Electromagnetic Fields in Multi-Sheet Spacetime: Sheet-Dependent Field Ratios, Charge Quantisation, and a New Experimental Prediction from Extended Lorentz Transformations

Electromagnetic Fields in Multi-Sheet Spacetime: Sheet-Dependent Field Ratios, Charge Quantisation, and a New Experimental Prediction from Extended Lorentz Transformations We present a new derivation of the transformation law of the electromagnetic field tensor \ (F_\) under the Extended Lorentz Transformations (ELT), which replace the standard Lorentz boost when the worldline of a physical system is non‑injective – i. e. when a single ultra‑relativistic worldline intersects a constant‑time hypersurface in \ (N>1\) distinct spatial points. This multi‑sheet structure is a necessary and sufficient condition for the emergence of finite holographic spacetime, as proved in our companion paper on worldline non‑injectivity. Unlike the standard Lorentz boost, the ELT contains a topological phase offset \ (ₙ = ² v (ₙ - ₁) \) that depends on the proper time of each sheet. We show that this offset is not constant: its derivatives \ (_ₙ\) with respect to position and time are non‑zero when the velocities of the sheets differ (as they must for a genuine non‑injective worldline). Consequently, the Jacobian of the ELT acquires a gradient term that modifies the transformation law of the Faraday tensor. Our main physical prediction is that: Two observers in the same inertial reference frame but on different topological sheets measure different ratios of electric to magnetic field components. This is impossible in standard special relativity, where the field depends only on the frame, not on a sheet index. The prediction is derived explicitly, and the corrections to the electric and magnetic fields are expressed in terms of \ (ₜₙ\) and \ (ₓₙ\). From the same multi‑sheet structure, we derive **charge quantisation**: the Ontological Identity Principle forces every sheet to carry the same elementary charge \ (q\), and the observable charge on any single sheet is the topological average \ (q₎₁ₒ = (1/N) ₙ qₙ = q\), independently of the sheet number \ (N\). This provides a geometric explanation for why all elementary particles have charges that are integer multiples of a fundamental unit. To make the prediction experimentally testable, we introduce the **De Giuseppe Photonic Crystal (DGPC) ** – a silicon‑on‑insulator photonic crystal with lattice constant \ (a = 440\, nm\) that realises an effective multi‑sheet metric for photons, with \ (N = 4\) degenerate flat bands. Applying an external electric field via an electro‑optic modulator induces a differential phase shift \ (₂ - ₁ 0. 04\) rad between the two selected modes. This shift is detectable with a balanced homodyne detector (sensitivity \ (10^-3\) rad) and is predicted to vanish in control experiments (single‑mode crystal, field polarity reversal, mode‑selective detection). The experiment therefore provides a direct, falsifiable test of worldline non‑injectivity and the multi‑sheet structure of spacetime. Finally, the paper extends the universal topological cancellation identity \ (N () ^d-2 = O (1) \) to the fifth level of physical theory: after holographic gravity, classical electromagnetic self‑energy, quantum mechanics, and thermodynamics, we now include the transformation of electromagnetic fields. All five levels are unified under the single principle of worldline non‑injectivity. This work is part of the TPST–DGQ programme and is self‑contained: all necessary concepts (non‑injectivity, ELT, Ontological Identity Principle, DGPC) are defined and derived from first principles. The paper includes explicit calculations, numerical estimates for the experiment, and three control measurements to distinguish the multi‑sheet effect from standard single‑sheet artefacts. 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