Rotation Field of the Cosmic Microwave Background — Dual-Domain Coherence & Boundary Geometry (v2.9)

Summary Version 2.9 of the 22 Blue rotation-field analysis examines whether the Δℓ ≈ 109 standing wave—established in earlier releases (v2.0–v2.8)—is supported by a coherent geometric structure in the rotation field α(n̂). This version uses only locked NSIDE=16 inputs from v2.6–v2.8 and performs no new α reconstructions, no smoothing, and no harmonic-space modeling. All tests operate strictly in pixel space and θ-space. Prior versions established the following empirical facts:(1) α(ℓ) contains an intrinsic periodicity Δℓ ≈ 109;(2) α(n̂) exhibits a corresponding angular scale θ₀ ≈ 3.3°;(3) the top 10% ∇α² boundary mask (H-mask) forms a non-random topology consisting of two giant antipodal domains plus six minor fragments (v2.7–v2.8). Version 2.9 asks a single question: Does the Δℓ ≈ 109 standing wave reside on a coherent, antipodal dual-domain structure inside α(n̂)? To answer this, v2.9 evaluates: antipodal geometry, same-sign mean-field plateaus, inter-lobe α-texture correlation, boundary thickness, and boundary-perimeter structure via 1D FFT. All tests were recomputed fresh inside v2.9 using locked inputs. Locked InputsAll inputs are inherited from v2.6–v2.8 and treated as read-only:alpha_nside16.npy — downsampled α(n̂) (v2.6)boundary_proxy_nside16.npy — ∇α² proxy (v2.6)H_mask_nside16.npy — top 10% boundary mask (v2.7)domain_label_map.npy — 8-domain decomposition (v2.8)The v2.9 pipeline uses only these four files. Methodology and DefinitionsThe rotation field α(n̂) is analyzed at NSIDE=16. Boundary pixels are defined via the gradient-squared proxy:proxy(p) = (1 / N_nbr) Σ_q ( α(p) – α(q) )²The H-mask contains pixels exceeding the 90th percentile of this proxy. Connected components of the H-mask define domains. For each domain D, the centroid vector c = (1/N) Σ_{p∈D} n_p is used to evaluate antipodal alignment, where n_p is the unit vector of pixel p. Inter-lobe coherence is measured by pairing each pixel p in domain D1 with its antipodal pixel p′ in domain D2 and computing angular mismatch Δθ, Pearson correlation r between α(D1) and α(D2), and the linear relation α2 ≈ m α1 + b. Interior thickness is evaluated by defining interior pixels as |α(p)| ≤ |⟨α⟩_D| / e and dist(p, ∂D) ≥ 1 pixel. Boundary-perimeter FFT is computed by tracing the ordered boundary loop α(s) around each domain and evaluating its 1D spectrum. 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. Results(A) Antipodal Geometry: The two dominant domains (each 136 pixels) have centroids separated by exactly 180°. The dot product is numerically –1.0. Antipodality is exact at NSIDE=16 resolution. (B) Same-Sign Plateau Means: Both domains exhibit elevated, same-sign plateau values: Domain 8 mean 2.77 × 10⁻²⁰, Domain 1 mean 3.24 × 10⁻²⁰, Bulk mean –4.03 × 10⁻²². Domain means exceed bulk magnitude by ~70×. (C) Inter-Lobe α-Texture Correlation: Antipodal pixel matching shows extremely small angular mismatch (mean Δθ ≈ 7.5×10⁻⁸ degrees). The α-textures are tightly linear with Pearson r = 0.9999266 and α_D2 ≈ 1.118 α_D1 + 1.39×10⁻²¹. (D) Boundary Thickness: Under the formal 1/e interior criterion, no interior pixels are found. All pixels within the large domains lie at or near the boundary. This indicates a thin-walled plateau with no gradual interior rolloff. (E) Boundary-Perimeter FFT: FFT of the ordered boundary reveals dominant broad-scale modes (k ≈ 14–22) and a fine-scale component corresponding to θ₀ ≈ 3.3°. Thus the same characteristic angular scale identified in earlier versions appears directly in the boundary geometry. Conclusions (Interpretation-Free)v2.9 provides a geometry-only, reconstruction-free confirmation that:• The two largest boundary domains are perfectly antipodal.• They share the same-sign elevation in α.• Their internal α-textures are almost perfectly correlated (r ≈ 0.99993).• They have no interior structure under the v2.9 definition.• Their boundaries contain the characteristic 3.3° scale previously detected in harmonic space.These findings establish that the Δℓ ≈ 109 standing wave is supported by a coherent, antipodal, dual-domain structure in the α(n̂) field at NSIDE=16. Version 2.9 is strictly geometric and prepares the foundation for v2.10 (field geometry) and v2.11 (phase-structure analysis). Condit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Dual-Domain Coherence & Boundary Geometry (v2.9).* 22 Blue — The Heartbeat of the Universe. https://doi.org/10.5281/zenodo.17621871Derived from Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background — Topology of the Δℓ ~ 109 Boundary Network (v2.8). Zenodo. https://doi.org/10.5281/zenodo.17620605Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background — Angular Locality of the Δℓ ~ 109 Standing Wave (v2.6). Zenodo. https://doi.org/10.5281/zenodo.17613348Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background — Spectral Surgery on the Δℓ ~ 109 Harmonic (v2.5). 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