Rotation Field of the Cosmic Microwave Background — Sky-Local Origin of the Δℓ ≈ 109 Standing Wave (v2.7)

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. NSIDE=16 Analysis Note For geometric investigations presented in this release, the reconstructed rotation field was represented using a locked NSIDE=16 HEALPix grid. This representation emphasizes the largest angular-scale features of the field while reducing sensitivity to small-scale fluctuations, allowing coherent domains, boundaries, and large-scale geometric structure to be studied directly. 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 PUBLICATION RECORD PREDECESSOR PUBLICATION (separate record)1. Sep 20, 2025 (v1.0) -- Harmonic Phase Alignments in Planck 2018 CMB -- DOI:10.5281/zenodo.17167268 MAIN RESEARCH SERIESConcept DOI:10.5281/zenodo.173173972. Oct 10, 2025 (v1.0) -- Scale-Dependent Anisotropic Birefringence: Initial Detection -- DOI:10.5281/zenodo.173173983. Oct 20, 2025 (v1.1) -- Scale-Dependent Anisotropic Birefringence: Validation Dataset -- DOI:10.5281/zenodo.173964284. Oct 21, 2025 (v1.2) -- Two-Harmonic Extension -- DOI:10.5281/zenodo.174107645. Oct 28, 2025 (v1.3) -- Two-Harmonic Dipole Verification -- DOI:10.5281/zenodo.174689886. Nov 1, 2025 (v1.4) -- MASTER-Calibrated Dipole -- DOI:10.5281/zenodo.175007917. Nov 1, 2025 (v1.41) -- Extended MASTER Calibration and Robustness -- DOI:10.5281/zenodo.175089088. Nov 7, 2025 (v1.42) -- Dependence-Aware Joint Validation -- DOI:10.5281/zenodo.175538299. Nov 8, 2025 (v1.43) -- Phase Model Validation -- DOI:10.5281/zenodo.1756131310. Nov 8, 2025 (v1.44) -- Axis + Frequency + Half-Mission Validation -- DOI:10.5281/zenodo.1756176811. Nov 9, 2025 (v1.5) -- Multipole Structure and Model Selection -- DOI:10.5281/zenodo.1756296512. Nov 9, 2025 (v1.6) -- Phenomenology and Physical Interpretation -- DOI:10.5281/zenodo.1756619713. Nov 9, 2025 (v1.7) -- Prediction and Experiment Overlays -- DOI:10.5281/zenodo.1756687014. Nov 9, 2025 (v1.8) -- Model Rejection and Delta-l Persistence -- DOI:10.5281/zenodo.1756724115. Nov 10, 2025 (v2.0) -- Intrinsic Periodicity in l-space -- DOI:10.5281/zenodo.1757404816. Nov 10, 2025 (v2.1) -- Physical Origin of Delta-l Modulation -- DOI:10.5281/zenodo.1757708617. Nov 11, 2025 (v2.2) -- Universe-Model Evaluation -- DOI:10.5281/zenodo.1758541918. Nov 12, 2025 (v2.3) -- Domain Geometry and Topological Inference -- DOI:10.5281/zenodo.1759415719. Nov 13, 2025 (v2.4) -- Real-Space Correlation of Birefringence Field -- DOI:10.5281/zenodo.1759753720. Nov 13, 2025 (v2.5) -- Spectral Surgery on Delta-l ~109 Harmonic -- DOI:10.5281/zenodo.1760498221. Nov 14, 2025 (v2.6) -- Angular Locality of Delta-l = 109 Standing Wave -- DOI:10.5281/zenodo.1761334822. Nov 15, 2025 (v2.7) -- Sky-Local Origin of Delta-l ~109 Standing Wave -- DOI:10.5281/zenodo.1762002923. Nov 15, 2025 (v2.8) -- Domain Topology of Delta-l ~109 Standing Wave -- DOI:10.5281/zenodo.1762060524. Nov 16, 2025 (v2.9) -- Dual-Domain Coherence and Boundary Geometry -- DOI:10.5281/zenodo.1762187125. Nov 17, 2025 (v2.10) -- Boundary Sequence Structure on Dual-Domain Loop -- DOI:10.5281/zenodo.1763581126. Nov 19, 2025 (v2.11) -- Boundary Standing-Wave and Phase-Structure Analysis -- DOI:10.5281/zenodo.1764803327. Nov 21, 2025 (v2.12) -- Boundary Universality and Standing-Wave Fingerprints -- DOI:10.5281/zenodo.1767637728. Nov 23, 2025 (v2.13) -- Interior Propagation and Boundary-Driven Structure -- DOI:10.5281/zenodo.17693540 Contact email: [e-mail] 22 Blue - The Heartbeat of the Universe