Rotation Field of the Cosmic Microwave Background — Waveform Geometry of the Calibrated Interior Propagation Field (V2.34)

Summary Version 2.34 characterizes the physical-space waveform geometry of the calibrated Cosmic Microwave Background (CMB) birefringence rotation field α(n̂). Previous releases within the 22 Blue Research Series established that the Domain 1/8 boundary contains a coherent harmonic structure dominated by Δℓ≈109 and that this boundary fingerprint propagates inward through successive adjacency shells. Version 2.34 addresses the next question: what is the geometric form of that propagation in physical space? Using only locked products generated by previous releases, the calibrated rotation field is analyzed as a four-shell radial propagation system containing 136 independent radial chains extending from the Domain 1/8 boundary into the interior. These chains are characterized to determine their common waveform, geometric coherence, curvature, antipodal structure, parity symmetry, amplitude evolution, and robustness. Principal findings Global phase coherence is independently verified before any geometric analysis. Reproduction of the calibrated patch-phase pipeline on the locked calibrated map yields R = 0.9886 across 40 retained sky patches (Rayleigh p = 3.51 × 10⁻¹⁶), reproducing the Version 2.33 result and confirming that all geometric analyses are performed on the same coherent phase structure. The locked shell system forms a closed four-shell propagation geometry. Shell 4 and all subsequent candidate shells returned zero pixels under every tested expansion rule, including saved adjacency expansion, raw HEALPix neighbor expansion, and relaxed exterior expansion. The four-point shell sequence is not a truncated analysis choice but the complete physical extent of the calibrated propagation system. Every reconstructed radial chain exhibits directed monotonic strengthening from the Domain 1/8 boundary toward the deepest interior shell. Mean reverse correlation across all 136 chains is −0.9727 (median −0.9820), ruling out a mirror-wave or standing-wave interpretation. The propagation geometry is fundamentally directional. The normalized waveform is highly reproducible across independent chains. Mean normalized amplitudes increase from 0.0000 at Shell 0 to 0.2758 at Shell 1, 0.6648 at Shell 2, and 1.0000 at Shell 3. Quadratic curvature provides the best aggregate description of the waveform, with the quadratic model selected as the best fit for 64 of 136 chains by Bayesian Information Criterion. Chain-to-chain coherence is exceptionally strong. Mean pairwise Pearson correlation is 0.9812 across all 136 chain pairs, with mean Spearman correlation of exactly 1.000. Principal component analysis demonstrates that the first principal component explains approximately 99.1% of total waveform variance. After removal of the common mean waveform, the residual variation separates into two structured modes (80.2% and 19.8%), confirming that even residual variation is geometrically organized. All 136 radial propagation chains form 68 exact antipodal pairs with no missing or ambiguous matches. Mean antipodal Pearson correlation is 0.9998 and mean Spearman correlation is 1.000. The mean normalized absolute difference between antipodal waveforms is 0.00723, corresponding to approximately 0.7% of the normalized waveform range. Antipodal symmetry established at the boundary in Version 2.31 is preserved throughout the complete four-shell propagation geometry. Even- and odd-indexed chains exhibit statistically indistinguishable waveform geometry. Differences between the even and odd mean waveforms are on the order of 2–3 × 10⁻⁵ in normalized units. Independent-sample t-tests, Mann–Whitney U tests, and Kolmogorov–Smirnov tests all failed to identify statistically significant differences at any shell. The radial waveform is parity invariant. Shell amplitude distributions evolve systematically with propagation depth. Mean amplitude exhibits a near-perfect monotonic relationship with shell index (r = 0.9981), as do median (r = 0.9960), standard deviation (r = 0.9732), interquartile range (r = 0.9584), and total range (r = 0.9606). Every chain undergoes a positive amplitude change between successive shells, with positive transition fractions of 1.000 at all three shell-to-shell transitions. Boundary amplitudes contain detectable predictive information about interior amplitudes. Shell 0 predicts Shell 1 with Pearson r = 0.4322 and Spearman ρ = 0.5853. The direct linear relationship between Shell 0 and Shell 3 is weak (Pearson r = 0.0567), but rank ordering is preserved (Spearman ρ = 0.5518), indicating nonlinear amplitude evolution while the organizational structure established at the boundary remains detectable throughout the interior. Random-chain null testing confirms that the observed geometry is specific to the locked shell structure. A null ensemble of 1,000 randomly assembled chain realizations produced a mean PC1 variance of 62.0% (null maximum 65.1%) compared with the observed 99.1%. Mean pairwise Pearson in the null ensemble was approximately zero (maximum 0.0422) compared with the observed 0.9812. Antipodal organization is entirely absent from the null ensemble: none of the 1,000 realizations produced a single valid antipodal pairing, compared with 68 observed pairs. All metrics yield empirical p = 0.000999. Controlled geometry perturbation demonstrates dose-response degradation of waveform coherence. Replacing 25% of shell positions reduces mean pairwise Pearson from 0.9812 to 0.0673. Complete replacement reduces it to −0.0004. PC1 variance falls from 99.1% to 59.6% under full randomization. Coherence decreases continuously and proportionally to geometric disruption, confirming that the observed waveform organization depends upon preservation of the locked radial shell geometry. Relationship to prior releases Version 2.34 builds directly upon Version 2.31 (antipodal topology of the Domain 1/8 boundary), Version 2.32 (ordering-dependent spectral organization and progressive-disorder robustness), and Version 2.33 (global phase coherence and model comparison). The physical-space waveform geometry characterized here is the spatial expression of the field organization established across those prior releases. No new CMB maps, calibration procedures, or shell definitions are introduced; all analyses operate exclusively on previously validated locked products. Locked inputs All analyses use the calibrated α(n̂) rotation map established in Version 2.30, the locked NSIDE=16 shell geometry and Domain 1/8 boundary definition from Version 2.13, and the validated calibrated phase products from Version 2.33. No modifications were made to any locked input during any stage of the analysis. Package contents This release includes the complete manuscript, all publication figures (Figures 1–8), statistical summary tables, final conclusion reports for all twelve analyses (Tests 1–12), intermediate analysis products, null-distribution outputs, geometry perturbation outputs, chain-level data products, complete machine-readable metadata, SHA-256 integrity hashes, and reproducibility documentation. Version numbering note This release is published as Version 2.34 within the complete 22 Blue Research Series. Intermediate version numbers correspond to unrelated analyses published in parallel. This release is part of an iterative research series in which analysis methods, masks, multipole selections, and calibration procedures were progressively refined. The all-versions concept DOI for the 22 Blue Research Series is https://doi.org/10.5281/zenodo.17317397. PUBLICATION 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 Jun 20, 2026 (v2.31) — Rotation Field of the Cosmic Microwave Background — Physical Origin of Boundary-to-Interior Propagation — DOI: 10.5281/zenodo.20777435 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.20787307 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