DOI: 10.5281/zenodo.17620605
title: Rotation Field of the Cosmic Microwave Background — Domain Topology of the Δℓ ≈ 109 Standing Wave (v2.8)

description: 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.

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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?**

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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).

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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)

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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.

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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.

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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**.

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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.

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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.

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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.

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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  

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 birefringence  v1.1 — Oct 20 • Statistical validation and reproducibility  v1.2 — Oct 21 • Two-harmonic extension  v1.3 — Oct 28 • Robustness tests and injections  Dataverse Mirror — Oct 29 • DOI: 10.7910/DVN/PTDG20  v1.4 — Nov 1 • MASTER-calibrated α(ℓ)  v1.41–1.44 — Nov 1–8 • SMICA/NILC splits; parity and axis tests  v1.5 — Nov 9 • Model selection; Δℓ ≈ 454.7  v1.6 — Nov 9 • Interpretation  v1.7 — Nov 9 • Forward prediction of α(n̂) and EB(ℓ)  v1.8 — Nov 9 • Harmonic persistence and rejection tests  v2.0 — Nov 10 • Intrinsic Δℓ ≈ 108 discovered  v2.1 — Nov 10 • ξ(θ) physics; θ₀ ≈ 3.3°  v2.2 — Nov 11 • Universe-model evaluation  v2.3 — Nov 12 • Domain geometry inference  v2.4 — Nov 13 • Direct ξ(θ) confirmation  v2.5 — Nov 13 • Spectral surgery: Δℓ ≈ 109 necessary & phase-coherent  v2.6 — Nov 14 • Angular locality  v2.7 — Nov 15 • Sky-local origin  v2.8 — Nov 15 • Domain topology of the Δℓ ≈ 109 boundary network  

22 Blue — The Heartbeat of the Universe  Twitter/X: @Project22Blue