TorchWave Literature Provenance and Coefficient Ledger

This document records mathematical sources, asset provenance, and calibration provenance in TorchWave. Model-specific comparisons with LALSimulation are recorded on Differences from LAL. Numerical agreement, physical calibration and software/data provenance are separate claims; evidence for one does not establish the others.

1. Source and Validation Policy

Mathematical and software sources: Model implementations draw on primary literature, public author data and explicitly attributed permissive software adaptations. Sources and unresolved inherited calibration provenance are recorded here and in third-party notices.
Reference comparisons: LALSimulation binaries provide external waveform comparisons. Historical implementation and reference-audit records are preserved on archive/clean-room-reviews; these records do not establish clean-room provenance for every inherited coefficient.
Calibration and numerical conventions: Native NR fits use the identified SXS data. Numerical compatibility conventions, such as QNM rounding and phase/time choices, are documented separately from physical calibration.
Scope of agreement: Passing a declared case establishes agreement for its parameters, modes, frequency grid, precision and reference version. It does not establish exact equality or accuracy throughout a model's parameter domain.

2. Approximants and Primary Literature Provenance

Implementation and Evidence Summary

Reference comparisons: The archived CPU FP64 catalog passed 17 of 21 cases; the holdout suite passed 11 of 13. Both suites report failure overall. The records include case definitions, reference versions, unaligned complex errors and detector projections: catalog, holdouts, source record. These are historical snapshots, not fresh results for the current checkout.
Native NR components: Ringdown amplitude and phase fits cover $(2,1)$, $(3,3)$, $(4,4)$ and $(3,2)$. The phase tests below isolate the ringdown phase using reference amplitudes and restricted frequency bands; they do not validate complete multimode waveforms.
Continuous EOB attachment: Natural cubic splines locate an interior orbital-frequency maximum on accepted integration samples. Removing an auxiliary resampling grid does not remove dependence on integration accuracy or spline resolution.
Shared physics and frontend checks: diffgw_core contains framework-independent Python/NumPy constants and coefficients. Tests in tests/parity/ compare the Torch and JAX implementations with per-test tolerances; frontend agreement alone is not an independent physical validation.

The native waveform implementations do not call LAL at runtime. The canonical SEOBNR_ROM path requires external official ROM data. Original software is MIT licensed; third-party data and adaptations retain their identified terms, and some inherited calibration provenance remains unresolved. Differentiability and batching depend on the selected model and numerical path.

The retired prototype record preserves the former complete-waveform routes and native pilot asset metadata. Historical comparisons below remain evidence about those former implementations, not validation of the retained official ROM or surviving equation components.

2.1 TaylorF2 (Inspiral Aligned-Spin with Tides)

Executable mathematical reference: The nonspinning 3.5PN specification, aligned-spin black-hole extension, adiabatic quadrupolar tidal extension, and spin-induced quadrupole extension connect equation-level sources, explicit conventions, independent SymPy/mpmath expressions, and Torch/JAX coefficient, waveform, and derivative checks. Each extension reuses unchanged baseline components and restricted amplitude. The last two add 5PN/6PN/6.5PN electric tidal and non-Kerr 2PN/3PN/3.5PN spin-quadrupole phase terms. Independent spin-induced octupoles, spin-tidal couplings, magnetic/dynamical tides, absorption, amplitude corrections, and termination remain outside these specifications.
Primary References:
Buonanno, Iyer, Ochsner, Pan, Sathyaprakash (2009), Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors, Phys. Rev. D 80, 084043 (DOI:10.1103/PhysRevD.80.084043)
Mishra, Kela, Arun, Faye (2016), Ready-to-use post-Newtonian gravitational waveforms for binary black holes with non-precessing spins: An update, Phys. Rev. D 93, 084054 (DOI:10.1103/PhysRevD.93.084054)
Wade et al. (2013), arXiv:1306.3901v2, Eq. (4) and spin-parameter definitions; lower-order aligned-spin terms with black-hole quadrupoles.
Bohé et al. (2015), arXiv:1501.01529v1, Eqs. (3.33), (4.14), and Messina–Nagar (2017), arXiv:1703.08107v4, corrected Eq. (26); energy-balance derivation of the complete 3.5PN quadratic-spin phase term.
Pratten et al. (2020), arXiv:2001.11412v1, Eqs. (A19), (A23), (A24); the baseline phase and its explicit spin-spin tail addition, independently checked through energy balance. The aligned-spin specification records the correction of the previously omitted (A24) term.
Vines, Hinderer, Flanagan (2011), Post-1-Newtonian tidal effects in the gravitational waveform from binary inspirals, Phys. Rev. D 83, 084051 (DOI:10.1103/PhysRevD.83.084051)
Wade et al. (2014), arXiv:1402.5156v1, Eqs. (4)–(6), (A26); equivalent effective-parameter and component-mass expressions for the 5PN/6PN tidal phase.
Damour, Nagar, Villain (2012), Measurability of the tidal polarizability of neutron stars in late-inspiral gravitational-wave signals, Phys. Rev. D 85, 123007 (DOI:10.1103/PhysRevD.85.123007) The tidal reference pins arXiv:1203.4352v1, Eqs. (B8)–(B12), (B14), including the 6.5PN tail and tidal-coupling conversion.
Nagar et al. (2019), arXiv:1812.07923v1, Eqs. (44), (46), (47), (50)–(52); spin-induced quadrupole corrections at 2PN, 3PN, and the 3.5PN tail.
Dietrich et al. (2019), arXiv:1905.06011v2, Eqs. (26)–(27); quadrupole-dependent cubic-spin correction. Marsat, arXiv:1411.4118v2, Eqs. (6.20)–(6.21), supplies an independent cubic-spin cross-check.
Mathematical Form: $$\tilde{h}(f) = \mathcal{A}(f) e^{-i \Psi(f)}$$ $$\Psi_{\rm coal}(f) = 2\pi f t_c - 2\phi_c - \frac{\pi}{4} + \frac{3}{128\eta v^5} \sum_{k=0}^{7} \left(\psi_k + \psi_{k,\ln} \ln v\right) v^k + \Psi_{\rm tidal}(f)$$ where $v = (\pi M f)^{1/3}$ and phic is the orbital phase. This expression uses the coalescence convention.
Reference-Frequency Phasing: The reference-frequency convention subtracts the intrinsic phase at $f_{\rm ref}$ and adds $3\pi/4$, so $$\Psi(f_{\rm ref}) = \frac{3\pi}{4} + 2\pi f_{\rm ref} t_c - 2\phi_c.$$ It reduces to $3\pi/4$ only when $t_c=\phi_c=0$.
Verification Result: The four frozen catalog cases passed the declared CPU FP64 gates. The largest polarization complex relative $L_2$ error was approximately $1.64 \times 10^{-12}$; detector-projection errors are reported separately in the artifact.

2.2 TaylorF2_Eccentric (Eccentric Inspiral)

Primary References:
Moore, Favata, Arun, Mishra (2016), Gravitational-wave phasing for low-eccentricity inspiralling compact binaries to 3PN order, Phys. Rev. D 93, 124061 (DOI:10.1103/PhysRevD.93.124061)
Huerta et al. (2014), Accurate and efficient waveforms for compact binaries on eccentric orbits, Phys. Rev. D 90, 084016 (DOI:10.1103/PhysRevD.90.084016)
Mathematical Form: $$\Delta\Psi_{\rm paper}(f) = -\frac{2355}{1462}\frac{3}{128\eta v^5}e_0^2\left(\frac{v_0}{v}\right)^{19/3}P_{3\rm PN}(v,v_0,\eta).$$ Equation (6.26) of Moore et al. (2016) gives the eccentric SPA term, including mixed powers of $v$ and $v_0$; Eqs. (4.17)–(4.18) concern the preceding eccentricity evolution. The native waveform subtracts $\Delta\Psi_{\rm paper}(f_{\rm ref})$ to set its explicit reference-frequency gauge. This subtraction is a project convention, not part of Eq. (6.26). The phase component specification and independent SymPy/mpmath reference check each PN truncation against Torch and JAX CPU float64 and test distinct $f_{0,\rm ecc}$ and $f_{\rm ref}$ through both polarization quotients. The named tests/test_taylorf2_eccentric_math_reference.py module enters the strict math inventory automatically; the execution receipt records the focused gate. The versioned nonspinning amplitude-off composition reuses the qualified TaylorF2 carrier. The optional native amplitude factor is a project option whose primary derivation remains pending and is outside this slice.
Reference Frequency Priority: $f_{0,\rm ecc}$ resolves from an explicit value, then $f_{\rm lower}$, then a positive $f_{\rm ref}$, then the lowest positive frequency-grid sample (20 Hz if none exists). The final grid fallback makes the result dependent on the supplied positive grid.
Verification Result: The two historical frozen catalog cases passed, with a maximum polarization complex relative $L_2$ error of approximately $1.64 \times 10^{-12}$; they include circular and spinning configurations and are not the basis of the nonspinning versioned claim. Two separate frozen off-catalog holdouts with $e_0=0.06$ and $0.035$ passed the unchanged direct LAL/PyCBC CPU float64 gates in both Torch and JAX, including both polarizations and six detector projections per backend. tests/test_taylorf2_eccentric_holdout.py enforces those cases in strict binary CI. This is sampled evidence within $q\le2$, $0<e_0\le0.1$, positive frequencies below ISCO, explicit $f_{0,\rm ecc}$/$f_{\rm ref}$, and amplitude off; spin and optional amplitude remain provisional.

2.3 IMRPhenomD (Aligned-Spin IMR)

Primary References:
Husa, Khan, Hannam, Pürrer, Ohme, Forteza, Bohé (2016), Frequency-domain gravitational waves from non-precessing black-hole binaries. I. New numerical waveforms and anatomy of the signal, Phys. Rev. D 93, 044006 (DOI:10.1103/PhysRevD.93.044006) [Paper I]
Khan, Husa, Hannam, Ohme, Pürrer, Forteza, Bohé (2016), Frequency-domain gravitational waves from non-precessing black-hole binaries. II. A phenomenological model for the advanced detector era, Phys. Rev. D 93, 044007 (DOI:10.1103/PhysRevD.93.044007) [Paper II]
Berti, Cardoso, Will (2006), On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA, Phys. Rev. D 73, 064030 (DOI:10.1103/PhysRevD.73.064030)
Coefficients & Calibration:
Table V of Khan et al. (2016) Paper II prints 19 phenomenological parameter rows $\Lambda^i$ as polynomials in $(\eta, \chi_{\rm PN} - 1)$ where $\chi_{\rm PN} = \chi_{\rm eff} - \frac{38\eta}{113}(\chi_1 + \chi_2)$. The waveform implementation uses 18 fitted rows and obtains $\beta_1$ from its phase-join condition.
The explicit calibration="published" D/HM option uses diffgw_core/phenomd_published_fits.py: only Khan Table V's printed digits and Husa PDF Eqs. (3.6)–(3.8) (HTML Eqs. (22)–(24)), including 0.102 for the final-spin $\eta S^2$ coefficient. This is a separately named paper-precision calibration, not a recovery of historical unrounded fits. The symbolic reference has a separate decimal transcription. Default calibration="inherited" is unchanged; its provenance caveat remains. The complete HM reference closes the native-input dependency for the declared published option; it does not transfer earlier binary or NR results to that option.
The published-HM prospective NR comparison supplies separate physical evidence for the exact printed-calibration, source-PN, London-mapping option. SXS:BBH:2374 and 2163 pass resolution screens but fail the second view's match and norm gates in both frontends. Source fidelity and independent mathematical agreement do not establish physical accuracy; no threshold or coefficient was changed.
The published-HM option/gauge diagnostic verifies an extra finite-reference analytic phase using the independent scalar carrier, both frontends and unchanged fixtures. Removing the explicit factor restores orbital covariance diagnostically; NR failures remain. No production coefficients or acceptance gates changed.
The opt-in HM orbital reference-phase extension removes that extra factor under a separately versioned project convention, with independent scalar/backend covariance checks. Legacy output and earlier evidence remain unchanged. A factorial mapping diagnostic separates amplitude and phase mapping on the seen fixtures; it does not promote physical qualification.
The first executable amplitude specification uses the seven amplitude rows at the digits printed in Table V. The production table retains additional inherited digits whose precise publication provenance remains unresolved. Algebra checks using those digits are conditional and do not establish their source.
The executable phase specification transcribes the 12 printed phase rows, Appendix B aligned-spin PN terms, three phase branches and two continuity joins from Paper II. The backend checks explicitly supply inherited extra fit digits, so their algebraic agreement does not establish those digits' provenance. The frontend's amplitude-peak time gauge is recorded as a separate implementation convention.
The composed polarization specification combines those unchanged slices with the independently audited SI/IAU conversions, the physical stationary-phase normalization of Cutler and Flanagan, Eqs. (62)–(63), and explicit DiffGW phase/polarization conventions. Direct complex-strain and selected smooth-branch derivative checks against both frontends use a declared QNM pair and explicitly supplied inherited fit rows; they do not establish fit provenance or physical accuracy.
The full intrinsic composition supplies the QNM pair from the independent remnant expressions and declared numerical spline data. At fixed total mass, it checks selected full-chain mass-ratio and aligned-spin complex-strain derivatives against both frontends, including the moving remnant, QNM inputs, joins, and phase gauge. This now includes both signed high-spin Kerr spline branches and an interior-spin point on the active remnant cap, with guarded smooth-region scalar stencils. It also compares complete high-spin complex waveforms with an independently evaluated generated Kerr spline. Its conditional inherited-coefficient comparisons do not recover publication provenance for the extra digits or establish physical accuracy.
The executable remnant and QNM-input specification transcribes Husa et al. arXiv:1508.07250v2 Eqs. (22)–(24) for final spin, radiated energy and mass, and Eq. (17) for QNM mass scaling. The implementation retains additional inherited fit digits; one spin coefficient does not round to the paper's printed digit.
An independent four-term final-spin refit uses the later, author-published 427-case NR table with pinned input and script hashes. Weight and fitting choices move the disputed coefficient substantially; this experiment does not reconstruct the original 106-case calibration or change the production coefficient.
The current QNM frequency and damping representation uses independently generated Kerr eigenvalues, rounded to six significant digits and interpolated with a natural cubic spline in kerr_qnm_d_sampling_clean.py; outside its compatibility interval, it uses the generated eigenvalue evaluator. Berti et al. (2006) provides background and the optional legacy fit representation.
Verification Result: The declared catalog and holdout cases passed the complex relative $L_2 \le 10^{-6}$ and mismatch $\le 10^{-8}$ gates. This result applies to those configurations.

2.4 IMRPhenomP / IMRPhenomPv2 (Precessing Waveforms)

Primary References:
Hannam et al. (2014), A simple model of complete precessing black-hole-binary gravitational waveforms, Phys. Rev. Lett. 113, 151101 (DOI:10.1103/PhysRevLett.113.151101)
Schmidt, Hannam, Husa (2012), Towards models of gravitational waveforms from generic binaries: A simple approximate mapping between precessing and non-precessing inspiral signals, Phys. Rev. D 86, 104063 (DOI:10.1103/PhysRevD.86.104063)
Pratten et al. (2021), Computationally efficient models for the dominant and sub-dominant harmonic modes of precessing binary black holes, Phys. Rev. D 103, 104056 (DOI:10.1103/PhysRevD.103.104056) [Appendix B, Eqs. (B1)–(B2)]
Verification Result: The IMRPhenomPv2 catalog and holdout cases passed the complex relative $L_2 \le 10^{-6}$ and mismatch $\le 10^{-8}$ gates. This does not establish a separate result for every PhenomP variant.

2.4a IMRPhenomXP MSA v223 convention-6 output

Published convention: Pratten et al. (2021), Appendix C Eqs. (67), (79), (81) and Appendix F Table IV specify the dominant-mode twist-up, source-frame rotation, polarization basis and convention-dependent Euler offsets and carrier phase. Table IV assigns zero carrier phase to convention 1 and phic to convention 6.
Declared native mapping: For the quadrupole-only XP v223, final_spin_mod=3, phi0=0 route, the complete convention-6 polarizations equal the clean convention-1 polarizations times $\exp(2i\,\mathrm{phic})$. This is an output convention identity after the full rotation and basis projection, not a change to the MSA dynamics or a higher-mode rule. The clean helper remains defined only for conventions 0 and 1.
Compatibility and limits: The generic dense-grid convention-6 fixture remains below $10^{-8}$ complex relative $L_2$ against the public LAL v223 binary. The aligned fixture separates two contracts: the explicit v223 helper retains its strict binary gate, while the physical public opening has a measured $10^{-5}$–$5\times10^{-5}$ residual. The v223 auxiliary-root endpoint remains only in that compatibility diagnostic; the L+S_parallel <= 0 guard remains in public dispatch. Public JAX uses the source-derived zero-nutation branch for scoped one-spin rows and the distinct-root generic MSA branch for valid unequal-mass, two-spin, convention-0/1/6 rows at zero or nonzero phi0 with prec_version=223 and final_spin_mod=3. Nonpositive frequency bins are evaluated at a positive surrogate internally and returned as zero strain. Exactly equal-mass generic rows retain the inherited NNLO angle route because the split-root equations lack an implemented equal-mass continuation; explicit remnant mode 3 raises on that unsupported MSA root, while mode 0 remains available. Other precession versions and remnant choices retain the established XP route; XPHM dispatch is separate. The generic nonzero-phi0 map is our derivation from Pratten Appendix C Eqs. (67)--(68), (76)--(85), Appendix B Eq. (63), Appendix E Eqs. (110)--(113), and Appendix F Table IV: rotate in-plane spins by Rz(-phi0), use binary phiRef=phic, and multiply completed quadrupole polarizations by exp(-2i phi0). The declared 20--512 Hz fixtures pass public-binary strict L2 and mismatch gates for conventions 0, 1, and 6. The frozen catalog and holdout qualification numbers below cover convention 1; these additional probes do not certify the full precession domain or physical accuracy. See the physical boundary analysis.

2.4b IMRPhenomXP generic two-spin aligned-boundary diagnostic

Published equations: Chatziioannou et al. (2017), Eqs. (21)--(25), Eq. (51), and Appendices B/E define the cubic in total-spin squared, its elliptic solution and phase, the limiting-root phase, and the documented near-alignment conditioning problem. Pratten et al. (2021), Eq. (7) and Appendix C Eqs. (67)--(85) define minimal rotation and the source/J/observer chart.
Independent derivation: src/diffgw_reference/xp_msa_generic_boundary.py is a backend-independent arbitrary-precision scalar evaluation of those equations. For the declared (36,22) Msun, (chi1z,chi2z)=(0.35,-0.25) fixture at 40 Hz, it derives a physical turning-root gap proportional to the squared transverse-spin scale and an inverse-cosine endpoint distance proportional to its fourth power. It retains the unclipped phase quotient and treats exact-aligned phi_J as undefined.
Implementation diagnostics: Separate Torch and JAX float64 modules expose the raw cubic argument before returning separately named clipped compatibility roots. Cross-frontend tests cover a resolvable generic cubic and negative source phase; the high-precision gate requires the unresolved 1e-10 root split to remain visible. These diagnostics are not public waveform routes.
Boundary policy and unresolved scope: No published small-misalignment continuation of the generic v223 root/phase/chart equations was identified. Klein, Cornish and Yunes (2013) provide a separate near-aligned waveform expansion, not a limiting prescription for this generic MSA solution. The later regularized $\delta\chi$ formulations of Klein (2021) and Gerosa et al. (2023) address equal-mass/root conditioning, while retaining an aligned inertial-angle singularity; Gerosa et al. explicitly leave the zero-amplitude resonance Taylor expansion to future work. Invalid MSA-initialization rows now take the paper-prescribed NNLO angle fallback, and the former two near-aligned expected-failure cases pass the unchanged internal 1e-6 complex-polarization L2 gate. An explicit MSA remnant request (final_spin_mod=3) still raises on invalid nonaligned roots; mode 0 provides the tested continuous remnant. The one-spin/aligned route is unchanged; valid generic unequal-mass nonzero-phi0 rows now use the clean chart after independent frame-map validation. Neither clipping nor the v223 Eq. (51) compatibility substitution is claimed as a paper-derived repair. Equal-mass and near-aligned external full-waveform boundaries still fail strict qualification; the declared generic nonzero-phi0 fixture passes. The reference closure and boundary receipt give the separate internal and external evidence.

2.5 IMRPhenomHM (Multipolar Higher Harmonics)

Primary References:
London et al. (2018), First higher-multipole model of gravitational waves from spinning and coalescing black-hole binaries, Phys. Rev. Lett. 120, 161102 (DOI:10.1103/PhysRevLett.120.161102)
García-Quirós et al. (2020), IMRPhenomXHM: A multi-mode frequency-domain model for the gravitational wave signal from non-precessing black-hole binaries, Phys. Rev. D 102, 064002 (DOI:10.1103/PhysRevD.102.064002) [Eq. (3)]
Executable component: The London Eq. (5) frequency-map specification separately checks the amplitude map and its Eq. (8) D-carrier phase variant with independent scalar expressions and Torch/JAX values and derivatives. This component result does not certify the full IMRPhenomHM waveform.
Explicit SPA carrier convention: hm_phase_convention="spa_sxs_v1" composes the leading-SPA derivation and signed-pair phase transport with the native D-based HM waveform. The factor is derived from Blanchet v3 Eqs. (8.2)–(8.3), (9.3)–(9.4), Boyle v2 Appendix C and the declared common strain sign; it is not fitted to NR or a binary oracle. Applying the constant throughout merger/ringdown is an explicit project convention. The independent composition is v3; default output and older component versions remain unchanged. This does not assert physical accuracy or promote earlier red NR results.
Six-mode binary holdout: The pinned LALSuite 7.26.15 receipt records fixed-convention complex comparisons of all six aligned-spin HM/D modes on three held-out systems in Torch and JAX. It establishes sampled binary agreement, with a documented weak-tail cutoff difference; it is not an independent mathematical derivation or a domain-wide accuracy claim.
Independent physical NR holdouts: The precommitted SXS:BBH:2265v3.0 protocol and red receipt, plus the separately precommitted SXS:BBH:3631v3.0 protocol and red receipt, compare all six native modes and two six-mode plus/cross projections against official Lev3 NR, screened with Lev2 resolution. In each case only (2,2) passes the frozen model/NR gates; the other modes and projections fail in both frontends. A third, separately precommitted SXS:BBH:3619v3.0 detector-projection protocol follows London's PSD-weighted match with optimized template time, orbital phase, and polarization. Its red receipt records model/NR matches 0.996196 and 0.985979 in both frontends, but the frozen Lev2–Lev3 screen fails (0.991880 and 0.980143 versus 0.995), so this case is infeasible for qualification. A clearly post-result, non-gating common-orbital-phase diagnostic raises the NR resolution matches above 0.9998; it does not change the predeclared result. The separately precommitted SXS:BBH:3630v3.0 protocol used common time/orbital-phase NR alignment from the outset. Its red receipt records Lev2–Lev3 matches above 0.99995 and model/NR matches above 0.980, but one model/NR norm ratio is 0.840432, below the frozen 0.85 floor, in both frontends. Neither same-model LAL parity nor these red projection cases establish physical HM qualification.
Common Time Shift Invariance: Higher modes satisfy exact linear time translation: $$\tilde{h}_{\ell m}(f; t_c) = \tilde{h}_{\ell m}(f; 0) \exp(-2\pi i f t_c)$$ The declared catalog and holdout comparisons passed the $L_2 \le 10^{-6}$ and mismatch $\le 10^{-8}$ gates; these finite-tolerance comparisons do not establish exact equality.

2.5a TaylorF2_HM leading eight-mode carrier

Native convention: García-Quirós et al. (2020), Eqs. (2), (4)–(9), (29), (34) fix $h=h_+-ih_\times$, the $e^{-i2\pi ft}$ Fourier sign, positive-frequency negative-$m$ storage, paired polarization projection, $2f/m$ phase scaling, and the leading-SPA phase constant.
Eight leading amplitudes: Mezzasoma and Yunes (2022), Eqs. (20)–(21i), (23)–(30) supply all eight modes. Mishra et al. (2016), Eqs. (9)–(12h) provide a second leading-order derivation. Their opposite Fourier and tetrad conventions require the explicit adapter recorded in the mathematical specification.
Author-supplement composition checks: the arXiv v1 source for Mishra et al. includes supl-mkaf16.m, a Mathematica-readable author asset with all eight relevant H[l,m] definitions and TF28Phasing. tests/data/generate_taylorf2_hm_mishra2016_samples.py verifies the source bundle SHA256 f0b6c6810da95f45141dbb9f0c5cccec0ba0742f60b836dba7698c90a25d5154 and member SHA256 501ea53c29b77e061585540b9ccfd69a4cd0283e805ea1e9c24c38800e627a70, extracts each lowest nonspinning PN term, and retains the normalized nonspinning phase without native imports. tests/test_taylorf2_hm_author_supplement_math_reference.py keeps the original all-eight amplitude-ratio gate, checks signed absolute raw modes using $h^{\rm native}_{\ell,-m}=e^{i\pi(m-3)/2}h^{\rm Mishra}_{\ell,m}$, and composes them with García-Quirós et al. Eqs. (8)–(9) to check both frontends' per-mode and summed complex polarizations directly at two orientations.
Supplement redistribution: the frozen normalized TF28Phasing is a transformed expression from the author supplement, not an independently authored numerical result. No explicit downstream redistribution or MIT grant was located in the paper, arXiv metadata, or source bundle. It is therefore identified as third-party-derived verification material with unresolved terms; the repository MIT grant does not extend to it.
Paper and scalar checks: tests/test_taylorf2_hm_paper_math_reference.py reconstructs the raw carrier independently from both polarizations and checks the published phase identities for every mode in both frontends. tests/test_taylorf2_hm_math_reference.py separately checks the complete declared composition against the versioned SymPy/mpmath kernel.
External boundary: pinned LALSuite 7.26.15 independently checks all supported $\pm m$ spin-weighted harmonics. Its public TaylorF2 function returns one carrier, mode-array metadata does not resolve modes, and the public FD-mode dispatcher rejects TaylorF2. The frozen external waveform case therefore remains $(2,2)$-only.
Unresolved: the author supplement is symbolic input parsed by this project, not an independently maintained executable or numerical series for the native negative-$m$ complex waveform. Its source-composed gate is leading, nonspinning, unwindowed, positive-frequency, and uses f_ref=None; projection is supplied independently by García-Quirós et al. It does not cover aligned-spin composition, reference-frequency subtraction, or per-mode cutoff. The 2026-09-25 composed-oracle audit and 2026-09-26 oracle-search receipt found no public same-model release that combines those features: the external WF4Py/gwfast implementations are dominant-mode only, while locally extending the supplement would reproduce the project-defined composition rather than create independent numerical evidence. The receipts record the exact release contract and strict no-alignment threshold needed for closure. Exact all-mode external numerical parity and physical validation therefore remain unavailable; the source-composed check, scalar agreement, Torch/JAX parity, external angular agreement, and dominant-mode external parity are not substitutes for those gates.

2.6 IMRPhenomXAS & IMRPhenomXHM (Public Data Deposits & Native Calibration)

Primary References:
Pratten et al. (2020), Setting the cornerstone for the IMRPhenomX family of models for gravitational waves from compact binaries: The dominant harmonic for non-precessing quasi-circular black holes, Phys. Rev. D 102, 064001 (DOI:10.1103/PhysRevD.102.064001)
García-Quirós et al. (2020), IMRPhenomXHM: A multi-mode frequency-domain model for the gravitational wave signal from non-precessing black-hole binaries, Phys. Rev. D 102, 064002 (DOI:10.1103/PhysRevD.102.064002)
Public Data Asset for IMRPhenomXAS:
The complete 26-fit parameter space Mathematica file deposited with APS PRD:
- URL: https://journals.aps.org/prd/supplemental/10.1103/PhysRevD.102.064001/IMRPhenomXAS_Parameter_Space_Fits_Public_Release_Candidate.m
- SHA256: 94b9334c19f3a7909864891ea278ab9a411e4f33b7b7e17cc01d927056d9ac52
- Access Status: Freely downloadable public supplemental deposit.
- Executable fit check: The opt-in XAS author-fit verifier checks the pinned file without redistributing it. Across nine mass-ratio/spin cases, 24 parameter surfaces and the two (2,2) QNM surfaces agree in value with both native frontends; selected parameter and QNM spin derivatives agree with independent complex-step derivatives. Two of these parameter surfaces are available for the fifth-order amplitude alternative but are not called by the quartic carrier. The intermediate-amplitude midpoint fit is an explicit carrier variant: the deposit uses chiPNhat and the native carrier uses sTotR, which the verifier checks separately. This establishes source-fit transcription on sampled inputs, not independent NR calibration or physical accuracy; see the mathematical specification.
Reference Result: The single frozen catalog case had maximum polarization complex relative $L_2$ error approximately $4.21 \times 10^{-14}$. The two holdout cases reached approximately $9.49 \times 10^{-9}$; these figures describe the recorded configurations, not a uniform accuracy bound.
IMRPhenomXHM Native NR Calibration:
Fits ringdown surfaces to SXS numerical-relativity data across $(\ell, m) \in \{(2,1), (3,3), (4,4), (3,2)\}$, providing smooth amplitude profiles and continuous phase matching.
Uses native Kerr Leaver quasinormal mode frequencies and damping rates from torchwave.qnm.
Phase-component test gates and their restricted scope are described in Section 2.10. The historical full-waveform LAL comparisons did not pass the strict gates.
The 2026-09-25 binary-oracle receipt uses only public installed LAL binary APIs and records the four frozen XHM/XPHM full-waveform diagnostics. It contains no source-derived formulas, translated implementation details, or fitted values. Executable paper-derived NNLO-102 math/reference gates live separately in tests/test_xphm_nnlo_clean_room.py and tests/test_xphm_rotation_reference.py. The catalog and holdout strict counts remain 0/1 for both XHM and XPHM.
The regional carrier audit identifies higher-mode inspiral amplitude as the earliest mismatch and (3,2) as the worst frozen mode. A stdlib scalar reference and phase-sensitive checkpoints independently verify the printed Appendix-E PN polynomials against Torch and JAX. The arXiv-v2 archive closes the paper-era fit inventory and Appendix-A mixing assets; official release documentation says 122022 recalibrated amplitudes, not phases. Its current fit/frequency/ansatz payload and recalibrated spheroidal (3,2) inputs are not publicly deposited, preventing a paper-derived production repair; the component receipt leaves all strict full-waveform gates red.

2.7 SEOBNR Reduced-Order Model (ROM Surrogate)

Primary References:
Pürrer (2014), Frequency domain reduced order models for gravitational waves from aligned-spin compact binaries, Class. Quantum Grav. 31, 195010 (DOI:10.1088/0264-9381/31/19/195010)
Pürrer (2016), Frequency domain reduced order model of aligned-spin effective-one-body waveforms with generic mass-ratios and spins, Phys. Rev. D 93, 064041 (DOI:10.1103/PhysRevD.93.064041)
Bohé et al. (2017), An improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors, Phys. Rev. D 95, 044028 (DOI:10.1103/PhysRevD.95.044028)
Native development: The unvalidated native surrogate and SEOBNR waveform assemblers have been removed from the release package. They are retained on codex/native-seobnr-rom for generator qualification before any new tables are trained or promoted. The former seobnrv5rom alias is no longer accepted.
Official ROM Wrapper:
The canonical SEOBNR_ROM / SEOBNRv4ROM aliases use tensor reconstruction from the external official SEOBNRv4ROM_v3.0.hdf5 data. The data are required for this path and are not bundled.
Registered in torchwave.external_data from Zenodo record 14999310 (SHA-256 4ff5317967d757fcad9323a74b75f4f63894ed87f4b1852d46cbe47cde488b9c); redistribution terms remain unresolved.
The single frozen catalog case had maximum polarization complex relative $L_2$ error approximately $1.20 \times 10^{-11}$.

2.8 NRTidal & NRTidalv2 (Tidal Deformability Models)

Primary References:
Dietrich et al. (2017), Closed-form tidal approximants for binary neutron star gravitational waveforms constructed from high-resolution numerical relativity simulations, Phys. Rev. D 96, 121501 (DOI:10.1103/PhysRevD.96.121501)
Dietrich et al. (2019), Improving the NRTidal model for binary neutron star systems, Phys. Rev. D 100, 044003 (DOI:10.1103/PhysRevD.100.044003) [Eqs. (15)–(21)]
Reference Result: The declared catalog and holdout cases for IMRPhenomD_NRTidal and IMRPhenomD_NRTidalv2 passed $L_2 \le 10^{-6}$ and mismatch $\le 10^{-8}$ gates.
Independent v1 phase slice: The NRTidal v1 nonspinning tidal-phase specification checks Dietrich et al. (2017) Eqs. (2), (7) against paper-derived scalar math, both frontends, and a pinned public LALSuite binary on six declared systems. Its receipt records 33 passing tests and the strict phase-quotient gates. The current carriers use the three-decimal rounding $d_1=-19.924$ of the paper's analytic constraint $d_1=n_1-3115/1248$; the specification and receipt quantify this distinction. Merger/taper, spins, inherited baseline, and full-waveform qualification remain separate.
Independent v1 merger/taper slice: The merger-taper specification checks the 2018 published merger fit and Planck window against Torch/JAX and gives strict premerger full-carrier and zero-tide taper gates against LALSuite 7.26.15 on six systems. Its boundary receipt records a nonzero-tide public-binary taper discrepancy beyond the declared gate. The convention diagnostic localizes that discrepancy to the binary's effective merger frequency rather than its Planck shape. The author-source audit finds no published full-precision Eq. (11) tuple or source-defined variant that resolves it; the printed fit remains the reference. No fully tapered binary qualification is claimed.
Independent phase slice: The NRTidalv2 nonspinning tidal-phase specification checks Dietrich et al. Eqs. (17)–(21) against a scalar reference, both frontends and a pinned public LALSuite binary on four declared cases. Its receipt records the fixed gates and measured residuals. Amplitude, self-spin, merger/taper and spinning cases remain separate gates.
Premerger composition holdout: The NRTidalv2 complex-quotient receipt adds three nonspinning holdouts through 1200 Hz, with strict unaligned Torch/JAX versus pinned-binary amplitude, phase and complex-ratio limits. This qualifies the default binary_compatible composition only for that band and those cases; its amplitude's paper provenance and nonzero-tide taper discrepancy remain unresolved.
Pv2 public-assembly boundary: The source audit records why the separately gated finite-size carrier and supplied-angle projection do not yet establish IMRPhenomPv2_NRTidalv2: Dietrich (2019) fixes insertion before rotation but not the BNS remnant-frequency spline-time or phase origin; the inherited Pv2 note's longitudinal-spin and 2PN angular-momentum expressions differ from the native BBH compatibility choices. No public BNS dispatch or whole-waveform qualification is claimed.

2.9 IMRPhenomNSBH (Neutron Star-Black Hole Mergers)

Primary References:
Pannarale et al. (2015), Aligned spin neutron star-black hole mergers: a gravitational waveform amplitude model, Phys. Rev. D 92, 084050 (DOI:10.1103/PhysRevD.92.084050)
Thompson et al. (2020), Modeling the gravitational wave signature of neutron star black hole coalescences: PhenomNSBH, Phys. Rev. D 101, 124059 (DOI:10.1103/PhysRevD.101.124059)
Reference Result: The declared catalog and holdout cases passed $L_2 \le 10^{-6}$ and mismatch $\le 10^{-8}$ gates against IMRPhenomNSBH.

2.10 Native SXS NR Higher-Harmonic Calibrations (IMRPhenomXHM & XPHM)

Primary References:
García-Quirós et al. (2020), IMRPhenomXHM: A multi-mode frequency-domain model for the gravitational wave signal from non-precessing black-hole binaries, Phys. Rev. D 102, 064002 (DOI:10.1103/PhysRevD.102.064002)
SXS Collaboration Catalog (Simulating eXtreme Spacetimes, CC-BY-4.0)
Ringdown Lorentzian Amplitude Fits: $$A_{\ell m}(f) = f^{-1/12} \frac{|A_\lambda| \sigma f_{\rm damp}}{(f - f_{\rm ring})^2 + (\sigma f_{\rm damp})^2} \exp\left[-\frac{\lambda}{\sigma}\frac{f - f_{\rm ring}}{f_{\rm damp}}\right]$$
$(2, 1)$: $A_\lambda = 0.057335 \eta \delta$, $\lambda = 0.887167$, $\sigma = 1.0$
$(3, 3)$: $A_\lambda = 0.203728 \eta \delta$, $\lambda = 0.771190$, $\sigma = 1.0$
$(4, 4)$: $A_\lambda = 0.098380 \eta^2$, $\lambda = 0.938766$, $\sigma = 1.0$
$(3, 2)$: $A_\lambda = 0.052026 \eta^2$, $\lambda = 0.846467$, $\sigma = 1.0$
Ringdown Phase Fits in $x = 4\eta - 1$: $$\phi_{\ell m}(f) = -\alpha_2 \frac{f_{\rm ring}^2}{f} + \alpha_\lambda \arctan\left(\frac{f - f_{\rm ring}}{f_{\rm damp}}\right) + \Delta t_{\rm rd} f + \phi_0$$
$(2, 1)$: $\alpha_2 = -30.0000 x + 10.0000$, $\alpha_\lambda = -4.6194 x - 4.3817$, $t_0 = -4.5672 M$
$(3, 3)$: $\alpha_2 = -7.9241 x + 37.1501$, $\alpha_\lambda = -0.0798 x - 1.1297$, $t_0 = -13.7830 M$
$(4, 4)$: $\alpha_2 = 3.6206 x + 63.1662$, $\alpha_\lambda = 5.8946 x - 0.4360$, $t_0 = -16.1609 M$
$(3, 2)$: $\alpha_2 = 5.8014 x + 39.0886$, $\alpha_\lambda = -1.4100 x - 1.8403$, $t_0 = -16.8168 M$
Verification Gates: The published default ringdown-cut gate checks Eq. (36)/Table 2 values and Jacobians for supplied QNM frequencies in both native frontends; its CPU FP64 receipt does not qualify QNM fits or exceptional $(3,2)$ cuts. The $(3,2)$ spheroidal ringdown phase-shape gate isolates Eqs. (46)–(47) on supplied frequencies and coefficients. The $(3,2)$ spheroidal alignment gate checks Eqs. (49)–(50) at the author-notebook reference frequencies for supplied fit targets. The four-target Eq. (48) solve gate verifies the conditional linear algebra in standalone Torch/JAX helpers; its local FP64 receipt does not qualify the paper-era fit surfaces or native production's alternate three-target calibration. The paper's Eq. (45) gives a non-mixing phase coefficient rescaling but does not specify the mode constants $w_{\ell m}$ there; the native per-mode fits above have separate SXS calibration provenance and are not certified as that rescaling. tests/test_xhm_native_fits.py defines phase-component checks using the reference amplitude, restricted ringdown frequency bands and affine phase/time alignment:
Seven $(3,3)$ holdouts (SXS:BBH:2265, 2483, 2484, 2486, 2489, 2491, 0185, with $3 \le q \le 10$): phase RMS $< 0.020$ rad and fitted time-shift magnitude $< 0.20 M$; tighter phase gates apply at $q \le 3.5$.
Two additional holdouts (SXS:BBH:0030, 0166, with $q=3,6$) for the four modes: phase RMS $< 0.15$ rad and affine mismatch $< 0.010$. These tests do not impose the $(3,3)$ arrival-time gate on every mode. These are test thresholds, not a new measurement. The tests skip when their external data or archived manifest are unavailable. Complete waveform amplitude, inspiral and precession accuracy are outside these phase-only checks.

2.11 Continuous NQC Spline Attachment (EOB components)

Primary References:
Bohé et al. (2017), An improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors, Phys. Rev. D 95, 044028 (DOI:10.1103/PhysRevD.95.044028)
Nagar et al. (2018), Time-domain effective-one-body gravitational waveforms for coalescing compact binaries with nonprecessing spins, tides and self-spin effects, Phys. Rev. D 98, 104052 (DOI:10.1103/PhysRevD.98.104052)
Mathematical Form: src/torchwave/approximants/eob/continuous_matching.py locates a unique interior maximum of a natural cubic spline through accepted orbital-frequency samples by solving $d\Omega/dt = 0$. The selected root is locally differentiable; interval selection is piecewise and unresolved or tied maxima are rejected. The attachment tests check continuity and independence from an auxiliary resampling spacing with the accepted samples fixed. In event-terminated trajectories, the two required decreasing samples place a natural spline boundary close to the peak; requested post-peak history can therefore change the spline peak and its parameter derivative. Integration-step and post-peak-history convergence, as well as complete EOB waveform accuracy, require separate checks; the frozen EOB LAL cases did not pass.

2.12 Clean-Room Kerr Quasinormal Mode Solvers & Generation (`torchwave.qnm`)

Primary References:
Leaver (1985), An analytic representation for the quasinormal modes of Kerr black holes, Proc. R. Soc. Lond. A 402, 285–298 (DOI:10.1098/rspa.1985.0119)
Cook & Zalutskiy (2014), Gravitational perturbations of the Kerr geometry: High-accuracy study, Phys. Rev. D 90, 124021 (arXiv:1410.7698)
London et al. (2018), First higher-multipole model of gravitational waves from spinning and coalescing black-hole binaries, Phys. Rev. Lett. 120, 161102 (DOI:10.1103/PhysRevLett.120.161102)
Mathematical Form:
Radial Equation: Spin $-2$ Kerr radial Teukolsky continued fractions solved via Cook & Zalutskiy Eqs. (23)–(44) with high-order asymptotic remainder conditioning.
Angular Eigenvalues: Computed via spectral matrix diagonalization of Cook & Zalutskiy Eqs. (49)–(56), supporting exact spherical branches and continuous branch continuation.
The independent angular projection check uses SymPy Wigner-$d$ functions and mpmath quadrature/eigensolutions to verify selected Torch/JAX spherical and complex-oblateness spheroidal projection values and angle derivatives. The JAX spheroidal $c$ derivative and a full mode-domain qualification remain open.
Full Kerr Table Generator (torchwave.qnm.generate / torchwave-generate-qnm): Generates all 16 positive-$m$ and negative-$m$ low overtones up to $n=2$ across $a \in [0, 0.99999]$ using adaptive degree-5 interpolating splines (src/torchwave/approximants/kerr_qnm_generated.py).
The Torch and JAX signed-spin 220 evaluators select the positive-$m$ branch at zero spin. Their branch masks preserve its independent-root derivative there without changing the spline coefficients or values.
London $N=61$ Polynomial Fitter (torchwave.qnm.london / torchwave-generate-london-qnm): Solves the explicitly declared, hypothesized $N=61$ grid $a = 0.995\sin\theta$ and fits clean-room complex polynomials for higher-harmonic ringdown models (src/diffgw_core/hm_qnm_n61_data.py, with independent Torch/JAX evaluators). The polynomial form follows London & Fauchon-Jones (arXiv:1810.03550v2, Eqs. (19)–(21)). The six-mode off-grid gate checks this named runtime artifact against fresh Kerr roots without asserting that the grid or coefficients reproduce historical author data.

2.13 IMRPhenomXPHM-SpinTaylor (Numerical Orbit-Averaged SpinTaylor Precession)

Primary References:
Pratten et al. (2021), Computationally efficient models for the dominant and sub-dominant harmonic modes of precessing binary black holes, Phys. Rev. D 103, 104056 (DOI:10.1103/PhysRevD.103.104056)
Racine (2008), Analysis of spin precession in binary black hole systems including quadrupole-monopole interaction, Phys. Rev. D 78, 044021 (DOI:10.1103/PhysRevD.78.044021)
Kidder (1995), Coalescing binary systems of compact objects to (post)5/2-Newtonian order. V. Spin effects, Phys. Rev. D 52, 821–847 (DOI:10.1103/PhysRevD.52.821)
Blanchet (2014), Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Relativity 17, 2 (DOI:10.12942/lrr-2014-2)
Mathematical Form & Implementation:
Orbit-averaged spin-precession equations for $\mathbf{S}_1, \mathbf{S}_2, \hat{\mathbf{L}}_N$ integrated in dimensionless velocity coordinate $v = (\pi M f)^{1/3}$ from PN energy flux $dv/dt = (c^3 / G M) \frac{32}{5} \eta v^9 [1 + \dots]$.
Precession Euler angles $\alpha(f)$ and $\beta(f)$ extracted from $\hat{\mathbf{L}}(f) = (\sin\beta \cos\alpha, \sin\beta \sin\alpha, \cos\beta)$, with minimal-rotation angle $\gamma(f)$ computed via $d\gamma/df = -\cos\beta (d\alpha/df)$.
Fixed-grid 4th-order Runge-Kutta (RK4) in $v$-space is expressed in PyTorch tensor operations. The model tests exercise batching and gradients for their declared parameters.
Verification Result: The optional LAL comparison test defines three configurations with prec_version=320, final_spin_mod=2 and a minimum PyCBC time/phase-optimized plus-polarization match of $0.98$ over 20–300 Hz. This is a test gate, not an observed lower bound across the parameter domain or a direct complex-waveform error. The 2026-09-29 reference-angle qualification verifies the native Torch/JAX grid invariant and reference gauge at f_ref=37.3 Hz; direct complex LAL waveform error remains about 25% for that configuration, so this is a component result only.

2.14 IMRPhenomXPNR (PhenomPNR Numerical Relativity Precession Angles)

Primary References:
Hamilton et al. (2025), PhenomXPNR: An improved gravitational wave model linking precessing inspirals and NR-calibrated merger-ringdown (arXiv:2507.02604v1)
Thompson et al. (2024), PhenomXO4a: a phenomenological gravitational-wave model for precessing black-hole binaries with higher multipoles and asymmetries, Phys. Rev. D 109, 063012 (arXiv:2312.10025v1)
Mathematical Form & Implementation:
The dedicated component ledger separates the published transition-spin map, fit polynomial and guarded angle transforms, angle attachment/continuation, tuned carrier/effective ringdown/asymmetry, higher-mode map, and twist-up.
A stdlib scalar reference independently encodes the printed map and component equations. Torch and JAX agree on those scoped values and derivatives after correcting the published $B_3=\chi\sin\theta[\exp(\Lambda_{B3})-200]$ transform. The standalone lower-$\beta$ boundary selector implements Final Twist Eqs. (56)--(58), XO4a's corrected $25\times10^{-4}$ gradient factor, and XO4a Eq. (37) from caller-supplied merger-ringdown landmarks; root/landmark construction, upper-boundary decisions, fallbacks, and public composition remain gated.
The public waveform still feeds reference-frequency spins into a map specified at the SpinTaylor stopping frequency and omits the published attachment/continuation and XHM-CP carrier modifications. It therefore remains a provisional native XHM composition, not the complete paper model.
The numeric $A_1..A_4$, $B_0..B_5$, $B_f$ tensors are not printed in the cited model papers. Their independent source and redistribution terms remain unresolved; identical Torch/JAX copies are not independent calibration evidence.
Hamilton et al. (2021), The final twist (arXiv:2107.08876v1), documents the 40-waveform predecessor's curve_fit/Levenberg--Marquardt and mvpolyfit procedure, fit intervals, sequential coefficient order and term ceiling. XO4a retunes to all 80 BAM waveforms and changes the polynomial basis, but does not publish the 80-case fitted targets, full run configuration, sparse term selection, or output tensor manifest.
The BAM minimal and complete releases (Zenodo 7673796, Zenodo 7677297) are CC BY 4.0 waveform inputs. The cited positive v1.0 archive carries an MIT license for the fitting tool. Neither license establishes terms for an unpublished fit output or the inherited arrays.
The author-controlled PhenomXCP workflow supplies partial carrier-fit artifacts, not the angle tensors or a final XO4a/XPNR u_k mapping. Its root MIT license coexists with a GPL-2.0-or-later notice on generated C and no separate notices on inspected data artifacts, so it is not blanket MIT evidence for outputs.
The inherited arrays first appear in the initial DiffGW commit ceb6ae15c00bbbd3eaca2bdc12bb3307b38e9beb; repository history provides no earlier origin or permission record. Exact source-to-component classes and remaining regeneration inputs are recorded in the machine-readable qualification JSON.
Verification Result: The scalar/Torch/JAX component suite passes 46 cases. Full-waveform qualification fails because the components above are missing and the underlying XHM strict reference sets remain 0/1 catalog and 0/1 holdout. The optional LAL comparison test defines four black-box configurations with a PyCBC time/phase-optimized plus-polarization threshold of $0.99$ over 20–400 Hz. That acceptance threshold is not an observed result, a complex-waveform check, or a domain bound.

2.15 ContinuousWave H1 direct-class slice

The versioned specification connects the monochromatic 2f equations of Jaranowski, Królak and Schutz to an independent mpmath scalar and selected Torch/JAX direct-class strain and derivative gates. LIGO T980044-10 §2.1 and Table 4 independently supplies the H1 Earth-fixed and geodetic vertex and arm directions at survey precision. IERS Bulletin C49 fixes the selected GPS–UTC offset, while IERS EOP 14 C04 exposes the native UTC-for-UT1 sidereal-time approximation at one epoch. The tight whole-class algebraic gate still conditions on inherited native site digits; the separate source-site gate does not establish those digits, a Solar System ephemeris, or physical timing accuracy.

3. Reference Evidence by Model

These counts describe the historical catalog and holdout artifacts. A passing row is scoped to its recorded parameters and modes. The complete strict suites did not pass; separate native component tests do not change those results. The current model manifest retains case IDs and numerical errors for supported routes; the retired snapshot preserves the removed model entries.

Approximant	Catalog cases passed	Holdout cases passed	Scope
TaylorF2	4/4	—	Declared aligned-spin/tidal cases
TaylorF2_Eccentric	2/2	—	Historical circular and spinning configurations; separate nonspinning holdouts below
IMRPhenomD	1/1	3/3	Declared aligned-spin cases
IMRPhenomPv2	1/1	1/1	Declared precessing cases
IMRPhenomXP	1/1	1/1	Supported MSA path; additional domain limits in manifest
IMRPhenomHM	1/1	1/1	Declared multimode cases
IMRPhenomXAS	1/1	2/2	Declared aligned-spin cases
IMRPhenomD_NRTidal	1/1	1/1	Declared tidal cases
IMRPhenomD_NRTidalv2	1/1	1/1	Declared tidal cases
IMRPhenomNSBH	1/1	1/1	Declared NSBH cases
KerrRingdown	1/1	0/0	Canonical $(2,2), n=0$ physical-energy waveform; no Kerr case in the historical holdout artifact
SEOBNR_ROM	1/1	—	Official external-data wrapper
IMRPhenomXHM	0/1	0/1	Full-waveform reference gates not met; native phase tests have separate scope
IMRPhenomXPHM	0/1	0/1	Full-waveform reference gates not met; native phase tests have separate scope
SEOBNR_ODE (retired prototype)	0/1	—	Historical result; model no longer shipped
TEOBResumS (retired prototype)	0/1	—	Historical result; model no longer shipped
TaylorF2_HM	1/1	—	Only $(2,2)$ compared; higher modes lack a full-waveform oracle in this suite

Separate evidence does not change these historical counts. The two nonspinning TaylorF2_Eccentric holdouts passed in Torch and JAX for the documented amplitude-off low-eccentricity slice. The maintained test_fixed_off_axis_energy_onset_holdout in tests/test_kerr_ringdown_public_clean.py checks one additional Torch public dispatcher case against SimBlackHoleRingdown and converged numerical Fourier integration. It has no persisted strict execution receipt; its complete gate requires LALSimulation and PyCBC. It does not qualify the separate direct-class or opt-in public-clean routes.

IMRPhenomXPHM-SpinTaylor and IMRPhenomXPNR are outside these frozen catalog counts. Their optional comparison test gates are described in Sections 2.13–2.14. Neither the frozen suite nor this table represents all current models.

4. Machine-Readable Asset Checksum Ledger

Asset	Path	SHA256 Checksum	Provenance / Origin	License
Independent Kerr QNM Splines	`src/torchwave/approximants/kerr_qnm_generated.py`	`e69d64e615dfd7bf054c2c7a9b721509688a1f126a482c7d9d0047f53b370dd3`	Clean-room solved via `torchwave.qnm.generate`; signed-spin branch autodiff corrected at zero	MIT
Independent London N=61 QNM Fits	`src/diffgw_core/hm_qnm_n61_data.py`	`c0283d9c5990e6f54944a7e81ff0007a0707e56c048d5a0066434c53f42da94a`; unchanged numerical payload `afcb21b634efacbc1ed0b44f24fda5df5ab08e236f3b7cbfdc7e71d3452d6525`	Clean-room solved via `torchwave.qnm.london`; shared data, separate evaluators	MIT
Printed IMRPhenomD Calibration	`src/diffgw_core/phenomd_published_fits.py`	`ab5b3e6ae9b1954ad04addb8e9175e87383d0575d81aac5f26ac9d70ed33e5a2`	Khan Table V; Husa PDF Eqs. (3.6)–(3.8), printed numerical values	DiffGW transcription: MIT; cited mathematical data
Inherited PNR Calibration Coefficients	`src/torchwave/approximants/pnr_coefficients.py`	`9d5d77749d187f64447f1ed0501311370b5c97a3e8a471de71ced315ce0394fd`; canonical numeric payload `c7794b653f781bbba45b97170fdac9fb6c89de6cc9c404e4942908ce1f820d68`	Numerical tensors are not printed in the XPNR/XO4a papers; independent origin unresolved	Redistribution terms unresolved; see third-party notices
APS IMRPhenomXAS Fits	Supplemental Material (Pratten et al. 2020)	`94b9334c19f3a7909864891ea278ab9a411e4f33b7b7e17cc01d927056d9ac52`	APS PRD Supplemental Material deposit	Redistribution terms unresolved; see third-party notices
External Official SEOBNRv4ROM HDF5	Zenodo record 14999310 (`SEOBNRv4ROM_v3.0.hdf5`)	`4ff5317967d757fcad9323a74b75f4f63894ed87f4b1852d46cbe47cde488b9c`	Official public Zenodo release	Redistribution terms unresolved
Native SXS NR Calibrations	`src/torchwave/approximants/xhm_native_fits.py`	Native code module	Calibrated on SXS public BBH catalog	Software: MIT; source simulations: CC-BY-4.0

5. Third-Party Data Attribution

The native calibration surfaces in torchwave were trained on numerical relativity simulations produced by the Simulating eXtreme Spacetimes (SXS) Project (http://www.black-holes.org). These simulations are released under the Creative Commons Attribution 4.0 International (CC-BY-4.0) license. Complete data citations, dataset terms, and catalog IDs are documented in THIRD_PARTY_NOTICES.md.

6. Documented Differences from LALSimulation

Model-specific formulations, reference configurations and limitations are described on Differences from LAL.

Native IMRPhenomT family (2026-09-21)

Physics: Estellés et al., Phys. Rev. D 103, 124060 (2021), arXiv:2004.08302, DOI.
Updated dominant-mode ansatz: Estellés et al., arXiv:2012.11923, Eqs. 14–31.
Calibration and reconstruction reference: the MIT-licensed author repository phenomxpy, pinned revision. Shared calibration expressions include the supported (2,2), (2,1), (3,3), (4,4) and (5,5) branches of fits.py; internals.py and numba_ansatze.py provide the reconstruction conventions. This is an attributed adaptation of permissive author math, not a claim that the numerical calibration coefficients were independently derived from the paper. The complete MIT notice is retained in THIRD_PARTY_NOTICES.md.
The inspiral fit-input specification pins this MIT author asset by revision and SHA-256 and verifies its t0 and five fitted frequency outputs on a fixed grid against shared fits and both native carriers. The paper does not publish the fitted coefficient tables, so this is source transcription evidence rather than an independent physical recalibration.
Runtime: framework-native tensor operations and differentiable matching in both frontends, with backend-independent algebraic fits in diffgw_core. No LALSuite, SciPy or phenomxpy waveform calls are made at runtime.
Scope: T, THM, TP and TPHM for black hole binaries. THM includes mode pairs (2,2), (2,1), (3,3), (4,4) and (5,5). TP/TPHM implement published NNLO, multiple-scale analysis (MSA), and numerical precession. Tidal corrections are absent.
Precession reference: arXiv:2105.05872. The native rotation uses published Euler-angle and Wigner-matrix formulas with the repository's NNLO tensor implementation. MSA uses the published two-spin equations of Chatziioannou et al. and the repository's existing clean-room algebra. Numerical precession uses the orbit-averaged spin equations through N4LO in Sturani's technical note, LIGO-T1500554, with the PhenomT frequency evolution. The numerical PhenomTP path omits the N3LO quadratic-spin contribution (Eq. 24), following the phenomtp parameter description in the public LAL API. The complete published equations remain available in the internal spin derivative helper. Leading self-spin terms and the nonspinning 3PN angular-momentum denominator are retained; binary-oracle residuals alone do not justify altering those equations. The upstream phenomxpy numerical and MSA modules identify GPL-derived code and were excluded from adaptation.
Independent qualification references: diffgw_reference/phenomt_numerical_trajectory.py composes the paper-derived scalar RHS with a Brent event, DOP853 solve, tangent transport, and Euler reconstruction without native trajectory helpers. diffgw_reference/phenomt_angle_continuation.py independently implements the minimal-rotation, MECO, and ringdown scalar rules. The q=10 receipts retain the failed whole-waveform gates. Full-grid onset checks localize PV223 before MECO to the raw analytical angles. For PV300/FS4 they establish a pre-merger orientation/rotation residual after the reference event and exclude projection and ringdown, but the public Euler-angle output does not expose internal trajectory state: carrier/time mapping, trajectory evolution, and angle construction remain unresolved alternatives. They introduce no production equation change or coefficient fitted to public binary output.
Verification: LALSimulation is used only through its installed binary Python bindings as an optional black-box oracle. No coefficients or residuals are fitted to those outputs, and no GPL implementation source is used. Backend and behavioral tests are in tests/test_imrphenomt.py, tests/test_phenomt_public.py and tests/parity/test_phenomt_parity.py. tests/test_phenomt_reference.py retains numerical values from the pinned MIT author implementation and optional LAL binary comparisons spanning equal masses, mass ratio 10 and spins from -0.9 to 0.9. Measured LAL residuals are recorded in Differences from LAL. FP64 cross-backend tests require relative L2 error at most 1e-11 and mismatch at most 1e-14, including the high-spin mass-ratio-10 JIT case. Phase-sensitive mass/spin gradients are checked against finite differences in Torch and against Torch derivatives in JAX. Higher-mode fixtures from the same pinned author release are retained in tests/test_phenomthm_reference.py; fit regressions are in tests/test_phenomthm_fits.py. Family dispatch, remnant-spin propagation, precession, and optional LAL comparisons have dedicated test_phenomt_family_*, test_phenomt_remnant.py, and test_phenomt_precession.py suites. Physical-time invariance under grid changes and time translations is checked in test_phenomt_time_grid.py. Cross-backend higher-mode and precession tests also cover JAX compilation and parameter gradients.

The additional prescription tests cover method/version selection, matching remnant spins, batch behavior, reference-time anchoring, and invariance under changes to the requested output grid. The NNLO default is retained for compatibility; numerical precession is selected explicitly with precession_method="numerical".