DOI: 10.5281/zenodo.17676377
title: Rotation Field of the Cosmic Microwave Background – Boundary Universality and Standing-Wave Fingerprint Analysis (v2.12)

description: 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.

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 ≈ 156–165Fingerprint 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

VERSION HISTORYSep 20 2025 – Harmonic phase alignments discoveredv1.0 Oct 10 – First quantitative detectionv1.1 Oct 20 – Statistical validationv1.2 Oct 21 – Two-harmonic extensionv1.3 Oct 28 – Robustness testsDataverse Oct 29 – DOI 10.7910/DVN/PTDG20v1.4 Nov 1 – MASTER calibrated spectrav1.41 to 1.44 – SMICA/NILC splits and parity testsv1.5 Nov 9 – Model selectionv1.6 Nov 9 – Interpretationv1.7 Nov 9 – Forward predictionv1.8 Nov 9 – Persistence testsv2.0 Nov 10 – Intrinsic Delta-ell ~108 discoveredv2.1 Nov 10 – xi(theta) physicsv2.2 Nov 11 – Universe model evaluationv2.3 Nov 12 – Domain geometry inferencev2.4 Nov 13 – Real-space confirmationv2.5 Nov 13 – Spectral surgeryv2.6 Nov 14 – Angular localityv2.7 Nov 15 – Sky localityv2.8 Nov 15 – Domain topologyv2.9 Nov 16 – Field coherencev2.10 Nov 17 – Boundary sequence structurev2.11 Nov 19 – Boundary standing-wave and phase structurev2.12 Nov 21 – Boundary universality and standing-wave fingerprint analysis

22 Blue – The Heartbeat of the Universe