Canonical Fourier Filtration and the Obstruction to a Spatial Continuum Limit

The companion Foundation paper establishes that the admissible fibre carries the Weil representation of Heis₃(Z/qZ) at a fixed non-trivial central character, and defines the canonical filtration of the fibre as the orbit spans Ωₙ = span\ρ(g)v₀: g ∈ Bₙ\ 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, Ωₙ is precisely the toric Fourier window span2π i b x/q: |b| ≤ n\, of dimension min(2n+1, q); every fixed toric mode is captured once the published saturation depth n₁(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∂ₓ² on L²(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.