 each had a maximum qualified joining-frequency relative error of \(4.371\times10^{-16}\), below the frozen \(2\times10^{-13}\) limit. This qualifies the high/no-torus, high/torus, and low/torus rows, plus low/no-torus \(\widetilde f_1\). Table I gives no \(\widetilde f_2\) for either low row, so the native value there has no paper-literal comparator.

The low/no-torus \(\widetilde f_0\) is a measured exception. Table I prints \([(q-1)\widetilde f_{\rm RD}+\epsilon_{\rm ins}f_{\rm tide}]/q\); the native carrier applies one more outer \(\epsilon_{\rm ins}\) to this entire center. Its maximum relative mismatch against the printed formula was \(0.1505\) in both backends on the frozen grid. The existing compatibility adapter is unchanged. This row and the four-row joining component remain conditional, rather than paper-qualified.

The next amplitude corrections also have source-variant blockers. The same paper's Eq. (26) prints \(\gamma'_1=1-0.5\Lambda-0.25\Lambda^2\) for \(\Lambda\le1\) and \(1.25\) for \(\Lambda>1\). The native small-\(\Lambda\) factor has plus \(0.5\Lambda\) and multiplies the PhenomC \(\gamma_1\). At \(\Lambda=1\), the printed small-branch value is \(0.25\) while the native factor is \(1.25\); the paper does not provide an unambiguous source for this adapter. Eq. (29) prints \((A/2)\omega^-_{x_3,d_3}\) for \(\Lambda>1\), with \(A/2=b_0=0.81248\); the native helper uses \(b_0\omega^+_{x_3,d_3}\). The opposite window sign is material away from its center: for \(x=(f_{\rm tide}-\widetilde f_{\rm RD})/\widetilde f_{\rm RD}=x_3+d_3/4\), the printed value is \(0.09684999\) and the native value is \(0.71563001\) (absolute difference \(0.61878002\)). Neither correction is included in this gate or retuned to binary output.

On len (Linux x86-64 CPU), from /home/xangma/tmp/diffgw-nsbh-joins-20260927, the command JAX_ENABLE_X64=1 PYTHONPATH=src python -m pytest -q -s tests/test_nsbh_joining_frequencies.py tests/test_nsbh_branch_amplitude_fits.py tests/test_nsbh_branch_reference.py tests/test_nsbh_torus_reference.py tests/test_nsbh_disruption_reference.py tests/test_nsbh_clean_amplitude.py tests/test_clean_room_nsbh.py tests/parity/test_nsbh_parity.py passed 104/104 in 79.39 s with no skips or expected failures. The log is pytest.log in that scratch directory. Environment: Python 3.12.3, Torch 2.14.0+cpu, JAX 0.11.2, NumPy 2.5.3, mpmath 1.3.0, pytest 9.1.1. The joining test was subsequently renamed to tests/test_nsbh_joining_frequencies_math_reference.py for the strict CI gate; the command above records the name used for this run. Remnant/QNM, Eq. (26)/(29) variants, amplitude assembly, phase, and the whole waveform remain provisional.