Lorentz Transformations beyond Injectivity: The Ziegelstein Gedankenexperiment and the Emergence of Multi-Sheet Spacetime: From the Bricks Paradox to Multi-Sheet Spacetime Structure

How it all started: My intuition A 21st-century Gedankenexperiment in the purest Einsteinian tradition, revealing that the 'Bricks Paradox' is not a kinematic accident, but the geometric engine of spacetime formation. Overview This paper formalizes the Extended Lorentz Transformations (ELT), providing the rigorous kinematic foundation for the De Giuseppe Qubit (DGQ) and the theory of emergent holographic spacetime. While standard special relativity implicitly assumes a bijective mapping between proper time and coordinate time, this work explores the physical and mathematical consequences of breaking that assumption. The Ziegelstein Gedankenexperiment (The Bricks Paradox) At the heart of this derivation lies a new relativistic thought experiment: the Bricks Paradox. In a scenario involving ultra-relativistic transport (= 30), a single set of physical objects (bricks) is shown to exist simultaneously in two distinct states: in transit aboard a spacecraft and already integrated into a construction at the destination. The paradox demonstrates that for a Lorentz factor exceeding a critical threshold (> ₂ₑ₈ₓ), a worldline can intersect a fixed-time hypersurface ₜ at multiple spatial points (N > 1). This "Bricks Paradox" is not a violation of causality, but a manifestation of Worldline Non-Injectivity. Key Mathematical Innovations Extended Lorentz Boosts: The paper derives a set of N transformation branches: x'ₙ = (x - vt) + ₙ. Topological Phase Offset (ₙ): A new physical quantity is introduced, defined as ₙ: = ² v (ₙ - ₁). This offset accounts for the "proper-time gap" between intersections, translating temporal separation in the rest frame into macroscopic spatial separation in the boosted frame. Ontological Identity Principle: A formal framework is established to treat the N appearances as a single physical entity. This principle ensures that energy-momentum and charge conservation laws are preserved via sheet-averaging. Implications for Quantum Gravity and Computation The transition from injective to non-injective kinematics (N 1 limit) acts as the bridge between classical relativity and emergent quantum structures: DGQ Foundation: The N kinematic branches directly generate the multi-sheet Hilbert space required for the De Giuseppe Qubit. Holographic Regularization: Non-injectivity is proven to be a necessary condition for finite holographic entanglement entropy, providing a topological mechanism to cancel UV divergences. Topological Protection: The phase offsets ₙ act as invariants, providing the DGQ with intrinsic protection against decoherence. Conclusion By extending the Lorentz transformations beyond the injective regime, this work reveals that the "Bricks Paradox" is the geometric engine of spacetime formation. The standard 1905 transformations are recovered as a special case (N=1) valid only at lower velocities, while the extended version provides the necessary tools for the next generation of topological quantum field theories. **Note on this preprint (April 2026) ** A small notational error appears in Section 6, Definition 1 (eq. 29) and Section 6. 2 (eq. 32). In those equations, the topological phase offset is written as: Φₙ: = γ v (τₙ - τ₁) The correct definition, consistent with the derivation in Section 5 (eq. 26) and the numerical value in Section 5. 4 (eq. 31), is: Φₙ: = γ² v (τₙ - τ₁) The error is purely formal. It does not affect any physical or mathematical conclusion of the paper. All numerical results (e. g. , Φ₂ = 150 light-years) remain correct because they use γ² v Δτ. A corrected version will be uploaded shortly. Alex De Giuseppe