This paper presents a rigorous, self-contained derivation of dynamical dark energy from worldline non-injectivity, a new principle of spacetime geometry discovered within the TPST–DGQ framework. The central idea is deceptively simple. In standard special relativity, the worldline of a moving body is assumed to be injective: each moment of proper time corresponds to exactly one moment of coordinate time. This assumption, embedded in all of physics for over a century, breaks down when the body's Lorentz factor exceeds a critical threshold. At velocities sufficiently close to the speed of light, the worldline intersects the observer's simultaneity foliation in \ (N > 1\) distinct spatial points. The body appears to be in multiple places at the same instant. This is not an optical illusion. It is a geometric fact — worldline non-injectivity — and it forces a multi-sheet structure of spacetime. The physical content of this structure is captured by the Ontological Identity Principle: the \ (N\) spatial intersections are not \ (N\) distinct particles, but \ (N\) simultaneous appearances of the same entity. Any physical observable is therefore the topological average over all \ (N\) sheets. This averaging mechanism is not an ad hoc prescription. It is already established, in the companion paper on non-injectivity, as the unique internal regulator of ultraviolet divergences that preserves four fundamental properties: the absence of free parameters, the Observer-Geometry Identity, the entanglement first law, and the no-signalling constraint. It operates identically at nine distinct levels of physical theory, from the Ryu–Takayanagi entropy of holographic gravity to the Coulomb self-energy of classical electrodynamics, the Born rule of quantum mechanics, and the electromagnetic field tensor. Applying this principle to the vacuum energy yields a decisive result. The bare cosmological constant, calculated from quantum field theory, is not wrong. It is simply distributed over \ (N\) topological sheets, and what we observe is the average: \ (₎₁ₒ = ₁₀ₑ₄ / N\). Since the intersection multiplicity scales as \ (N () ^- (d-2) \) with the dimensionless UV cutoff \ (= P / LH (t) \), where \ (P\) is the Planck length and \ (LH (t) \) is the Hubble radius, and the Hubble radius evolves with cosmic expansion, the number of sheets grows with time. In a universe that is \ (10^122\) Planck lengths across, there are \ (N 10^122\) sheets. The bare vacuum energy and the intersection multiplicity are controlled by the same geometric parameter, and their ratio is finite: the smallness of the cosmological constant is a necessity, not a fine-tuning. But there is more. Since \ (LH (t) = c/H (t) \) and \ (H (t) \) decreases as the universe expands, \ (N (t) \) increases with cosmic time. This makes \ (₎₁ₒ (t) H (t) ²\) a genuinely dynamical quantity. Dark energy evolves. It is not a constant. And this evolution is not introduced by hand through a new scalar field, a modification of general relativity, or an arbitrary parametrisation. It falls out of the same geometric structure that already regularises ultraviolet divergences at every other level of the theory. From this geometric starting point we derive the effective dark energy equation of state. Using only the observed cosmological parameters \ (ₘ 0. 31\) and \ (_ 0. 69\) from the Planck 2018 results, the framework predicts: ₄₅₅ -0. 78 0. 08, \ where the uncertainty reflects the approximations used in the derivation rather than any free parameter. This is a parameter-free prediction. It is compatible with the DESI 2024 measurement \ (w₀ = -0. 76 0. 20\) within \ (0. 1\). The framework also predicts a specific CPL parametrisation with \ (wₐ -0. 09\), and is consistent with the DESI 2024 preference for \ (w -1\) at the \ (2. 5\) level. Beyond the equation of state, the framework makes three further falsifiable predictions. First, the growth of cosmic structure is suppressed by approximately \ (7\%\) relative to the \ (\) CDM prediction, a consequence of the modified expansion history. Second, the distance–redshift relation receives a topological correction of order \ (2. 2\%\) at redshift \ (z = 1\), encoded in the modified Friedmann equation. Third, the time variation of \ (\) contributes an effective anisotropic stress that modifies the cosmic microwave background power spectrum at large angular scales (\ (30\) ) at the level of approximately \ (1\%\). All three predictions are testable with upcoming surveys from the Euclid satellite, the Vera C. Rubin Observatory, and the CMB-S4 experiment. Two original Gedankenexperimente — the Cable Paradox and the Fibre-Optic Paradox — are introduced in this paper to bridge the gap between the abstract geometric principle and its cosmological consequences. In the Cable Paradox, a steel cable connecting Earth to an ultra-relativistic spacecraft reveals that different portions of the cable have different intersection multiplicities, producing a non-uniform multi-sheet structure. The oxidation rate, mechanical tension, and even the chemical state of the cable become multi-sheet quantities, with observable inter-sheet fluctuations. In the Fibre-Optic Paradox, an optical pulse propagating in a fibre attached to the spacecraft exhibits a distance ambiguity that depends on the reference frame, revealing that the refractive index itself becomes a multi-sheet quantity in the non-injective regime. These thought experiments are not merely pedagogical illustrations. They show that the time-dependent UV cutoff \ ( (t) = P / LH (t) \) is not an artefact of the cosmological formalism. It is a physical property of any system in which the relevant macroscopic scale evolves with time. For the fibre, that scale is the coherence length of the optical pulse. For the cable, it is the local Lorentz factor. For cosmology, it is the Hubble radius. The transition from the laboratory to the cosmos is the transition \ (L₋₀₁ LH (t) \). Everything else follows from the same geometric principle. The framework introduces no new fields, no free parameters, and no modifications of general relativity. The Master Equation of the TPST–DGQ framework — established in the companion paper on holographic extension — already contains the dynamical cosmological constant as a functional of the observer's quantum state. This paper shows that the same functional, when evaluated over cosmic time, yields the evolution \ ( (t) H (t) ²\) and the equation of state \ (w₄₅₅ -0. 78\). The two descriptions are not competing. They are the same equation viewed from two perspectives: the local perspective of the observer at a fixed instant, and the global perspective of the expanding universe. Dark energy is dynamical because spacetime becomes more topologically complex as the universe expands. The universe is not filled with a mysterious substance. It is filled with sheets of reality, and what we measure is their average. This is the tenth level of the universal cancellation identity \ (N () ^d-2 = O (1) \) that unifies quantum mechanics, gravity, electromagnetism, and cosmology under a single geometric principle — 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. 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