IMRPhenomXHM higher-mode amplitude closure

Scope and status

This component closes only the printed post-Newtonian baseline of the XHM higher-mode inspiral amplitude. Its stable IDs are xhm.amplitude.inspiral.appendix-e.v1 and xhm.amplitude.inspiral.pn-normalized.v1. An independent scalar reference, the clean Torch complex implementation, and the production Torch and JAX magnitudes agree for modes (2,1), (3,2), (3,3), and (4,4).

This is not a complete inspiral amplitude model. It excludes fitted collocation surfaces, collocation vetoes, release-dependent transition cuts, intermediate and ringdown calibration, and (3,2) mode mixing. It therefore does not promote XHM or XPHM.

Inputs, conventions, and options

The scalar reference accepts a positive dimensionless frequency Mf, eta, positive-mass-ordering delta, and the aligned-spin combinations chi_s and chi_a. It supports only the four displayed positive-m modes listed below. It intentionally exposes no release, fit-provider, cutoff, veto, or mode-mixing option.

Mathematics and components

The source is García-Quirós et al., arXiv:2001.10914v2, Sec. IV A and Appendix E. With

\[ v_m = \left(\frac{2\pi M f}{m}\right)^{1/3}, \]

the scalar reference transcribes every displayed term through v_m^6 in the Appendix-E expressions for Hhat_21, Hhat_32, Hhat_33, and Hhat_44. The normalized PN baseline is

\[ \mathcal H^{\rm PN}_{\ell m}(f) = \left(\frac m2\right)^{7/6} \left|\widehat H_{\ell m}(v_m)\right|. \]

Writing u = f / f_ins, the paper's complete inspiral ansatz adds three pseudo-PN terms,

\[ \mathcal H_{\ell m}(f) = \mathcal H^{\rm PN}_{\ell m}(f) + \alpha_{\ell m} u^{7/3} + \beta_{\ell m} u^{8/3} + \gamma_{\ell m} u^3, \]

whose coefficients are fixed from fitted values at [0.5, 0.75, 1.0] f_ins. The printed equations specify the solve after those values are supplied; they do not publish a complete current-release parameter-space calibration.

Implementation mapping

Component	Scalar	Torch	JAX	Claim
complex Appendix-E polynomial	`printed_hhat`	`hhat_lm_printed`	magnitude only	printed equations
normalized PN magnitude	`normalized_pn_amplitude`	production `_pn_amplitude_ratio`	production `_pn_amplitude_ratio`	printed equations
pseudo-PN collocation solve	equation helper	clean helper	not routed	supplied nodes only
fitted current-release nodes	unavailable	inherited native surfaces	inherited native surfaces	no independent asset

The scalar module has no Torch, JAX, LAL, fit-table, or waveform-runtime dependency. Frozen complex checkpoints retain phase-sensitive information before either production implementation takes the magnitude. The frontend test also covers Torch frequency gradients and JAX JIT.

Evidence and limitations

Earliest mismatch

The binary-only regional diagnostic evaluates public installed LAL mode outputs without inspecting source or using their values as fit targets. On the exact stacked base 576f325f8da292ecb550ffc7ebde90e71399bc07, the earliest material discrepancy is higher-mode inspiral amplitude, and (3,2) is worst in both frozen cases:

Mode	Catalog inspiral amplitude relative L2	Holdout inspiral amplitude relative L2
`(2,1)`	0.21601	0.02318
`(3,2)`	0.47847	0.10595
`(3,3)`	0.17294	0.08986
`(4,4)`	0.39970	0.08534

By contrast, higher-mode inspiral phase RMS after an independent affine phase/time gauge removal is 4.54e-5--1.33e-4 rad across these rows. This does not qualify the complete phase: intermediate phase residuals are 6.24--28.80 rad, while intermediate amplitude relative L2 errors are 0.77--7.04. Ringdown residuals also remain nonzero, and the finite 512 Hz grid does not reach the catalog (4,4) ringdown region. Exact values and environment versions are in the component receipt.

Why production is unchanged

The new scalar checks show that both production frontends already implement the printed Appendix-E baseline. Changing those coefficients would contradict the cited equations and would not supply the missing fitted collocation data. The maintained public-API diagnostic also distinguishes the installed default release 122022 from the explicit author-era 122019 control; their amplitude outputs are not interchangeable. A 20-row clean-assembly run records result SHA-256 57230acd...1826 and raw-array SHA-256 a4379774...0f32 in the component receipt. The full arXiv v2 source archive was checked before changing production. Its SHA-256 is cc05e7dbff7bf0c42a10b7da1f4840eefc55b252168d6bfb085155a7bdc5318e; it contains the author amplitude/phase notebooks, 37 amplitude .m tables (21:10, 32:9, 33:9, 44:9), and 33 phase .m tables (21:7, 32:12, 33:7, 44:7), eight mixing-coefficient files, and the fit README. Its README describes those as the fits for the paper implementation, but the archive has no release manifest tying them to default release 122022, its all-new amplitude option, or a complete current-release collocation provider. Existing checksum-pinned fixtures verify selected paper-era surfaces only. Paper-era (3,3) files are present; their mapping to the recalibrated 122022 provider, together with the other modes' current-default amplitude calibration, remains unresolved. No complete standalone XHM 122022 calibration asset is registered in external data.

Consequently, routing the existing earlier-provider nodes, compatibility sign branch, or older transition prescription into production would create an unvalidated hybrid. A justified fix requires an author-published, checksum- pinned current-release fit asset or an independently reproducible calibration dataset and procedure. Neither is available here, so this change makes no production waveform modification.

Paper-era versus release-era boundary

The author-published arXiv v2 archive is a coherent paper-era asset: its checksum-pinned 218-member source bundle contains 37 amplitude-fit files, 33 phase-fit files, 8 complex mixing-coefficient files, 10 QNM files, and the author notebooks. Appendix A, Eqs. (55)--(57), publishes the geometric spheroidal-to-spherical transform for (2,2) and (3,2) and states that its mixing coefficients depend only on final spin. These sources are sufficient to define a clean paper-era mixing reference; they are not evidence that a current-release (3,2) spheroidal amplitude is the same object.

The official LALSuite 7.26 API documentation draws the release boundary more sharply. It identifies 122022 as the default after a higher-mode amplitude recalibration, retains 122019 as the old release, separates amplitude region/collocation frequencies, fit surfaces, and ansatz versions, and states that 122022 did not modify phases. The corresponding author merge request !2128, merged 2023-04-06 at commit 9eaa3722, says the recalibration propagates to XPHM. Its linked review/calibration wiki redirects unauthenticated readers to sign-in and no checksum-pinnable 122022 calibration deposit was found in this bounded search.

Therefore the exact missing public fields are:

release-122022 amplitude fit surfaces or coefficients for inspiral, intermediate, and ringdown collocation values for (2,1), (3,2), (3,3), and (4,4);
their region-cut and collocation-frequency maps, veto/branch rules, and amplitude-ansatz version mapping;
a hashed, licensed manifest tying those assets to release 122022; and
for the complete current (3,2) mode, the recalibrated spheroidal amplitude inputs used before the published basis transform.

The paper-era phase equations, phase assets, and mixing geometry remain the best public map because the release notes say phases were unchanged, but no public release manifest proves file-for-file identity. During this audit a broad web search unexpectedly exposed generated-source search snippets. No snippet, LAL source formula, coefficient, or implementation detail was used; the mapping above comes only from the papers, author archive, official API documentation, public merge metadata, and existing binary-output receipts. The arXiv page identifies its manuscript/source distribution license, while the inspected fit-asset README gives no separate redistribution terms for the tables. No accessible 122022 calibration deposit with version and reuse terms was found. Any future clean-room implementation of the 122022 delta must start in a fresh context with an independently published calibration asset.

The separate release-boundary diagnostic checks the public 122019 and 122022 binary interfaces on both frozen parameter cases. The (2,1), (3,3), and (4,4) spherical phases, and the (3,2) spheroidal phase, are bitwise equal on its grid. The (3,2) spherical phase is not: the recalibrated spheroidal amplitude enters the published Appendix-A mixing transform. After removing one reference-phase offset, its wrapped phase-shape RMS changes by 0.1771 and 0.03565 radians for the catalog and holdout cases. Thus “phases unchanged” identifies the underlying phase prescription, not invariance of the observable mixed mode. This is a binary-output diagnosis only; it supplies no missing amplitude fit.

Intermediate, ringdown, and native calibration boundary

The paper supplies analytic ansätze for intermediate and ringdown regions, but their mode-dependent coefficients are calibrated to hybrid/NR data. The native implementation intentionally uses its independently maintained SXS fit surfaces and native Kerr-Leaver QNM frequencies. Those choices are physical calibration paths, not numerical reconstructions of the installed public LAL release; agreement with the printed functional form therefore does not imply binary parity. Mode (3,2) additionally requires spherical-spheroidal mixing.

The component receipt is descriptive evidence, not a new acceptance gate. The strict frozen full-waveform catalog/holdout results remain 0/1 for XHM and 0/1 for XPHM in both sets, hence red.

The exact PR-base qualification rerun at 4814835f7744fe00a5863ed70eb590d41a7e3ac3 is recorded in the release-qualification receipt. It reproduced the (3,2) inspiral-amplitude relative L2 values 0.47847072820479275 (catalog) and 0.10594593969797324 (holdout). The strict whole-complex-mode holdout also fails: direct complex relative L2 is 1.5613045346149474 and flat mismatch is 0.617904191209656, against 1e-6 and 1e-8 limits. The qualification verdict is therefore no-release, with no production waveform, transition, or gate change.

Reproduce the checks

Run the paper/scalar/frontend checks with:

JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu PYTHONPATH=src \
  pytest -q tests/test_xhm_inspiral_amplitude_math_reference.py

The regional decomposition above is a historical, one-off binary diagnostic; its replay script is not maintained. With the pinned optional LAL Python bindings installed, run the maintained frozen binary gate and clean inspiral assembly diagnostic with:

JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu PYTHONPATH=src \
  python scripts/qualify_xhm_xphm_frozen_binary.py

PYTHONPATH=src python scripts/diagnose_xhm_inspiral_assembly_clean.py