IMRPhenomXAS mathematical specification

Definition field	Catalog value
Model ID	IMRPhenomXAS
Reference composition	imrphenomxas-composed-polarizations-carrier/v1
Scope	Aligned-spin (2,2) carrier composition with supplied fit and QNM outputs.
Domain and modes	frequency; (2, 2)
Native backends	torch, jax
Scoped verification claims	0

Option	Default	Native choices	Reference-only choices
intermediate_amplitude	carrier_quartic	carrier_quartic	paper_quartic, paper_fifth_order
intermediate_phase	carrier_sign	carrier_sign	printed_sign
branch_joins	lower_branch_at_join	lower_branch_at_join	None

Scope and status

This page covers the aligned-spin dominant-mode amplitude, phase, and complex polarizations. Generate IMRPhenomXAS with either backend through the frequency-domain API. Higher-mode and precessing X-family models have different scope. The reference distinguishes paper-literal formulas, current Torch/JAX carrier conventions, and selected checks using a separately supplied author fit asset.

Inputs, conventions, and options

The public carrier uses component masses in solar masses, aligned spins, distance in Mpc, inclination and orbital phase in radians, time shift in seconds, and an Hz frequency grid. Intrinsic formulas use the dimensionless frequency u=M_seconds*f_Hz. Calibrated fit outputs, QNM frequencies, and join locations are supplied to the independent scalar composition. The generated table above identifies the native quartic/sign choices and the paper alternatives that remain reference only. SI scaling, phase origin, polarization signs, and the f<=0 mask are specified in composed complex polarizations. For direct Torch/JAX model calls, f_ref defaults to absent. Both frontends accept window_type, phasing_convention, and reference_frame for API compatibility but do not use them to choose a window, phase prescription, or frame. Once an explicit frequency grid is supplied, direct calls also ignore f_lower, f_final, and delta_f; the public waveform wrapper can use frequency limits when constructing that grid.

Mathematics and components

Quartic intermediate amplitude

Specification imrphenomxas-intermediate-amplitude-quartic/v1 covers one component of the nonprecessing dominant-mode model. Its source is Pratten et al., arXiv:2001.11412v1, Sec. VI.B, Eqs. (67), (71), (79)–(83), and Table 2. The paper generally uses the fifth-order alternative, Eq. (72), for its reported default results.

Let f denote dimensionless Mf and f1 < f3 the intermediate-region boundaries. Define t=(f-f1)/(f3-f1) and

P(t)=c0+c1*t+c2*t²+c3*t³+c4*t⁴.

Eq. (71), with Eq. (67), then reads A(f)=A0(f)/P(t) where A0(f)=K*f^(-7/6) and K is independent of frequency. t is only a change of polynomial basis from the paper's powers of f; it does not change the quartic family. Define B(f)=A(f)/A0(f)=1/P(t). The Table-2 conditions are

B(f1)=v1, B((f1+f3)/2)=v2, B(f3)=v3, B'(f1)=d1, and B'(f3)=d3.

For y_i=1/v_i, h=f3-f1, and s_i=-h*d_i/v_i², the five linear constraints on P are P(0)=y1, P(1/2)=y2, P(1)=y3, P_t(0)=s1, and P_t(1)=s3. The SymPy kernel solves this matrix system with exact scalar inputs. For physical amplitude conditions from Eqs. (79)–(83), it first applies B=A/A0 and B'=(A'+7*A/(6*f))/A0. The domain is 0<f1<f3, positive collocation amplitudes, and positive K; no clamp or extrapolation is specified by the paper. Derivatives use Mf, so slopes in hertz require a total-mass conversion.

The paper's Eq. (81) labels the right endpoint f4, while Eq. (83) and Table 2 use f3. This component follows Table 2's three-node quartic definition. It does not use the APS parameter-fit supplement or any LAL source or constants.

Component ID	Evidence	Torch/JAX relationship	Maintained check
phenomxas.amp.intermediate.newtonian_factor.v1	Eq. (67)	Waveform amplitude prefactor; full complex assembly is checked below	test_paper_quartic_and_physical_slope_constraints
phenomxas.amp.intermediate.quartic.v1	Eq. (71)	solve_intermediate_amplitude_quartic in Torch and JAX, algebraic helper only	test_frontend_quartic_against_scalar_matrix_solve
phenomxas.amp.intermediate.collocation.v1	Eqs. (68)–(69), (79)–(84), Table 2	Full carrier boundary values and slopes from the independent inspiral/ringdown shapes, then the quartic solve; fitted parameters are supplied inputs	test_full_frontend_intermediate_boundary_construction, test_midpoint_coefficient_derivative
phenomxas.amp.intermediate.constant_normalization.v1	Algebraic variant; pointwise LALSimulation binary check below, not the printed Eq. (71)	Full Torch/JAX carrier assembly	test_observed_quartic_and_normalization_distinction, test_full_frontend_intermediate_amplitude_shape, test_external_binary_intermediate_amplitude_fixture

The maintained tests check the paper equation and five physical constraints with a known quartic, then compare each native helper against the independent scalar matrix solve and its midpoint derivative. Separate tests evaluate each full Torch/JAX carrier at interior frequencies against the observed scalar convention and three pinned binary amplitude samples. These check assembly and sampled calibrated behavior, not whether the fits follow the printed paper equation.

The complete carrier boundary adapter starts from the Eq. (68)–(69) inspiral shape S_ins and Eq. (84) ringdown shape S_RD, with native fitted parameters supplied. At f1=f_T^A and f3=f_peak, its constant-normalized physical endpoint values are A1/K=f1^(-7/6) S_ins(f1) and A3/K=f3^(-7/6) S_RD(f3); their slopes are the derivatives of those full expressions. The independently supplied midpoint fit is A2/K=v_A. These five conditions determine P through the exact scalar solve. The maintained test checks the resulting coefficients against each carrier at four mass/spin points and checks all five reconstructed constraints. This closes the boundary-value algebra, conditional on the supplied calibration; it does not turn the carrier's K/P convention into the printed Eq. (71).

Normalization investigation. The waveform assembly supplies the quartic helper with endpoint values of scaled_amp/f^(7/6) and reconstructs scaled_amp_int=f^(7/6)/P. Multiplication by the physical Newtonian prefactor gives A_int=K/P, where K is independent of frequency. Its physical boundary slope is A'/K; its midpoint fit supplies A/K. The printed Eqs. (67) and (71) instead give A_int=K*f^(-7/6)/P, whose boundary conversion is B'=(A'+7*A/(6*f))/A0. These two quartic families are not interchangeable by changing constant coefficients: equal P would yield amplitudes in the ratio f^(-7/6).

The implementation convention has direct external evidence. With the pinned LALSuite 7.21 binary (LALSimulation 5.3.1), CPU FP64, default XAS settings, m1=m2=30 M_sun, zero spins, f_ref=f_low=20 Hz, f_high=512 Hz, distance=500 Mpc, face-on, and a 0.125 Hz grid, Torch and LAL agree across the 1,373 bins between Mf=0.02731805 and 0.07802608: magnitude relative L2 is 2.13e-14 and maximum pointwise relative error is 3.24e-14. At the midpoint Mf≈0.05267, the fitted collocation value is A/K≈29.14; multiplying it by f^(7/6) gives the waveform's Newtonian-normalized shape A/A0≈0.94. Applying the printed prefactor to the same polynomial would multiply the midpoint amplitude by about 31. This comparison identifies the implementation's normalization; it does not establish that the paper has a typographical error or validate the waveform against hybrid/NR data. Three raw binary amplitudes at grid indices 812, 1426, and 1760 are retained as an external-output fixture in the maintained test, separate from the paper-derived scalar reference.

The executable reference therefore retains both functions with separate names and evidence. The printed equation remains a literature reference; the K/P function is a binary-checked implementation reference. Whether the authors intended a different normalization for Eq. (71), or used a distinct convention during calibration, remains unresolved without further author clarification or independent calibration evidence. Neither the published parameter-space fits nor any LAL source or constants have been copied into this component.

Fifth-order paper option

Specification imrphenomxas-intermediate-amplitude-fifth-order/v1 records the paper's Eq. (72), Eqs. (73)–(78), and Table 1 as a reference-only component. The paper uses this option for its reported results, but the current Torch and JAX carriers implement the quartic option above. No frontend or LAL parity is claimed for the fifth-order component.

For four supplied nodes 0<f1<f2<f3<f4, define t=(f-f1)/(f4-f1) and P5(t)=c0+c1*t+c2*t²+c3*t³+c4*t⁴+c5*t⁵. The printed equation is A(f)=K*f^(-7/6)/P5(t). Let B=A/(K*f^(-7/6))=1/P5. Given the four physical amplitudes A(fi)=ai and endpoint slopes A'(f1)=d1, A'(f4)=d4, the scalar reference converts them to B(fi) and B'(fj)=(A'(fj)+7*A(fj)/(6*fj))/(K*fj^(-7/6)) for j=1,4. It solves the resulting six linear constraints on P5 exactly in SymPy. The maintained test recovers a known fifth-degree polynomial from all six physical constraints, including both derivatives.

Component ID	Evidence	Reference and frontend status	Maintained check
phenomxas.amp.intermediate.fifth_order.v1	Eqs. (67), (72)	paper_amplitude in the scalar reference; reference-only	test_paper_fifth_order_and_physical_slope_constraints
phenomxas.amp.intermediate.fifth_order_collocation.v1	Eqs. (73)–(78), Table 1	collocation_coefficients and physical_collocation_coefficients; reference-only	test_paper_fifth_order_and_physical_slope_constraints, test_fifth_order_reference_only_evidence

The caller supplies the four nodes and collocation amplitudes. The printed Table 1 labels f1 as f1^W and prints its two interior-frequency formulas in terms of f_T^A and f_peak; this component does not reinterpret those notations or import parameter-space fits. As with the quartic paper equation, this is a literal literature specification, not evidence that the production carrier uses the same frequency prefactor. Physical validation remains pending.

Inspiral and merger-ringdown amplitude

Specification imrphenomxas-inspiral-ringdown-amplitude/v1 independently transcribes Pratten et al., arXiv:2001.11412v1 into the SymPy reference. Frequency is dimensionless f=Mf. Eq. (67) gives A0=sqrt(2 eta/(3 pi^(1/3))) f^(-7/6). Eqs. (68)–(69) give A_Ins/A0=sum_{i=0}^6 A_i(pi f)^(i/3) + sum_{i=1}^3 rho_i(pi f)^((6+i)/3). The seven A_i are exact rational expressions in eta, ordered spins, and delta=(m1-m2)/(m1+m2) from Appendix A.1, Eqs. (120)–(126). The rho_i are calibrated inputs; this component does not infer their physical values from the paper or native fit constants. The physical domain used for the scalar comparisons has f>0, 0<eta<=1/4, and |chi1|,|chi2|<=1; the carriers order the masses so delta>=0. Ringdown requires f_damp>0, sigma>0, and a positive join frequency. The simple stationary-peak expression assumes 0<lambda<1.

The printed Eq. (70) collocation nodes are {1/2,3/4,1} f_MECO. The Torch and JAX carriers instead pass f_T^A=f_MECO+(f_ISCO-f_MECO)/4 (Eq. (52)) to the same three-node helper. These endpoints are generally different. Tests check exact recovery at the paper's nodes and helper algebra for a supplied endpoint. They do not claim the carrier endpoint agrees with printed Eq. (70).

Eq. (84) prints A_MR=a_R Gamma/((f-f_RD)^2+Gamma^2) exp[-lambda(f-f_RD)/Gamma], where Gamma=sigma f_damp. The reference also solves a_R from a supplied amplitude at one node. Differentiating Eq. (84) gives the stationary point f_RD+Gamma(sqrt(1-lambda^2)-1)/lambda when 0<lambda<1 and the result is positive. Printed Eq. (59) instead contains sqrt(1-lambda). The reference keeps both formulas separate; the carriers use the stationary formula with numerical guards. This does not establish which expression was intended in the paper's calibration.

The carriers use Eq. (84) as a scaled ringdown shape, multiplying it by f^(-7/6) in full waveform assembly. That differs from printed Eq. (84) as a physical amplitude. The maintained tests check the PN coefficients and d/d(Mf) against both native helpers, the collocation solve, the ringdown stationary point and one-node scale, and full carrier shape for inspiral and ringdown at four binary/spin points. The full-carrier checks use native fitted parameters as supplied inputs; they establish equation implementation, not independent calibration or physical validation against hybrid/NR waveforms. Phase, tapering, cutoff, and full waveform composition remain outside this component.

Component ID	Published evidence	Native relationship	Maintained check
phenomxas.amp.inspiral.pn.v1	Eqs. (67), (69), (119)–(126)	IMRPhenomXAS.pn_amp_func in Torch/JAX	test_frontend_pn_shape_and_derivative
phenomxas.amp.inspiral.pseudo_pn.v1	Eq. (68)	Full inspiral carrier assembly	test_full_frontend_inspiral_and_ringdown_shapes
phenomxas.amp.inspiral.collocation.v1	Eqs. (52), (68), (70)	solve_inspiral_residual_collocation; carrier endpoint differs from Eq. (70)	test_collocation_endpoint_distinction_and_exact_recovery, test_frontend_collocation_solve
phenomxas.amp.ringdown.lorentzian.v1	Eq. (84)	Ringdown expression in both carriers, then extra f^(-7/6)	test_ringdown_stationary_peak_and_scale, test_full_frontend_inspiral_and_ringdown_shapes
phenomxas.amp.ringdown.peak_printed.v1	Eq. (59)	Paper-literal reference only	test_ringdown_stationary_peak_and_scale
phenomxas.amp.ringdown.peak_stationary.v1	Derived from Eq. (84)	Carrier join frequencies within the guard-free domain	test_ringdown_stationary_peak_and_scale

Inspiral phase

Specification imrphenomxas-inspiral-phase/v1 records Pratten et al., arXiv:2001.11412v1, Eqs. (85)–(88) and Appendix A.2 Eqs. (129)–(138), in an independent SymPy reference. Its f is dimensionless Mf, with f>0, eta>0, aligned component spins, chi_s=(chi1+chi2)/2, and chi_a=(chi1-chi2)/2. The intrinsic phase uses v=(pi f)^(1/3) and 3/(128 eta v^5) sum_k phi_k v^k-pi/4, followed by the Eq. (85) four-coefficient default [sigma0+sigma1 f+(3/4)sigma2 f^(4/3)+(3/5)sigma3 f^(5/3) +(1/2)sigma4 f^2]/eta. Its derivative is checked symbolically. Coalescence time and phase are separate affine gauges. All fitted values are supplied inputs; this reference does not independently reproduce the parameter-space calibration.

Eqs. (86) and (129) display a sum through k=7, while Appendix A.2 also prints the 4PN spin-orbit tail phi8 in Eq. (138). Both native canonical carriers include phi8; the reference exposes both the printed truncated sum and the carrier's Appendix-extended sum. The existing aligned-spin TaylorF2 reference includes an additional 3.5PN spin-spin tail from Appendix A.3 that is absent from this XAS canonical Eq. (137), so it is not reused as an unchanged base here.

The four collocation nodes printed in Eq. (88) omit the lower-frequency offset at the two middle nodes: {fL, delta/4, 3 delta/4, fH}, where fL=0.0026, fH=1.02 f_MECO, and delta=fH-fL. Both carriers instead use {fL, fL+delta/4, fL+3 delta/4, fH}. The reference names both sets and solves the four-point basis 1, f^(1/3), f^(2/3), f exactly. The native values are {v3+d13, v3+d23, v3, v3+d43} and its coefficient scale is -5/(128 pi^(5/3)). This scale and the supplied fitted values are recorded as carrier conventions; they are not derived solely from the published Eq. (87) residual definition. sigma1 is restated by the derivative join to the intermediate phase and sigma0 enforces its value join.

The maintained tests compare the Appendix phase and analytic derivative with both Torch and JAX at zero, equal, and unequal spins, recover collocation coefficients, check fitted carrier coefficients, and test the inspiral phase and C1 join at four binary points. They remove the final carrier time and constant phase gauge before comparison. This verifies implementation of the published intrinsic equations and explicitly labeled carrier choices, not calibration or physical accuracy against hybrid/NR waveforms. Full waveform composition remains a separate component.

Component ID	Published evidence	Native relationship	Maintained check
phenomxas.phase.inspiral.pn.v1	Eqs. (86), (129)–(137)	pn_phase_canonical_*, d_pn_phase_canonical_* in Torch/JAX	test_native_pn_phase_and_analytic_slope
phenomxas.phase.inspiral.spin_tail_4pn.v1	Eq. (138)	Both canonical PN helpers include this term	test_printed_pn_tail_and_pseudo_derivative, test_native_pn_phase_and_analytic_slope
phenomxas.phase.inspiral.pseudo_pn.v1	Eq. (85)	Both full inspiral carrier branches	test_printed_pn_tail_and_pseudo_derivative, test_native_collocation_and_inspiral_join
phenomxas.phase.inspiral.collocation.v1	Eqs. (87)–(88)	Both carrier four-point solves with the shifted nodes and supplied fits	test_printed_and_carrier_node_distinction_and_exact_solve, test_native_collocation_and_inspiral_join
phenomxas.phase.inspiral.join.v1	Eq. (85), Sec. VII.1 continuity statement	Both carriers' sigma0/sigma1 and full inspiral branch	test_native_collocation_and_inspiral_join

Intermediate and merger-ringdown phase

Specification imrphenomxas-intermediate-ringdown-phase/v1 transcribes Pratten et al., arXiv:2001.11412v1, Eqs. (90) and (92)–(97), into an independent SymPy reference. Its frequency f means dimensionless Mf, with f>0, eta>0, and f_damp>0. All collocation values, ringdown frequencies, and fitted coefficients are supplied inputs. The reference gives both phase derivatives and their explicit antiderivatives; additive and affine phase gauges are kept separate.

Eq. (96) has a constant, powers f^(-1/3), f^(-2), and f^(-4), and a Lorentzian proportional to c0 a_phi. The five nodes of Eq. (97) begin at f_T^phi, put the fourth node at f_RD, and end at f_RD+1.25 f_damp. Eq. (90) has five inverse-power coefficients and a broadened Lorentzian with width 2 f_damp; its five nodes follow Eq. (92). Independent linear solves recover coefficients from supplied values of eta phi'(f) at those nodes. Eqs. (93)–(95) define the intermediate fitted-value offsets and the weighted second value used by the native carriers. The prose after Eq. (90) lists b5 among “five” coefficients; the displayed equation contains only b0 through b4, which is the basis specified here.

There is a convention difference: printed Eq. (90) has a minus sign before the intermediate Lorentzian, whereas the Torch and JAX carriers use a plus sign. The reference retains the printed expression and a separately named carrier-sign branch. In the native code the factor a_phi is absorbed into the fitted c0 supplied to both regions. The paper excerpt does not resolve the sign discrepancy; matching the carrier branch does not establish that it matches the printed equation.

The maintained tests check both primitives by symbolic differentiation, the sign difference, five-node collocation recovery, all five carrier ringdown nodes and the four intermediate nodes beyond the inspiral boundary, phase values for four binary/spin points against both frontends, and the value and derivative at their join. They remove the carrier's overall time and phase gauge before comparison. Inspiral TaylorF2 terms, the inspiral pseudo-PN fit, the calibrated parameter-space fits, and independent physical validation are outside this component.

For the native intermediate solve, the fifth fitted derivative is evaluated at f_T^phi, although the fifth collocation node is f_T^phi + delta_R/2. The maintained check preserves and exposes that choice; it does not derive it from Eq. (92). The fitted collocation values remain supplied inputs from the existing author-code-derived fit functions.

Component ID	Published evidence	Native relationship	Maintained check
phenomxas.phase.ringdown.ansatz.v1	Eq. (96)	Both carrier ringdown phase branches	test_printed_ansatze_primitives_and_sign_distinction, test_full_frontend_phase_branches_and_join
phenomxas.phase.ringdown.collocation.v1	Eqs. (96)–(97)	Both carriers' five-node solve	test_collocation_nodes_and_exact_coefficient_recovery, test_native_collocation_values
phenomxas.phase.intermediate.printed.v1	Eq. (90)	Paper-literal minus-sign reference	test_printed_ansatze_primitives_and_sign_distinction
phenomxas.phase.intermediate.collocation.v1	Eqs. (90), (92)–(95)	Both carriers' five-node solve with native sign	test_collocation_nodes_and_exact_coefficient_recovery, test_native_collocation_values
phenomxas.phase.intermediate.carrier_sign.v1	Eq. (90), sign distinction recorded above	Both carrier intermediate phase branches	test_printed_ansatze_primitives_and_sign_distinction, test_full_frontend_phase_branches_and_join
phenomxas.phase.intermediate_ringdown.join.v1	Antiderivatives of Eqs. (90), (96)	Both carriers' affine ringdown adjustment	test_full_frontend_phase_branches_and_join

Composed complex polarizations

Specification imrphenomxas-composed-polarizations-carrier/v1 combines the independently recorded amplitude and phase branches in the scalar reference. It accepts the carrier's calibrated fit outputs, QNM frequencies, and join locations as supplied inputs. No fitted parameter-space rows or native waveform implementation are imported into the reference. The preceding sections distinguish paper-literal from carrier conventions; this composition selects the carrier intermediate-amplitude normalization K/P, the carrier's plus sign for the intermediate phase Lorentzian, and the carrier's phase-fit normalization. Therefore it is a verification specification for the present Torch/JAX waveform, not a claim that those disputed choices follow from the printed paper.

Pratten et al., Sec. II.2, Eqs. (6), (14)–(16) define the positive-frequency mode h_2,-2=A exp(-i phi) and the angular factors (1+cos² iota)/2 and -i cos iota for plus and cross. Eqs. (66)–(67) give the dimensionless Newtonian mode amplitude. Restoring Fourier strain units with M_sec²/D_sec and the harmonic factor gives the common physical scale sqrt(5 eta/24) M_sec^(5/6) f_Hz^(-7/6)/(pi^(2/3) D_sec). This restoration is a dimensional derivation, not an equation printed in the paper. M_sec and D_sec use the independently audited exact SI/IAU definitions in the units specification; u=M_sec f_Hz selects the dimensionless intrinsic branches.

For f_Hz>0, the carrier assembles Psi=phi_raw(u)-phi_raw(u_ref)+tshift_sec*(f_Hz-f_ref) +2 pi f_Hz t_c-2 phi_c+3 pi/4 when f_ref is supplied. Without f_ref, the two reference terms are absent. The assembled polarizations are h_+=((1+cos² iota)/2) A_phys exp(-i Psi) and h_x=-i cos(iota) A_phys exp(-i Psi). Both are zero for f_Hz<=0. The exact f_ref subtraction, tshift_sec, and fixed 3 pi/4 are carrier gauges. The XHM companion paper, Appendix C documents related reference-frequency, orbital-phase, and waveform-time conventions; its mode-phase relation is distinct from the assembled polarization phase. The arbitrary upper reconstruction cutoff discussed in XAS Sec. V (Eqs. (64)–(65), \(Mf=0.3\)) is not an intrinsic cutoff in this carrier; public wrapper f_lower/f_final limits are outside this component.

The maintained test compares absolute complex plus/cross strain and unwrapped phase with both FP64 frontends at equal-mass zero-spin and unequal aligned-spin points. It samples both sides of every amplitude and phase transition, the ringdown, nonpositive frequencies, and f_ref absent or placed in each phase branch, with nonzero distance, inclination, time, and orbital phase. It also checks the exact polarization algebra, and Torch/JAX automatic derivatives against the scalar reference for four extrinsic parameters and smooth-branch frequency at inspiral, intermediate, and ringdown points. It does not validate parameter-space calibration, physical agreement with NR/hybrid waveforms, the fifth-order paper amplitude option, the public wrapper's user masks, or full-domain intrinsic derivatives. Selected mass-ratio and spin tangents are checked below.

The maintained pinned-binary whole-polarization gate compares Torch and JAX directly with the public LALSimulation 7.26.15 IMRPhenomXAS waveform-sequence API over every positive-frequency sample below the paper's \(Mf=0.3\) upper limit. It checks absolute complex \(h_+\) and \(h_\times\), without phase/time alignment, over a catalog case and two frozen unequal-mass spin holdouts. The public binary zeroes bins above that limit; the native carrier continues to evaluate there, so the gate explicitly tests the public cutoff and does not count its zero tail as intrinsic carrier error.

Component ID	Evidence	Native relationship	Maintained check
phenomxas.waveform.branch_selection.v1	XAS Eqs. (52), (71), (85), (90), (96); carrier join choice	Both XASCarrierState.eval_waveform implementations	test_full_complex_polarizations
phenomxas.waveform.physical_scale.v1	XAS Eqs. (14)–(16), (66)–(67), dimensional restoration; units	SI frequency/mass/distance adapter	test_full_complex_polarizations, test_polarization_and_extrinsic_algebra
phenomxas.waveform.phase_adapter.v1	XAS mode convention Eq. (14); carrier gauge	Reference subtraction, time/phase offset in both carriers	test_full_complex_polarizations
phenomxas.waveform.polarizations.v1	XAS Eqs. (15)–(16), azimuth fixed in carrier	Positive-frequency plus/cross signs	test_full_complex_polarizations, test_polarization_and_extrinsic_algebra
phenomxas.waveform.nonpositive_mask.v1	XAS positive-frequency mode Eq. (14); carrier mask	Both carriers output zero strain for f<=0	test_full_complex_polarizations
phenomxas.waveform.extrinsic_derivatives.v1	Product rule on the preceding equations	Closed partials at fixed intrinsic inputs	test_polarization_and_extrinsic_algebra, test_native_extrinsic_gradients
phenomxas.waveform.frequency_derivatives.v1	Product rule and chain rule on the preceding equations	Smooth-branch complex d(h+,hx)/df_Hz at fixed fits and f_ref	test_native_complex_frequency_gradient

Total-mass similarity and derivative

Specification imrphenomxas-total-mass-scaling/v1 extends the unchanged composed-polarization specification. Pratten et al., Secs. II.1 and V–VII, Eqs. (1), (14)–(16), (52)–(67) parameterize intrinsic fits and transitions by mass ratio, dimensionless spins, and u=M_sec*f_Hz. The SI amplitude factor above has mass dependence M_sec^(5/6) at fixed physical frequency and distance. The carrier's tshift_sec=M_sec*tshift is a separately identified phase-gauge choice. At fixed mass ratio and spins, calibrated dimensionless fits, QNM coordinates, and joins are unchanged when both component masses scale by lambda>0.

Consequently, away from user frequency limits and with distance, inclination, and orbital phase fixed, the Fourier strain obeys the exact similarity h(lambda*m1,lambda*m2;f/lambda,f_ref/lambda,lambda*t_c) =lambda²*h(m1,m2;f,f_ref,t_c) for each polarization. Omit both f_ref arguments when no reference frequency is supplied. This identity follows from the dimensional adapter and is not a fitted relation.

For a derivative instead holding physical f, f_ref, t_c, and distance fixed, set a=a(u) for the selected amplitude shape and phi=phi(u) for the selected raw phase. With S the SI amplitude scale above and C either polarization's inclination factor, the scalar reference computes

d h/d ln M = C*S*exp(-i Psi) * [(5/6)*a + u*a' - i*a*Psi_M],

where Psi_M=u*phi'(u)-u_ref*phi'(u_ref) +tshift_sec*(f-f_ref). The reference terms are absent when f_ref is absent. At f=f_ref, Psi_M=0 exactly. A prime here means differentiation with respect to dimensionless u. These formulas apply only to smooth positive-frequency branch interiors; neither f nor f_ref may lie on a moving join. The maintained test checks the similarity for both frontends at equal and unequal masses, and compares the closed full-carrier mass tangent against Torch/JAX automatic derivatives through carrier construction at unequal mass, across amplitude branches and all reference-frequency phase branches. This does not test mass-ratio or spin derivatives of the calibrated fit surfaces, or establish independent calibration or physical accuracy.

Component ID	Evidence	Native relationship	Maintained check
phenomxas.waveform.mass_scaling.v1	XAS Eqs. (1), (14)–(16), (52)–(67); dimensional similarity and carrier time gauge	Both XASCarrierState constructors and eval_waveform	test_total_mass_similarity
phenomxas.waveform.mass_derivatives.v1	Chain rule on the composed model and mass similarity	Full complex d(h+,hx)/d ln M through both native constructors	test_native_log_total_mass_gradient, test_mass_phase_tangent_cancels_at_reference

The same scalar composition also checks selected intrinsic tangents through the entire carrier. At q=5, chi1=0.5, chi2=-0.3, fixed total mass and physical frequency, the intrinsic derivative test uses centered shifts in each of q, chi1, and chi2 to rebuild the scalar waveform from the carrier's named fit/collocation outputs. It compares complex plus/cross and unwrapped-phase tangents with Torch and JAX automatic derivatives in the inspiral, intermediate, and ringdown regions, with a fixed reference frequency in the intermediate phase. The separate author-asset check below independently verifies the underlying fit-surface derivatives. This composition check supplies carrier state outputs at each shifted point; it is therefore a check of waveform assembly and automatic differentiation, not an independent derivation of every state-construction derivative or a full-domain intrinsic accuracy bound.

Component ID	Evidence	Native relationship	Maintained check
phenomxas.waveform.intrinsic_composition_derivatives.v1	Chain rule on the composed amplitude and phase; APS fit-surface derivatives checked separately below	Full complex and unwrapped-phase tangents through both native constructors	test_intrinsic_composition_derivatives

Implementation mapping

The component tables above map stable IDs and source locations to the independent scalar functions, both native carriers, and maintained checks. The full carrier selects the quartic K/P amplitude and carrier-sign phase; the printed quartic normalization, fifth-order amplitude, and printed phase sign have separate reference expressions. Torch and JAX use their own tensor implementations. The composition names the supplied calibration and QNM inputs; neither frontend evaluates through the scalar reference.

Evidence and limitations

The component tables and their linked tests distinguish source transcription, sampled scalar-to-native comparison, and external binary agreement. They do not establish independent calibration or physical accuracy. The following asset check applies only when the pinned author file is supplied.

Author fit-surface cross-check

The opt-in external-fit verifier reads the APS Mathematica companion file at run time, verifies its SHA-256 94b9334c19f3a7909864891ea278ab9a411e4f33b7b7e17cc01d927056d9ac52, and evaluates the selected source versions independently of either frontend. The file and its formula rows are excluded from this repository and package; its redistribution terms remain unresolved. Run with python scripts/verify_xas_author_fits.py --asset /path/to/IMRPhenomXAS_Parameter_Space_Fits_Public_Release_Candidate.m in an environment with both FP64 frontends and the reference dependencies.

The verifier maps all 25 non-precessing parameter-fit methods exposed by XASParameterFits to named source functions and explicit version selectors. The current quartic carrier calls 23 of them; fit_int_amp_v2 and fit_int_amp_v3 are available for the paper's fifth-order alternative but are not used by the current carrier. The deposited InsPhased53Fit surface is not used or implemented by the current inspiral phase. The verifier also checks the two (2,2) QNM rational fits. The separate fit_final_spin_nnlo method is a precession helper and is not called by this aligned-spin carrier. A structural check fails if either frontend adds a fit method without updating the map. Values are compared at nine mass-ratio/spin points, parameter derivatives (d/d eta, d/d chi1, d/d chi2) at two unequal-mass points, and QNM spin derivatives at five points. The scaled error divides by max(1, |source|); the gates are 2e-10 for values and 2e-8 for derivatives. On the pinned asset, 24 parameter surfaces and both QNM functions agree with Torch and JAX; the largest derivative scaled error is 3.05e-12 for the QNM real fit.

The remaining midpoint amplitude fit, IntAmpvAFit["104"], uses chiPNhat in the deposited file, whereas both binary-compatible carriers use sTotR. The source/carrier value difference reaches 0.042 on the declared cases. Re-evaluating the same deposited expression with the carrier's declared sTotR argument agrees in value and all three derivatives. This is an identified model choice, not agreement with the published source fit. The check verifies the deposited surfaces and their local derivatives; it does not reproduce the original NR calibration or prove physical accuracy.

Diagnostic evidence	Source	Native relationship	Maintained check
Parameter-fit surfaces	Pinned APS companion file, explicit function/version map	24 matching surfaces and one declared midpoint spin-argument variant in both frontends	verify_xas_author_fits.py (opt-in external asset)
QNM fit surfaces	Pinned APS companion file, (2,2) real/imaginary functions	Both carriers' QNM fit functions	verify_xas_author_fits.py (opt-in external asset)

These opt-in comparisons are diagnostic evidence for the supplied calibration outputs. They are outside the executable scalar composition and do not add calibration components or verification claims to the public catalog.

Reproduce the checks

From the repository root with the optional reference, Torch, and JAX dependencies installed, run JAX_ENABLE_X64=1 JAX_PLATFORMS=cpu python -m pytest tests/test_imrphenomxas*math_reference.py --reference-device=cpu -ra. For qualification, run the full strict CPU reference gate. The opt-in author-asset verifier command and required hash are given in the cross-check; the maintained CI gate runs without that external file.