22 Blue — v2.3: Domain Geometry and Topological Inference

Summary  

Version 2.3 extends the verified birefringence pattern found in Versions 2.0–2.2 into a parametric investigation of the Universe’s angular-domain geometry.  The goal is to determine what finite-domain or boundary-mode configuration could reproduce the locked harmonic spacing  Δℓ ≈ 109  and  θ₀ ≈ 3.3°,  which together describe the fundamental periodicity of the CMB rotation field α(ℓ).

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Theoretical BackgroundIf the cosmic birefringence field is spatially coherent, its residual power spectrum after dipole- and quadrupole-subtraction will exhibit harmonic modulation:  

α(ℓ) = A exp[−(ℓ−ℓ₀)² / (2σ²)] cos(2πℓ / Δℓ + φ) + C.  

In real space this corresponds to a damped-cosine correlation function  ξ(θ) = A e^{−θ / τ} cos(2πθ / θ₀ + φ) + C,  where θ₀ ≈ 360° / Δℓ.  A persistent pair (Δℓ, θ₀) implies a repeating geometric domain or resonant boundary structure on the celestial sphere.

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What This Version Tests  1. Domain-size fitting — Fit ξ(θ) to determine whether a finite angular segmentation can sustain the observed periodicity.  2. Harmonic-phase alignment — Measure phase coherence between the empirical α(ℓ) residual and synthetic damped-cosine templates.  3. Boundary-mode (holographic) models — Evaluate whether standing-wave modes at radius ≈ 3.3° reproduce the same harmonic sequence.  4. Stability tests — Verify that the 3.3° scale and Δℓ ≈ 109 spacing persist under the same apodization masks used in v2.1.

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Why These Tests  These tests distinguish whether the locked periodicity arises from:  • Finite-domain topology — a tessellated polarization field with repeating angular domains;  • Boundary-mode resonance — a standing-wave interference pattern at a fixed boundary radius;  • Random field — baseline ΛCDM case with no harmonic structure.  

Only a coherent domain or boundary geometry can reproduce both the Δℓ and θ₀ values simultaneously.

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Key Calculations and Results  • Input residual : alpha_residual_used_v20DETFOUND_20251110_235343.npy (SHA-256 verified)  • Parameter scan Δℓ ∈ [106, 112] → best-fit Δℓ = 108.35 (θ₀ = 3.323°)  • Damping scale τ = 1.0°  • Amplitude, phase, offset A = 0.726, φ = –0.222 rad, C = 0.326  • χ² / dof = 0.044; AICc = –138 145  • Phase coherence R = 0.553 (R² = 0.306)  

These values quantitatively define a self-consistent angular domain geometry producing the observed harmonic modulation without invoking a physical interaction model.

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Citation  

Condit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Domain Geometry and Topological Inference (v2.3).*  

*22 Blue — The Heartbeat of the Universe.* Zenodo. https://doi.org/10.5281/zenodo.17594157

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Version History (22 Blue Dataset Series)  Separate publication — Sep 20 2025 • Evidence for Harmonic Phase Alignments in Planck 2018 CMB polarization data (Δℓ ≈ 320)  v1.0 — Oct 10 2025 • First quantitative detection of anisotropic cosmic birefringence (8.5 σ isotropic + 4–5 σ directional)  v1.1 — Oct 20 2025 • Full statistical validation (TB/EB diagnostics, covariance)  v1.2 — Oct 21 2025 • Two-harmonic extension (Δℓ ≈ 180 and 360)  v1.3 — Oct 28 2025 • Robustness tests (calibration, simulation, injection/recovery)  Harvard Dataverse Mirror — Oct 29 2025 • DOI: 10.7910/DVN/PTDG20  v1.4 — Nov 1 2025 • MASTER-calibrated rotation field α(ℓ)  v1.41 — Nov 1 2025 • SMICA/NILC + half-mission splits  v1.42 — Nov 7 2025 • Joint parity + harmonic significance (165 000 null realizations, ≈ 4.38 σ)  v1.43 — Nov 8 2025 • Parity invariant under MASTER coupling  v1.44 — Nov 8 2025 • Axis + frequency + half-mission validation  v1.5 — Nov 9 2025 • Multipole model selection; dipole-only rejected; Δℓ ≈ 454.7  v1.6 — Nov 9 2025 • Phenomenology & physical interpretation  v1.7 — Nov 9 2025 • Forward prediction (LiteBIRD / SO / CMB-S4 overlays)  v1.8 — Nov 9 2025 • Harmonic persistence & model rejection  v2.0 — Nov 10 2025 • Intrinsic Δℓ discovered (≈ 108) and artifact Δℓ (~454) isolated as low-ℓ leakage  v2.1 — Nov 10 2025 • Physical origin of intrinsic Δℓ ≈ 109 harmonic · ξ(θ) feature at θ ≈ 3.3° ± 0.3° · Legendre → Bessel dual confirmation · Locked/hashes-only workflow  v2.2 — Nov 11 2025 • Universe-model evaluation · Domain/topological framework favored · No reanalysis · Fixed priors Δℓ = 109 ± 3, θ = 3.3° ± 0.3°  v2.3 — Nov 12 2025 • Domain geometry and topological inference · Finite-domain angular scale θ₀ ≈ 3.3° · Phase coherence R = 0.553 (R² = 0.306) · Fully validated JSON and public upload bundle  

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