Rotation Field of the Cosmic Microwave Background — Angular Locality of the Δℓ = 109 Standing Wave (v2.6)

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. 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. 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 This release is part of an iterative research series in which analysis methods, masks, multipole selections, and calibration procedures were progressively refined. PUBLICATION RECORD 1. Sep 20, 2025 (v1.0) — Harmonic Phase Alignments in Planck 2018 CMB — DOI:10.5281/zenodo.17167268 MAIN RESEARCH SERIES Concept DOI:10.5281/zenodo.17317397 1. Oct 10, 2025 (v1.0) — Scale-Dependent Anisotropic Birefringence: Initial Detection — DOI:10.5281/zenodo.173173982. Oct 20, 2025 (v1.1) — Scale-Dependent Anisotropic Birefringence: Validation Dataset — DOI:10.5281/zenodo.173964283. Oct 21, 2025 (v1.2) — Two-Harmonic Extension — DOI:10.5281/zenodo.174107644. Oct 28, 2025 (v1.3) — Two-Harmonic Dipole Verification — DOI:10.5281/zenodo.174689885. Nov 1, 2025 (v1.4) — MASTER-Calibrated Dipole — DOI:10.5281/zenodo.175007916. Nov 1, 2025 (v1.41) — Extended MASTER Calibration and Robustness — DOI:10.5281/zenodo.175089087. Nov 7, 2025 (v1.42) — Dependence-Aware Joint Validation — DOI:10.5281/zenodo.175538298. Nov 8, 2025 (v1.43) — Phase Model Validation — DOI:10.5281/zenodo.175613139. Nov 8, 2025 (v1.44) — Axis + Frequency + Half-Mission Validation — DOI:10.5281/zenodo.1756176810. Nov 9, 2025 (v1.5) — Multipole Structure and Model Selection — DOI:10.5281/zenodo.1756296511. Nov 9, 2025 (v1.6) — Phenomenology and Physical Interpretation — DOI:10.5281/zenodo.1756619712. Nov 9, 2025 (v1.7) — Prediction and Experiment Overlays — DOI:10.5281/zenodo.1756687013. Nov 9, 2025 (v1.8) — Model Rejection and Δℓ Persistence — DOI:10.5281/zenodo.1756724114. Nov 10, 2025 (v2.0) — Intrinsic Periodicity in ℓ-Space — DOI:10.5281/zenodo.1757404815. Nov 10, 2025 (v2.1) — Physical Origin of Δℓ Modulation — DOI:10.5281/zenodo.1757708616. Nov 11, 2025 (v2.2) — Universe-Model Evaluation — DOI:10.5281/zenodo.1758541917. Nov 12, 2025 (v2.3) — Domain Geometry and Topological Inference — DOI:10.5281/zenodo.1759415718. Nov 13, 2025 (v2.4) — Real-Space Correlation of the Birefringence Field — DOI:10.5281/zenodo.1759753719. Nov 13, 2025 (v2.5) — Spectral Surgery on the Δℓ ≈ 109 Harmonic — DOI:10.5281/zenodo.1760498220. Nov 14, 2025 (v2.6) — Angular Locality of the Δℓ = 109 Standing Wave — DOI:10.5281/zenodo.1761334821. Nov 15, 2025 (v2.7) — Sky-Local Origin of the Δℓ ≈ 109 Standing Wave — DOI:10.5281/zenodo.1762002922. Nov 15, 2025 (v2.8) — Domain Topology of the Δℓ ≈ 109 Standing Wave — DOI:10.5281/zenodo.1762060523. Nov 16, 2025 (v2.9) — Dual-Domain Coherence and Boundary Geometry — DOI:10.5281/zenodo.1762187124. Nov 17, 2025 (v2.10) — Boundary Sequence Structure on the Dual-Domain Loop — DOI:10.5281/zenodo.1763581125. Nov 19, 2025 (v2.11) — Boundary Standing-Wave and Phase-Structure Analysis — DOI:10.5281/zenodo.1764803326. Nov 21, 2025 (v2.12) — Boundary Universality and Standing-Wave Fingerprints — DOI:10.5281/zenodo.1767637727. Nov 23, 2025 (v2.13) — Interior Propagation and Boundary-Driven Structure — DOI:10.5281/zenodo.1769354028. Jun 18, 2026 (v2.29) — Rotation Field of the Cosmic Microwave Background — Interior Propagation Audit & Harmonic Normalization — DOI:10.5281/zenodo.2075303729. Jun 19, 2026 (v2.30) — Calibrated Interior Propagation Validation — DOI:10.5281/zenodo.2075533030. Jun 20, 2026 (v2.31) — Rotation Field of the Cosmic Microwave Background — Physical Origin of Boundary-to-Interior Propagation — DOI:10.5281/zenodo.2077743531. Jun 21, 2026 (v2.32) — Rotation Field of the Cosmic Microwave Background — Antipodal Specificity, Boundary Ordering, and Cross-Scale Spectral Organization — DOI:10.5281/zenodo.2078730732. Jun 24, 2026 (v2.33) — Model Comparison and Global Phase Coherence of the Cosmic Birefringence Rotation Field — DOI:10.5281/zenodo.20825890 22 Blue - The Heartbeat of the Universe