Mirror Reflection in Multi-Sheet Spacetime: Anticipatory Images from Extended Lorentz Transformations and Worldline Non-Injectivity

This is the revised and finally corrected version of my previous "Quantum Anticipatory Reflection Paradox" paper. Mirror Reflection in Multi-Sheet Spacetime: Anticipatory Images from Extended Lorentz Transformations and Worldline Non-Injectivity Description: This paper reinterprets the classic Einstein train‑and‑lightning gedankenexperiment through the lens of **worldline non‑injectivity** — a previously overlooked kinematic regime where a single worldline can intersect a constant‑time hypersurface in multiple distinct points. This occurs when the Lorentz factor γ exceeds a critical threshold γcrit, a condition already demonstrated in the companion Ziegelstein Gedankenexperiment (the “Bricks Paradox”). In the standard relativistic analysis, a ground observer sees the rear lightning strike in the past, the front strike in the future, and the reflected images from a central mirror arrive after both events have occurred — no anticipation. However, when the implicit assumption of injectivity (τ → X⁰ (τ) is one‑to‑one) is relaxed, the usual Lorentz boost must be replaced by the **Extended Lorentz Transformations (ELT) **: x'ₙ = γ (xₙ – vt) + Φₙ, Φₙ = γ²v (τₙ – τ₁). These transformations generate a **multi‑sheet spacetime**. One sheet reproduces the standard causal order; a second sheet carries an **anticipatory image** of the future event. We derive the anticipation time Δtₐnt and the precise threshold condition v > c/√2 for the effect to occur. For ultra‑relativistic velocities (γ ≫ 1), the anticipation grows as γ², opening a macroscopic temporal window. Crucially, we prove that the **Sagnac time shift** in a rotating fiber loop is exactly equal to the topological phase offset of the ELT — not an analogy, but a direct equivalence derived from first principles. This allows us to propose a concrete laboratory implementation: a **Recirculating Sagnac Fiber Loop (RSFL) **. By recirculating light N ≈ 5×10⁹ times, the tiny single‑loop Sagnac shift (≈ 7×10⁻¹² s) accumulates to a macroscopic anticipatory margin of tens of milliseconds, well within the reach of current fiber‑optic technology (fibre length 100 km, rotor radius 0. 5 m, angular velocity 2000 rad/s, EDFA gain 30 dB per loop). The work further reveals that the same topological mechanism regularises ultraviolet divergences in holographic entanglement entropy (through the identity N (ε) ·ε^d-2=O (1) ) and underlies the **De Giuseppe Qubit (DGQ) ** — a multi‑sheet topological qubit with emergent parallelism and decoherence suppression. Thus worldline non‑injectivity emerges as a **universal geometric principle** operating at the deepest levels of fundamental physics (holography, causality, singularities), quantum information (topological qubits), and experimental optics. This paper is fully self‑contained but references the companion works on worldline non‑injectivity, extended Lorentz transformations, holographic TPST, and the DGQ. It offers both a conceptual breakthrough (causality as a single‑sheet property) and a testable experimental pathway to access future information via engineered multi‑sheet spacetime. 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.