TaylorF2_HM leading eight-mode mathematical specification

Definition field	Catalog value
Model ID	TaylorF2_HM
Reference composition	taylorf2-hm-leading-carrier/v1
Scope	Eight leading nonspinning SPA mode amplitudes with inherited aligned-spin TaylorF2 phase and negative-m polarization projection.
Domain and modes	frequency; (2, 2), (2, 1), (3, 1), (3, 2), (3, 3), (4, 2), (4, 3), (4, 4)
Native backends	torch, jax
Scoped verification claims	0

Option	Default	Native choices	Reference-only choices
spin_order	7	-1, 0, 1, 2, 3, 4, 5, 6, 7, 8	None
f_isco_cutoff	true	false, true	None
smooth_cutoff	false	false, true	None

Scope and status

Specification: taylorf2-hm-leading-carrier/v1. This scalar SymPy/mpmath reference covers the native frequency-domain carrier's eight modes \((2,2),(2,1),(3,1),(3,2),(3,3),(4,2),(4,3),(4,4)\). It composes leading nonspinning mode amplitudes, the separately versioned aligned-spin TaylorF2 phase, and positive-frequency polarizations. The mode amplitudes are not full-PN or spin-corrected mode expansions. The mathematical domain is positive frequencies and positive component masses; slow inspiral is the physical regime in which the approximation is motivated.

Inputs, conventions, and options

The public masses and distance are in nominal solar masses and Mpc. At the reference boundary, \(M=(m_1+m_2)T_\odot\) and \(D=D_{\rm Mpc}\,\mathrm{Mpc}/c\) are seconds, \(\delta=(m_1-m_2)/(m_1+m_2)\), and \(\eta=(1-\delta^2)/4\). \(f\) and optional \(f_{\rm ref}\) are Hz; \(t_c\) is seconds; \(\iota\), \(\phi_c\), and observer azimuth \(\phi_0\) are radians. Spins \(\chi_1\), \(\chi_2\) are aligned dimensionless projections. The scalar kernel has no production imports or unit constants.

The stored positive-frequency mode is \(h_{\ell,-m}(f)\), and the Fourier carrier is \(e^{-i\Psi_{\ell m}}\). Set \(Y_{\ell m}={}_{-2}Y_{\ell m}(\iota,\phi_0)\) with \(Y_{22}=\sqrt{5/(64\pi)}(1+\cos\iota)^2e^{2i\phi_0}\). The scalar reference constructs these harmonics with Wigner \(d\) matrices, independently of the frontends' trigonometric polynomials. With the native positive-frequency convention,

\[ d_+={Y_{\ell,-m}+(-1)^\ell Y_{\ell m}^*\over2},\qquad d_\times={i\over2}\bigl(Y_{\ell,-m}-(-1)^\ell Y_{\ell m}^*\bigr),\qquad (h_+,h_\times)_{\ell m}=(d_+,d_\times)h_{\ell,-m}. \]

This identification is not inferred from frontend agreement. García-Quirós et al., Eqs. (2), (4)–(9) use \(h=h_+-ih_\times\) and the LAL Fourier sign \(\tilde h(f)=\int h(t)e^{-i2\pi ft}dt\), show that \(m>0\) modes are concentrated at negative frequency, and define the positive-frequency object \(\tilde h_{\ell,-m}=A_{\ell m}e^{-i\Phi_{\ell m}}\) before deriving the two projection factors above. The public mode dictionary retains positive \((\ell,m)\) labels for these paired polarization contributions; it does not expose raw positive-\(m\) Fourier modes.

The default native spin_order=7 selects SO3, SS4, SO5, SO6, SS6, SO7 in the aligned-spin option map. The scalar formulas can also select other named term sets, including the full set used by native spin_order=8. The direct complex comparison uses f_isco_cutoff=False, lambda1=lambda2=None; only the Schwarzschild hard cutoff threshold is additionally checked. Native Kerr ISCO, half-cosine taper, and tidal terms are outside this reference slice. Native hard-cutoff masks are piecewise and have no derivative claim at the boundary.

Mathematics and components

Let \(A_0(f)\) be the inherited restricted TaylorF2 face-on amplitude, \(v=(\pi Mf)^{1/3}\), and \(I(v)\) its aligned-spin intrinsic phase with the declared term set. Define \(q_m=2/m\) and

\[ \Psi_{22}^{(m)}=I\bigl((\pi Mq_m f)^{1/3}\bigr) +2\pi q_m f t_c-2\phi_c-\pi/4, \quad \Psi_{\ell m}={m\over2}\Psi_{22}^{(m)} +{3\pi\over4}\left(1-{m\over2}\right), \quad h_{\ell,-m}=\sqrt{16\pi/5}\,A_0(f)R_{\ell m}(v,\delta) e^{-i\Psi_{\ell m}}. \]

The phase adapter follows García-Quirós et al., Eqs. (29), (34): the low-frequency SPA relation is \(\Phi_{\ell m}\simeq(m/2)\Phi_{22}(2f/m)\) and its convention-dependent constant is \(3\pi(1-m/2)/4\). Consequently \(t_c\) remains a common \(e^{-i2\pi f t_c}\) shift while \(\phi_c\mapsto\phi_c+\Delta\phi_c\) multiplies the native carrier by \(e^{im\Delta\phi_c}\). Eqs. (30)–(32) of that paper add a frequency-dependent phase for complex higher-PN amplitudes; omitting it is justified only for this declared leading real-amplitude slice.

When f_ref is supplied, subtract \(I((\pi Mq_m f_{\rm ref})^{1/3})\) inside \(\Psi_{22}^{(m)}\) and replace \(-\pi/4\) by \(+3\pi/4\). This is the inherited project phase origin and scales the reference frequency by \(q_m\). The normalized real leading-amplitude ratios are

Mode	\(R_{\ell m}\)
22	\(1\)
21	\(\delta v/(3\,2^{1/3})\)
31	\(\delta v/(12\sqrt7\,2^{5/6})\)
32	\(\sqrt{5/7}(1-3\eta)v^2/3\)
33	\((3/4)\sqrt{5/7}(3/2)^{5/6}\delta v\)
42	\(\sqrt5(1-3\eta)v^2/63\)
43	\((3/4)\sqrt{3/35}(3/2)^{1/6}\delta(1-2\eta)v^3\)
44	\((8/9)\sqrt{5/7}(1-3\eta)v^2\)

These ratios use the observed-mode \(v\). In the independent kernel, the printed \(\mathcal H_{\ell m}(x)\) of Eq. (21) is evaluated at \(x_m=(2/m)^{2/3}v^2\) and multiplied by \((m/2)^{2/3}\) after dividing Eq. (30) by its \((2,2)\) Newtonian amplitude. A fixed unit-complex factor for each mode converts the source's complex coefficient to the declared real-amplitude carrier convention. Mezzasoma and Yunes, Eqs. (20)–(21i), (23)–(30) provide an all-eight-mode leading-amplitude construction; Mishra et al., Eqs. (9)–(12h) give a second all-eight-mode frequency-domain construction. Their arXiv v1 source bundle also contains the author-published Mathematica supplement supl-mkaf16.m, whose H[l,m] definitions cover every mode in this specification through 2PN amplitude order. Both papers use the opposite Fourier sign, so their printed complex modes cannot be compared directly with the stored negative-\(m\) mode without the Fourier, tetrad, and per-mode phase adapter described above. García-Quirós et al. fix the native convention and give seven of the mode amplitudes in Appendix E; the \((4,2)\) ratio uses the first two all-eight sources. García-Quirós et al. also report a higher-PN amplitude disagreement with the earlier calculation, beyond this leading-order slice.

For the optional Schwarzschild hard mask, mode \(m\) has \(f_{\rm cut,m}=m/(2\,6^{3/2}\pi M)\); the native carrier retains \(0<f\le f_{\rm cut,m}\). This is a project truncation, not a source amplitude.

The stable IDs are tf2.hm.ratio_LM.v1 and tf2.hm.projection_LM.v1 for every listed mode, plus tf2.hm.mode_phase.v1, tf2.hm.polarizations.v1, and tf2.hm.cutoff_frequency.v1. All are executable in the stated carrier scope. The ratio, projection, phase, and polarization comparisons cover both frontends; the cutoff ID covers the threshold expression and hard-mask samples.

Implementation mapping

The scalar functions amplitude_ratio, spin_weighted_harmonic, projection, mode_phase, mode_expr, waveform_expr, and cutoff_frequency_expr live in taylorf2_hm.py. Native mode amplitudes, phases, and projections live in the separate torchwave and jaxwave implementations. return_modes=True exposes per-mode projected \((h_+,h_\times)\), which the reference checks before testing their sum. The scalar reference imports neither native implementation; the native implementations import no scalar reference.

Evidence and limitations

The evidence is intentionally separated:

Evidence layer	Maintained check	Establishes	Does not establish
Author-supplement composition	test_taylorf2_hm_author_supplement_math_reference.py	Checksum-pinned leading $	h_{\ell m}/h_{22}
Paper fidelity	test_taylorf2_hm_paper_math_reference.py	All-eight native \(h_{\ell,-m}\) reconstruction, Eqs. (8)–(9), the \(2f/m\) map, SPA constant, and \(t_c/\phi_c\) signs	An independent executable waveform
Scalar-reference agreement	test_taylorf2_hm_math_reference.py	All-eight leading amplitudes, composed aligned-spin phase, complex polarizations, selected derivatives and hard cutoffs	External parity or physical accuracy
Torch/JAX parity	tests/parity/test_hm_parity.py	Cross-backend values, individual modes, JIT and autodiff at sampled points	Independence from shared equations
External angular/API boundary	test_taylorf2_hm_external_api.py	Public LAL spin-weighted harmonics for every supported \(\pm m\); public TaylorF2 does not return resolved higher modes	Any higher-mode TaylorF2 waveform value
External waveform parity	frozen TaylorF2_HM_diagnostic	Dominant \((2,2)\) carrier only	The other seven modes
Physical validation	None	—	Accuracy against NR, EOB, or observations

The scalar-reference test compares all eight complex per-mode polarizations and their sum at two frequencies, two mass ratios, both zero and aligned spin, two orientations, and both phase origins, in FP64 CPU Torch and JAX against an 80-digit scalar evaluator. It also checks each mode's azimuth derivative, a component-mass derivative, equal-mass suppression of odd modes, and Schwarzschild hard-mask thresholds. The per-mode normalized complex tolerance is \(4\times10^{-10}\) and the derivative tolerances are stated in the test. These are sampled implementation checks, not a uniform error bound.

The author-supplement fixture is generated without native imports from the lowest nonspinning power of each pinned H[l,m]. At common observed frequency, its explicit adapter is \(V_m=(2/m)^{1/3}v\) and \(|h_{\ell m}/h_{22}|=|H_{\ell m}^{\rm LO}(V_m)|(V_m/v)^{-7/2}\). The source bundle SHA256 is f0b6c6810da95f45141dbb9f0c5cccec0ba0742f60b836dba7698c90a25d5154; the supplement SHA256 is 501ea53c29b77e061585540b9ccfd69a4cd0283e805ea1e9c24c38800e627a70. The project stores only the generated fixture and provenance, not the author source asset. The fixture's normalized TF28Phasing is nevertheless a transformed author-supplement expression, not an independently authored numerical result; its downstream redistribution terms remain unresolved.

The same pinned supplement contains TF28Phasing. Its nonspinning part is an orbital SPA phase with prefactor \(3/(256\eta v^5)\); the factor of \(m\) in Mishra et al. Eq. (11) makes it identical to the native \((m/2)I(V_m)\) phase. This also resolves the factor-of-two ambiguity introduced by the \(3/(128\eta v^5)\) prefactor printed in the paper's Eq. (4): the maintained component gate follows the checksum-pinned author expression used in Eq. (11). The generator retains the normalized SymPy expression, and the test combines it with the signed leading \(H_{\ell m}\) term and Eq. (11) normalization. The fixed raw-mode convention adapter is \(h^{\rm native}_{\ell,-m}=e^{i\pi(m-3)/2}h^{\rm Mishra}_{\ell,m}\). The composition gate multiplies those author-derived modes by the independent Wigner-\(d\) evaluation of García-Quirós et al. Eqs. (8)–(9), then sums all eight complex \(h_+\) and \(h_\times\) contributions. It checks native per-mode polarizations and both final polarizations directly, without using the native projection to construct its expectation or inverting native modes. Its per-mode errors divided by the raw mode magnitude and summed errors divided by the sum of projected-mode magnitudes are below \(8\times10^{-10}\) at the three binaries, three frequencies each, and two orientations in FP64. This is a source-composed symbolic check restricted to positive-frequency, leading nonspinning, unwindowed modes with f_ref=None. The projection comes from García-Quirós et al., not from the author supplement.

For each of the three binaries, at its predeclared middle frequency and both orientations, the same source-composed modes are multiplied by analytic inclination and azimuth derivatives of García-Quirós et al. Eqs. (8)–(9). The eight contributions are summed before comparison with Torch and JAX autodiff of the public complex \(h_+\) and \(h_\times\). Errors divided by the sum of the magnitudes of the derivative contributions are below \(8\times10^{-10}\). These sampled angular derivatives do not qualify intrinsic parameter derivatives or the full aligned-spin option set.

Exact all-mode external numerical parity remains unprovable from the available public sources. The supplement is symbolic author input parsed by this project, not an independently maintained executable or numerical series. It supplies full 2PN amplitudes and a phase expression while the maintained gate selects the leading nonspinning amplitudes and composes the projection from a second published source. Neither source fixes the native aligned-spin composition, f_ref subtraction, or per-mode cutoff as one independently evaluated series. No author-published numerical complex arrays fix that complete option set. The bounded independent-oracle search and its exact acceptance fields are recorded in the 2026-09-26 evidence receipt. The 2026-09-27 minimal two-mode probe uses an unequal-mass (2,2)+(3,3) candidate and confirms that the public TaylorF2 mode-array API still returns the same single carrier. Even this smaller nonspinning multimode external gate remains open; the existing author-source composition already checks all eight modes more broadly within its stated, non-external evidence category. Public LAL TaylorF2 returns one frequency series and its public FD-mode dispatcher has no TaylorF2 generator. IMRPhenomXHM is a different full-IMR model and gwfast's TEOBResumSPA uses different EOB dynamics and amplitude corrections, so neither is a binwise same-model oracle. Higher-PN mode amplitudes, spin-amplitude corrections, generic tidal matter, merger, Kerr cutoffs, taper derivatives, FP32, and full-domain accuracy remain unqualified.

The dated composed-oracle audit also checked the public WF4Py and gwfast TaylorF2 implementations and exact code-search matches. Both external implementations are restricted dominant-mode models. The audit did not inspect or translate LAL implementation source. Extending the author supplement locally would not create the missing independence: its unmodified \(v^7\) spin content matches neither native selector, it has no arbitrary \(f_{\rm ref}\) input, and its harmonic Theta factors belong to a different full-PN polarization expansion. Selecting the native spin terms, adding reference-frequency subtraction, projecting the leading modes, and applying the native hard mask would reproduce the project-authored composition already checked by the scalar reference; freezing that recomposition would not turn it into an external numerical oracle.

The exact next external-parity gate is an author-supplied numerical table or an independently maintained, checksum-pinned executable under suitable terms that exposes all eight complex \(h_{\ell,-m}(f)\) values (or both complex polarizations for every mode contribution and their sum) for a fully specified binary. The release must state its Fourier, tetrad, mode-label, spin-term, \(f_{\rm ref}\), and cutoff conventions and provide bins below, exactly at, and above each mode cutoff. Direct comparison must use no time, phase, amplitude, or frequency alignment, normalized complex FP64 error at most \(4\times10^{-10}\) for every unmasked mode and sum in both frontends, and exact zeros in masked bins. A dominant-mode series or angular basis alone cannot satisfy that gate.

Reproduce the checks

Install python -m pip install -e '.[reference,torch,jax,dev]', then run python -m pytest -q tests/test_taylorf2_hm_author_supplement_math_reference.py tests/test_taylorf2_hm_math_reference.py tests/test_taylorf2_hm_paper_math_reference.py for the mathematical slice. With pinned LALSuite 7.26.15 installed, run python -m pytest -q tests/test_taylorf2_hm_external_api.py for the public angular/API boundary. To regenerate the frozen supplement fixture, download https://export.arxiv.org/e-print/1601.05588v1 and run python tests/data/generate_taylorf2_hm_mishra2016_samples.py --archive PATH; the generator rejects either source hash before parsing and imports no native waveform code. Run the complete CPU gate with the required reference inventory. That gate requires both frontends and reference dependencies and rejects skips, expected failures, and deselection. Run python scripts/generate_waveform_library_table.py --check to verify the catalog-derived sections. Equation, convention, or cutoff changes require a new component/specification version and a reviewed source and test update.