Rotation Field of the CMB — Oscillating Antipodal Parity Mode (v2.36)

Summary The 22 Blue rotation field series has, in earlier releases, examined the calibrated CMB birefringence field α(n̂) for structured, non-random organization. Following the identification of an antipodal two-component shell topology in Version 2.31 — each shell splitting into exactly two connected components related by point reflection through the sky, with strong correlation between them — this release examines the internal structure of that antipodal relationship itself. v2.36 decomposes each shell's antipodal pixel pairs into even (symmetric) and odd (antisymmetric) parity channels and tests whether the odd channel's dominant single-cycle (k=1) angular mode, present at every tested shell depth, constitutes a statistically genuine, non-random feature of the calibrated field. This is tested against four independent lines of evidence: random-orientation rotation surrogates, noise-matched-geometry surrogates, real Planck half-mission instrument noise at three frequencies, and injection-recovery calibration of the detection method itself. A fifth, independent test examines whether the geometry-construction method underlying the antipodal shell structure itself has any inherent bias toward producing the observed structure, regardless of input. Locked inputs - Calibrated birefringence rotation field α(n̂): alpha_lowL_v230_calibrated_rad.fits (NSIDE=512), unchanged from prior releases. - Uncalibrated proxy rotation field: alpha_lowL_combined_v141.fits, the map originally used to construct the domain/shell geometry in Versions 2.6 through 2.8; used exclusively for the geometry construction validation test in this release. - Analysis mask: mask_qmaskproxy_fsky0467_apod1deg_NSIDE512.fits. - Antipodal shell component indices: v2_31_connected_components.json, locked and unchanged from Version 2.31 (shells 0 through 3, two components per shell). - Half-mission noise-difference maps at 100, 143, and 217 GHz (HM1, HM2 pairs), used exclusively for real-instrument-noise validation, never as an analysis input. All ten locked inputs are bundled with this release and verified against a SHA-256 hash manifest at notebook runtime; no external dependency on any other package or shared file structure is required. Definitions - Even parity: even = 0.5(a + b), the symmetric average of each antipodal pixel pair (a, b) within a shell. - Odd parity: odd = 0.5(a − b), the antisymmetric difference of each antipodal pixel pair. - k=1 fractional power: for a shell's ordered ring of N values y, with Fourier transform F, the fraction |F[1]|² / Σ_{j≥1} |F[j]|² — the share of total non-DC power carried by the first (single-cycle) harmonic. - Cross-shell phase coherence R: the resultant vector length |mean(exp(i·φ))| of the odd-channel k=1 phase φ measured independently at each of the four shells; R=1 indicates identical phase at every shell, R→0 indicates random, unrelated phases. - Shells 0–3: the boundary shell and three interior adjacency layers surrounding the Domain 1/8 boundary, as constructed and locked in Versions 2.6 through 2.9 and reused unchanged through Version 2.31 into this release. Methods All shell and domain geometry is reused unchanged from Version 2.31. No domain construction, boundary identification, or shell assignment is performed in this release; only the parity decomposition and its harmonic content are new. Before any new analysis, the notebook first reproduces the exact odd- and even-channel k1_frac values reported in Version 2.31 from the locked shell components and calibrated map, confirming an exact match at all four shells as an integrity check. For each shell, the calibrated map was downsampled to NSIDE=16 (matching the resolution at which shell geometry was originally constructed), the antipodal component pairs were extracted, and the even and odd channels were computed. The k=1 fractional power and phase of each channel were computed via direct FFT decomposition around the shell's pixel ordering. Five independent validation procedures were applied: 1. Rotation surrogates (2000 trials): the calibrated map's spherical harmonic representation (lmax=47) was rotated by random Euler angles, re-downsampled to NSIDE=16, and the identical fixed shell geometry was used to recompute the odd-channel k=1 statistic. This tests whether the observed value depends on the real sky's actual orientation, holding geometry fixed. 2. Noise-matched-geometry surrogates (500 trials): the calibrated map was replaced with independent Gaussian random values, variance-matched to the real field, at the identical shell pixel indices. This isolates whether the shell/domain geometry's shape alone — independent of any real sky content — could produce the observed statistic. 3. Real instrument noise: half-mission noise-difference maps ([HM1−HM2]/2, containing no common sky signal by construction) at 100, 143, and 217 GHz were downsampled and passed through the identical parity decomposition and k=1 statistic, for direct comparison against the real signal. 4. Injection-recovery calibration (300 trials per amplitude level): a synthetic k=1 signal was injected into pure random noise on the identical shell geometry at 0.5×, 1.0×, 1.5×, and 2.0× the real observed amplitude, and the fraction of trials correctly flagged by a 95th-percentile noise threshold was recorded. This calibrates whether the detection method has genuine statistical power at the amplitude scale actually observed, rather than assuming significance implies power. 5. Cross-shell phase-coherence test (500 trials each, rotation and noise-matched-geometry): the same rotation and noise surrogates used in (1) and (2) were used to test whether the near-constant phase observed across all four real shells is itself unlikely under chance, independent of any single shell's individual significance. Procedures (1) and (2) were also applied to the even channel, using the same trial counts, for direct comparison against the odd channel's results. A sixth, separate procedure validated the geometry-construction method itself: the gradient-based boundary detection, percentile thresholding, and connected-component clustering method used in Versions 2.6 through 2.8 was independently reconstructed, verified to exactly reproduce the original historical pipeline (threshold value matched to 15+ significant figures, component structure matched exactly), then applied to 50 trials of pure Gaussian random noise on the uncalibrated v141 map's own scale, to test whether the construction process has any inherent tendency to produce the observed antipodal structure independent of input. Key Findings Odd-channel k=1 fractional power and significance, all four shells: Shell 0: k1 = 0.4237 | rotation p = 0.0015 (2000 trials) | noise-geometry p = 0.0020 Shell 1: k1 = 0.4243 | rotation p = 0.0045 (2000 trials) | noise-geometry p = 0.0020 Shell 2: k1 = 0.4216 | rotation p = 0.0050 (2000 trials) | noise-geometry p = 0.0020 Shell 3: k1 = 0.3355 | rotation p = 0.0125 (2000 trials) | noise-geometry p = 0.0060 Real instrument noise comparison (k1 fractional power, F100 / F143 / F217), all consistently far below the real signal with one noted exception: Shell 0: 0.0015 / 0.0506 / 0.0039 Shell 1: 0.0022 / 0.0165 / 0.0242 Shell 2: 0.0019 / 0.0037 / 0.0171 Shell 3: 0.0131 / 0.1571 / 0.0035 Odd-channel phase across shells: −84.16° (Shell 0), −87.66° (Shell 1), −86.01° (Shell 2), −80.04° (Shell 3) — a total drift of approximately 8° from the boundary to the deepest interior shell tested. Cross-shell phase coherence R=0.9988, itself significant against rotation surrogates (p=0.0080) and noise-matched-geometry surrogates (p=0.0020). Even-channel k=1 fractional power and significance, reported for direct, transparent comparison against the odd channel: Shell 0: k1 = 0.2887 | rotation p = 0.0160 | noise-geometry p = 0.0020 Shell 1: k1 = 0.2012 | rotation p = 0.0080 | noise-geometry p = 0.0020 Shell 2: k1 = 0.1943 | rotation p = 0.0120 | noise-geometry p = 0.0080 Shell 3: k1 = 0.3242 | rotation p = 0.0060 | noise-geometry p = 0.0200 Injection-recovery rates (fraction of 300 trials detected at the 95th-percentile noise threshold): Shell 0: 0.5×=0.13 1.0×=0.24 1.5×=0.43 2.0×=0.67 Shell 1: 0.5×=0.28 1.0×=0.77 1.5×=0.98 2.0×=1.00 Shell 2: 0.5×=0.25 1.0×=0.66 1.5×=0.94 2.0×=1.00 Shell 3: 0.5×=0.07 1.0×=0.14 1.5×=0.21 2.0×=0.25 Geometry construction validation: the real data's two largest boundary components capture 88.3% of all 308 boundary pixels, resolving into 8 total components. Across 50 independent noise trials using the identical, verified construction pipeline, the maximum top-two capture fraction achieved was 10.7%, with a mean of 7.3% and 124–158 total components typically resulting. Systematics and Robustness Injection-recovery testing reveals that statistical power at the observed amplitude is substantially lower for Shells 0 and 3 (13–24% recovery) than for Shells 1 and 2 (66–77% recovery). This does not invalidate the significance tests already passed by Shells 0 and 3 — both independently clear p<0.05 against rotation and noise-matched-geometry surrogates — but it means those two shells' detections rest on comparatively less statistical power, and a smaller fraction of equivalent-amplitude signals would be expected to be recovered by chance alone at those specific shells. This is disclosed transparently rather than omitted. Shell 3's comparison against F143 noise (p=0.1571) is the sole instance, among twelve real-instrument-noise comparisons across three frequencies and four shells, where the real signal is not clearly separated from noise. The remaining eleven comparisons show clear separation. Shell 3 has the smallest sample size of any shell tested (12 pixels per antipodal component), which is the most likely explanation for both its comparatively weaker injection-recovery power and this single noise-comparison exception. Dependency and Independence This analysis reuses shell and domain geometry from Version 2.31 unchanged. It has no dependency on the Δℓ≈109 periodicity claim or the patch-phase coherence method used elsewhere in this series. Version Context This release is built entirely on the antipodal shell/domain framework established in Version 2.31, which was independently verified against rotation and permutation testing and found to hold. That framework traces to the domain and shell geometry constructed in Versions 2.6 through 2.9 via gradient-based methods applied directly to the calibrated map. Geometry Construction Validation The gradient-based boundary detection, percentile thresholding, and connected-component clustering method used to construct the domain/shell geometry (established in Versions 2.6 through 2.8 and reused unchanged through this release) was independently reconstructed and verified to exactly reproduce the original historical pipeline: the calibrated threshold value matched archived metadata to 15+ significant figures, and the resulting component structure (136, 136, 17, 9, 4, 3, 2, 1 pixels) matched the real, published boundary construction exactly. This verified pipeline was then applied to 50 independent trials of pure Gaussian random noise, matched in variance to the real field, with no other change. The real data's two largest components capture 88.3% of all 308 boundary pixels, resolving into 8 total components. Across 50 noise trials, the maximum top-two capture fraction achieved was 10.7%, with a mean of 7.3% and 124–158 total components typically resulting (heavily fragmented). No noise trial approached the real data's structure. The geometry-construction method does not have an inherent bias toward producing a clean, dominant, antipodal-looking split. The real sky's boundary structure — its resolution into two large, nearly equal, connected components — is a genuine property of the actual calibrated field, not an artifact of the construction method applied to arbitrary input. Interpretation The odd-parity (antisymmetric antipodal-difference) component of the calibrated rotation field exhibits a genuine, statistically validated single-cycle oscillation in angular position around each of the four tested shells. This is a spatial pattern in the real-space geometry of the field, not a periodicity claim in multipole space, and it has been independently verified against random sky orientation, noise-matched geometry, and real instrument noise at three frequencies. The oscillation's phase remains nearly constant across all four shells, and this cross-shell stability is itself independently significant against the same two adversarial tests. This indicates a single, coherently-oriented oscillating structure that persists with depth from the boundary into the sky interior, rather than four unrelated single-shell coincidences. Statistical power, measured directly via injection-recovery calibration, varies meaningfully by shell and is weaker at Shells 0 and 3 than at Shells 1 and 2; this is disclosed as a property of the available sample size at each shell, not as a limitation on the validity of the individual significance tests already passed. No claim is made beyond what is directly tested here: this release establishes a real, oscillating, phase-coherent angular pattern in physical space at four shell depths, built on a geometric foundation independently shown not to be an artifact of its own construction method. Files in this v2.36 bundle - v2_36_oscillating_parity_mode_reproduction.ipynb: the full ten-step reproduction notebook. Self-contained; locates itself via sentinel file, works with the package uploaded directly to a Colab session or run locally, with Google Drive used only as an explicit fallback if nothing is found locally. - inputs/: all ten locked inputs (calibrated map, uncalibrated v141 map, mask, v2.31 shell components, six half-mission noise files), each verified by SHA-256 hash at notebook runtime. - hashes/locked_input_hashes.json: the hash manifest against which all inputs are checked before any analysis runs. - outputs/: JSON results for each of the notebook's ten steps, including the full validation battery, phase-stability test, injection-recovery calibration, extended-precision rotation trials, even-channel comparison, and geometry construction validation. - figs/v2_36_fig1_k1_null_distributions.png: per-shell histogram comparison of the real odd-channel k1 value against its rotation-surrogate and noise-matched-geometry null distributions. - figs/v2_36_fig2_phase_vs_shell_depth.png: odd-channel phase plotted against shell depth, showing the observed ≈8° total drift. - v2_36_standalone_package.zip: a single archive containing the notebook, all ten locked inputs, and the hash manifest (items above, bundled together) for convenience — download and extract this one file to get a complete, ready-to-run copy without assembling the individual pieces separately. How to Use This Notebook This release includes a complete, self-contained reproduction notebook (v2_36_oscillating_parity_mode_reproduction.ipynb). To run it: 1. Download all files in this record and keep them in a single folder, preserving the inputs/ and hashes/ subfolder structure alongside the notebook. 2. Open the notebook in Google Colab (upload the folder to the session, or place it in Google Drive) or in a local Jupyter environment. No manual configuration is required — the notebook locates its own files automatically via a sentinel marker. 3. Run all cells in order, top to bottom. Each step depends on variables established by the previous ones. 4. The notebook will install healpy automatically if not already present. All ten locked inputs are verified against SHA-256 hashes before any analysis runs; if verification fails, the notebook will stop rather than proceed on unverified data. 5. Outputs (JSON results for each step) and figures (two PNGs) are generated fresh in outputs/ and figs/ subfolders during the run, reproducing the same files included separately in this record. 6. Runtime is approximately 20–30 minutes end to end. No GPU is required. 7. Google Drive is used only as a fallback if the package is not found locally (e.g., in the current working directory or, in Colab, under /content) — it is not required to run the notebook. Version lineage (context) - v2.31 — antipodal two-component shell topology established; exact pixel correspondence and strong correlation confirmed via rotation and permutation testing. - v2.32 — antipodal specificity examined via value-ordering and boundary-geometry disentanglement. - v2.36 — this release: the odd-parity k=1 oscillating mode across the antipodal shell structure, independently validated, with the underlying geometry-construction method itself independently validated against noise. 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 Jun 24, 2026 (v2.33) — Model Comparison and Global Phase Coherence of the Cosmic Birefringence Rotation Field — DOI:10.5281/zenodo.20825890 Jun 27, 2026 (v2.34) — Rotation Field of the Cosmic Microwave Background — Waveform Geometry of the Calibrated Interior Propagation Field — DOI:10.5281/zenodo.20977739 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