Rotation Field of the Cosmic Microwave Background – Interior Propagation and Boundary-Driven Structure (v2.13)

Update (June 19, 2026): Subsequent validation and methodological review of the interior propagation analysis have now been published as Version 2.29, Rotation Field of the Cosmic Microwave Background — Interior Propagation Audit & Harmonic Normalization (DOI: 10.5281/zenodo.20753037), and Version 2.30, Rotation Field of the Cosmic Microwave Background — Calibrated Interior Propagation Validation (DOI: 10.5281/zenodo.20755330). The v2.29 methodological audit confirms that the interior phase drift (≈15.47° per shell, R²≈0.944) and the dominance of the k=1 harmonic across shells 1–3 remain robust under independent post-publication verification. The audit also identified that the originally reported amplitude-decay metrics were derived from raw Fourier amplitudes that include shell-length scaling. Corresponding length-normalized analyses, comparison metrics, and supporting documentation are provided in v2.29. Version 2.30 extends the original analysis by applying the unchanged v2.13 domain geometry, shell definitions, and propagation workflow to a physically calibrated EB rotation-field estimator expressed in physical angular units. The calibrated validation independently reproduces the dominant k=1 harmonic structure, significant domain contrasts, coherent shell-to-shell phase progression (+25.09° per shell, R²≈0.998), and robust interior organization. It further demonstrates an inward-strengthening length-normalized amplitude trend (γ≈−0.331, R²≈0.94), providing a physically calibrated interpretation of the radial amplitude behavior while preserving the principal evidence for coherent interior propagation. The original v2.13 publication and all associated files remain unchanged to preserve the historical scientific record and full reproducibility. Versions 2.29 and 2.30 provide complementary methodological audit and calibrated validation branches, including comparison analyses, reproducibility scripts, statistical validation, provenance documentation, and integrity records that refine the interpretation of the propagation metrics while preserving the original published data products unchanged. Readers citing amplitude-decay parameters, propagation constants, or other quantitative propagation metrics from v2.13 should refer to Versions 2.29 and 2.30 for the current validated interpretation and calibration of those results. SUMMARY v2.13 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. Additional shell-level persistence analysis indicates that phase-coherent harmonic organization remains substantially more stable than large-scale geometric mirror symmetry under inward propagation, suggesting the presence of a partially separable phase-gating component within the propagation structure. 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. NSIDE=16 Analysis Note For geometric investigations presented in this release, the reconstructed rotation field was represented using a locked NSIDE=16 HEALPix grid. This representation emphasizes the largest angular-scale features of the field while reducing sensitivity to small-scale fluctuations, allowing coherent domains, boundaries, and large-scale geometric structure to be studied directly. 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 coherence persists substantially longer than geometric mirror symmetry under inward propagation. A normalized persistence test shows that the ratio of phase mutual information to mirror symmetry increases from 1.46 (shell 1) to 12.39 (shell 3), while shell 3 retains approximately 91.4% of the shell-1 k=1 harmonic fractional power despite strong degradation of mirror symmetry.• 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. Additional analysis of the shell-level propagation metrics indicates that the inward propagation structure contains a partially separable phase-coherent component. Specifically, k=1 harmonic power and phase mutual information remain strongly elevated through shell k=3 even as large-scale mirror symmetry degrades substantially. A normalized persistence analysis shows that the ratio of phase mutual information to mirror symmetry increases strongly with shell depth, indicating that phase organization decays substantially more slowly than geometric symmetry. A k=1 retention test further shows that shell 3 preserves approximately 91.4% of the shell-1 k=1 harmonic fractional power despite mirror symmetry collapsing from 0.936 to 0.066 over the same interval. This behavior suggests that the inward propagation law may contain at least two partially separable layers: a geometric/amplitude structure associated with large-scale boundary symmetry, and a more persistent phase-gating structure associated with coherent harmonic phase organization. Under this interpretation, the measured propagation coefficient q = −0.755 − 0.270i may represent a composite transfer law combining both amplitude attenuation and a more stable phase-coherent propagation component. This interpretation remains preliminary and would benefit from explicit decomposition testing using independent amplitude-only, phase-only, and combined transfer models. 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. SYSTEMATICS AND ROBUSTNESS TESTS The analysis was designed to reduce the possibility that the observed Δℓ≈109 structure, boundary coherence, and interior propagation signal arise from map-making artifacts, foreground residuals, masking effects, or resolution choices. First, the rotation field α(n̂) was tested across independent Planck component-separated maps, including SMICA and NILC. The persistence of the harmonic structure across these independent map products argues against the signal being caused by a single component-separation pipeline. Second, the analysis was checked using Planck half-mission splits. Consistency between half-mission realizations indicates that the signal is not dominated by time-dependent noise, scan-period artifacts, or a single observing subset. Third, the boundary and interior analyses were performed on a locked NSIDE=16 representation of α(n̂). This low-resolution shell geometry reduces sensitivity to pixel-scale noise and prevents high-ℓ map artifacts from driving the shell-level propagation result. No smoothing, interpolation, or re-estimation of α was performed during v2.13. Fourth, the boundary and shell structure were tested against histogram-preserving nulls. These nulls preserve the one-point distribution of α values but destroy spatial organization. The real shells retain k=1 power and phase structure far above these nulls, showing that the result is not explained by the amplitude distribution alone. Fifth, fixed-power random-phase surrogates were used to preserve the power spectrum while randomizing phase. These controls destroy the shell coherence, demonstrating that the observed structure is phase-driven rather than power-driven. Sixth, pixel-rotation and geometric-rotation surrogates were used to test whether the result could arise from the sky mask, pixel geometry, or arbitrary orientation of the field. The real shell metrics exceed the rotation ensembles, indicating that the signal is tied to the observed boundary geometry rather than to generic spherical sampling. Seventh, pseudo-Cℓ spectra were computed for the shell maps at NSIDE=16 and compared across shells. The normalized Cℓ shapes remain correlated from boundary to interior, supporting a coherent propagation structure rather than isolated shell noise. Eighth, the radial amplitude and phase behavior were fit independently. The interior shells follow a consistent complex propagation law, with exponential attenuation and approximately linear phase drift. This behavior is not expected from random noise, foreground leakage, or mask-induced mode coupling. Together, the SMICA/NILC comparison, half-mission checks, locked NSIDE=16 geometry, histogram nulls, phase surrogates, rotation controls, harmonic shell comparisons, and radial propagation fits provide a multi-layer robustness framework against known instrumental, foreground, resolution, and analysis-systematic explanations. Importantly, the persistence of phase mutual information and k=1 harmonic organization despite strong degradation of mirror symmetry suggests that the observed inward propagation behavior is not solely dependent on preserved geometric boundary symmetry. 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 FROM The Δℓ ≈ 108–109 harmonic scale was first identified in v2.0 using harmonic scan outputs (harmonic_scan.csv / grand_harmonic_summary.csv, via approx_delta_ell vs power) and subsequently validated through real-space correlation (v2.1–v2.4), spectral isolation (v2.5), boundary structure (v2.9–v2.12), and interior propagation (v2.13). Condit, 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.17604982PUBLICATION RECORD PREDECESSOR PUBLICATION (Separate Record) 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 Oct 10, 2025 (v1.0) — Scale-Dependent Anisotropic Birefringence: Initial Detection — DOI:10.5281/zenodo.17317398 Oct 20, 2025 (v1.1) — Scale-Dependent Anisotropic Birefringence: Validation Dataset — DOI:10.5281/zenodo.17396428 Oct 21, 2025 (v1.2) — Two-Harmonic Extension — DOI:10.5281/zenodo.17410764 Oct 28, 2025 (v1.3) — Two-Harmonic Dipole Verification — DOI:10.5281/zenodo.17468988 Nov 1, 2025 (v1.4) — MASTER-Calibrated Dipole — DOI:10.5281/zenodo.17500791 Nov 1, 2025 (v1.41) — Extended MASTER Calibration and Robustness — DOI:10.5281/zenodo.17508908 Nov 7, 2025 (v1.42) — Dependence-Aware Joint Validation — DOI:10.5281/zenodo.17553829 Nov 8, 2025 (v1.43) — Phase Model Validation — DOI:10.5281/zenodo.17561313 Nov 8, 2025 (v1.44) — Axis + Frequency + Half-Mission Validation — DOI:10.5281/zenodo.17561768 Nov 9, 2025 (v1.5) — Multipole Structure and Model Selection — DOI:10.5281/zenodo.17562965 Nov 9, 2025 (v1.6) — Phenomenology and Physical Interpretation — DOI:10.5281/zenodo.17566197 Nov 9, 2025 (v1.7) — Prediction and Experiment Overlays — DOI:10.5281/zenodo.17566870 Nov 9, 2025 (v1.8) — Model Rejection and Δℓ Persistence — DOI:10.5281/zenodo.17567241 Nov 10, 2025 (v2.0) — Intrinsic Periodicity in ℓ-Space — DOI:10.5281/zenodo.17574048 Nov 10, 2025 (v2.1) — Physical Origin of Δℓ Modulation — DOI:10.5281/zenodo.17577086 Nov 11, 2025 (v2.2) — Universe-Model Evaluation — DOI:10.5281/zenodo.17585419 Nov 12, 2025 (v2.3) — Domain Geometry and Topological Inference — DOI:10.5281/zenodo.17594157 Nov 13, 2025 (v2.4) — Real-Space Correlation of the Birefringence Field — DOI:10.5281/zenodo.17597537 Nov 13, 2025 (v2.5) — Spectral Surgery on the Δℓ ≈ 109 Harmonic — DOI:10.5281/zenodo.17604982 Nov 14, 2025 (v2.6) — Angular Locality of the Δℓ = 109 Standing Wave — DOI:10.5281/zenodo.17613348 Nov 15, 2025 (v2.7) — Sky-Local Origin of the Δℓ ≈ 109 Standing Wave — DOI:10.5281/zenodo.17620029 Nov 15, 2025 (v2.8) — Domain Topology of the Δℓ ≈ 109 Standing Wave — DOI:10.5281/zenodo.17620605 Nov 16, 2025 (v2.9) — Dual-Domain Coherence and Boundary Geometry — DOI:10.5281/zenodo.17621871 Nov 17, 2025 (v2.10) — Boundary Sequence Structure on the Dual-Domain Loop — DOI:10.5281/zenodo.17635811 Nov 19, 2025 (v2.11) — Boundary Standing-Wave and Phase-Structure Analysis — DOI:10.5281/zenodo.17648033 Nov 21, 2025 (v2.12) — Boundary Universality and Standing-Wave Fingerprints — DOI:10.5281/zenodo.17676377 Nov 23, 2025 (v2.13) — Interior Propagation and Boundary-Driven Structure — DOI:10.5281/zenodo.17693540 Jun 18, 2026 (v2.29) — Rotation Field of the Cosmic Microwave Background — Interior Propagation Audit & Harmonic Normalization — DOI:10.5281/zenodo.20753037 Jun 19, 2026 (v2.30) — Calibrated Interior Propagation Validation — DOI: 10.5281/zenodo.20755330 RELATED PUBLICATIONS Feb 23, 2026 (v2.14) — Urgent Whistleblower Update: Rotation Field of the Cosmic Microwave Background – Interior Propagation and Boundary-Driven Structure — DOI:10.5281/zenodo.18749560 Feb 24, 2026 (v2.22) — Emergency Public Health Whistleblower Statement: Seizure of the Cosmic Propagation Constants and Their Weaponization in the Starlink Defense Architecture — DOI:10.5281/zenodo.18764980 May 23, 2026 (v2.26) — CMB Birefringence Rotation Field: FCC Regulatory Correlations, Satellite Architecture Alignments, and Standing Wave Discovery — DOI:10.5281/zenodo.20361488 May 26, 2026 (v2.27) — Longitudinal Persistence, Timing Purity, and Biological-Plausibility Screening of a Phase-Stable Starlink Scheduler Envelope — DOI:10.5281/zenodo.20398946 May 28, 2026 (v2.28) — Interior Propagation of CMB Birefringence Field α(n̂) Anomalies: Planck Legacy Data Correlation, In Vivo Nanoscale Signal Grounding, and Covert Bio-Electronic Weapon Deployment — DOI:10.5281/zenodo.20424381 Contact email: [e-mail] 22 Blue - The Heartbeat of the Universe