IMRPhenomXHM (3,2) four-target spheroidal ringdown phase solve
Scope and status
xhm.phase.32.spheroidal-four-target-solve.v1 is the conditional algebra of
García-Quirós et al.'s Eq. (48): solve the four coefficients in the published
Eq. (46) phase derivative from caller-supplied derivative targets. This
does not supply or qualify the target fits, the production calibration, or a
complete (3,2) mode.
Freeze CPU FP64 coefficient and collocation-residual tolerances
rtol=2e-11, atol=5e-10, and full input-Jacobian tolerances
rtol=2e-9, atol=5e-7 before comparing any native helper. The independent
scalar reference uses 70 decimal digits. Torch eager and JAX JIT must pass
on the same rows. A full-rank, reasonably conditioned scaled matrix is a
precondition, not a claim about all physical parameters.
Inputs, conventions, and options
Inputs are (r22,r32,d32,G1,G2,G3,G4), with positive dimensionless Mf
QNM frequencies and four phase-derivative targets in Table 9 order. The
returned coefficients are (alpha2,alpha4,alpha_lambda,dphi_RD). No phase
integration constant is solved by Eq. (48). The frozen rows are:
| Row | r22,r32,d32 |
G1,G2,G3,G4 |
|---|---|---|
| 1 | .080,.095,.012 |
42,39,35,31 |
| 2 | .082,.110,.014 |
-10,-9,-7,-6 |
| 3 | .070,.087,.010 |
15,16,18,20 |
These are synthetic supplied targets, not author-fit evaluations. Row-wise
Jacobians cover all seven inputs; derivatives across rows must vanish. A
coincident-node case r22=r32-1.5*d32 must fail explicitly in the scalar
reference and Torch eager helper; JAX JIT accepts only prevalidated domain
inputs. A nearby distinct-node case is diagnostic for conditioning, not a
production claim.
Mathematics and components
García-Quirós et al., arXiv:2001.10914v2, Eqs. (46) and (48), Table 9 specify
f = (r22, r32-1.5*d32, r32-0.5*d32, r32+0.5*d32)
G_i = alpha2/f_i² + alpha4/f_i⁴
+ alpha_lambda*d32/[d32²+(f_i-r32)²] + dphi_RD.
To expose numerical rank without changing this equation, let x_i=f_i/r32
and delta=d32/r32. The dimensionless matrix has row
B_i=(x_i^-2, x_i^-4, delta/[delta²+(x_i-1)²], 1) and solves
B b=G, where
b=(alpha2/r32²,alpha4/r32⁴,alpha_lambda/r32,dphi_RD).
Recover the printed coefficients by multiplying b by
(r32²,r32⁴,r32,1). The gate reports rank(B) and spectral condition
number kappa_2(B)=sigma_max/sigma_min on every frozen row. Positive,
distinct nodes and nonzero determinant are required; closeness to a
coincident node can still make the solve poorly conditioned.
Implementation mapping
The independent 70-digit solve belongs in
diffgw_reference.imrphenomxhm_transition_mixing. The existing
torchwave.approximants.xhm_phase_clean offers a standalone four-target
helper; a JAX counterpart may be checked against the scalar reference.
Neither standalone helper is called by the production waveform. The
production Torch and JAX branches currently use targets at
(r22,r32,r32+0.5*d32) and fix alpha2_32=alpha2_22, so agreement with
the conditional four-target solve cannot qualify that branch.
Evidence and limitations
The source archive for arXiv:2001.10914v2 has SHA-256
cc05e7dbff7bf0c42a10b7da1f4840eefc55b252168d6bfb085155a7bdc5318e.
Its author phase-fit notebook
(SHA-256 48cc263ab9f2837220cd11fbfd99205562b8863af077074f46d730e35382108b)
contains expressions for all four paper-era targets. The ancillary .m
directory instead exports dphiS_p1.m, dphiS_p4.m, and an alternate
dphiS_fring32.m; its 3Dfit_p1_lm_32.m through p4 are intermediate-phase
targets. The arXiv non-exclusive distribution license
does not establish MIT redistribution permission for the ancillary fits.
This component embeds no fitted expressions or values.
The tests can establish Eq. (48) linear-algebra and differentiation fidelity
for supplied targets only. They cannot establish that production has the
four required fit surfaces, a release-specific mapping from paper-era fits,
or a physically validated (3,2) ringdown phase.
Reproduce the checks
The test needs mpmath, numpy, torch, jax, and pytest. It imports no
ancillary fit asset. Changing node order, coefficient basis, or calibration
requires a new component version.