Overview
The 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. Q5a version 3.0 identifies this filtration exactly and determines what it does and does not converge to.
The answer is negative in the direction Q5 hoped for, and exact. With the pipeline's initial vector, the filtration stage is precisely the toric Fourier window \(\mathrm{span}\{e^{2\pi i b x/q} : |b| \le n\}\): admissibility, as published, organises the fibre by frequency, not by position. The published admissibility form converges to the zero form on this filtration, and no common scalar normalisation produces a non-trivial toric differential operator.
Main results
- Exact identification (proved): with the pipeline's initial vector \(v_0\) (the uniform state), the canonical filtration stage is exactly the toric Fourier window \(\Omega_n = \mathrm{span}\{e_b : |b| \le n\}\), of dimension \(\min(2n+1, q)\). Every fixed toric mode is captured once the published saturation depth \(n_1(q)\) exceeds its index; balanced (line-scale) profiles are rejected at all measured primes.
- Zero-form theorem (proved): with the published prefactor \(q^{-2}\) and bounded weights, the admissibility form tends to zero uniformly on norm-bounded subsets of the filtration; its Mosco limit is the zero form.
- Normalisation no-go (proved under [H-depth] and the measured weight behaviour): if the saturation depth diverges and the modulation and translation weights have positive finite limits, then every common scalar normalisation \(\alpha_q\,\mathcal{E}_q\) yields, on toric targets, either the zero form, a bounded multiplication form with no derivative term, or the degenerate form \(+\infty\) off zero. No choice produces a non-trivial toric differential operator.
- Dual window limit (conditional): under the frequency blow-up \(u = b/n_1(q)\), the rescaled form converges to a Dirichlet quadratic form on \(L^2([-1,1])\) in the rescaled frequency variable, with a potential term controlled by the critical coverage \(x_1(q)\). This is a dual-space statement; it does not produce a spatial continuum.
What was withdrawn
Versions of Q5a up to 2.1 claimed, conditionally on a set of analytic hypotheses, a Hilbert inductive limit, representational convergence of rescaled Weil generators, and Mosco convergence of the admissibility forms to a second-order operator \(L_\Pi = -A\partial_x^2\) on \(L^2(\mathbb{R})\). Version 3.0 withdraws this framework: the exact identification of the filtration shows that the admissible spaces are frequency windows, and the zero-form theorem and the normalisation no-go rule out every derived route from the published form to a spatial differential operator.
Downstream papers consume the withdrawn framework in revised forms: Q5b (2.0) conditions every metric result on the explicit hypothesis [H-L]; Q9 (1.1) freezes the framework as [H-F] and proves the implication [H-F] \(\Rightarrow\) [H-lift]; Q11 (1.1) keeps its algebraic content with the metric reading conditional on [H-L].
Status of the statements
| Statement | Content | Status |
|---|---|---|
| Identification | \(\Omega_n\) is exactly the Fourier window \(|b| \le n\) | Proved |
| Zero form | Published form \(\to 0\) at rate \(q^{-2}\) | Proved |
| No-go | No common scalar normalisation gives a toric differential operator | Proved under [H-depth] + measured weights |
| Dual window limit | Dirichlet form in the rescaled frequency variable | Conditional (subsequential \(x_1\) limit) |
| [H-depth] | \(n_1(q) \to \infty\) | Hypothesis; numerically supported |
| \(x_1(q) \sim q^{-0.76}\) | Critical coverage decay | Numerical fit, not a law |
| Q5 (spatial continuum) | Emergence of a spatial \(L^2\) operator | Open |
Conceptual interpretation
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. Fixed toric modes need only depth (\(n_1 \ge |m|\), eventually true at every measured prime), while balanced profiles would need the physical bandwidth \(n_1 h = \sqrt{2\pi}\,(x_1/C_{\mathrm{Heis}})^{1/4}\) to stay bounded below — which the published critical-coverage data contradict.
Whether a spatial continuum can emerge from this structure is exactly the open content of Q5. The one identified conditional route — an oscillator-type operator in the rescaled frequency variable — would require the critical coverage \(x_1(q)\) to remain bounded away from zero, a corpus-internal question tied to the CC-Note's open programme.
Relation to the Cosmochrony programme
Q5a sits at the discrete-to-continuum interface of the emergent geometry sub-programme. Version 3.0 replaces the former analytic scaffolding with exact finite-\(q\) statements: the filtration is known in closed form, and the obstruction to a spatial limit is a theorem rather than a difficulty.
The relevant chain is now: Foundation establishes the admissible fibre and its canonical filtration; Q5a 3.0 identifies the filtration and proves the obstruction; the spatial limit input of Q5b, Q6b and the metric-coefficient papers is the explicit hypothesis [H-L]; establishing or replacing [H-L] is the open content of Q5.
Reference
Jérôme Beau. Canonical Fourier Filtration and the Obstruction to a Spatial Continuum Limit. Q5a of the Cosmochrony Spectral Geometry Programme, version 3.0, 2026.