Rotation Field of the Cosmic Microwave Background — Domain Topology of the Δℓ ≈ 109 Standing Wave (v2.8)

Summary Version 2.8 examines whether the Δℓ ≈ 109 standing wave—shown in earlier versions to be θ-local (v2.6) and sky-local on boundary pixels (v2.7)—arises from a coherent domain topology in the birefringence field α(n̂). This version performs a purely geometric analysis on the locked NSIDE=16 maps, with no new α reconstruction, no Cℓ estimation, and no Δℓ fitting. --- Purpose Past results established the structure of the Δℓ ≈ 109 mode: • v2.4: The real–space correlation function ξ(θ) peaks at θ₀ ≈ 3.3°. • v2.6: The mode is θ-local — removing the 3.3° band destroys the harmonic. • v2.7: The mode is sky-local — it lives entirely on a small fraction of “boundary” pixels identified by a ∇²-based proxy. v2.8 asks the next logical question: **Do those boundary pixels form a coherent, non-random domain skeleton?** --- Methods 1. Input maps A locked NSIDE=16 α-map and a locked ∇²-proxy map from v2.6 were used: • α₁₆(i) — birefringence values at NSIDE=16 • proxy(i) = |∇α|² (smoothed) Boundary pixels were defined exactly as in v2.7: the top 10% of proxy values. Let H denote the boundary set: H = { i | proxy(i) ≥ P90(proxy) } Total boundary pixels: |H| = 308 (out of 3072). 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. --- 2. Domain clustering Using NSIDE=16 HEALPix neighbor graphs, connected components of H were identified. Domains are defined by: Domain D_k = connected component of H Size(D_k) = number of pixels Perimeter ≈ number of boundary pixels adjacent to non-boundary pixels For each domain, the following statistics were computed: • size (# pixels) • mean α • variance of α • centroid (lon, lat) • perimeter and perimeter/area ratio • principal-axis orientation (via covariance of 3D unit vectors) --- 3. Boundary-boundary spacing test For all (i, j) ∈ H × H, angular separation θ_ij was computed using: cos θ_ij = n_i · n_j A histogram of all separations in 0–10° with 0.1° bins was produced. This tests the coarse geometric spacing of the boundary network (not ξ(θ)). The ξ(θ) correlation scale θ₀ = 3.3° is **not** expected to be the peak of the spacing histogram; it is a correlation property of α, not a pixel-separation statistic. --- 4. Topological null test A 200-realization null ensemble was generated by shuffling α values across all 3072 pixels while preserving geometry. For each null field: • The boundary mask H_null was re-computed using the same percentile rule. • Domain clustering was run again. • The size of the largest domain and total number of domains were recorded. This tests whether the observed domain skeleton could arise from a random field under the same mask/percentile definition. --- Findings 1. Domain topology v2.8 finds **8** connected boundary domains, with sizes: [136, 136, 17, 9, 4, 3, 2, 1] Two enormous antipodal domains dominate, each with 136 pixels (~44% of all boundary pixels). In contrast, the null ensemble produces: ⟨max_size⟩ = 7.38 ± 1.74 ⟨N_domains⟩ ≈ 198 ± 7 Thus: Z_max = (136 – 7.38) / 1.74 = 73.74 σ The real sky exhibits a **highly non-random, extremely coherent domain skeleton**. --- 2. Boundary spacing The H–H pair separation histogram shows: • Peak spacing ≈ 8.95° (coarse geometric mesh of the boundary network). • 3.0°–3.6° window contains 166 pairs (consistent presence of the ξ peak scale). Interpretation: The boundary grid spacing (~9°) reflects the domain structure; the Δℓ ≈ 109 mode’s correlation scale (3.3°) is **not** expected to dominate the spacing statistic. The findings confirm that the Δℓ mode rides *on top* of this domain skeleton rather than being produced by geometric spacing. --- 3. Interpretation The v2.8 results show: • The Δℓ ≈ 109 standing wave is carried by a boundary network that forms two giant polar domains. • The topology is not consistent with random-field geometry at the 70+σ level. • The correlation scale (3.3°) and the domain network scale (~9°) coexist naturally: – 9° ≈ coarse separation of boundary structures – 3.3° ≈ intrinsic oscillation scale of α(n̂) riding along those structures Thus v2.8 supports the interpretation that the Δℓ ≈ 109 harmonic is not an artifact of pixel geometry, noise, or random topology — it reflects a structured domain skeleton in the CMB rotation field. --- Instructions All inputs are locked NSIDE=16 products inherited from v2.6–v2.7. No α reconstruction, harmonic fitting, or smoothing should be re-run when reusing this dataset. All geometry statistics are fully reproducible from the included NPZ, JSON, and NPY files. --- CitationCondit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Topology of the Δℓ ~ 109 Boundary Network (v2.8).* 22 Blue — The Heartbeat of the Universe. Zenodo. https://doi.org/10.5281/zenodo.17620605 Derived from 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 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