A logarithmic descent relation for the aligned CMB quadrupole-octopole system

Abstract

Methods-and-measurement paper. A new observable for the CMB quadrupole–octopole alignment: in the axial frame of the aligned 2⊕3 system, the ring-averaged profile near a node of the odd multipole obeys D(z) = n·log2(1/|z|) + C — integer slope fixed by topology (the node order), intercept C = log2|A2/A3| carrying the physics (the axial amplitude ratio in bits). The chain is verified end-to-end in exact arithmetic (dyadic–triadic rationals, integer mantissa, zero rounding; lattice test: slope 2.9991 vs 3 exact for an engineered triple node). On Planck 2018 component-separated temperature maps (SMICA/NILC/SEVEM): slope +1.048508 identical to the sixth decimal across the three maps; C = -0.754 / -0.991 / -1.478 bits — the axial octopole dominates the axial quadrupole by about one bit, a frame-locked restatement of the low-quadrupole anomaly measured as the intercept of a straight line. Two independent arithmetic substrates agree on C to 0.2–6.1 thousandths of a bit. The two natural frames of the planar system (proper axis vs axial frame, mutually perpendicular) are disentangled with a conditioning argument; the 5.10-bit spread of C on the proper axis is a new diagnostic of separation-method sensitivity near a projection null. A closed-form prediction D(z) = log2(1/|z|) - log2(3) is stated with its falsification path. No cosmological model is fitted. [ES] Paper de metodo y medida. La relacion de descenso: pendiente entera (topologia), ordenada fisica (bits). Pendiente +1,048508 identica en los tres mapas; el octopolo axial domina al cuadrupolo por ~1 bit. Integrity: PDF SHA-256 1a7791ed4fb6a8a406e5051d7373606700cbbdf416730d8af51ca23b90eb1257 · source SHA-256 05989e67abfd5f8494863a88df68c6ccc2b563f367dbc1a8b32217054571f797; OpenTimestamps-anchored before deposit.