Rotation Field of the Cosmic Microwave Background — Boundary Standing-Wave & Phase-Structure Analysis (v2.11)

Summary This release (v2.11) completes the boundary-analysis branch of the 22 Blue project by studying the standing-wave structure, phase behavior, symbolic patterns, and residual grammar encoded along the dual-domain boundary discovered in earlier versions. The investigation is entirely map-free: all analysis is performed only on the locked 272-element boundary sequence inherited from v2.9 and v2.10. No recomputation or modification of earlier inputs occurs. The goal is to determine whether the boundary itself carries an interpretable physical signal. Versions v2.8 through v2.10 revealed that the sky contains a stable two-domain structure. The interface between these domains traces a closed loop around the sky. When alpha(n_hat) is sampled along this loop in the correct geometric order, the resulting sequence behaves like a one-dimensional measurement of the global Delta-ell ~ 109 standing wave. Version 2.10 established that the boundary sequence is not random: it contains repeating motifs, mirror patterns, and phase-linked structure. Version 2.11 takes the next step: treat the boundary sequence as a physical signal, similar to measuring displacement along a vibrating ring. The aim is to find out which structures are intrinsic to the chain and which collapse under null tests, shuffles, or synthetic replacements. This is the first release to bring all sequence-level tests into a single framework. MethodsThe chain x[i] is normalized and analyzed using several complementary approaches. (1) Global Fourier standing-wave test.The chain is expanded as a discrete Fourier series. The fraction of total power in the k = 1 mode,P1 = A1^2 / sum(Ak^2),is used to measure whether the boundary behaves like a dominant standing wave. (2) Mirror-structure scan.Sliding windows of length 2*m are split into left and right halves. The right half is reversed and compared to the left. The mismatch fraction is compared to a null ensemble of shuffled chains. (3) Harmonic finite-difference test.This test checks whether the chain behaves like a discrete harmonic function. Correlations between x, the negative first difference, and the leading eigenmode of the second-difference operator provide a strong check. (4) Local phase coherence.A sliding-window FFT extracts the local k = 1 phase. Phase dispersion is compared to null sequences. (5) Scramble robustness.Three types of controlled shuffles are tested: block permutations, local pixel shuffles, and full randomization. The goal is to separate geometric effects from global-order effects. (6) Pure k = 1 surrogate comparison.A synthetic cosine mode is fitted and analyzed to determine which features of the real chain cannot be explained by a single standing wave. (7) Residual structure.After subtracting the k = 1 mode, r[i] = x[i] - k1[i], the residuals are symbolized and tested for memory using entropy and null ensembles. (8) Phase–symbol locking.The Fourier phase is binned and compared to symbol categories. Mutual information measures whether symbol patterns encode phase. 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. FindingsThe chain exhibits a dominant standing wave.P1 = 0.8334, with amplitude A1 = 5.961e-20.Correlation between the chain and the pure k = 1 mode is 0.9129.This confirms the Delta-ell ~ 109 standing wave is directly expressed along the boundary. Mirror structure is unusually strong.Perfect mirrors occur in long windows, with null comparisons reaching ~6.7 sigma. Finite-difference tests show harmonic-string behavior.Differences between real and null ensembles reach very high significance (for example, z ~ -36). Local phase coherence is real but spatially variable.Average windowed power is about half the global value. Phase coherence is weak but significantly above random. Scramble tests show the structure is geometric.Local shuffles preserve the mode and mirror patterns, but global shuffles destroy them. The pure k = 1 surrogate does not reproduce the chain.Mismatch ~0.26 and failures in symbolic grammar show additional structure is present. Residuals are highly structured.Entropy differs from null by roughly 92 sigma, indicating organized substructure beyond the global standing wave. Phase–symbol locking is extremely strong.Mutual information = 1.42 bits, compared to a null of about 0.13 bits. Scientific InterpretationThe domain boundary in the CMB rotation field is not a passive or noisy object. It carries a coherent standing-wave signature, strong mirror symmetry, symbolic structure, and multiple layers of harmonic and residual behavior. The Delta-ell ~ 109 harmonic is directly expressed along the boundary, not only globally but as a structured, rule-governed boundary signal. The boundary acts as the geometric locus where the global harmonic becomes locally measurable. InstructionsAll analysis runs solely on the locked boundary sequence. Reproduction requires the NSIDE=16 domain maps from v2.9, the boundary sequence from v2.10, and the v2.11 analysis code. All random seeds are stored in the JSON files. CitationCondit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Boundary Standing-Wave and Phase-Structure Analysis (v2.11). 22 Blue – The Heartbeat of the Universe. https://doi.org/10.5281/zenodo.17648033 Derived FromCondit, 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 ell approx 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 ell approx 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 ell approx 109 Harmonic (v2.5). https://doi.org/10.5281/zenodo.17604982 PUBLICATION RECORD PREDECESSOR PUBLICATION (separate record)1. Sep 20, 2025 (v1.0) -- Harmonic Phase Alignments in Planck 2018 CMB -- DOI:10.5281/zenodo.17167268 MAIN RESEARCH SERIESConcept DOI:10.5281/zenodo.173173972. Oct 10, 2025 (v1.0) -- Scale-Dependent Anisotropic Birefringence: Initial Detection -- DOI:10.5281/zenodo.173173983. Oct 20, 2025 (v1.1) -- Scale-Dependent Anisotropic Birefringence: Validation Dataset -- DOI:10.5281/zenodo.173964284. Oct 21, 2025 (v1.2) -- Two-Harmonic Extension -- DOI:10.5281/zenodo.174107645. Oct 28, 2025 (v1.3) -- Two-Harmonic Dipole Verification -- DOI:10.5281/zenodo.174689886. Nov 1, 2025 (v1.4) -- MASTER-Calibrated Dipole -- DOI:10.5281/zenodo.175007917. Nov 1, 2025 (v1.41) -- Extended MASTER Calibration and Robustness -- DOI:10.5281/zenodo.175089088. Nov 7, 2025 (v1.42) -- Dependence-Aware Joint Validation -- DOI:10.5281/zenodo.175538299. Nov 8, 2025 (v1.43) -- Phase Model Validation -- DOI:10.5281/zenodo.1756131310. Nov 8, 2025 (v1.44) -- Axis + Frequency + Half-Mission Validation -- DOI:10.5281/zenodo.1756176811. Nov 9, 2025 (v1.5) -- Multipole Structure and Model Selection -- DOI:10.5281/zenodo.1756296512. Nov 9, 2025 (v1.6) -- Phenomenology and Physical Interpretation -- DOI:10.5281/zenodo.1756619713. Nov 9, 2025 (v1.7) -- Prediction and Experiment Overlays -- DOI:10.5281/zenodo.1756687014. Nov 9, 2025 (v1.8) -- Model Rejection and Delta-l Persistence -- DOI:10.5281/zenodo.1756724115. Nov 10, 2025 (v2.0) -- Intrinsic Periodicity in l-space -- DOI:10.5281/zenodo.1757404816. Nov 10, 2025 (v2.1) -- Physical Origin of Delta-l Modulation -- DOI:10.5281/zenodo.1757708617. Nov 11, 2025 (v2.2) -- Universe-Model Evaluation -- DOI:10.5281/zenodo.1758541918. Nov 12, 2025 (v2.3) -- Domain Geometry and Topological Inference -- DOI:10.5281/zenodo.1759415719. Nov 13, 2025 (v2.4) -- Real-Space Correlation of Birefringence Field -- DOI:10.5281/zenodo.1759753720. Nov 13, 2025 (v2.5) -- Spectral Surgery on Delta-l ~109 Harmonic -- DOI:10.5281/zenodo.1760498221. Nov 14, 2025 (v2.6) -- Angular Locality of Delta-l = 109 Standing Wave -- DOI:10.5281/zenodo.1761334822. Nov 15, 2025 (v2.7) -- Sky-Local Origin of Delta-l ~109 Standing Wave -- DOI:10.5281/zenodo.1762002923. Nov 15, 2025 (v2.8) -- Domain Topology of Delta-l ~109 Standing Wave -- DOI:10.5281/zenodo.1762060524. Nov 16, 2025 (v2.9) -- Dual-Domain Coherence and Boundary Geometry -- DOI:10.5281/zenodo.1762187125. Nov 17, 2025 (v2.10) -- Boundary Sequence Structure on Dual-Domain Loop -- DOI:10.5281/zenodo.1763581126. Nov 19, 2025 (v2.11) -- Boundary Standing-Wave and Phase-Structure Analysis -- DOI:10.5281/zenodo.1764803327. Nov 21, 2025 (v2.12) -- Boundary Universality and Standing-Wave Fingerprints -- DOI:10.5281/zenodo.1767637728. Nov 23, 2025 (v2.13) -- Interior Propagation and Boundary-Driven Structure -- DOI:10.5281/zenodo.17693540 Contact email: [e-mail] 22 Blue - The Heartbeat of the Universe