Evidence for Dihedral D₃ Symmetry in the Planck CMB Temperature Anisotropy

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 lmax = 150 with NMC = 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 ≤ 2 × 10⁻⁴), with even-l entirely null (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 Cl 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. Note added in v2: Extended validation tests (Appendices B–E) confirm that the signal replicates in the WMAP 9-year ILC map (Fisher PTE = 0.0025), is uniquely selected among dihedral groups D₃–D₆, is frequency-independent across Planck HFI channels (100–143 GHz ∆ f_A2 correlation r = 0.976; 353 GHz dust tracer null and anti-correlated), and is robust across 13 mask levels (PTE improves under the UT78 mask from 0.008 to 0.005).