This manuscript proposes a new perspective on the structure of time in general relativity and quantum gravity. The central hypothesis is that the assumption of global temporal injectivity — the idea that time is a single‑valued coordinate everywhere — is not a logical necessity but a postulate that can be removed without violating local causality. Removing it leads to a multi‑sheet spacetime in which the time coordinate is a section of a sheaf over a compact base, rather than a global function. The paper develops this idea in four main directions. **1. Sheaf‑theoretic formulation. ** The multi‑sheet structure is formalised as the étalé space of a sheaf of local inverses over the coordinate‑time circle. This étalé space is a Riemann surface ramified over the critical points of the worldline. The generic stalk is isomorphic to ZN, where N is the number of temporal sheets, and is canonically identified with the computational basis of the sheet Hilbert space H ₒ₇₄₄ₓₒ = ² (ZN). **2. Monodromy and Weyl algebra. ** The topological phase offsets that distinguish the sheets are identified as the monodromy invariants of the Riemann surface. The deck transformations of the covering generate a finite‑dimensional Weyl algebra on H ₒ₇₄₄ₓₒ, satisfying UV = e^2 i/NVU. This algebra is derived solely from the topology of the covering and provides a discrete precursor of the canonical commutation relations of quantum mechanics. **3. Universal UV regularisation. ** The topological average of any UV‑divergent quantity over the N sheets cancels the leading divergence exactly for observables that scale as 1/, where is the UV cutoff. The number of temporal sheets is derived from the Weyl algebra and the existence of a physical UV cutoff, yielding N () T/, where T is the circumference of the base time circle. This mechanism provides a kinematic regularisation of linear UV divergences, including the Coulomb self‑energy in four dimensions and the energy density near spacetime singularities. The regularisation requires no exotic matter, no energy conditions are violated, and no fine‑tuning is invoked. **4. Realisation in Taub–NUT spacetime. ** The framework is shown to be concretely realised in the Taub–NUT solution of the vacuum Einstein equations. The Misner periodicity t t + 8 defines a time sheaf whose monodromy is precisely the NUT charge. Closed timelike curves are artefacts of the single‑sheet description; passing to the universal cover removes them entirely, exemplifying the general principle that multi‑sheet descriptions resolve single‑sheet pathologies. The NUT charge is thus identified, for the first time, as the monodromy of a time sheaf. The paper is mathematically self‑contained. The core results — the sheaf construction, the monodromy‑Weyl correspondence, and the Taub–NUT realisation — are derived with full rigour. The UV regularisation mechanism is presented as a kinematic consequence of the multi‑sheet structure, and its limitations are honestly acknowledged. The work opens several directions for future investigation, including the extension to non‑abelian coverings, the dynamical generation of temporal topology in a cosmological setting, and the non‑perturbative quantisation of the sheet Hilbert space. This manuscript is current in Official Peer Review. Not final version. Copyright©2026 Alex De Giuseppe. All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact: degiuseppealex@gmail. com The hash files that determine ownership have been created
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