IMRPhenomHM six-mode projection gate — 2026-09-27

The bounded hm.projection.london-eq1.v1 gate checks the projection of six public positive-frequency HM mode carriers into complex plus/cross. It does not validate the intrinsic carriers or qualify the complete waveform.

Source derivation: London et al., Eq. (1) sets h=h+−ih×=Σh_lm {}₋₂Y_lm; García-Quirós et al., Sec. II, Eqs. (2), (4)–(9) fix the exp(-2πift) Fourier kernel and positive-frequency negative-m mode-pair formula. SymPy Wigner matrices generate the harmonics; mpmath evaluates the coefficients at 50 digits. The native public carrier adapter and azimuth sign are stated in the component specification.

The frozen specification preceded the reference/test implementation in commit b8a8e5f, based on integration commit 7ecf8ab. The test reads public return_unprojected_modes and return_modes values from both frontends. At three frequencies and four angles (face-on, edge-on, and two generic), it checks each of six complex plus/cross contributions and their sum. It also checks the pole selection and odd-ℓ sign, azimuth/orbital-phase/time covariance, body-exchange (-1)^m parity, and symmetric odd-mode zero.

Host len; working copy /home/xangma/tmp/diffgw-hm-projection-gate-20260927; Python /home/xangma/.cache/diffgw-taylorf2-hm-venv/bin/python; Torch 2.14.0+cpu, JAX 0.11.2, SymPy 1.14.0, mpmath 1.3.0, pytest 9.1.1.

PYTHONPATH=src JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu \
  /home/xangma/.cache/diffgw-taylorf2-hm-venv/bin/python -m pytest -q \
  tests/test_imrphenomhm_projection_math_reference.py \
  --require-math-reference --reference-device=cpu -ra

The initial focused run passed 13/13 in 44.77 s. The 50-digit mpmath rerun passed 13/13 in 43.49 s. The strict gate passed 13/13, with zero skips, expected failures, or deselections, in 40.89 s; its output is projection-pytest-strict.log in the working copy. No red projection defect was observed, so no native implementation was changed. These results do not establish mode calibration, QNM accuracy, negative-frequency reconstruction, LAL mode agreement, or physical accuracy.