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

Current Zenodo release: version 1.1 (2026-10-26). Official title: Semiclassical Consistency of the Weil Representation on Heis3(Z/qZ): Quantitative Sinc Embedding, Generator Convergence, and Aliasing Control.

Overview

Version 2.0 of Q5a (Zenodo record 19897055) formulated a large-$q$ limit theorem, its Theorem 7.1, under the hypotheses [H-$\Pi$1], [H-$\Pi$2], [H2], [H-w] and [H-E1], with a spectral tightness statement [C] for its form-convergence part. [H2] is its Hypothesis 4.3: the rescaled Weil generators $\hat{X}_q$ and $\hat{P}_q$ of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, transported by the sinc embedding, converge strongly to $x$ and $-i\partial_x$ on $L^2(\mathbb{R})$.

H2 proves that statement, for the isometric sinc embedding. The proof combines a quantitative Poisson aliasing lemma, controlling the $L^2$ error of periodic aliasing with an explicit rate $O(q^{-1})$ on Schwartz data, and a discrete Sobolev identity relating $\hat{P}_q$ to a finite difference whose strong limit is $-i\partial_x$.

Scope. The framework of Q5a version 2.0 is not that of the current Q5a, which proves that, for uniformly bounded weights, the published admissibility form has zero-form Mosco limit on the canonical Fourier filtration. H2 discharges no hypothesis of the current Q5a and supplies no spatial limit operator. In the version-2.0 framework, [H-w] is open (W1 establishes no proof of it), and the Q5a-O2 Fourier analysis rules out [H-E1] at the $q^{-1}$ scale and supersedes [C].

Core contributions

Strong convergence of the rescaled generators: $\hat{X}_q \to x$ and $\hat{P}_q \to -i\partial_x$ in the strong operator topology on $L^2(\mathbb{R})$, uniformly on bounded subsets of Schwartz space and by density on $L^2(\mathbb{R})$.
Quantitative Poisson aliasing lemma: explicit $L^2$ error bound for the periodic aliasing introduced by embedding discrete functions into $L^2(\mathbb{R})$ via sinc interpolation.
Discrete Sobolev identity: an exact identity relating the discrete momentum operator to a finite-difference derivative.
Quasi-isometry of the sinc embedding: the norm defect is $O(q^{-1})$. An identification of an effective metric constant $A$ in terms of the Born–Infeld constant would use such a bound as one ingredient, together with a supplied power-law profile that is not established (W1, Section 4.2); H2 supplies the bound, not the identification.

Relation to the Cosmochrony programme

H2 belongs to the emergent-geometry sub-programme as a paper of the former Q5a framework. Its result is a statement about generators: it does not make the large-$q$ limit of the admissible structure a continuum quantum mechanics or a spatial geometry. The current Q5a leaves the spatial limit as the hypothesis that Q5b names [H-L].

W1 version 2.0.1 establishes no proof of [H-w]; its retained conditional result concerns a partial-sum proxy, with no established bridge to Q5a response weights.

References

Jérôme Beau. Semiclassical Consistency of the Weil Representation on Heis₃(Z/qZ): Quantitative Sinc Embedding, Generator Convergence, and Aliasing Control. Preprint. doi:10.5281/zenodo.19962051