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HM amplitude factors and conditioning audit — 2026-09-28

The intrinsic-mode diagnostic found a large 2374 (2,1) deficit and 2163 (4,4) excess. This follow-up checks the amplitude construction independently and tests sensitivity to NR conditioning on those same seen cases. The checked factors agree with both frontends; the published alternative factor does not resolve these discrepancies. No production waveform, frozen fixture, threshold or qualification claim changes.

Source equation and controlled alternative

Let \(x=Mf\), \(g=f_{22}^{A}(x)\), and \(H_{\ell m}(y)=V_m(y)^{-7/2}\hat H_{\ell m}(V_m(y))\), with \(V_m(y)=(2\pi y/m)^{1/3}\). The common dimensional SPA prefactor cancels. Mishra et al., Eqs. (11), (12) supply the envelopes; existing references specify the retained 21, 33, 32, 44 and 43 terms. In particular, the 21 coefficients agree with Eq. (12b), and the leading 44 envelope is proportional to \((1-3\eta)V_4^{-3/2}\), including the SPA power.

London et al., Eq. (4) and its following paragraph give two distinct amplitude prescriptions:

\[ B=\left|\frac{H_{\ell m}(g)}{H_{22}(g)} \frac{H_{\ell m}(x)}{H_{\ell m}(2x/m)}\right|, \qquad B_{\rm alt}=\left|\frac{H_{\ell m}(x)}{H_{22}(g)}\right|. \]

The scratch alternative multiplies each existing complex mode by

\[ \frac{B_{\rm alt}}B= \left|\frac{H_{\ell m}(2x/m)}{H_{\ell m}(g)}\right|. \]

The positive ratio preserves phase and support. Symbolic checks establish the ratio identity, equality when \(g=2x/m\), and unchanged 22. All denominators are nonzero and mapped frequencies positive on these grids. This is not a general prescription at amplitude zeros or symmetric odd-mode limits.

The Eq. (5) amplitude map, 22 carrier, QNMs, cutoffs and mode phases remain fixed within each comparison. Both london_2018 and phenomd_runtime maps are tested separately; only the former uses the paper's amplitude transition. No detector alignment, amplitude normalization, optimizer or coefficient fit enters this comparison. No external implementation source was inspected.

Factor checks

The audit evaluates restricted symbolic envelopes on all 995 frequency bins, 30–1024 Hz at 1 Hz spacing, for all six modes, both cases and both maps. Separate 50-digit scalar evaluations check the map, factor, carrier and composed amplitude at fixed frequencies and near mapped joins. The scalar composition independently derives the published carrier/remnant and uses the declared QNM data. This reuses the maintained reference components; it is not an independent rederivation of their calibration assets.

Check Maximum error
Full symbolic SPA envelopes versus native factors, relative \(9.94\times10^{-16}\)
Eq. (4) amplitude factor, relative \(1.81\times10^{-15}\)
Scalar map frequency, absolute in \(Mf\) \(5.56\times10^{-17}\)
Scalar QNM frequency, absolute in \(Mf\) \(2.78\times10^{-17}\)
Sampled independent 22 carrier amplitude, relative \(1.24\times10^{-14}\)
Sampled independent composed amplitude, relative \(1.36\times10^{-14}\)
Native amplitude recomposed from its factors, relative \(2.09\times10^{-15}\)
Torch/JAX complex modes, relative L2 \(6.28\times10^{-13}\)

Relative factor errors use maximum absolute difference divided by maximum reference magnitude over the checked bins, not the maximum pointwise relative error. Phase-convention amplitude agreement is within \(4.12\times10^{-16}\); the legacy and SPA conventions consequently share the amplitude table below. All assertions passed. These sampled checks found no transcription discrepancy; they do not establish physical accuracy or complete parameter-domain coverage.

NR amplitude results

Use the prior independently recovered intrinsic NR carriers, fixed masses, distance, signed-mode conventions and aLIGO PSD. Define \(E_A=\|\,|P|-|R|\,\|/\|R\|\) with \(\|a\|^2=4\sum_f |a(f)|^2\Delta f/S_n(f)\). This measures amplitude shape and scale without phase alignment. It is not a detector mismatch or an existing qualification threshold.

Case Mode London baseline \(E_A\) London alternative \(E_A\) Runtime baseline \(E_A\) Runtime alternative \(E_A\)
2374 21 0.5193 0.5126 0.5104 0.5185
2374 33 0.0866 0.0667 0.0933 0.0679
2374 32 0.3764 0.2504 0.3761 0.2482
2374 44 0.2312 0.2053 0.2344 0.2056
2374 43 0.4185 0.3830 0.4186 0.3832
2163 21 0.1557 0.2203 0.1499 0.2245
2163 33 0.0928 0.0795 0.0964 0.0794
2163 32 0.6374 0.5552 0.6385 0.5553
2163 44 0.4303 0.4041 0.4318 0.4037
2163 43 0.4339 0.4465 0.4335 0.4481

22 is unchanged: \(E_A=0.009113\) and \(0.013835\) for 2374 and 2163. For London mapping, the 2374 21 norm ratio changes from 0.5294 to 0.5044; the 2163 44 ratio changes from 1.3613 to 1.3386. Neither discrepancy is removed. The 2163 21 ratio improves from 0.9339 to 0.9881 even though its amplitude error rises from 0.1557 to 0.2203. A norm close to one can hide an incorrect frequency dependence. Runtime mapping gives the same qualitative outcome for these three comparisons.

The alternative improves several other mode amplitudes, but worsens 2163 21 and 43 for both mappings. It cannot justify a general replacement or a qualification claim. Phase residuals are untouched; total projected overlap was deliberately not reoptimized for this amplitude-only diagnostic.

Independent conditioning sensitivity

The raw-reader calculation decodes the higher-resolution NR files and reconstructs the 22, 21 and 44 carriers using the frozen time interval and 22-peak clock. It first reproduces the frozen carriers to below \(10^{-10}\) relative complex L2. It then changes one setting at a time:

  • Start taper: 250M or 1000M instead of 500M.
  • End taper: 50M or 200M instead of 100M.
  • Sampling: 16384 Hz instead of 8192 Hz.
  • Interpolation: real/imaginary PCHIP instead of cubic splines.

All other settings, raw files, PSD and comparison frequencies stay fixed. The taper is the same product of squared-sine ramps used by the frozen runner. There is no extrapolation or clock realignment. The baseline end taper begins 45.46M after the 22 peak in 2374 and 50.40M after it in 2163. The 200M stress test starts at −54.54M and −49.60M, respectively, attenuating merger. Its large effect must not be presented as ordinary late-ringdown uncertainty.

Let \(\delta=\|\,|R'|-|R|\,\|/\|R\|\) and \(\rho=\|R'\|/\|R\|\). The triangle inequalities give

\[ \frac{\max(0,E_A-\delta)}{\rho} \le \frac{\|\,|P|-|R'|\,\|}{\|R'\|} \le \frac{E_A+\delta}{\rho}. \]

These bounds require no new model evaluation. Across the five variations other than the merger-attenuating 200M stress test:

Case Mode Maximum \(\delta\) Enclosing London baseline \(E_A\) bounds
2374 22 0.000371 [0.00874, 0.00949]
2374 21 0.000800 [0.51852, 0.52013]
2374 44 0.000883 [0.23027, 0.23205]
2163 22 0.000488 [0.01334, 0.01433]
2163 21 0.000724 [0.15496, 0.15642]
2163 44 0.000725 [0.42957, 0.43103]

Norm changes are below \(8.05\times10^{-5}\); complex spectral changes can reach 0.00232, so the sub-0.09% statement applies only to amplitudes. The targeted deficits/excesses therefore survive these conditioning variations. For completeness, the 200M stress-test deltas for (22,21,44) are (0.1262, 0.2219, 0.1226) for 2374 and (0.1040, 0.1680, 0.0958) for 2163. Even this test leaves lower London error bounds of 0.3759 for 2374 21 and 0.3530 for 2163 44. Its changed waveform is not substituted into any historical score.

This is a bounded sensitivity study, not a full NR systematic-error budget. It does not vary extrapolation order, extraction method, simulation physics or resolution. The historical resolution screens are reused. 32/43 conditioning has not been tested here; low-power frequency bands can have much larger relative changes than the integrated values above.

Decision and remaining work

Retain the current declared amplitude prescription. The independently checked formula and these conditioning variations do not explain away the two targeted discrepancies. The results support investigating source-model approximation error, without proving it is the unique cause. The paper's alternative remains a scratch diagnostic, not a new production option or fitted correction.

The next discriminating amplitude check is to replace only the mapped \(|D_{22}(g)|\) carrier by recovered \(|NR_{22}(g)|\) for 21/44, keeping each factor and map fixed. It must compare identical mapped-frequency support, report excluded modal power, and avoid extrapolation or a fitted gain. The small integrated 22 error does not bound its error under the different frequency weighting of a higher mode. This separates mapped-carrier error from the combined higher-mode factor/map approximation.

That mapped-carrier check is now complete. Direct NR22 substitution leaves the 2374 21 deficit and most of the 2163 44 excess intact. The 44 comparison excludes 17–24% of NR power, so its result is explicitly restricted to common support. No fitted correction follows.

The frequency-dependent 32/43 phase residual remains separate: a source-backed spherical/spheroidal and phase-map comparison is needed before attributing it to mixing. If a different model is pursued to improve physical accuracy, it needs a separate mathematical specification and prospectively frozen validation on untouched data. Repeated changes on these seen cases cannot close qualification. All prior failed gates remain visible.

Execution and reproducibility

Both jobs completed on len in /tmp/diffgw-hm-amplitude-audit-20260928. The amplitude job used source revision 1db24fb1417e7b4f8a84de53498ec2d039496e00, archived in repo/, with unchanged native source hashes from the prior intrinsic audit. PID 4126673 ran from 14:02:45 to 14:03:44 UTC on 2026-09-28; conditioning PID 4046395 ran from 13:59:48 to 13:59:53 UTC. Both receipts say completed and qualification_promoted=false; no scientific job remains active.

The amplitude environment was /tmp/diffgw-waveform-library-1315f6e/venv-latest: Python 3.12.3, NumPy 2.3.5, SciPy 1.16.3, PyCBC 2.11.0, Torch 2.14.0, JAX/jaxlib 0.11.2, SymPy 1.14.0. The raw-reader environment was /tmp/diffgw-hm-sxs-2265-20260927/.venv: Python 3.12.3, NumPy 2.5.3, SciPy 1.18.1, SXS 2026.0.6. Existing environments were retained for comparison; no packages changed.

Replay requires the prior intrinsic driver and its three unchanged helpers, the frozen source/data directories, and psd-2374.npy/psd-2163.npy extracted with the frozen runner. The scripts verify PSD content hashes, manifests, raw files, sidecars, fixtures and prior evidence. Use fresh output paths:

cd /tmp/diffgw-hm-amplitude-audit-20260928
export OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1
/tmp/diffgw-waveform-library-1315f6e/venv-latest/bin/python \
  hm_amplitude_audit.py --repo "$PWD/repo" \
  --source-revision 1db24fb1417e7b4f8a84de53498ec2d039496e00 \
  --output /tmp/new-hm-amplitude.json
/tmp/diffgw-hm-sxs-2265-20260927/.venv/bin/python \
  hm_amplitude_conditioning.py /tmp/new-hm-conditioning.json
Artifact SHA-256
amplitude.json 11e236d9a39ed64022b0d9824ca1c8b44e77057baa9d8ed786fbef6d9a7d1daa
hm_amplitude_audit.py 48959e61b42af601a3ad14d3062df3318802b24fad521805c60a7725292da567
conditioning.json ca6a48aaedd7d948ba80bf915f13073f38d68f95bbbd59c14126ca344c6ab2f0
hm_amplitude_conditioning.py 5158a05290aba8d7770301b1c10d393eaf69449c380857f5f908c41d55d0ac51
Prior intrinsic JSON 12114e33b00c2f68b18002bd0710db5ab4b3789652d8a484c7263ca340aee85f

Logs are amplitude.log and conditioning.log; complete per-band and component results remain in the external JSON artifacts. This maintained receipt preserves the derivation and findings, not permanent availability of the scratch files. No numerical regression suite is required by this documentation-only change. The first generated-library check imported the environment's older installed catalog; it was rerun with the archived checkout's src on PYTHONPATH. Both generated-document checks and strict MkDocs passed on len against that checkout.