Rotation Field of the Cosmic Microwave Background — Physical origin of the intrinsic Δℓ modulation (v2.1)

Summary Version 2.1 resolves the open question from prior releases:**Where does the intrinsic periodicity in the CMB rotation field α(n̂) come from?** Earlier versions (v1.0–v1.8) identified a harmonic modulation in Cℓ(αα).v2.0 isolated the *intrinsic* periodicity at Δℓ ≈ 108 and showed that a larger period (~454) was an artifact of low-ℓ leakage. v2.1 goes from detection → interpretation: > The Δℓ harmonic originates from a real angular correlation structure in α(n̂). METHOD (v2.1) Inputs (read-only):• locked residual Cℓ(αα) spectrum from v2.0 (dipole + traceless quadrupole removed, identical mask + apodization)• loaded by SHA-256 pointer lock (no Q/U recomputation, no overwrites) Procedure: 1. Detrend the residual Cℓ using a Savitzky–Golay window ≈ 0.5·Δℓ to isolate the oscillatory component without altering phase. 2. Project into real space using Gauss–Legendre quadrature: ξ^{αα}(θ) = Σℓ [(2ℓ+1)/(4π)] · Cℓ · Pℓ(cosθ) 3. Fit ξ(θ) with a damped cosine: ξ(θ) = A · exp(-θ/τ) · cos((2πθ / θ₀) + φ) + C 4. Independently refit Δℓ directly in ℓ-space using linear regression on {cos(2πℓ/Δℓ), sin(2πℓ/Δℓ)} basis. All inputs and outputs are checksum-verified (SHA-256 manifest).No file is ever overwritten. Reproducibility is guaranteed. RESULTS A statistically significant oscillation is found in real space: θ₀ = 3.3° ± 0.3° Using the Legendre→Bessel small-angle duality(narrowband assumption, small-angle approximation): Δℓ ≈ 360° / θ₀ → Δℓ = 109 ± 3 matching the intrinsic harmonic from v2.0 (Δℓ = 108). Thus: > The Δℓ periodicity in multipole space is the spherical (Legendre–Bessel) dual> of a real angular correlation scale in α(n̂). WHY THIS MATTERS v2.1 shows that the CMB rotation field α(n̂) is not featureless: • in ℓ-space: harmonic spacing Δℓ ≈ 108• in θ-space: real angular scale θ ≈ 3.3° This eliminates explanations such as:• mask ringing,• apodization artifacts,• binning / band-limit distortions. Any physical model of cosmic birefringence must now explainthe existence of a preferred angular scale. REPRODUCIBILITY & PROVENANCE • SHA-256 manifest for every file• pointer-locked residual input• zero-overwrite architecture• all ξ(θ) vectors, Δℓ scans, figures, fit JSON, and predictions included in ZIP CITATION Condit, Amy. Rotation Field of the Cosmic Microwave Background — Physical origin of the intrinsic Δℓ modulation (v2.1).22 Blue — The Heartbeat of the Universe. DOI: 10.5281/zenodo.17577086 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