Rotation Field of the Cosmic Microwave Background — Spectral Surgery on the Δℓ ≈ 109 Harmonic (v2.5)

Summary Version 2.5 performs a targeted “spectral surgery” on the locked birefringence spectrum Cℓ(αα) to determine whether the intrinsic harmonic at Δℓ ≈ 109—identified in v2.0–v2.1 and confirmed in v2.4—is the source of the 3.3° real-space correlation ridge observed in ξ(θ). This workflow constructs and compares four strictly ℓ-space spectra (observed, Δℓ-removed, Δℓ-only, and phase-scrambled), and evaluates their ξ(θ) peaks using only ℓ-domain and θ-domain operations (no maps, no alm, no healpy). The goal is to test whether the Δℓ ≈ 109 structure is necessary, nearly sufficient, and phase-coherent for producing the 3.3° feature. Scientific Motivation v2.0 revealed an intrinsic Δℓ ≈ 109 harmonic in the birefringence spectrum. v2.1 showed that this same harmonic naturally produces a 3.3° feature in ξ(θ). v2.4 directly measured that 3.3° peak in real space using Legendre projection. v2.5 asks the structural question: Does the 3.3° ridge genuinely originate from the Δℓ ≈ 109 harmonic—or could it be incidental, diffuse, or dependent on non-harmonic structure? Locked Inputs Residual spectrum (unchanged since v2.0–v2.4): alpha_residual_used_v20DETFOUND_20251110_235343.npy ℓ range: 100–1500 Locked priors: Δℓ = 109.0, θ₀ = 3.3° All methods restricted to ℓ-space and θ-space. No α(n̂), no alm, no healpy, no recalibration. Methods 1. Harmonic Model Hℓ(A, φ, ℓ_d, C) = A cos(2πℓ/109 + φ) exp(−ℓ/ℓ_d) + C Fitted parameters (from v2.5): A = 0.7164216634 φ = −0.0983750160 ℓ_d = 513.0066225 C = 0.3264039017 2. Constructed Spectra Observed spectrum: Cℓ^obs Δℓ-removed spectrum: Cℓ^noΔℓ = Cℓ^obs − Hℓ Δℓ-only spectrum: Cℓ^harm = Hℓ Phase-scrambled spectrum: Cℓ^scrambled = Hℓ with randomized φℓ (fixed seed) 3. Real-Space ξ(θ) Reconstruction ξ(θ) = Σℓ (2ℓ+1)/(4π) · Cℓ · Pℓ(cos θ) θ grid: 2°–5° with 0.01° spacing. Findings Observed ξ(θ): θ = 3.350°, ξ = 1551.36 Δℓ-removed: θ = 2.840°, ξ = 206.781 Δℓ-only: θ = 3.350°, ξ = 1451.36 Scrambled phases: θ = 2.250°, ξ = 258.65 Amplitude ratios (relative to the observed ξ peak): R_noΔℓ = 0.133 R_Δℓ-only = 0.936 R_scrambled = 0.167 Interpretation Necessary: Removing the Δℓ ≈ 109 harmonic collapses the ξ(θ) amplitude to 13% and shifts the peak away from 3.3°, demonstrating that the ridge cannot form without this mode. Nearly Sufficient: Keeping only the Δℓ ≈ 109 harmonic reproduces 94% of the observed ξ(θ) amplitude at exactly the same angular location, showing that the intrinsic standing wave accounts for nearly the full structure. Phase-Coherent: Using the same power spectrum but scrambling the phases destroys the coherence, reducing the signal to 17% and shifting the peak to 2.25°. The ridge is therefore a phase-coherent standing-wave, not a random power arrangement. These three results together show that the Δℓ ≈ 109 harmonic is the structural origin of the 3.3° real-space feature. Derived from Condit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Real-Space Correlation of the Birefringence Field (v2.4).* *22 Blue — The Heartbeat of the Universe.* Zenodo. https://doi.org/10.5281/zenodo.17597537 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