Worldline Non-Injectivity and Celestial Holography: The BMS Algebra as the N Limit\\ of the Multi-Sheet Framework

This paper establishes a precise, quantitative connection between the TPST–DGQ framework of worldline non-injectivity and the celestial holography programme. The TPST–DGQ framework is a geometric extension of special relativity in which a timelike worldline X^ (), at Lorentz factors ₂ₑ₈ₓ 2. 2 10⁴, becomes non-injective with respect to the simultaneity foliation \ₜ\ of any inertial observer. This non-injectivity generates a multi-sheet structure: the worldline intersects each constant-time hypersurface ₜ in N > 1 distinct spatial points, which are interpreted, via the Ontological Identity Principle, as N simultaneous appearances of a single physical entity. The intersection multiplicity satisfies the universal scaling N () ^- (d-2), where is the dimensionless UV cutoff, = P/LH, with P the Planck length and LH the Hubble radius. For d=4, N 10^122. The N sheets generate a discrete Hilbert space Hₒ₇₄₄ₓₒ = ² (ZN) with canonical Weyl operators U, V satisfying UV = e^2 i/NVU. In the companion paper on noncommutative spacetime, this Weyl algebra was shown to reduce to the -Minkowski algebra and the Generalized Uncertainty Principle in the limit N. In the present work, we show that the same structure, in a different but equally rigorous limit, generates the Bondi–Metzner–Sachs (BMS) algebra, the infinite-dimensional symmetry group of asymptotically flat quantum gravity. The connection is established through three theorems. **Supertranslations. ** The topological phase offsets ₙ = ² v (ₙ - ₁) of the Extended Lorentz Transformations, which measure the proper-time separation between sheets, become smooth functions () on the celestial circle in the continuum limit N. Their Fourier modes Gₖ commute in this limit, Gₖ, G₊' 0, reproducing the commutative algebra of BMS supertranslations. For finite N, the commutator scales as 1/N, providing a finite-N regularisation of the infinite-dimensional supertranslation space. **Superrotations. ** The Virasoro generators are constructed from the Weyl phase operator V and the angular derivative on the sheet space: Lₘ: = -V^m+1_. We prove rigorously that these generators satisfy the classical Witt algebra Lₘ, Lₙ = (n-m) L₌+₍, which is the centerless BMS superrotation algebra. The central extension, proportional to the celestial central charge c, is not computed here and is left for future work; the relevant normal-ordering prescription on ² (ZN) is outlined. **Mixed commutator. ** The cross commutator is computed explicitly: Lₘ, Gₖ = -ik\, G₌+₊, establishing that the full BMS algebra structure is recovered from the multi-sheet framework in the N limit. A UV/IR correspondence emerges naturally from the scaling N ^-2. The minimum angular momentum step on the discretized celestial sphere is _ = 2/N ², yielding a minimum graviton energy _ ²/LH. The product ₔₕ _ LH O (1) shows that the UV cutoff of the non-injective framework and the IR cutoff of celestial holography are dual. The topological projection P₀ = 1N/N, which extracts the k=0 Fourier mode and recovers standard Lorentz physics, is identified with the leading soft graviton sector of the Weinberg theorem. The projection onto the trivial angular mode corresponds to the leading soft factor S^ (0), establishing a structural equivalence between the topological average and soft theorems. Finally, we note a structural convergence with the recent extension of the connected wedge theorem to asymptotically flat spacetimes by Arayath and Pasterski. Their result—that bulk geometric connectivity is equivalent to boundary entanglement—is precisely the theorem proved in our companion paper on entanglement (N > 1 S₄₍ₓ > 0 geometric connectivity), now extended from AdS to flat space through the BMS connection established here. Two open questions remain: the explicit computation of the celestial central charge from the normal-ordered Weyl algebra on ² (ZN), and the embedding of celestial scattering amplitudes as matrix elements of the multi-sheet operators. The multi-sheet framework thus provides a finite-N regularisation of the BMS algebra, with N 10^122 discrete modes replacing the continuous infinity of supertranslations. Celestial holography is the N limit of the multi-sheet framework. The same geometric principle that regularises holographic entanglement entropy, the Coulomb self-energy, and the cosmological constant also generates the symmetry algebra of flat-space quantum gravity. 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