Fourier Structure of the Pre-Saturation Admissible Pipeline: What the Three-Coordinate Projection Does and Does Not Measure

Abstract

The numerical pipeline of the spectral admissibility programme stores, for each conjugate Weil pair $(c, q-c)$, two distinct objects: a full Gram–Schmidt basis of ${\mathbb{C}}^q$ reaching rank $q$ at saturation, and a three-coordinate array of projections onto the first three selected basis vectors. This paper establishes the exact Fourier structure of the pre-saturation pipeline and the epistemic status of the three-coordinate reduction. The positive results are analytic: every BFS fingerprint evaluated at the uniform initial vector is a pure Fourier mode, with frequency given by the O12 displacement formula, and the Gram–Schmidt procedure preserves this purity, so every selected basis vector of the pre-saturation window is itself a pure Fourier mode. In the pre-saturation regime $n_*(q) = o(q)$ the realised displacements satisfy $|b_j| \le n_*(q) = o(q)$; for block parameters held fixed independently of $q$ the realised frequencies are $o(q)$, while the implemented pipeline samples the block parameters uniformly modulo $q$, so per-$q$ frequencies of macroscopic size occur and no $q$-independent limit frequency is defined by the block structure in either regime. The delimiting results follow. First, the three-coordinate array is a truncation fixed by the pipeline constant HEFF\_DIM = 3; every quantity computed from it lives, by construction, in the span of the first three selected vectors, so no such computation can measure the rank of the admissible sector or exclude further directions. Second, a uniform coercivity bound ${\mathcal{E}_q}(f,f) \ge c\,q^{-1}\|f\|^2$ is ruled out: the admissibility form is $O(q^{-2})$ on the whole unit sphere of ${\mathcal{C}_q}$, a direct consequence of its $q^{-2}$ normalisation prefactor, consistent with the exact identification of the canonical filtration as a growing Fourier window $\Omega_n$ of dimension $\min(2n+1, q)$ whose published admissibility form converges to the zero form; the spectral-gap question for the unnormalised form is untouched by this statement. The hypotheses [H-$\mathcal{E}$1] and [C] of the original Q5a framework are therefore not closed by the finite-rank route; their disposition on the canonical filtration is the one established in.