MemoryWaveform direct-class dominant-pair memory

Definition field	Catalog value
Model ID	MemoryWaveform22Direct
Reference composition	memory-dominant-22-direct-class/v1
Scope	Only Torch/JAX MemoryWaveform.evaluate_22 with supplied physical-strain h22, symmetric mirror, and relative memory offset on selected CPU float64 grids.
Domain and modes	time; (2, 0), (4, 0)
Native backends	torch, jax
Scoped verification claims	1

Option	Default	Native choices	Reference-only choices
call_route	evaluate_22	evaluate_22	None
mirror	equatorial_symmetry	equatorial_symmetry	None
grid_rule	gradient_trapezoid	gradient_trapezoid	None

Scope and status

MemoryWaveform22Direct, reference memory-dominant-22-direct-class/v1, covers only MemoryWaveform.evaluate_22 in Torch and JAX. It consumes a supplied oscillatory physical-strain \(h_{22}(t)\) and assumes its equatorial-symmetry mirror \(h_{2,-2}=\bar h_{22}\). It returns relative \(h_{20}^{\rm mem}\) and the full dominant-pair plus/cross pattern. The legacy class call (forward or __call__) is a distinct polarization-only diagnostic proxy and has no catalog claim. See the component derivation.

Inputs, conventions, and options

times is a one-dimensional, strictly increasing grid in SI seconds with at least two samples; h22 is complex dimensionless strain on that grid, at explicit positive distance_m in meters. Inclination is in radians from the orbital angular momentum. The mode convention is \(h=h_+-ih_\times=\sum h_{\ell m}{}_{-2}Y_{\ell m}\); \(h_\times^{\rm mem}=0\). For times shaped (N,), h22 has shape (..., N). distance_m must broadcast to the leading batch shape (...): a scalar applies to every waveform, and a (B,) distance array supplies one distance per waveform for h22 shaped (B, N), including when B == N. Distances have no time axis; per-sample distance arrays are unsupported. h20 retains h22's shape. The first returned memory sample is zero, fixing an integration constant only. The only choices are call_route=evaluate_22, mirror=equatorial_symmetry, and grid_rule=gradient_trapezoid; all run natively in both frontends. No external fit or asset is needed. Autodiff is checked for selected supplied-mode coefficients and inclination at fixed grid.

Mathematics and components

Favata Sec. 6.2, Eq. (21) gives, in SI seconds, for the symmetric dominant pair:

\[ h_+^{\rm mem}=\frac{R}{192\pi c}\sin^2\iota(17+\cos^2\iota) \int|\dot h_{22}|^2dt,\qquad h_\times^{\rm mem}=0. \]

Projection yields \(h_{20}^{\rm mem}=R\sqrt{5/(6\pi)}\int|\dot h_{22}|^2dt/(7c)\). The full plus pattern reconstructed from \(h_{20}\) retains the \(\ell=4,m=0\) contribution. Talbot Eqs. (1)–(9) supplies the strain-mode convention. The numerical rule takes endpoint secants and unequal-grid quadratic-interpolant interior derivatives, squares their complex magnitudes, and accumulates adjacent trapezoids.

Component ID/version	Source	Variant	Status	Scalar reference	Torch	JAX	Gate
memory.dominant_22.v1	Favata Eq. (21); Talbot Eqs. (1)–(9)	published dominant pair	executable	h20_coefficient, plus_from_flux_integral	compute_memory_h20_22, memory_polarizations_22	same	tests/test_memory_dominant_22_math_reference.py
memory.sampled_flux.v1	Project unequal-grid rule	sampled adapter	executable	sampled_h20	MemoryWaveform.evaluate_22	same	same

Implementation mapping

The independent diffgw_reference.memory_dominant_22 module uses mpmath and does not call either frontend. Torch uses torch.gradient on actual time coordinates. JAX explicitly evaluates the same unequal-grid three-point derivative, including endpoint secants; both integrate trapezoids and project the full plus pattern. evaluate_22 returns (h_plus, h_cross, h20) and does not use the constructor's legacy scale_factor.

Evidence and limitations

The strict CPU float64 gate checks the 70-digit angular projection, exact linear-mode values and derivatives on a four-sample nonuniform grid at four inclinations, a two-sample minimum-grid case, and one nonlinear complex polynomial on that grid against an independent scalar sampled evaluator. It checks Torch and JAX values, zero cross strain, selected derivatives, and JAX JIT at relative tolerance 5e-13 and strain absolute tolerance 1e-35. This is sampled implementation agreement, not a continuous-domain numerical error bound. The separate strict expected failure records the legacy proxy's coefficient mismatch.

Physical accuracy of the supplied \(h_{22}\), earlier memory offset, general mode-pair coupling, precession, nonlinear iteration, other class routes, same-model external comparison, and numerical-relativity validation remain unqualified. In particular, no LAL comparison or physical validation is claimed for this direct-class slice.

Reproduce the checks

With pytest, mpmath, CPU Torch and JAX available, run PYTHONPATH=src python -m pytest -q tests/test_memory_dominant_22_math_reference.py tests/test_memory_diagnostic.py. The latter includes one strict expected failure. Catalog and generated-doc checks run with tests/test_waveform_definitions.py and python scripts/generate_waveform_library_table.py --check. A change to the source law, mode convention, or numerical rule requires a new version and requalification.