A joint planarity tensor for the low-ℓ CMB

Abstract

Methods paper with negative results declared. Two no-go theorems are proved for the estimator family behind the CMB 'Axis of Evil': (i) a multipole's own eigenframe cannot measure its own planarity (P2≡1 identically); (ii) maximizing zonal power over real axes locates the plane of a planar structure, not its axis — measured 84.5°–88.7° from the literature axis on four maps. The corrected instrument — a locked 3×3 joint planarity tensor M = M(2)/I2 + M(3)/I3, principal axis computed exactly by matrix squaring — reproduces the twenty-year-old alignment axis to 6.7°–10.4° on Planck 2018 SMICA/NILC/SEVEM and WMAP ILC (two missions). On the locked frame: a participation statistic Sℓ2 of the angular power distribution with an exact Gaussian asymptote of 3/4 (analytic null, no Monte Carlo); a golden-ratio solvability lemma enabling verification in exact integer arithmetic; and a mask forward-model calibration ('customs gauge') that propagates exactly planted hypotheses through the full pipeline. Two independent software pipelines (exact rational arithmetic vs healpy) agree on the axis to 0.01° and on all statistics to ≤0.0023. No detection is claimed. The pre-registered sign observable on the pair (7,8) is shown to have no statistical power (fair-coin null); the P7 elevation, the S72>3/4 concentration and the (7,8) pair jump all fail look-elsewhere accounting. One sharp hypothesis survives unrefuted: P6 = 0 (an exactly planar-power-free ℓ=6), which mask forward-modeling cannot distinguish from the data (per-map likelihood ratios ×4–5 against isotropy) and whose depression is independently replicated by WMAP, excluding a Planck-specific artifact. A pre-registered E-mode test is sealed and timestamped; the E data remain unexamined and the outcome will be reported whichever way it falls. [ES] Paper de método con negativos declarados. Dos teoremas de imposibilidad sobre los estimadores clásicos del Eje del Mal, el tensor conjunto de planitud corregido (reproduce el eje histórico en cuatro mapas de dos misiones), un nulo gaussiano exacto (3/4), la calibración de la máscara por modelado directo, y todos los p-valores con look-elsewhere. Sin detección; sobrevive sin refutar la ley P6=0, replicada por WMAP. El test en polarización E quedó prerregistrado antes de mirar el dato. Integrity: PDF SHA-256 047cf00936c18ad7938197eac2d613b627ba07561d486a83273a548f01b3c0d1 · source SHA-256 38acf149c04aaa36b87faf823c541bfd37bf8b3b946d198217ca49761c530fd6; both OpenTimestamps-anchored before deposit; .ots proofs archived with this record.