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

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.

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.

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.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