The Depth of Almost Nothing II: Near-Planar Geometry in a Prime–Archimedean System

This is the second in a series studying an explicit prime–Archimedean spectral construction. Where the first volume examined a near-null eigenspace of this system, the present volume studies a different, related structure attached to the same aggregate. We show that a specific twelve-generator functional family, built from the aggregate's response to a continuous one-parameter perturbation, lies extraordinarily close to a two-dimensional subspace of an exactly three-dimensional ambient space — a phenomenon we call near-planarity, measured by the scale-free ratio ε = σ₃/σ₂ of the family's singular values — and that this near-planar family has an essentially unique spectral conormal. An independently constructed algebraic conormal, built from just two aggregate objects, aligns with it to high numerical accuracy and near-annihilates, via an exact closed-form expansion valid at every frequency, the twelve generators governing the aggregate's entire continuous response — not merely a finite set of arithmetically realized samples of it. Both the near-planarity and the conormal's near-annihilation are shown to be arithmetically selective: they hold, by many orders of magnitude, at the real prime-power weighting of the construction, and are destroyed under a fixed permutation null that preserves every atom's own geometry and the full multiset of weights while breaking only their specific correspondence. We then give a partial algebraic explanation of the effect: two exact identities, following from elementary properties of a cross product, account for a strong relative suppression in eleven of the twelve harmonic channels underlying the near-annihilation; the twelfth channel is reduced to a single, sharper near-orthogonality that is itself confirmed arithmetically selective under the same null, but is not derived from the two identities. All results concern a single fixed finite configuration, verified to fifty or more digits of working precision. No asymptotic statement and no implication for the Riemann Hypothesis is made or intended. This work is the direct continuation of "The Depth of Almost Nothing: Near-Null Geometry in a Prime–Archimedean System."