Evidence for Dihedral D₃ Symmetry in the Planck CMB Temperature Anisotropy

Evidence for Dihedral D₃ Symmetry in the Planck CMB Temperature Anisotropy | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Evidence for Dihedral D₃ Symmetry in the Planck CMB Temperature Anisotropy Robert This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9307498/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We report a statistically significant detection of dihedral D₃ symmetry in the Planck PR3 temperature anisotropy data, validated across all four independent component-separation pipelines (SMICA, NILC, SEVEM, Commander). At a single optimized axis (ℓ, b) = (50.3°, −64.9°), the power fraction in the A₂ (reflection-antisymmetric) irreducible representation exceeds isotropic expectations with a two-tier structure: a dense cluster at l ≤ 15 (Fisher PTE = 4.2 × 10⁻³ to 1.2 × 10⁻² across maps), driven by three multipoles significant in all four pipelines, with l = 3 serving as the axis-registration multipole (f_A₂ = 0.94, z > 4.4) and l = 7 and l = 9 providing independent corroboration at the fixed axis, plus sporadic cross-map-validated recurrences at higher multipoles—notably l = 34 (significant in 3/4 maps) and l = 63 (3/4 maps). The A₂ excess draws power specifically from the E (rotation-doublet) irrep with anti-correlation r = −0.81, while the A₁ (trivial) irrep is decoupled. Extension to l_max = 150 with N_MC = 10,000 simulations shows that the aggregate high-l Fisher PTE is consistent with isotropy (PTE > 0.91), but individual multipoles punctuate this null background. Among the nine strongest cross-map-consistent peaks, none belongs to the l ≡ 2 (mod 3) residue class (p ≈ 0.02 under uniformity), consistent with the C₃ selection rule. Cross-map correlations of f_A₂(l) exceed r = 0.93 for all pipeline pairs (SMICA–NILC: r = 0.997), ruling out component-separation artifacts. A null test on E-mode polarization at the same axis returns Fisher PTE = 0.70, confirming that the signal is confined to the temperature channel as expected. The irrep redistribution is sharply parity-gated: all four maps confine the A₂ collecting signal to odd-l multipoles (Fisher p 0.97). Crossing parity with residue class produces a six-cell grammar dominated by a single cell (odd, l ≡ 0 mod 3), with step-function onset at l = 3. Singular-value decomposition reveals that this 2 × 3 grammar admits an approximate rank-1 factorization into a binary parity selector and a D₃ residue routing vector, recovered independently by all four pipelines (rank-1 fraction > 94% in three of four maps). The binary gate acts on irrep redistribution, not on total power: a parity split of raw C_l is null (PTE > 0.61) in every map. The signal morphology—dense at large angular scales with isolated resonances at smaller scales—is consistent with a parity-gated boundary condition on the acoustic eigenvalue problem whose geometry is fully resolved only at l ≲ 15 (θ ≳ 12°). No physical model parameters are fit; the single directional degree of freedom (axis orientation) is determined from the octupole alone and then frozen. Astrophysics and Cosmology CMB anomalies dihedral symmetry D₃ Weyl group SU(3) irreducible representations spherical harmonics Planck large-angle anomalies parity asymmetry cosmic topology boundary conditions selection rules information-pressure theory cosmological constant Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9307498","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":616870972,"identity":"5305b545-d4ae-472e-93ac-c87fedeb5249","order_by":0,"name":"Robert","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4ElEQVRIiWNgGAWjYPACGwYDZiDF2MAgA+IeIEJLGlALM1gLD7FaDjMYMCBpwQt023sfPvjZdj5xOzv/wce8O+x4+BmYH+K1xezMcWPD3rbbiTubmZmNec8k80g2sBng13IjjU2Ct+127obDzGzSuW3MPED1BLTcf8b+82/bOZAW9t+5bfU89gfYPxCwhY2NmbftANgW5ty2wzwGDDwEbDmTxiwtcy65HqjFWPpv23EeicM8Bfi1HD/G+PFNmZ2xwfmDDz/ObKuW429v3/wBnxYsgJlE9aNgFIyCUTAKMAEA/m9Kkkz5fawAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0004-6576-5028","institution":"University of Calgary","correspondingAuthor":true,"prefix":"","firstName":"","middleName":"","lastName":"Robert","suffix":""}],"badges":[],"createdAt":"2026-04-03 01:02:55","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9307498/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9307498/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106208550,"identity":"d311164a-b5c8-47b8-b18e-e8a9d49a0ee2","added_by":"auto","created_at":"2026-04-06 06:30:12","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1528706,"visible":true,"origin":"","legend":"","description":"","filename":"d3cmbv2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9307498/v1_covered_8f066f5a-7e8a-4caf-8313-1fba6329776d.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eEvidence for Dihedral D₃ Symmetry in the Planck CMB Temperature Anisotropy\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"CMB anomalies, dihedral symmetry, D₃, Weyl group, SU(3), irreducible representations, spherical harmonics, Planck, large-angle anomalies, parity asymmetry, cosmic topology, boundary conditions, selection rules, information-pressure theory, cosmological constant","lastPublishedDoi":"10.21203/rs.3.rs-9307498/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9307498/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe report a statistically significant detection of dihedral D₃ symmetry in the Planck PR3 temperature anisotropy data, validated across all four independent component-separation pipelines (SMICA, NILC, SEVEM, Commander). At a single optimized axis (ℓ, b) = (50.3°, −64.9°), the power fraction in the A₂ (reflection-antisymmetric) irreducible representation exceeds isotropic expectations with a two-tier structure: a dense cluster at l ≤ 15 (Fisher PTE = 4.2 × 10⁻³ to 1.2 × 10⁻² across maps), driven by three multipoles significant in all four pipelines, with l = 3 serving as the axis-registration multipole (f_A₂ = 0.94, z \u0026gt; 4.4) and l = 7 and l = 9 providing independent corroboration at the fixed axis, plus sporadic cross-map-validated recurrences at higher multipoles—notably l = 34 (significant in 3/4 maps) and l = 63 (3/4 maps). The A₂ excess draws power specifically from the E (rotation-doublet) irrep with anti-correlation r = −0.81, while the A₁ (trivial) irrep is decoupled. Extension to l_max = 150 with N_MC = 10,000 simulations shows that the aggregate high-l Fisher PTE is consistent with isotropy (PTE \u0026gt; 0.91), but individual multipoles punctuate this null background. Among the nine strongest cross-map-consistent peaks, none belongs to the l ≡ 2 (mod 3) residue class (p ≈ 0.02 under uniformity), consistent with the C₃ selection rule. Cross-map correlations of f_A₂(l) exceed r = 0.93 for all pipeline pairs (SMICA–NILC: r = 0.997), ruling out component-separation artifacts. A null test on E-mode polarization at the same axis returns Fisher PTE = 0.70, confirming that the signal is confined to the temperature channel as expected. The irrep redistribution is sharply parity-gated: all four maps confine the A₂ collecting signal to odd-l multipoles (Fisher p \u0026lt; 2 × 10⁻⁴), with even-l entirely null (p \u0026gt; 0.97). Crossing parity with residue class produces a six-cell grammar dominated by a single cell (odd, l ≡ 0 mod 3), with step-function onset at l = 3. Singular-value decomposition reveals that this 2 × 3 grammar admits an approximate rank-1 factorization into a binary parity selector and a D₃ residue routing vector, recovered independently by all four pipelines (rank-1 fraction \u0026gt; 94% in three of four maps). The binary gate acts on irrep redistribution, not on total power: a parity split of raw C_l is null (PTE \u0026gt; 0.61) in every map. The signal morphology—dense at large angular scales with isolated resonances at smaller scales—is consistent with a parity-gated boundary condition on the acoustic eigenvalue problem whose geometry is fully resolved only at l ≲ 15 (θ ≳ 12°). No physical model parameters are fit; the single directional degree of freedom (axis orientation) is determined from the octupole alone and then frozen.\u003c/p\u003e","manuscriptTitle":"Evidence for Dihedral D₃ Symmetry in the Planck CMB Temperature Anisotropy","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-06 06:30:01","doi":"10.21203/rs.3.rs-9307498/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0eef60a0-9791-45ba-99f2-8d6c55bd98ee","owner":[],"postedDate":"April 6th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":65646890,"name":"Astrophysics and Cosmology"}],"tags":[],"updatedAt":"2026-04-06T06:30:01+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-06 06:30:01","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9307498","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9307498","identity":"rs-9307498","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}