Tidal Forces, the Equivalence Principle, and the Emergence of the Einstein Field Equations from Worldline Non-Injectivity in de~Sitter Spacetime

This paper derives three foundational results of general relativity — tidal forces, the equivalence principle, and the Einstein field equations with cosmological constant — from a single geometric principle: worldline non-injectivity. A timelike worldline with Lorentz factor above a critical threshold intersects constant-time hypersurfaces in N > 1 distinct spatial points, generating a multi-sheet structure of spacetime. The gravitational field is encoded in the proper-time distribution across sheets. The main results are as follows. Tidal forces emerge from proper-time gradients across extended bodies: the deformation condition ΔL/L = −Δτ/τ produces a strain tensor equal to the electric part of the Weyl tensor (Theorem 4. 1). The Einstein field equations are derived — not assumed — from the topological average of the most general Lorentz-scalar Lagrangian built from the proper-time field with at most two derivatives (Theorem 5. 1). The cosmological constant emerges as Λₒbs = Λ₀/N₀, finite and independent of the UV cutoff for d = 4, via the explicit scaling Λbare ~ ε^-2 and N (ε) ~ ε^-2: no fine-tuning, no anthropic argument, no supersymmetry. The equivalence principle is derived in two logically distinct steps: classically, as the theorem that non-injectivity is locally removable for any smooth worldline (using only differential geometry, without ℏ) ; and as a quantum correction with minimum scale δₘin = λ̄C/ (πc), where λ̄C is the reduced Compton wavelength. The quantum correction predicts a violation of the Weak Equivalence Principle of order Δa/g ~ 10^-13 for proton-electron comparisons — not excluded by MICROSCOPE (10^-15) and directly testable by the STE-QUEST mission (target 10^-17). This is the main falsifiable prediction of the paper. The analysis is performed in de Sitter spacetime, the physically correct background for our accelerating universe, extending the TPST-DGQ framework from Anti-de Sitter to de Sitter and showing that the cancellation identity N (ε) ·ε^d-2 = O (1) holds universally, independent of the sign of Λ. Newton's constant G is the single external input, fixed by the Newtonian limit, in full analogy with Sakharov (1967), Jacobson (1995), and Verlinde (2011). The paper is fully self-contained: an appendix provides the derivations of N (ε) ~ ε^- (d-2), ℏ from fold stability, and the sheet-dependent metric correction δg^ (n) ⏛⏜ ~ ε^d-2 from the Extended Lorentz Transformations, identical to those in the companion papers of the TPST-DGQ programme. This is the ninth paper in the TPST-DGQ framework, which unifies holographic gravity, quantum mechanics, thermodynamics, and electromagnetism under the single principle of worldline non-injectivity. 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