§VII.6.3.1 Prediction-Verification Scope — T-vs-E AoE Differential

Date: 2026-05-16 Spike: #26 Phase 1 (this artifact); Phase 2 = analysis dispatch to follow Branch: feat/spike-26-aoe-t-e-differential Notebook anchor: mfo_spectral_research_notebook.md §VII.6.3.1 — "Precession-fit: kinematic ruled out, bundle-projection reconfiguration candidate" Working-note origin: PR #437 Part VI — 73.3° AoE-Auger directional separation (the bundle-projection-reconfiguration evidence anchor)

Candidate framing only, not endorsed over the standard cosmology + posterior-selection baseline (Bennett et al. 2011). Per [[feedback_no_lineage_claims_in_notebook]], no claim that this resolves the AoE anomaly is being advanced — only that the precession-fit question is mathematically answerable and produces a clean channel separation between two readings, one closed (kinematic-precession, ruled out by ~10 orders of magnitude against Saadeh+ 2016) and one open (bundle-projection reconfiguration, consistent with extant matter-frame constraints by operating outside their scope).

The prediction in one sentence

Under MFO §VII.6.3.1 bundle-projection-reconfiguration at the 138°/unit-f_RD rate, the temperature-anchored AoE preferred direction and the polarisation-anchored AoE preferred direction should differ by a small angular differential — predicted range ≈ 0.5°–3°, anchored at ≈ 2° for the brief's Δz ≈ 10 / Δf_RD ≈ 0.014 reading. This is the falsifiable prediction the kinematic-precession reading does not make (under kinematic precession the matter frame is the matter frame regardless of which photon population is read).

Phase 2 analysis steps

Step 1 — Acquire Planck 2018 PR3 sky maps

Use the Spike #26 Phase 1 catalog (cmb_low_ell_maps) to enumerate the 4 component-separated map products (Commander/NILC/SEVEM/SMICA) plus 3 mask products. Fetch FITS bytes via the catalog's download_url_pla (PLA aio/product-action endpoint) or download_url_irsa (IRSA mirror). Each FITS is ≈ 168 MB; total fetch ≈ 700 MB. Compute SHA-256 of each fetched file and write into the catalog's expected_response_sha256 field (currently "pending") via a T1 collector refresh, OR via a Phase-2 calibration step.

Recommended: cache to ~/.cache/mlehaptics/cmb_low_ell_maps/ using T2 local-kernel overlay; do NOT commit FITS bytes to repo.

Step 2 — Read I,Q,U + apply masks (HEALPix)

Per PR3 distribution shape: - Stokes I at Nside=2048, 5 arcmin FWHM - Stokes Q,U at Nside=2048 in same FITS (re-downgrade to Nside=1024 if pursuing canonical-polarization-mode resolution) - Nested ordering, galactic coordinates

Apply COM_Mask_CMB-common-Mask-Int_2048_R3.00.fits to the I map before anafast. Apply COM_Mask_CMB-common-Mask-Pol_2048_R3.00.fits to the (Q,U) maps. For multipole-vector work at ℓ ≤ 8, inpainting (via COM_Mask_CMB-Inpainting-Mask-Int_2048_R3.00.fits + constrained-Gaussian-realisation fill, e.g., per Bucher & Louis 2012) is preferred over hard masking because masked-product power leaks into low-ℓ multipole vectors.

Tools: healpy.read_map, healpy.ma, healpy.anafast (or synfast-with-constraint for inpainting).

Step 3 — Extract a_ℓm at ℓ ≤ 8 (T-mode AND E-mode)

Per de Oliveira-Costa et al. 2004 (DOI 10.1103/PhysRevD.69.063516), the multipole-vector decomposition of ℓ=2 and ℓ=3 (the AoE multipoles) is:

a_ℓm  ←  healpy.map2alm(I_masked_inpainted, lmax=8)            [T-mode]
(a_E_ℓm, a_B_ℓm)  ←  healpy.map2alm_spin((Q, U), spin=2, lmax=8)  [E and B modes]

E-mode is the divergence-like polarization combination; B-mode is curl-like and is sub-dominant at large scales (Planck has only upper limits at low ℓ). The AoE T-mode and AoE E-mode preferred-axis comparison uses the T a_ℓm and E a_ℓm respectively.

Step 4 — Multipole-vector preferred direction extraction

The de Oliveira-Costa method computes an axis-of-symmetry direction for each ℓ:

n̂_ℓ  =  argmax_{n̂ on S²}  Σ_{m=-ℓ}^{ℓ} |a_ℓm(n̂)|² · cos(2 m φ_n̂)

where a_ℓm(n̂) is the a_ℓm coefficient in a frame whose z-axis points along n̂. Equivalent reformulations (Schwarz et al. 2004, Land & Magueijo 2005) use Maxwell-vector multipole-vector representations. For ℓ=2 there are 2 Maxwell vectors; for ℓ=3 there are 3.

Practical implementation: rotate the spherical-harmonic basis over a fine grid on S² (HEALPix Nside_grid = 256 or higher, ~786,000 directions, well-distributed), compute the maximand at each direction, find the argmax. The "AoE axis" is the ℓ=2 axis-of-symmetry; alignment with the ℓ=3 axis (and with the ecliptic plane / CMB dipole) is the canonical AoE signature.

Compute separately: - n̂_T_ℓ=2 from T-mode (temperature) a_ℓm - n̂_E_ℓ=2 from E-mode (polarisation) a_E_ℓm - (same for ℓ=3 if desired for stability cross-check)

Step 5 — Differential angle

The Phase 2 deliverable:

Δθ_TE  =  arccos(n̂_T_ℓ=2 · n̂_E_ℓ=2)

Compute Δθ_TE for each of the 4 pipelines (Commander/NILC/SEVEM/SMICA). Cross-pipeline agreement is the systematic-robustness check; the four pipelines use different component-separation strategies, so Δθ_TE matching across pipelines is evidence that the differential is signal-not-systematic.

Step 6 — Null-hypothesis significance via Monte Carlo

Generate N ≈ 10,000 ΛCDM-Gaussian CMB sky realisations (using healpy.synfast against the Planck 2018 best-fit C_ℓ from cmb_power_spectrum catalog). For each, extract n̂_T_ℓ=2 and n̂_E_ℓ=2 via the same pipeline and compute Δθ_TE_sim. The fraction of simulations with Δθ_TE_sim within the framework's predicted range (0.5°–3°) gives the consistency p-value; the fraction below the observed Δθ_TE gives the falsification p-value.

The kinematic-precession null is Δθ_TE_kinematic ≈ 0 (matter frame is matter frame). Any observed Δθ_TE > a few times the ΛCDM-Monte-Carlo standard error supports bundle-projection-reconfiguration over kinematic-precession; agreement at Δθ_TE_observed ≈ 0 supports kinematic-precession (already ruled out by Saadeh+ 2016) OR rules out the bundle-projection reading at the corresponding confidence level.

Visibility-function modelling

Standard cosmology Δz between T and E visibility peaks

The Thomson-scattering visibility function g(z) = (dτ/dz) × exp(-τ(z)) determines where photons last interacted with electrons. For temperature anisotropies, g(z) is the canonical visibility function and peaks at the textbook z ≈ 1090 (recombination); FWHM Δz_FWHM ≈ 80–100.

For polarisation, the visibility function is the polarisation visibility: g_pol(z) is g(z) weighted by the local quadrupole moment of the radiation. Because the quadrupole builds up through the partial-ionisation tail (electrons need free-streaming photons from elsewhere to generate the local quadrupole), g_pol peaks slightly later than g (i.e., at lower z). References:

• Hu & White 1997, "A CMB Polarisation Primer", New Astronomy 2:323 (arXiv:astro-ph/9706147) — canonical pedagogical treatment of g_pol vs g.
• Kosowsky 1996, "Cosmic microwave background polarization", Annals Phys 246:49 (arXiv:astro-ph/9501045) — earlier formalism.
• Hu & Dodelson 2002 (Annu. Rev. Astron. Astrophys 40:171) — Δz between T and E peaks discussed.

Conservative reading: Δz_T–E ≈ 10–30 between the centroids; with the same FWHM ≈ 80–100, the visibility windows substantially overlap but their peaks differ.

Conversion: Δz → Δf_RD

§VII.6.1 anchors f_RD ≈ 0.42 at z ≈ 1090, f_RD ≈ 0.949 at z = 0. Linear-local approximation near recombination:

df_RD / dz |_{z≈1090}  ≈  -(0.949 - 0.42) / (0 - 1090)  ≈  -4.85 × 10⁻⁴  per z-unit  (negative; f_RD increases as z decreases)

Therefore: - Δz = 10 → Δf_RD ≈ 4.85 × 10⁻³ → Δθ_predicted = 138.6° × 4.85 × 10⁻³ ≈ 0.67° - Δz = 14.4 → Δf_RD ≈ 7 × 10⁻³ → Δθ ≈ 1.0° - Δz = 30 → Δf_RD ≈ 1.45 × 10⁻² → Δθ ≈ 2.0°

The brief's "~2°" arises from Δf_RD ≈ 0.014 ≈ Δz≈29 OR from a non-linear (early-universe-rapid) cosmic-time relation that shifts the conversion factor. Either reading sits in the 0.5°–3° range.

Refinement opportunity (Phase 2 internal extension)

Phase 2 may improve the Δf_RD model by computing g(z) and g_pol(z) directly from CAMB or CLASS using Planck 2018 best-fit cosmology parameters (cmb_power_spectrum catalog has those parameters indirectly via the binned C_ℓ; Planck 2018 VI Table 2 has them explicitly). Replace the linear-local approximation with the true cosmic-time-vs-z relation:

Δf_RD  =  (f_RD(z_E,vis) - f_RD(z_T,vis))

where f_RD(z) is anchored to §VII.6.1's loop-down trajectory parameterisation. This refinement may shift the central prediction by a factor of ≈ 2.

Predicted differential — quantitative summary

Δz (Hu-White centroid offset)	Δf_RD (linear-local approx)	Predicted Δθ_TE @ 138°/unit-f_RD
10 (conservative)	4.85 × 10⁻³	0.67°
14.4 (mid range)	7.0 × 10⁻³	1.0°
30 (less conservative)	1.45 × 10⁻²	2.0°
50 (upper end)	2.42 × 10⁻²	3.4°

Central reading: Δθ_TE ≈ 1°–2°. Degrees-not-tens-of-degrees.

Falsifier

If multipole-vector extraction gives Δθ_TE_observed < 0.1° to within ΛCDM Monte-Carlo expectation, the bundle-projection-reconfiguration reading is disfavoured. This is the falsifiable channel separation §VII.6.3.1 advertises.

Conversely, if Δθ_TE_observed > 5° at high significance, the framework's 138°/unit-f_RD rate is too small to explain the observation (a different rate would be needed, weakening the §VII.6.1 f_RD-trajectory anchoring).

Open questions for user / conductor (fermata)

1. Phase 2 venue. Should Phase 2 dispatch land its analysis script in docs/srmech/notes/ (spike-style) or in docs/antikythera-maths/research-mfo/ (research-mfo style)? Spike #24 used both; the brief's "research-note artifact" lives in research-mfo (this file). Phase 2's script venue is unclear.

2. healpy Phase 2 dependency. Adding healpy as a Phase 2 dependency adds an LAPACK-class native dependency to any package that runs the analysis. Options: (a) Phase 2 lives as a one-off docs/antikythera-maths/research-mfo/vii_6_3_1_phase2_script.py outside the package surface; (b) catalog gets a "consumer" example showing how healpy would be wired; © defer entirely to a sister repo. Recommendation: option (a) — research script, not package surface.

3. Inpainting vs hard masking. The brief mentions mask handling; explicit choice between constrained-Gaussian-realisation inpainting (more rigorous for multipole vectors but expensive) and hard masking with apodisation (cheaper, more common in early-stage analyses) needs user direction for Phase 2.

4. Existing AMSC citation issue in cmb_anomalies. The descriptor cites Planck 2018 VII (isotropy/statistics) but uses the DOI 10.1051/0004-6361/201833881, which actually resolves to Planck 2018 IV (diffuse component separation). Planck 2018 VII's correct DOI is 10.1051/0004-6361/201935201. This is a separate fix outside Spike #26 Phase 1 scope — flagging for conductor disposition.

Cross-references

• §VII.6.3.1 — the prediction's home (lines ~1060–1110 of mfo_spectral_research_notebook.md)
• §VII.6.1 — loop-down completion f_RD trajectory anchoring 138°/unit-f_RD rate
• §VII.4.1.1 — Hopf-bundle / spherical-compression substrate reading
• §VIII.7 — fractal-shadow allegory (sister shadow-stance for the bundle-projection mechanism)
• Saadeh, Feeney, Pontzen, Peiris, McEwen 2016, "How isotropic is the Universe?", PRL 117 131302, arXiv:1605.07178, DOI 10.1103/PhysRevLett.117.131302 — verified per [[feedback_pdf_extraction_citation_discipline]]
• de Oliveira-Costa, Tegmark, Zaldarriaga, Hamilton 2004, "Significance of the largest scale CMB fluctuations in WMAP", Phys. Rev. D 69:063516, arXiv:astro-ph/0307282 — multipole-vector method
• Planck 2018 IV (Diffuse maps): Akrami et al. 2020, A&A 641:A4, arXiv:1807.06208, DOI 10.1051/0004-6361/201833881
• Planck 2018 V (Likelihoods): Aghanim et al. 2020, A&A 641:A5, arXiv:1907.12875, DOI 10.1051/0004-6361/201936386
• Planck 2018 VII (Isotropy): Akrami et al. 2020, A&A 641:A7, arXiv:1906.02552, DOI 10.1051/0004-6361/201935201
• Working-note origin: PR #437 Part VI — 73.3° AoE-Auger separation (bundle-projection-reconfiguration evidence anchor)
• Memories: [[user_stance_fiber_as_spatially_absent_encoding]], [[user_stance_dark_sector_ring_down_age]], [[user_stance_fractal_shadow]], [[reference_autonomous_validation_tos_landscape]], [[feedback_pdf_extraction_citation_discipline]]

Phase 1 deliverables (this dispatch)

• PLA TOS verdict: non-blocking. ESA Open Access policy applies; PLA + IRSA both serve component-separated CMB maps via CDN without authentication. Citation requirement is the standard Planck acknowledgment + per-paper DOI references. No autonomous-access prohibition surfaced.
• Data product chosen: PR3 component-separated sky maps (Nside=2048, 168 MB per FITS, 4 pipelines + 3 masks = 7 catalog rows). a_ℓm coefficients are NOT directly distributed by PLA; they are computed from sky maps in Phase 2 via HEALPix anafast / map2alm_spin.
• Catalog source key: cmb_low_ell_maps. 7 rows. Lives at docs/antikythera-maths/ephemerides-spectral/python/ephemerides_spectral/_research/attested/cmb_low_ell_maps/.
• Test ratchet: test_attested_collector.py updated 20→21 (total) and 17→18 (curated filter; literature_curated specific filter).

Phase 2 next-action checklist (for conductor when dispatching)

• Resolve fermata 1 (Phase 2 venue: docs/srmech/notes/ vs docs/antikythera-maths/research-mfo/).
• Resolve fermata 2 (healpy dependency: research script vs package surface vs sister repo).
• Resolve fermata 3 (inpainting vs hard masking).
• Optional: fix fermata 4 (separate small PR for cmb_anomalies DOI mismatch).
• Dispatch Phase 2 with: fetch 4 SMICA + Commander + NILC + SEVEM FITS, apply common-mask, extract a_ℓm and a_E_ℓm at ℓ ≤ 8, compute n̂_T_ℓ=2 and n̂_E_ℓ=2 multipole-vector axes, compute Δθ_TE, run 10⁴-sim ΛCDM null Monte Carlo, report significance.