DOI: 10.5281/zenodo.17613348
title: Rotation Field of the Cosmic Microwave Background — Angular Locality of the Δℓ = 109 Standing Wave (v2.6)

description: Summary

Version 2.6 of the 22 Blue project tests the angular locality of the locked Δℓ ~ 109 standing wave in the CMB rotation field. Previous releases (v2.4 and v2.5) established that the 3.3° real-space peak in ξ(θ) is the visible imprint of the Δℓ harmonic identified in v2.0–v2.1. Version 2.6 answers a deeper structural question: Is the Δℓ = 109 mode generated uniformly across all angular separations, or does it originate specifically from the narrow 3.3° region?

This release performs θ-domain isolation and exclusion tests using only the locked residual Cℓ(αα) spectrum and Legendre transforms, with no maps and no healpy. The baseline ξ(θ) is reconstructed from the frozen ℓ = 100–1500 spectrum using: ξ(θ) = Σ_ℓ [(2ℓ + 1) / (4π)] C_ℓ P_ℓ(cos θ).

The peak occurs at: θ_peak ~ 3.353°, ξ_peak ~ 1.55 × 10³.

Gaussian windows centered at 3.3° (σ = 0.15°, 0.30°, 0.60°) isolate the θ-region believed to drive the signal. After applying W(θ), the windowed correlation function ξ_win(θ) is projected back into ℓ-space using the inverse Legendre integral: C_ℓ^(win) = 2π ∫ ξ_win(θ) P_ℓ(cos θ) sin θ dθ.

Each reconstructed spectrum is compared to the fixed Δℓ = 109 standing-wave template: T_ℓ = cos(2πℓ / 109). The template-projection amplitude A_proj = Σ_ℓ [C_ℓ T_ℓ] / Σ_ℓ [T_ℓ²] is used as the stable, linear measure of harmonic strength. The ratio A_win / A_base quantifies harmonic survival.

Results: Gaussian isolation preserves most of the Δℓ = 109 mode: σ = 0.15° → 0.723, σ = 0.30° → 0.832, σ = 0.60° → 0.884. Removing only the band 3.0°–3.6° destroys ~85% of the Δℓ power: A_excl / A_base = 0.155. This demonstrates that the 3.3° region is the high-leverage source of the Δℓ = 109 standing wave. The harmonic is θ-localized, not θ-uniform.

Map-locality preparation

To prepare for sky-locality tests in v2.7, the locked α(n̂) map was downsampled from NSIDE = 512 to NSIDE = 16, and a neighbor-variance boundary proxy was constructed: proxy(p) = mean[(α_neighbor − α_p)²]. This field is not analyzed in v2.6 but will be used in v2.7 to correlate local boundary strength with local harmonic amplitude.

Findings

• The Δℓ = 109 mode is strongly θ-localized around 3.3°.  • The narrow 3.3° band carries the majority of the harmonic coherence.  • Removing this band collapses the Δℓ signal by ~85%.  • A full sky-locality grid (NSIDE = 16) is prepared for v2.7.  • All locked inputs remain unchanged; no α reconstruction and no Δℓ fitting were performed.

Citation

Condit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Angular Locality of the Δℓ ~ 109 Standing Wave (v2.6).* *22 Blue — The Heartbeat of the Universe.* Zenodo. https://doi.org/10.5281/zenodo.17613348

Derived from

Condit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Spectral Surgery on the Δℓ ~ 109 Harmonic (v2.5).* *22 Blue — The Heartbeat of the Universe.* Zenodo. https://doi.org/10.5281/zenodo.17604982

Version History (22 Blue Dataset Series)

Separate publication — Sep 20 2025 • Evidence for harmonic phase alignments in Planck 2018 CMB polarization data (Δℓ ~ 320)  v1.0 — Oct 10 2025 • First quantitative detection of anisotropic cosmic birefringence  v1.1 — Oct 20 2025 • Statistical validation and reproducibility  v1.2 — Oct 21 2025 • Two-harmonic extension  v1.3 — Oct 28 2025 • Robustness tests and injections  Dataverse Mirror — Oct 29 2025 • DOI: 10.7910/DVN/PTDG20  v1.4 — Nov 1 2025 • MASTER-calibrated α(ℓ)  v1.41–1.44 — Nov 1–8 2025 • SMICA/NILC + HM splits; parity and axis checks  v1.5 — Nov 9 2025 • Model selection; Δℓ ~ 454.7  v1.6 — Nov 9 2025 • Phenomenology and interpretation  v1.7 — Nov 9 2025 • Forward prediction of α(n̂) and EB(ℓ)  v1.8 — Nov 9 2025 • Harmonic persistence and rejection tests  v2.0 — Nov 10 2025 • Intrinsic Δℓ ~ 108 discovered  v2.1 — Nov 10 2025 • Physical interpretation; ξ(θ) at 3.3°  v2.2 — Nov 11 2025 • Universe-model evaluation  v2.3 — Nov 12 2025 • Domain geometry inference  v2.4 — Nov 13 2025 • Direct real-space ξ(θ) confirmation  v2.5 — Nov 13 2025 • Spectral surgery: Δℓ ~ 109 necessary, nearly sufficient, and phase-coherent  v2.6 — Nov 14 2025 • Angular locality: isolation vs exclusion of the 3.3° region; Δℓ = 109 shown to be θ-local

22 Blue — The Heartbeat of the Universe  Twitter/X: @Project22Blue