DOI: 10.5281/zenodo.17621871
title: Rotation Field of the Cosmic Microwave Background — Dual-Domain Coherence & Boundary Geometry (v2.9)

description: 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.

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

Version History (22 Blue Dataset Series)Separate publication — Sep 20 2025 Evidence for harmonic phase alignments in Planck 2018 polarization data (Δℓ ~ 320)v1.0 — Oct 10 First quantitative detection of anisotropic cosmic birefringencev1.1 — Oct 20 Statistical validation and reproducibilityv1.2 — Oct 21 Two-harmonic extensionv1.3 — Oct 28 Robustness tests and injectionsDataverse Mirror — Oct 29 DOI 10.7910/DVN/PTDG20v1.4 — Nov 1 MASTER-calibrated α(ℓ)v1.41–1.44 — Nov 1–8 SMICA/NILC splits; parity and axis testsv1.5 — Nov 9 Model selection; Δℓ ≈ 454.7v1.6 — Nov 9 Interpretationv1.7 — Nov 9 Forward prediction of α(n̂) and EB(ℓ)v1.8 — Nov 9 Harmonic persistence and rejection testsv2.0 — Nov 10 Intrinsic Δℓ ≈ 108 discoveredv2.1 — Nov 10 ξ(θ) physics; θ₀ ≈ 3.3°v2.2 — Nov 11 Universe-model evaluationv2.3 — Nov 12 Domain geometry inferencev2.4 — Nov 13 Direct ξ(θ) confirmationv2.5 — Nov 13 Spectral surgery Δℓ ≈ 109 necessary and phase-coherentv2.6 — Nov 14 Angular localityv2.7 — Nov 15 Sky-local originv2.8 — Nov 15 Domain topology of the Δℓ ≈ 109 boundary networkv2.9 — Nov 16 Field coherence of the Δℓ ≈ 109 dual-domain structure

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