Semiclassical Consistency of the Weil Representation on Heis3(Z/qZ): Quantitative Sinc Embedding, Generator Convergence, and Aliasing Control

Abstract

We prove that the rescaled Weil generators ${\hat{X}_q}$ and ${\hat{P}_q}$ of the discrete Heisenberg group ${\mathrm{Heis}_3}({\mathbb{Z}/q\mathbb{Z}})$ converge strongly to the position operator $x$ and the momentum operator $-i\partial_x$ as $q \to \infty$, in the sense of Hypothesis [H2] of the Cosmochrony $Q5a$ programme. The proof rests on two structural lemmas. First, a discrete Sobolev estimate controls the error between the finite-difference operator and the continuous derivative at the $\ell^2$ level. Second, a quantitative Poisson aliasing lemma controls the transition from $\ell^2({\mathbb{Z}/q\mathbb{Z}})$ to ${L^2(\mathbb{R})}$ with an explicit rate $O(q^{-1})$ for data in ${\mathcal{S}(\mathbb{R})}$. As a consequence we obtain a quasi-isometry lemma for the sinc embedding ${\iota_q}$ with the explicit rate \[ \abs{\norm{{\iota_q} f}_{L^2}^2 - \norm{f}_{\ell^2}^2} \leq \frac{C}{q}\,\mathcal{N}(\psi), \] which is a reusable structural estimate for the identification of the effective metric constant $A$ in terms of the Born–Infeld saturation constant $c_{\mathrm{BI}}$ (open problem $Q5a$-O5 and ). The proof of [H2] closes the generator-convergence hypothesis of the $Q5a$ large-$q$ limit theorem, complementing the geometric emergence sub-programme of. The hypotheses [H-E1] and [C] are not closed: the Q5a-O2 Fourier analysis rules out [H-E1] at the $q^{-1}$ scale and supersedes [C], the published admissibility form having zero-form Mosco limit on the canonical filtration.