A pre-registered test of multipole alignment structure in the CMB: the (7,8) pair and an inverse-signature observable

Pre-registration. No data has been analysed. This document is deposited before a single spherical-harmonic coefficient is computed from any real map, so that the result may be reported whichever way it falls. The multipoles under test are not selected by inspection of the data. Requiring the conventional band-power weight ℓ(ℓ+1) to be 3-smooth, and noting that ℓ and ℓ+1 are coprime, Størmer's theorem on consecutive smooth numbers yields a finite and complete solution set: ℓ = 1, 2, 3, 8. Excluding the dipole leaves exactly two consecutive pairs — (2,3), which coincides with the documented quadrupole–octopole alignment, and (7,8), which is untested. The look-elsewhere effect is therefore bounded by construction: the rule admits no further multipole at any ℓ. Two observables are fixed in advance: an angular alignment statistic evaluated over the eight consecutive pairs ℓ=2…10, and an inverse-signature statistic O(ℓ,ℓ+1) = sgn(𝒫̂ℓ+1 − 𝒫̂ℓ), which measures not an angle but the direction in which azimuthal power concentration flows as one ascends the ladder. Predictions are frozen here: O(2,3) = +1 (control) and O(7,8) = −1. Joint probability under isotropy 0.25; combined with the eight-pair angular pattern (1/544), the full configuration has probability 1/2176. The estimator's own bias is reported. The uncorrected statistic has ⟨Pℓ⟩ = 1/ℓ under isotropy, so a random sky reproduces the (7,8) prediction 54.0% of the time while the control demands an outcome of probability 1/3 — the test as first drafted favoured its own hypothesis. The per-multipole normalisation 𝒫̂ℓ = ℓ·Pℓ removes this; measured over 200,000 isotropic realisations the (7,8) null becomes 0.4997 / 0.5003. Two kill conditions are declared before unblinding: if the control fails the pipeline is deemed non-functional and nothing is interpreted; if O(7,8) = +1 the framework is reported as refuted, with no substitution of pair, multipole, estimator or threshold. [ES] Prerregistro. Ningún dato ha sido analizado. Los multipolos los fija una regla aritmética (Størmer) antes de mirar el cielo, lo que acota el look-elsewhere por construcción; las predicciones y las dos condiciones de refutación quedan congeladas aquí, y el sesgo del propio estimador se declara y se corrige en el documento. Integrity: source SHA-256 fb7daec63486c5413811e1fce31fb10140186bc6ded27ec73b7cc6fc0d6dad30 · PDF SHA-256 5f65abf5b847fb5ecd4e8ef281207879594ceb495713c9c428014f4c9c394f14; both OpenTimestamps-anchored before deposit.