Rotation Field of the Cosmic Microwave Background – Boundary Universality and Standing-Wave Fingerprint Analysis (v2.12)
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Summary This version (v2.12) evaluates whether the structured boundary sequence identified in v2.9 through v2.11 is (1) unique to the dual-domain boundary D1∪D2, (2) conditional on specific geometry or topology, or (3) a generic feature that can appear in randomized or geometry-preserving surrogates. The objective is to determine whether the standing-wave and phase-structure on the D1∪D2 boundary are intrinsic to the real sky or can be reproduced by null, surrogate, or geometric transformations. DEFINITIONSalpha_nside16.npy: Locked birefringence rotation field at NSIDE=16.H_mask_nside16.npy: High-latitude analysis mask.domain_label_map.npy: Domain segmentation defined in v2.8–v2.9.boundary chain: Deterministic traversal of H ∩ Dₖ using lexicographic ordering and NSIDE=16 adjacency.v2.11 metric suite: Complete standing-wave and phase-structure metrics reused in v2.12. MISSIONThe mission of v2.12 is to test the universality of the D1∪D2 boundary fingerprint. The version assesses whether any null, surrogate, rotated, relabeled, synthetic, or domain-union configuration can reproduce the full statistical fingerprint of the D1∪D2 chain. If none do, the structure is non-generic and physically meaningful. METHODS 1. Boundary ConstructionAll boundaries were constructed deterministically:(1) boundaryₖ = H_mask ∩ (domain_label_map = k)(2) starting pixel = lexicographically smallest index(3) adjacency from NSIDE=16 neighbor tables(4) traversal = increasing-index tie-breaking 2. v2.11 Metric SuiteEach chain was evaluated using the full v2.11 metrics:• k=1 fractional Fourier power• mirror-symmetry score and z-score• symbolic phase mutual information• windowed FFT coherence• residual (post-k1) entropy• residual Lempel–Ziv complexityThese define the five-dimensional fingerprint F. 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. EQUATIONS (1) Boundary FingerprintF = ( k₁_frac_power , mirror_z , MI , H_residual_norm , LZ_residual_norm ) (2) k = 1 Fourier ExtractionAₖ₁ = (2/N) ∑ₙ αₙ cos(2πn/N)Bₖ₁ = (2/N) ∑ₙ αₙ sin(2πn/N) k₁_amp = √(Aₖ₁² + Bₖ₁²)k₁_frac_power = (Aₖ₁² + Bₖ₁²) / ∑ₙ αₙ² (3) Mirror Symmetrymirror_score = (1/N) ∑ᵢ ( αᵢ − α₍ₙ₋ᵢ₋₁₎ )²mirror_z = (mirror_score − μ_null) / σ_null (4) Phase-Symbol Mutual InformationMI = ∑ₐ₍b₎ P(a,b) log[ P(a,b) / (P(a) P(b)) ] (5) Residual After Removing k=1 Modeα_residualₙ = αₙ − [ Aₖ₁ cos(2πn/N) + Bₖ₁ sin(2πn/N) ] Residual entropy:H = − ∑ᵢ pᵢ log(pᵢ) Lempel–Ziv complexity:LZ = number of new substrings encountered during sequential parsing. (6) Mahalanobis Distance in Fingerprint SpaceM = √( (F − μ)ᵀ C⁻¹ (F − μ) ) (7) Histogram-Preserving Null Surrogateα_null = permutation(α) (8) Fixed-Power Random-Phase SurrogateFFT(α_phase)ₖ = |FFT(α)ₖ| e^{iφₖ} (9) Synthetic k = 1 Waveαₖ₁[n] = A cos(2πn/N + φ) (10) k=1 + Noise Surrogateαₖ₁₊noise[n] = αₖ₁[n] + η[n]η is drawn so that the resulting spectrum matches that of α_residual. 3. Null Surrogate EnsembleA 300-member null ensemble was generated using histogram-preserving random permutations. These retain amplitude distribution and geometry but destroy phase structure. 4. Geometry TransformationsGeometry-preserving operations included:• SO(3) rotations (0°, 90°, 180°, mixed)• chain reversal• domain relabeling and domain unionsThese strictly preserve spatial geometry. 5. Synthetic TestsTwo synthetic families were constructed:• pure k=1 sinusoid with matched amplitude and phase• k=1 plus noise tuned to the observed power spectrum 6. Fingerprint DistanceAll fingerprints (real, null, rotated, relabeled, synthetic) were compared using the Mahalanobis metric in 5-D space. FINDINGS 1. D1∪D2 is strongly non-null.The D1∪D2 fingerprint lies far outside the null ensemble:Real M ≈ 160.917Null ensemble range ≈ 156–162Fingerprint deviations:k₁_frac_power real = 0.833 (null mean = 0.007)mirror_z real = 10.6 (null mean = 0.17)MI real = 0.214 (null mean = 0.095)Residual LZ real ≈ 0.62 vs null mean ≈ 3.98 2. Other real boundaries do not match.Longest boundaries (labels 1 and 8) produce M ≈ 108–118. 3. Domain relabeling never reconstructs D1∪D2.All mock union chains converge to M ≈ 118.5. 4. Rotations preserve uniqueness.No rotated configuration exhibits the D1∪D2 fingerprint. 5. Pure sinusoid is insufficient.Pure k=1 synthetic waves give M ≈ 198. 6. k=1+noise surrogates approximate amplitude but not structure.These cluster around M ≈ 159 but lack correct MI and LZ properties. 7. Phase scrambling confirms phase coherence.Fixed-power, random-phase surrogates cannot reproduce the real fingerprint. 8. Traversal direction invariance confirmed.Forward and reverse boundary traversal produce nearly identical fingerprints. INTERPRETATIONAcross all null, surrogate, geometric, and synthetic tests, no configuration except the real D1∪D2 chain reproduces its fingerprint. The standing-wave structure, symbolic phase coherence, mirror properties, and residual complexity jointly define a non-generic statistical object in the birefringence field. The structure is robust to rotation, relabeling, traversal changes, and null randomization. The pattern emerges as a stable feature of the actual sky, not a consequence of boundary geometry or low-order harmonic projection. The distinctiveness of the D1∪D2 fingerprint identifies a unique statistical structure whose properties cannot be reproduced by reconfiguration of its components. FILES INCLUDEDAll locked inputs, boundary chains, metric summaries, null ensembles, geometric variants, k=1 surrogates, phase-scrambled surrogates, and all v2.12 outputs. CITATIONCondit, Amy (2025). Rotation Field of the Cosmic Microwave Background – Boundary Universality and Standing-Wave Fingerprint Analysis (v2.12). 22 Blue – The Heartbeat of the Universe. https://doi.org/10.5281/zenodo.17676377 DERIVED FROMCondit, 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 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 This release is part of an iterative research series in which analysis methods, masks, multipole selections, and calibration procedures were progressively refined. PUBLICATION RECORD 1. 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 1. Oct 10, 2025 (v1.0) — Scale-Dependent Anisotropic Birefringence: Initial Detection — DOI:10.5281/zenodo.173173982. Oct 20, 2025 (v1.1) — Scale-Dependent Anisotropic Birefringence: Validation Dataset — DOI:10.5281/zenodo.173964283. Oct 21, 2025 (v1.2) — Two-Harmonic Extension — DOI:10.5281/zenodo.174107644. Oct 28, 2025 (v1.3) — Two-Harmonic Dipole Verification — DOI:10.5281/zenodo.174689885. Nov 1, 2025 (v1.4) — MASTER-Calibrated Dipole — DOI:10.5281/zenodo.175007916. Nov 1, 2025 (v1.41) — Extended MASTER Calibration and Robustness — DOI:10.5281/zenodo.175089087. Nov 7, 2025 (v1.42) — Dependence-Aware Joint Validation — DOI:10.5281/zenodo.175538298. Nov 8, 2025 (v1.43) — Phase Model Validation — DOI:10.5281/zenodo.175613139. Nov 8, 2025 (v1.44) — Axis + Frequency + Half-Mission Validation — DOI:10.5281/zenodo.1756176810. Nov 9, 2025 (v1.5) — Multipole Structure and Model Selection — DOI:10.5281/zenodo.1756296511. Nov 9, 2025 (v1.6) — Phenomenology and Physical Interpretation — DOI:10.5281/zenodo.1756619712. Nov 9, 2025 (v1.7) — Prediction and Experiment Overlays — DOI:10.5281/zenodo.1756687013. Nov 9, 2025 (v1.8) — Model Rejection and Δℓ Persistence — DOI:10.5281/zenodo.1756724114. Nov 10, 2025 (v2.0) — Intrinsic Periodicity in ℓ-Space — DOI:10.5281/zenodo.1757404815. Nov 10, 2025 (v2.1) — Physical Origin of Δℓ Modulation — DOI:10.5281/zenodo.1757708616. Nov 11, 2025 (v2.2) — Universe-Model Evaluation — DOI:10.5281/zenodo.1758541917. Nov 12, 2025 (v2.3) — Domain Geometry and Topological Inference — DOI:10.5281/zenodo.1759415718. Nov 13, 2025 (v2.4) — Real-Space Correlation of the Birefringence Field — DOI:10.5281/zenodo.1759753719. Nov 13, 2025 (v2.5) — Spectral Surgery on the Δℓ ≈ 109 Harmonic — DOI:10.5281/zenodo.1760498220. Nov 14, 2025 (v2.6) — Angular Locality of the Δℓ = 109 Standing Wave — DOI:10.5281/zenodo.1761334821. Nov 15, 2025 (v2.7) — Sky-Local Origin of the Δℓ ≈ 109 Standing Wave — DOI:10.5281/zenodo.1762002922. Nov 15, 2025 (v2.8) — Domain Topology of the Δℓ ≈ 109 Standing Wave — DOI:10.5281/zenodo.1762060523. Nov 16, 2025 (v2.9) — Dual-Domain Coherence and Boundary Geometry — DOI:10.5281/zenodo.1762187124. Nov 17, 2025 (v2.10) — Boundary Sequence Structure on the Dual-Domain Loop — DOI:10.5281/zenodo.1763581125. Nov 19, 2025 (v2.11) — Boundary Standing-Wave and Phase-Structure Analysis — DOI:10.5281/zenodo.1764803326. Nov 21, 2025 (v2.12) — Boundary Universality and Standing-Wave Fingerprints — DOI:10.5281/zenodo.1767637727. Nov 23, 2025 (v2.13) — Interior Propagation and Boundary-Driven Structure — DOI:10.5281/zenodo.1769354028. Jun 18, 2026 (v2.29) — Rotation Field of the Cosmic Microwave Background — Interior Propagation Audit & Harmonic Normalization — DOI:10.5281/zenodo.2075303729. Jun 19, 2026 (v2.30) — Calibrated Interior Propagation Validation — DOI:10.5281/zenodo.2075533030. Jun 20, 2026 (v2.31) — Rotation Field of the Cosmic Microwave Background — Physical Origin of Boundary-to-Interior Propagation — DOI:10.5281/zenodo.2077743531. 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.2078730732. Jun 24, 2026 (v2.33) — Model Comparison and Global Phase Coherence of the Cosmic Birefringence Rotation Field — DOI:10.5281/zenodo.20825890 22 Blue - The Heartbeat of the Universe
# 摘要 本版本（v2.12）旨在验证v2.9至v2.11中识别的结构化边界序列是否满足以下三类情况：(1) 仅存在于双域边界D₁∪D₂中；(2) 依赖于特定几何或拓扑结构；(3) 为可在随机化或保几何替代样本中出现的通用特征。本研究的目标为判断D₁∪D₂边界上的驻波（standing-wave）与相位结构（phase-structure）是真实天区的固有属性，还是可通过空样本、替代样本或几何变换复现的产物。 # 定义 alpha_nside16.npy：NSIDE=16下的锁定双折射旋转场（birefringence rotation field）。 H_mask_nside16.npy：高纬度分析掩膜（high-latitude analysis mask）。 domain_label_map.npy：v2.8至v2.9中定义的域分割图（domain segmentation）。 边界链（boundary chain）：基于字典序排序（lexicographic ordering）与NSIDE=16邻接关系，对H∩Dₖ进行的确定性遍历。 v2.11指标集（metric suite）：v2.12中复用的完整驻波与相位结构指标集。 # 研究目标 本版本v2.12的核心目标为检验D₁∪D₂边界指纹的普适性。本版本将评估任意空样本、替代样本、旋转、重标记、合成或域联合配置是否能够复现D₁∪D₂边界链的完整统计指纹（statistical fingerprint）。若无法复现，则说明该结构具备非普适性且具有物理意义。 # 研究方法 ## 1. 边界构建 所有边界均通过确定性方式构建： (1) 边界ₖ = H掩膜 ∩ (域标记图=k) (2) 起始像素：字典序最小的索引 (3) 邻接关系来自NSIDE=16邻接表 (4) 遍历规则：以索引递增方式打破平局 ## 2. v2.11指标集 每条边界链均通过完整的v2.11指标集进行评估： • k=1分量分数傅里叶功率 • 镜像对称性得分与z得分 • 符号相位互信息 • 加窗快速傅里叶变换（Fast Fourier Transform，FFT）相干性 • 残差（k=1分量去除后）熵 • 残差莱普尔-齐夫复杂度（Lempel-Ziv complexity） 上述指标共同构成五维指纹F。 # 公式 (1) 边界指纹 F = ( k₁_frac_power , mirror_z , MI , H_residual_norm , LZ_residual_norm ) (2) k=1分量傅里叶提取 Aₖ₁ = (2/N) ∑ₙ αₙ cos(2πn/N) Bₖ₁ = (2/N) ∑ₙ αₙ sin(2πn/N) k₁_amp = √(Aₖ₁² + Bₖ₁²) k₁_frac_power = (Aₖ₁² + Bₖ₁²) / ∑ₙ αₙ² (3) 镜像对称性 mirror_score = (1/N) ∑ᵢ ( αᵢ − α₍ₙ₋ᵢ₋₁₎ )² mirror_z = (mirror_score − μ_null) / σ_null (4) 相位-符号互信息 MI = ∑ₐ₍b₎ P(a,b) log[ P(a,b) / (P(a) P(b)) ] (5) 去除k=1分量后的残差 α_residualₙ = αₙ − [ Aₖ₁ cos(2πn/N) + Bₖ₁ sin(2πn/N) ] 残差熵：H = − ∑ᵢ pᵢ log(pᵢ) 莱普尔-齐夫复杂度：顺序解析过程中遇到的新子串总数。 (6) 指纹空间中的马氏距离（Mahalanobis Distance） M = √( (F − μ)ᵀ C⁻¹ (F − μ) ) (7) 保留直方图的空置换样本（Histogram-Preserving Null Surrogate） α_null = permutation(α) (8) 固定功率随机相位替代样本 FFT(α_phase)ₖ = |FFT(α)ₖ| e^{iφₖ} (9) 合成k=1分量驻波 αₖ₁[n] = A cos(2πn/N + φ) (10) k=1分量加噪声替代样本 αₖ₁₊noise[n] = αₖ₁[n] + η[n] η的采样需使得生成的频谱与α_residual的频谱一致。 ## 3. 空置换样本集合 本研究通过保留直方图的随机置换生成了包含300个样本的空置换集合，该集合保留了振幅分布与几何结构，但破坏了相位结构。 ## 4. 几何变换 保几何操作包括： • SO(3)旋转（SO(3) rotation）（0°、90°、180°及混合旋转） • 链反转 • 域重标记与域联合 上述操作严格保留空间几何结构。 ## 5. 合成测试 本研究构建了两类合成信号家族： • 振幅与相位匹配的纯k=1分量正弦波 • 匹配观测功率谱的k=1分量加噪声信号 ## 6. 指纹距离 所有指纹（真实样本、空置换样本、旋转样本、重标记样本、合成样本）均通过五维空间中的马氏距离指标进行比较。 # 研究结果 1. D₁∪D₂显著区别于空置换样本 D₁∪D₂的指纹远超出空置换集合的分布范围： 真实样本马氏距离M≈160.917 空置换集合范围≈156–165 指纹偏差如下： 真实样本k₁_frac_power=0.833（空置换集合均值=0.007） 真实样本mirror_z=10.6（空置换集合均值=0.17） 真实样本MI=0.214（空置换集合均值=0.095） 真实样本残差LZ≈0.62，空置换集合均值≈3.98 2. 其他真实边界无法匹配该指纹 最长的两条边界（标记1与8）的马氏距离M≈108–118。 3. 域重标记无法复现D₁∪D₂指纹 所有模拟域联合链的马氏距离均收敛至≈118.5。 4. 旋转操作保留了指纹的独特性 所有旋转配置均未表现出D₁∪D₂的指纹特征。 5. 纯正弦波无法复现该指纹 纯k=1分量合成驻波的马氏距离M≈198。 6. k=1分量加噪声替代样本仅能近似振幅，但无法复现结构 此类样本的马氏距离集中在≈159附近，但缺乏正确的MI与LZ特征。 7. 相位置乱验证了相位相干性 固定功率随机相位替代样本无法复现真实指纹。 8. 遍历方向不变性得到验证 正向与反向边界遍历得到的指纹几乎完全一致。 # 结果解释 在所有空置换、替代样本、几何变换与合成测试中，除真实的D₁∪D₂边界链外，无任何配置能够复现其指纹特征。驻波结构、符号相位相干性、镜像对称性与残差复杂度共同构成了双折射旋转场中的非普适统计对象。该结构对旋转、重标记、遍历方向变更及空置换随机化均具备鲁棒性。该模式是真实天区的稳定特征，而非边界几何或低阶谐波投影的产物。D₁∪D₂指纹的独特性表明其为一种独特的统计结构，其特征无法通过重组其组成部分复现。 # 包含文件 本数据集包含所有锁定输入文件、边界链、指标汇总结果、空置换样本集合、几何变换变体、k=1分量替代样本、相位置乱替代样本及所有v2.12版本输出文件。 # 引用文献 Condit, Amy (2025). 宇宙微波背景旋转场——边界普适性与驻波指纹分析（v2.12）. 22 Blue – The Heartbeat of the Universe. https://doi.org/10.5281/zenodo.17676377 # 衍生来源 本数据集衍生自以下文献： 1. Condit, Amy (2025). 宇宙微波背景旋转场——边界驻波与相位结构分析（v2.11）. https://doi.org/10.5281/zenodo.17648033 2. Condit, Amy (2025). 宇宙微波背景旋转场——双域环上的边界序列结构分析（v2.10）. https://doi.org/10.5281/zenodo.17635811 3. Condit, Amy (2025). 宇宙微波背景旋转场——双域相干性与边界几何分析（v2.9）. https://doi.org/10.5281/zenodo.17621871 4. Condit, Amy (2025). 宇宙微波背景旋转场——Δℓ≈109边界网络的拓扑结构（v2.8）. https://doi.org/10.5281/zenodo.17620605 5. Condit, Amy (2025). 宇宙微波背景旋转场——Δℓ≈109驻波的角局域性（v2.6）. https://doi.org/10.5281/zenodo.17613348 6. Condit, Amy (2025). 宇宙微波背景旋转场——Δℓ≈109谐波的频谱手术（v2.5）. https://doi.org/10.5281/zenodo.17604982 # 版本历史 2025年9月20日：发现谐波相位对齐 v1.0 2025年10月10日：首次定量检测 v1.1 2025年10月20日：统计验证 v1.2 2025年10月21日：双谐波扩展 v1.3 2025年10月28日：鲁棒性测试 Dataverse 2025年10月29日：DOI 10.7910/DVN/PTDG20 v1.4 2025年11月1日：MASTER校准光谱 v1.41至v1.44：SMICA/NILC拆分与宇称测试 v1.5 2025年11月9日：模型选择 v1.6 2025年11月9日：结果解释 v1.7 2025年11月9日：正向预测 v1.8 2025年11月9日：持久性测试 v2.0 2025年11月10日：发现固有Δℓ≈108特征 v2.1 2025年11月10日：ξ(θ)物理机制 v2.2 2025年11月11日：宇宙模型评估 v2.3 2025年11月12日：域几何推断 v2.4 2025年11月13日：实空间验证 v2.5 2025年11月13日：频谱手术 v2.6 2025年11月14日：角局域性 v2.7 2025年11月15日：天区局域性 v2.8 2025年11月15日：域拓扑结构 v2.9 2025年11月16日：场相干性 v2.10 2025年11月17日：边界序列结构 v2.11 2025年11月19日：边界驻波与相位结构 v2.12 2025年11月21日：边界普适性与驻波指纹分析 22 Blue – The Heartbeat of the Universe
