Canonical Fourier Filtration and the Obstruction to a Spatial Continuum Limit

Abstract

The companion Foundation paper establishes that the admissible fibre carries the Weil representation of ${\mathrm{Heis}_3}({\mathbb{Z}}/q{\mathbb{Z}})$ at a fixed non-trivial central character, and defines the canonical filtration of the fibre as the orbit spans $\Omega_n = \mathrm{span}\{\rho(g)v_0: g \in B_n\}$ over breadth-first balls of the Cayley graph. This paper identifies that filtration exactly and determines what it does and does not converge to. Three results are established. First, an exact identification: with the pipeline's initial vector, $\Omega_n$ is precisely the toric Fourier window $\mathrm{span}\{e^{2\pi i b x/q}: |b| \le n\}$, of dimension $\min(2n+1, q)$; every fixed toric mode is captured once the published saturation depth $n_1(q)$ exceeds its index, while balanced (line-scale) profiles are rejected at all measured primes. Second, the published admissibility form converges to the zero form on this filtration, uniformly on norm-bounded sets, at rate $q^{-2}$. Third, a normalisation no-go: if the saturation depth diverges and the modulation and translation weights both remain positive and of order one — as the pipeline data indicate — then no common scalar normalisation of the form produces a non-trivial finite toric differential operator: preserving the derivative sector makes the modulation sector diverge, and preserving the modulation sector eliminates the derivative. The only non-trivial rescaled limit is a conditional Dirichlet operator in the rescaled frequency variable on the window itself, which does not provide a spatial continuum. Question Q5 of the Foundation programme therefore remains open, and the downstream identification of a flat spatial co-metric from a limit operator $-A\partial_x^2$ on $L^2({\mathbb{R}})$ rests on an input that is not established here. Interpretive status. The structural reading is that admissibility, as published, organises the fibre by frequency rather than by position: the emergent object is a growing Fourier window, not a discretised spatial line. Whether a spatial continuum can emerge from this structure is exactly the open content of Q5.