Cosmological Recombination Radiation:
a falsification history

Short conclusion

1. CRR itself is not falsified. Its spectrum is a public-model input here; no sky data test its existence. This work rejects or constrains particular proposed measurement advantages, not CRR as a physical signal.
2. The annual (orbital) term is rejected as a sensitivity booster. Under equal treatment it carries exactly beta_earth/beta_sun = 0.0805 of the solar-dipole signal — a factor 12.42 in amplitude noise, ~154 in exposure. It survives only as a time-dependent auxiliary consistency parameter.
3. The solar CRR dipole is rejected as a better channel than the CRR monopole. Total-signal detection would need ~297–486× lower effective noise; under equal smooth-foreground marginalization the five-parameter requirement is ~48–104×.
4. A conditional survival window of ~4.8–13× exists only when the monopole is handicapped with far more nuisance freedom than the dipole. That is a statement about a covariance asymmetry, not evidence that any real instrument supplies it.
5. Frequency-space differencing (FSD) is not an independent verifier. Applied to the same channels it costs 0.7–5.8% in statistical error and, by the data-processing inequality, cannot add information.
6. The one qualified survivor — a 14.0× bias reduction on omega_cdm — did not generalize. Against a preregistered PIXIE-like gain/bandpass basis (47 shapes), 0 of 43 material templates reached the tenfold threshold; the worst case made the bias 1.77× worse.
7. Net result: the factor 14 was a shape-specific near-cancellation of one idealized logarithmic gain slope, not a robust property of the preregistered, physically anchored PIXIE-like FTS basis tested here.

The tombstone metaphor used during the work is retained only here. Everything below is stated as auditable calculation.

The original question

1.1 Kinematic template

For an isotropic sky spectrum S(nu) observed from a frame moving with velocity beta = v/c, the first-order Doppler-induced anisotropy has the spectral shape

D(nu) = 3 S(nu) - nu dS(nu)/dnu  =  3 S(nu) - dS(nu)/dln(nu)

The two forms are identical; the code uses the logarithmic-derivative form (annual_crr_fisher.py, binned_templates). Because CRR is a line forest rather than a smooth continuum, D is not a small perturbation of S in shape: in the tested bands ||D||/||S|| is 5.12 (10–4990 GHz, 15 GHz bins) and 2.92 (1–20 GHz, 0.1 GHz bins). The dipole template is spectrally rich, which is why the idea looked promising before the amplitudes were put in.

1.2 The two velocities

1/sqrt(3) is the full-sky RMS of vhat · nhat under uniform coverage. The ratio beta_earth/beta_sun = 0.08052566 is the single number that governs Section 3.

The attraction of the annual term was never its amplitude. It was that the modulation is periodic and phase-known, so in principle it could ride above slowly varying systematics. The question tested here is whether that hoped-for systematics advantage is plausibly large enough to overcome a factor of 12.42 in amplitude.

Chronology of the investigation

Each row is one falsification round. "Escape route" is the loophole that survived that round and became the next round's target.

Audit of the annual CRR channel

Source: DEATH_CERTIFICATE.md, results.json, annual_crr_fisher.py.

3.1 Setup

• Input: public 6840-row CosmoSpec standard-ΛCDM CRR spectrum DI_CRR.dat.
• Three symmetry channels compared at identical white noise per effective sample: S(nu), beta_sun D(nu)/sqrt(3), beta_earth D(nu)/sqrt(3).
• Nuisance: Chebyshev polynomials in log nu of stated degree, plus the blackbody temperature derivative dB_nu/dT, all amplitudes freely marginalized by exact linear projection (SVD, rank-truncated at 1e-11).
• Bands: 10–4990 GHz in 15 GHz bins (332 bins); 1–20 GHz in 0.1 GHz bins (190 bins).

3.2 Results

The last column is a dimensionless same-band scaling, not a mission number. It may only be compared to a published S/N if that publication uses the same band, binning, coverage and covariance.

3.3 The exact invariance

Under symmetric nuisance treatment the ratio is exactly the velocity ratio:

(annual S/N) / (solar S/N) = beta_earth / beta_sun = 0.0805256611

verified to 1e-12 in results.json for both bands and both symmetric cases. This is structural: identical templates projected against identical nuisance spaces leave only the amplitude prefactor. Consequently the annual channel needs

• 12.42× smaller effective amplitude noise, or
• ≈154× more independent white-noise exposure,

merely to tie the solar CRR dipole — which has itself already been rejected against the monopole in Section 4.

3.4 The overfit diagnostic

A deliberately asymmetric scan asked how much smooth freedom the solar channel must lose before annual degree-3 wins. Answer: solar degree 68 (wide band) or 59 (low band). At those degrees the nuisance basis has rank 57 and begins absorbing CRR line structure itself. This is a measurement of the required asymmetry, not a foreground prescription.

3.5 Verdict

The annual term is dead as a generic sensitivity amplifier. It remains formally valid as a time-dependent auxiliary consistency parameter — a phase-locked null test — for an experiment that can demonstrate the order-of-magnitude covariance advantage in its own time-ordered systematics model. No such model is supplied here.

Solar dipole versus monopole

Source: SOLAR_DIPOLE_SURVIVAL.md, solar_parameter_results.json.

4.1 Setup

Public CRRfast fiducial table table_Taylor_coeff_1.dat (4828 frequencies). Five parameters fitted jointly as logarithmic responses p dI/dp: omega_b, Y_p, T_0, omega_cdm, N_eff (fiducial 0.02242, 0.2437, 2.7255 K, 0.11933, 2.99). No external priors; all five mutually marginalized. Bands: 1–20 GHz @ 0.1, 20–100 GHz @ 1, 100–600 GHz @ 5 GHz.

Why no per-species amplitudes. The public CRRfast release supplies the total reference spectrum and Taylor responses to those five parameters only. It does not ship independent H, He I and He II amplitude templates. Constructing such templates from frequency windows would ignore line overlap and interspecies photon feedback, so A_H, A_HeI, A_HeII were never fitted anywhere in this work.

4.2 Total CRR amplitude

So the popular "factor ~1000" framing is not exactly right, but the correct number is still several hundred.

4.3 Five parameters, equal degree-6 nuisance in both channels

Factor by which the dipole channel's effective amplitude noise must be lower to match the monopole's marginalized error:

No parameter-specific win exists under symmetric treatment.

4.4 The only survival window (deliberately dipole-favoured)

Monopole nuisance degree 20 (rank 21–22) versus dipole degree 1 (rank 3):

Over 20–600 GHz the requirement falls to 4.80–13.37. (SOLAR_DIPOLE_SURVIVAL.md rounds this to "5–13"; the JSON floor is 4.7968 for omega_b at 20–100 GHz.)

Read this correctly. It says: if absolute spectroscopy loses far more smooth spectral modes than differential dipole spectroscopy does, the kinematic channel becomes less hopeless than raw beta suppression suggests. It does not say the dipole wins, and it supplies no evidence that a real instrument produces that covariance asymmetry. It is an existence result about a matrix, obtained by handicapping one side.

"Absolute" spectroscopy versus frequency-space differencing

Source: DIFFERENTIAL_SPECTROSCOPY_AUDIT.md, bias_transfer_results.json.

5.1 The false dichotomy

A PIXIE-like Fourier transform spectrometer does not perform an absolute measurement in the naive sense. It interferes the sky against an external full-aperture blackbody reference and measures the calibrator null. It is already a differential instrument. Three architectures must therefore be kept distinct:

1. Calibrator-null monopole — sky vs external blackbody. The relevant "absolute" measurement. Full CRR response, no beta suppression.
2. Kinematic sky dipole — opposite sky directions at the same frequency. Suppressed by beta_sun. Section 4.
3. Frequency-space differential (FSD) — adjacent frequency channels differenced. Not the kinematic dipole; no beta suppression.

Only (3) is at issue here. Framing the comparison as "absolute vs differential" is a category error; the real comparison is between different null structures.

5.2 The information ceiling

If FSD is formed by post-processing the same channels, y = D x with D the adjacent-difference operator, then the correct covariance is

C_FSD = D C D^T          (dense, with off-diagonal correlations)

Treating differenced channels as independent is the error that manufactures a fake advantage. With the exact covariance, the error ratio Q = sigma_mono/sigma_FSD on the 20–600 GHz / 5 GHz grid (116 channels, 115 differences, 10 foreground modes) is:

All Q < 1, as the data-processing inequality requires. Post-processing cannot create information, and same-data FSD is therefore not an independent replication channel. The five-parameter statistical cost is 0.68%–5.77%.

5.3 When could hardware FSD win?

Only if the hardware difference cancels a monopole-only systematic floor comparable to or larger than white noise. Granting FSD perfect cancellation and zero extra cost:

• five smooth calibrator/thermal modes: Q saturates at 1.06–1.42 even as the residual amplitude s -> infinity; never reaches 5.
• plus twelve spectral-ripple modes: Q saturates at 1.97–4.58; still never 5.
• 39 independent coherent modes: Q reaches 5 only at s ≈ 7–40, i.e. a badly failed spectrometer whose coherent residual eigenmodes are many times its white noise.
• an uncorrelated monopole-only floor reaches Q = 5 at ≈5–5.6× white noise and Q = 13 at ≈14×.

These hardware-FSD thresholds are numerically similar to the 4.8–13.37 solar-dipole survival window in Section 4.4, but they are not the same calculation. Section 4.4 compares marginalized solar-dipole and monopole parameter errors under deliberately asymmetric nuisance models. This section asks how large a monopole-only hardware floor must be before an optimistically perfect FSD channel wins. Both expose demanding covariance asymmetries, but one must not be used as a derivation of the other.

The saturation values and mode counts above were re-checked against differential_survival_results.json in the complete directory. Published PIXIE-like simulations place residual systematics at a few percent of white noise or below, so the large-floor regime assumed by the optimistic hardware scan is not demonstrated.

FSD therefore survives as an alternative architecture, a cross-check, or a hardware-simplicity choice — not as a demonstrated 5–13× sensitivity booster.

The bias-transfer map: one qualified survivor

Source: BIAS_TRANSFER_MAP.md, bias_transfer_results.json.

Having lost the variance argument, the remaining hypothesis was narrower: FSD might be worth its ~1–6% variance penalty if it cancels systematic bias by an order of magnitude. The test propagates each systematic template through both analyses and reports the signed parameter bias per unit error, plus

R = | bias_mono / bias_FSD |

Pre-declared success rule: sigma_FSD/sigma_mono <= 1.1, material monopole bias >= 0.01 sigma per stated unit, and R >= 10.

6.1 The hit

So the requested conjunction — at most ~5.8% sensitivity loss and at least one tenfold bias improvement — did exist in this model.

6.2 What did not survive

Largest R for the other named templates: 1.25 (common CMB gain, N_eff), 1.10 (4.5 K thermal residual), 1.46 (CMB band-centre shift, omega_cdm), ≈1.02 (CRR-proportional gain or band-centre errors). A frequency-independent offset is annihilated algebraically by differencing, but its monopole projection fell below the pre-declared materiality threshold at 1 Jy/sr, so it was not counted.

6.3 The ripple scan is not a survivor

A period/phase scan found very large cancellations (R up to 3141 for omega_b, 550 for Y_p). These are not counted. Each maximum was selected after scanning 48 periods × 24 phases = 1152 combinations separately for every parameter. A hardware claim would require fixing the admissible ripple family and its prior before looking at parameter bias, then reporting worst-case or prior-averaged transfer. The T_0, omega_cdm and N_eff ripple entries are additionally flagged material_success: false in the JSON.

6.4 Status at that point

A robustness projection against one particular idealized spectral error shape, on FSD data built from the same monopole channels. Not an independent replication channel, not a measured error budget. The stated next falsification step was explicit: replace the idealized log slope with a preregistered gain/bandpass basis from a concrete spectrometer, and see whether the tenfold reduction persists without tuning the error shape to omega_cdm.

The preregistered instrument-basis audit (final)

Source: INSTRUMENT_BASIS_PREREGISTRATION.md (frozen before any result), INSTRUMENT_BASIS_AUDIT.md, instrument_basis_results.json.

7.1 What was frozen in advance

The preregistration fixed, before evaluation: 20–600 GHz in 5 GHz bins (116 channels); the same five CRRfast responses; the same ten foreground nuisance modes (synchrotron + index, free-free, AME, dust + index, CIB + index, zodiacal, CMB temperature); equal pre-difference channel white noise; exact D C D^T covariance; and the same success rule. It also fixed the decision procedure — every material template must reach R >= 10; a single material failure rejects maintenance over the range — and stated that no shape would be selected, removed or reweighted after seeing its ratio. The old logarithmic slope was declared a reference case, outside the pass/fail set.

7.2 The 47-shape basis

Odd Legendre modes are excluded because at first order they enter the imaginary/antisymmetric Fourier channel and are separately jackknifed.

7.3 Outcome for omega_cdm

The old idealized slope was reproduced inside the same run at R = 14.0045, confirming that nothing in the pipeline changed. Against the preregistered basis:

Per family:

7.4 The decisive case

Lowest even apodization perturbation, W -> W(1 + epsilon P2):

R = 0.56585: FSD makes the absolute bias 1.77× larger. The best apodization mode P8 reaches only 1.70874. The global frequency-scale error is also slightly worse under FSD (R = 0.91784); the frequency zero point improves by only 1.458×.

7.5 Mixtures cannot rescue it

For every parameter the monopole and FSD bias vectors are non-proportional (cosines 0.851–0.999), so arbitrary signed mixtures include cancellation directions with R below any positive bound. uniform_R_ge_10_for_arbitrary_signed_mixtures is false for all five parameters. A full-range guarantee is mathematically unavailable, not merely unattained. (This diagnostic mixes heterogeneous stated units and is algebraic only.)

7.6 Final verdict

The factor 14 was a shape-specific near-cancellation belonging to the logarithmic gain slope, not a robust tenfold bias projection over the error space of a concrete preregistered, physically anchored PIXIE-like FTS basis used in this forecast.

The 1.45% variance penalty remains real; there is no longer an order-of-magnitude bias rejection to buy with it.

What may and may not be claimed

Claimable (as calculation, within the stated model)

• Under symmetric nuisance treatment, the annual/solar CRR Doppler S/N ratio is exactly beta_earth/beta_sun = 0.0805257, verified to 1e-12.
• Total-CRR solar-dipole detection in this setup requires 297–486× lower effective dipole noise; the five-parameter requirement under equal degree-6 nuisance is 47.8–104.1×.
• Under a deliberately asymmetric nuisance assignment (monopole degree 20, dipole degree 1), the 20–600 GHz requirement falls to 4.80–13.37×.
• Same-data FSD with exact D C D^T covariance costs 0.68–5.77% in marginalized parameter error and cannot exceed monopole information.
• The idealized frequency-dependent CMB log gain slope gives R = 14.0045 for omega_cdm, reproduced in two separate executions/code paths using the same input model. This is an internal reproduction, not an independent experiment.
• Against the preregistered 47-shape PIXIE-like basis, 0/43 material templates reach R >= 10; R spans 0.5659–1.7087 with median 1.1315.

Not claimable

• No CRR detection. No data were analysed; CRR spectra are model inputs.
• No instrument was designed, and no mission sensitivity was forecast. The equal-white-noise Fisher normalization is not a noise budget: the ppm and MHz values at a given sigma threshold must not be read as requirements. The primary result is the dimensionless same-model ratio R.
• No new cosmology. No parameter constraint here supersedes any published one.
• Same-data FSD is not independent replication. It is correlated with the monopole by construction.
• The old 14× must not be quoted as the conclusion. Its generality was rejected by the later preregistered audit; it is a reference case only.
• No per-species A_H, A_HeI, A_HeII templates exist here — only the five released CRRfast responses.
• The 5 GHz grid is not PIXIE. It was retained for controlled comparability and is not the nominal ≈14.4 GHz synthesized channel width.
• The ripple maxima are not survivors. They are post-selected over 1152 period/phase combinations per parameter.

Everything is linearized, one-error-at-a-time, with no time-ordered data, no scanning strategy, no 1/f, no sidelobe or zodiacal residual amplitudes, and no FSD-specific relative gain, bandpass, beam or offset error.

What would revive each rejected branch

Reproduction, file map, and primary sources

9.1 File map

All inputs, scripts, JSON records, and tests required for the fixed calculations are present in this directory. CRRfast_reference.zip is retained as the downloaded source archive; the extracted Taylor table above is the file actually read by the calculations.

9.2 Original reproduction commands (as recorded by each round)

python -m unittest -v
python annual_crr_fisher.py --output results.json
python solar_crr_parameter_fisher.py --output solar_parameter_results.json
python differential_spectroscopy_survival.py --output differential_survival_results.json
python bias_transfer_map.py --output bias_transfer_results.json
python instrument_basis_audit.py --output instrument_basis_results.json

Historical test counts were 11 passing at the bias-transfer round and 18 passing at the instrument-basis round. The final README audit re-ran the current complete 18-test suite successfully (including reproduction of the old factor 14, exact differenced covariance, fixed template counts, and apodization quadrature convergence — 4097 vs 8193 phase-delay samples agree to 1e-6 relative).

9.3 Numerical discrepancies between reports and records

Where a Markdown report and a JSON record disagree, the JSON is authoritative.

1. SOLAR_DIPOLE_SURVIVAL.md states the survival window as "about 5–13". The JSON floor is 4.7968 (omega_b, 20–100 GHz) and the ceiling 13.3734 (N_eff, 20–100 GHz). "5–13" is a rounded restatement.
2. DEATH_CERTIFICATE.md's table omits the annual_favored_12_vs_3 case, which is present in results.json; it is restored in §3.2 above.
3. The idealized-slope R differs in the 12th significant digit between bias_transfer_results.json (14.004544824908427) and instrument_basis_results.json (14.00454482490506) — floating-point path differences, not a change of result. Both are quoted as 14.0045.
4. BIAS_TRANSFER_MAP.md reports the statistical penalty as "below 5.8%"; the exact maximum is 5.7701% (Y_p).

9.4 Primary sources

• CRR spectra and parameter responses: public CosmoSpec output (DI_CRR.dat) and the public CRRfast Taylor table (table_Taylor_coeff_1.dat).
• PIXIE device facts used to build the preregistered basis (±5 mK blackbody calibration excursion, 4 mm/s mirror phase velocity, ~7 ms detector time constant, W(x) = (1-x⁴)² apodization): the PIXIE calibration paper, arxiv.org/abs/2002.00976, and the PIXIE systematic-error study, arxiv.org/abs/2304.00091.

Verification performed for this README

The final audit used the complete local analysis directory. It did not overwrite the reference JSON records. Each calculation was re-run to a separate temporary directory:

python annual_crr_fisher.py --output <temp>/results.json
python solar_crr_parameter_fisher.py --output <temp>/solar_parameter_results.json
python differential_spectroscopy_survival.py --output <temp>/differential_survival_results.json
python bias_transfer_map.py --output <temp>/bias_transfer_results.json
python instrument_basis_audit.py --output <temp>/instrument_basis_results.json
python -m unittest -v

10.1 Full calculation reproduction

The current full test suite passed 18/18 in 1.98 s. This re-executes the input-table shape checks, three-band parameter calculations, exact adjacent-difference covariance, information-ceiling checks, named bias-transfer map, old-log-slope reproduction, preregistered template counts, primary pass/fail decision, and apodization quadrature convergence.

10.2 Supplemental record-consistency audit

The draft README arrived with a separate 105-check script and a recorded result of 0 failures. After redirecting its hard-coded upload path to this complete directory, it again returned 105/105. That script checks JSON identities and several analytic code identities, but it does not parse this README; it is therefore treated as a supplemental record-consistency check, not as proof that every prose statement or number in the README was automatically extracted and audited.

10.3 Verification boundary

The successful reproduction establishes consistency of this repository's fixed, linearized calculations with their saved records. It does not turn the forecast into an end-to-end mission simulation, validate the illustrative equal-white-noise model as an instrument noise budget, or make same-data FSD an independent measurement.