DOI: 10.5281/zenodo.17693540
title: Rotation Field of the Cosmic Microwave Background – Interior Propagation and Boundary-Driven Structure (v2.13)

description: SUMMARYv2.13 performs the first interior-propagation test of the CMB birefringence field α(n̂). Earlier versions (v2.9–v2.12) established that the dual-domain boundary D₁ ∪ D₂ contains an intrinsic standing-wave and phase-coherent structure dominated by Δℓ ≈ 109. This version tests whether that boundary fingerprint continues inward—whether interior pixels inherit the same harmonic and phase structure.

Using only locked inputs from prior versions (α₁₆, H_mask, domain_label_map, boundary chain, and v2.12 fingerprints), the sky interior is divided into adjacency shells k = 0…3. Each shell is analyzed for harmonic structure, phase coherence, k=1 power, surrogate significance, and radial transfer behavior. The analysis confirms that the boundary pattern is not isolated: interior shells preserve strong k=1 coherence, Δℓ≈109 signatures, and stable phase evolution. A radial propagation law is detected, with exponential amplitude decay and a consistent phase drift per shell.

DEFINITIONSα(n̂): Locked birefringence rotation field (v2.9), NSIDE=16.H_mask: High-latitude analysis mask.domain_label_map: Domain segmentation map.Boundary chain: Deterministic D₁ ∪ D₂ boundary pixels from v2.11.Shells: Interior sets of pixels defined by graph distance from the boundary.

EQUATIONSShell distance: dist(p) = min_b geodesic_graph_distance(p, b)k=1 fractional power: k1 = |FFT₁(α)[1]|² / Σ_j |FFT₁(α)[j]|²Complex shell coefficient: c_k = A_k · exp(i φ_k)Radial model: c_k ≈ C₀ · exp[q (k – 1)] where q = –γ + iκPseudo-Cℓ: Cℓ = hp.anafast(masked_shell, lmax=47)

METHODS1. Shell construction: BFS from the v2.11 boundary using NSIDE=16 adjacency (140 boundary pixels, shells up to k=3).2. Metric suite: k=1 power, mirror symmetry, phase mutual information, residual entropy, LZ complexity, windowed FFT.3. Surrogates:   – 200 histogram-preserving nulls   – 200 random-phase fixed-power surrogates   – 50 pixel rotation surrogates   – 50 geometric rotations4. Harmonic analysis: alm extraction, pseudo-Cℓ, normalized spectrum shapes, cross-shell spectral correlation.5. Radial propagation: amplitude decay fit, phase drift fit, complex exponential model, prediction tests.

KEY FINDINGS• k=1 fractional power remains extremely high in shells k=1–3 (null z-scores up to 34; phase/rotation surrogates excluded).  • Mirror symmetry remains significant through k=2.  • Phase mutual information remains far above nulls (z ≈ 7–15).  • Harmonic spectra maintain strong correlation with boundary spectra (0.93, 0.85, 0.67 for shells k=1,2,3).  • Cross-shell spectral correlation matrix reveals coherent propagation.  • Radial amplitude attenuation: e-fold ≈ 1.32 shells.  • Phase rotates inward by ≈ 15.5° per shell (R² ≈ 0.94).  • Complex propagation coefficient: q = –0.755 – 0.270i (R² ≈ 0.93).  • Predictive propagation from Shell 1 reproduces Shell 3 amplitude/phase.  • Boundary→interior prediction fails, showing the boundary is not the source but an exposed cross-section of deeper interior structure.

INTERPRETATIONThe standing-wave and Δℓ≈109 structure discovered on the dual-domain boundary propagates inward into the domain interiors. Interior shells inherit the same harmonic mode and phase pattern with predictable amplitude decay and phase drift. This indicates that α(n̂) contains a domain-level interior field pattern; the boundary reveals this structure but does not generate it.

INSTRUCTIONS• All inputs locked from v2.9–v2.12.  • No recomputation of α performed.  • All randomness fully seeded.  • Contains two run directories: the main interior-propagation analysis and the harmonic-environment extension.  • See manifest for full file listing and SHA checksums.

CITATIONCondit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Interior Propagation and Boundary-Driven Structure (v2.13). 22 Blue – The Heartbeat of the Universe. https://doi.org/10.5281/zenodo.17693540

DERIVED FROMCondit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Boundary Universality and Standing-Wave Fingerprint Analysis (v2.12). https://doi.org/10.5281/zenodo.17676377Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Boundary Standing-Wave and Phase-Structure Analysis (v2.11). https://doi.org/10.5281/zenodo.17648033Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Boundary Sequence Structure on the Dual-Domain Loop (v2.10). https://doi.org/10.5281/zenodo.17635811Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Dual-Domain Coherence and Boundary Geometry (v2.9). https://doi.org/10.5281/zenodo.17621871Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Topology of the Delta ℓ ≈ 109 Boundary Network (v2.8). https://doi.org/10.5281/zenodo.17620605Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Angular Locality of the Delta ℓ ≈ 109 Standing Wave (v2.6). https://doi.org/10.5281/zenodo.17613348Condit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Spectral Surgery on the Delta ℓ ≈ 109 Harmonic (v2.5). https://doi.org/10.5281/zenodo.17604982

VERSION HISTORYSep 20 2025 – Harmonic phase alignments discoveredOct 10 – v1.0 First quantitative detectionOct 20 – v1.1 Statistical validationOct 21 – v1.2 Two-harmonic extensionOct 28 – v1.3 Robustness testsOct 29 – Dataverse DOI 10.7910/DVN/PTDG20Nov 1 – v1.4 MASTER calibrated spectraNov 1–7 – v1.41 to v1.44 SMICA/NILC splits and parity testsNov 9 – v1.5 Model selectionNov 9 – v1.6 InterpretationNov 9 – v1.7 Forward predictionNov 9 – v1.8 Persistence testsNov 10 – v2.0 Intrinsic Δℓ≈108 discoveredNov 10 – v2.1 ξ(θ) physicsNov 11 – v2.2 Universe model evaluationNov 12 – v2.3 Domain geometry inferenceNov 13 – v2.4 Real-space confirmationNov 13 – v2.5 Spectral surgeryNov 14 – v2.6 Angular localityNov 15 – v2.7 Sky localityNov 15 – v2.8 Domain topologyNov 16 – v2.9 Field coherenceNov 17 – v2.10 Boundary sequence structureNov 19 – v2.11 Boundary standing-wave and phase structureNov 21 – v2.12 Boundary universalityNov 23 – v2.13 Interior propagation and boundary-driven structure

— 22 BlueThe Heartbeat of the Universe