DOI: 10.5281/zenodo.17620029
title: Rotation Field of the Cosmic Microwave Background — Sky-Local Origin of the Δℓ ≈ 109 Standing Wave (v2.7)

description: Purpose: To determine whether the Δl = 109 standing wave originates from specific regions of the sky by comparing high-gradient “boundary” pixels to low-gradient “bulk” pixels using only NSIDE = 16 pixel-space correlations and θ-space template projection.

Summary

Version 2.7 extends the real-space and harmonic-space results of v2.4–v2.6 into map-space locality. Earlier releases established that the 3.3° peak in ξ(θ) is the real-space imprint of the intrinsic Δl ≈ 109 mode discovered in v2.0–v2.1 and isolated cleanly in v2.5–v2.6. This version asks a structural question: Does the Δl = 109 harmonic arise uniformly across the sky, or does it originate preferentially from “boundary” regions of the rotation field α(n̂)?

This analysis uses only locked inputs:(1) α(n̂) downsampled to NSIDE = 16,(2) a boundary-strength proxy map derived from neighbor variance,(3) the fixed Δl = 109 harmonic prior, and(4) no healing, no alm, no map recomputation, and no Δl fitting.

Boundary vs Bulk Pixel SetsBoundary pixels (H) are defined as the top 10% of the boundary-proxy distribution; bulk pixels (L) are the lowest 60%. These sets do not overlap. Pairwise pixel separations and α_i α_j products are accumulated into 100 bins across 0°–10° to form ξ_H(θ) and ξ_L(θ).

Correlation FunctionFor each subset S ∈ {H, L}:  xi_S(theta_k) = (1 / N_k) * sum_{pairs (i,j) in bin k} [alpha_i * alpha_j]

Template ProjectionUsing the fixed 3.3° standing-wave template:  T(theta) = cos(2π * theta / 3.3°)

Template amplitudes are computed over the 1°–6° region:  A_S = sum_k [ xi_S(theta_k) * T(theta_k) * w_k ] / sum_k [ T(theta_k)^2 * w_k ]where w_k are pair-counts.

ResultsBoundary pixels show a dramatically stronger 3.3° coherence:  A_H = -2.3308e-40  A_L =  1.8166e-42  A_H / A_L = -128.3

The difference ΔA = A_H – A_L = -2.349e-40.

Null TestA shuffle null test (N = 200) preserves pixel positions and boundary strengths but randomly permutes α across the sky. All null realizations cluster around zero:  mean(ΔA_null) = -2.49e-42  std (ΔA_null) = 1.46e-41

Real data:  Z = -15.95  p ≈ 0 (no null realization approaches ΔA_data)

InterpretationThe Δl ≈ 109 standing wave is sky-localized: the overwhelming majority of its coherent power originates in boundary regions—areas of high α-gradient—rather than from smooth bulk zones. Combined with v2.6 (θ-locality), this shows dual locality: the harmonic is localized both in angular separation (3.3°) and on the sky (boundary domains).

This positions v2.8 to examine domain topology: whether these boundary regions form a coherent, connected structure across the sky.

CitationCondit, Amy (2025). Rotation Field of the Cosmic Microwave Background — Sky-Local Origin of the Δl ~ 109 Standing Wave (v2.7). 22 Blue — The Heartbeat of the Universe. Zenodo. https://doi.org/10.5281/zenodo.17620029

Derived fromCondit, 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

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 birefringencev1.1 — Oct 20 2025 • Statistical validation and reproducibilityv1.2 — Oct 21 2025 • Two-harmonic extensionv1.3 — Oct 28 2025 • Robustness tests and injectionsDataverse Mirror — Oct 29 2025 • DOI: 10.7910/DVN/PTDG20v1.4 — Nov 1 2025 • MASTER-calibrated α(l)v1.41–1.44 — Nov 1–8 2025 • SMICA/NILC + HM splits; parity and axis checksv1.5 — Nov 9 2025 • Model selection; Δl ~ 454.7v1.6 — Nov 9 2025 • Phenomenology and interpretationv1.7 — Nov 9 2025 • Forward prediction of α(n̂) and EB(l)v1.8 — Nov 9 2025 • Harmonic persistence and rejection testsv2.0 — Nov 10 2025 • Intrinsic Δl ~ 108 discoveredv2.1 — Nov 10 2025 • Physical interpretation; ξ(θ) at 3.3°v2.2 — Nov 11 2025 • Universe-model evaluationv2.3 — Nov 12 2025 • Domain geometry inferencev2.4 — Nov 13 2025 • Direct real-space ξ(θ) confirmationv2.5 — Nov 13 2025 • Spectral surgery: Δl ~ 109 necessary, nearly sufficient, and phase-coherentv2.6 — Nov 14 2025 • Angular locality: isolation vs exclusion of the 3.3° region; Δl = 109 shown to be θ-localv2.7 — Nov 15 2025 • Sky-local origin: boundary regions dominate the Δl = 109 harmonic

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