Large-q Limit of the Admissible Fibre: Representational and Operator Convergence

The companion paper establishes that the admissible fibre Fₙ carries the structure of the Weil representation V_ρ of Heis₃(Z/qZ) on L²(Z/qZ) (Foundation Theorem 5.6). The relationship between this discrete structure and continuous spacetime geometry in the large-q limit is identified as open in Foundation Remark 3.6. The present paper reduces this problem to a uniform coercivity estimate and a spectral tightness condition, and provides strong numerical evidence that the tightness condition holds. We formulate the large-q limit in the category of Hilbert spaces and operator algebras — not in the category of sets — and decompose it into three convergences: a Hilbert inductive limit (T1), a representational convergence of Weil generators to metaplectic generators (T2), and a Mosco convergence of admissibility forms to a second-order operator L_Π on L²(R) (T3). Conditions (1) and (3Y) of the embedding hypothesis [H1] are proved for the sinc embedding; [H1] is not needed in full generality: it holds in the admissible low-frequency regime relevant to T1–T3. Each result is stated with explicit hypotheses and proof status. The central remaining condition — Mosco tightness (Conjecture C, equivalently: spectral mass does not escape to high frequencies under admissibility constraints) — is formulated precisely, reduced to a Nash inequality programme, and confirmed quantitatively: for all tested primes q ∈ \29, 61, 101, 151\, the spectral tail mass of admissible test vectors satisfies Eq(q/3)/\|fq\|² < 2.1 × 10⁻⁵, with values decreasing toward machine precision as q grows. The continuum limit is shown to be a forced consequence of admissibility and spectral stability, not an additional assumption. The four-dimensional effective geometry and the Lorentzian signature are established in the companion paper Q5b.