IMRPhenomHM six-mode pinned-binary holdout — 27 September 2026

The maintained tests/test_imrphenomhm_lal_modes_binary.py compares the complex, unprojected (2,2), (2,1), (3,3), (3,2), (4,4), and (4,3) HM/D modes from Torch and JAX with the public LALSuite 7.26.15 binary API. All six tests (three systems × two backends), covering all 36 mode/backend/system comparisons, pass the criteria frozen before the holdout systems were run. The test is included in the CI maintained binary-oracle step with DIFFGW_REQUIRE_LAL_BINARY=1.

Source and fixed comparison contract

London et al., Eqs. (1)–(9) give the six-mode decomposition and frequency/phase constructions; García-Quirós et al., Sec. II, Eqs. (2), (4)–(9) state the Fourier sign and positive-frequency mode-pair convention used by the native projection specification. The oracle call uses only the documented public SimInspiralChooseFDModes and SphHarmFrequencySeriesGetMode API. No LAL source or numeric coefficient was copied into the test.

For this public IMRPhenomHM call, the binary returns all twelve ±m labels. Its positive-m complex series have nonzero samples at negative Fourier frequency and zero samples at positive frequency. An exploratory system outside the holdout (60, 20 M_sun, spins 0.5, -0.3, f_ref=31 Hz) established the single fixed adapter

\[ h^{\mathrm{native}}_{\ell m}(f) \quad\longleftrightarrow\quad (-i)^m\,\overline{h^{\mathrm{LAL}}_{\ell m}(-f)},\qquad f>0. \]

The conjugation and factor are an empirical public-API convention inference, not claimed as a derivation of LAL's internal orbital-phase origin from the paper. They were frozen before any holdout was evaluated. All systems use LAL phiRef=0, native phic=0, tc=0, inclination 0, aligned spins, default native D carrier, and no per-system or per-mode amplitude, constant-phase, or time alignment. Input masses are solar masses converted with lal.MSUN_SI; distances are Mpc converted with lal.PC_SI. LAL uses deltaF=1 Hz, f_min=20 Hz, f_max=512 Hz; both native backends and the comparison sample 20–400 Hz inclusive at 1 Hz spacing in CPU FP64. The unnormalized complex values retain the absolute distance and mode scale.

The three holdouts were declared in the test before their first binary run:

System	(m1,m2) in solar masses	(chi1,chi2)	f_ref (Hz)	Distance (Mpc)
Moderate mass ratio, mixed spins	(38,27)	(0.15,-0.25)	29	530
High mass ratio, negative primary spin	(58,14)	(-0.45,0.27)	33	760
Positive spins	(70,35)	(0.61,0.11)	23	850

For each mode, the predeclared bounds are complex relative L2 = ||native−reference||₂/||reference||₂ ≤ 10⁻³ over all 381 bins, and maximum complex pointwise relative residual ≤ 10⁻² where the reference amplitude is at least 1% of its per-mode peak. No residual is phase-aligned or magnitude-normalized before subtraction.

Observed complex residuals

Each pair below is relative L2 / maximum supported-bin pointwise residual; the latter evaluates 209–381 bins depending on system and mode. Values are rounded from the complete log.

System	Mode	Torch	JAX
Moderate	22	7.72e-14 / 5.04e-13	3.27e-14 / 1.81e-13
Moderate	21	3.07e-7 / 8.75e-7	3.07e-7 / 8.75e-7
Moderate	33	4.24e-7 / 1.93e-6	4.24e-7 / 1.93e-6
Moderate	32	9.91e-7 / 2.75e-6	9.91e-7 / 2.75e-6
Moderate	44	1.53e-7 / 6.73e-7	1.53e-7 / 6.73e-7
Moderate	43	2.05e-7 / 8.20e-7	2.05e-7 / 8.20e-7
High mass ratio	22	6.05e-13 / 3.86e-12	3.97e-13 / 2.51e-12
High mass ratio	21	2.56e-6 / 2.34e-5	2.56e-6 / 2.34e-5
High mass ratio	33	1.85e-6 / 2.20e-5	1.85e-6 / 2.20e-5
High mass ratio	32	8.71e-7 / 4.40e-6	8.71e-7 / 4.40e-6
High mass ratio	44	6.70e-7 / 6.25e-6	6.70e-7 / 6.25e-6
High mass ratio	43	2.39e-6 / 1.47e-5	2.39e-6 / 1.47e-5
Positive spins	22	2.71e-6 / 1.29e-12	2.71e-6 / 2.35e-14
Positive spins	21	3.80e-5 / 9.43e-6	3.80e-5 / 9.43e-6
Positive spins	33	9.46e-7 / 1.01e-5	9.46e-7 / 1.01e-5
Positive spins	32	8.44e-7 / 2.56e-6	8.44e-7 / 2.56e-6
Positive spins	44	8.44e-7 / 3.18e-6	8.44e-7 / 3.18e-6
Positive spins	43	1.78e-6 / 1.16e-5	1.78e-6 / 1.16e-5

In the positive-spin system, native 22 is exactly zero in the final 14 bins (387–400 Hz) and native 21 is exactly zero in the final 50 bins (351–400 Hz), while the LAL binary returns small nonzero tails. Their reference amplitudes at the first omitted bins are respectively 3.51e-6 and 6.35e-5 of the mode peak, below the frozen 1%-peak pointwise support. Those bins remain included in the full-band L2, explaining the larger 22/21 L2 values. All other system/mode comparisons have zero such bins. This is a real cutoff difference, not a phase or scale fit; the acceptance bounds were not changed after observing it.

Reproduce and scope

The run used host len, cwd /home/xangma/tmp/diffgw-hm-sixmode-holdout-20260927, Python 3.12.3, LALSuite 7.26.15, NumPy 2.5.3, SciPy 1.18.1, Torch 2.14.0+cpu, JAX/JAXlib 0.11.2, and pytest 9.1.1:

cd /home/xangma/tmp/diffgw-hm-sixmode-holdout-20260927
env JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu DIFFGW_REQUIRE_LAL_BINARY=1 \
  OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 PYTHONPATH=src:tests \
  /home/xangma/.cache/diffgw-taylorf2-hm-venv/bin/python -m pytest -q -s \
  tests/test_imrphenomhm_lal_modes_binary.py -ra

The full second-run log is hm_sixmode_binary.log in that cwd; it reports 6 passed in 16.56 s and every per-mode residual, support count, and native-zero count. The first run, before adding diagnostic-only output, is retained there as hm_sixmode_binary_initial.log (6 passed in 16.99 s). LALSuite is pinned so CI repeats this exact binary oracle; the len package index had DNS errors during a latest-version query, so no claim of package-index currentness is made.

This result supports sampled implementation agreement with the public same-model binary for the named HM/D contract. It does not establish the independently sourced fit coefficients, derivatives, all spin/mass/frequency limits, negative-frequency reconstruction, polarization projection under arbitrary orientation, or agreement with numerical relativity. The separate equation and projection receipts cover bounded pieces of those questions.