Emergent Spacetime Geometry from a Split-Quaternionic Scalar Progenitor: From Lorentz Invariance to Optical Gravity (Part VIII)

This paper represents Part VIII of the comprehensive research series on the Higher-Derivative Lee-Wick (HDLW) Theory, which reconstructs the Standard Model and Gravity from a single scalar progenitor governed by Split-Quaternionic Algebra ($\mathbb{H}_{\eta}$).In this installment, we address the unification of Quantum Mechanics and General Relativity by proposing that spacetime geometry is not a fundamental entity but an emergent optical phenomenon arising from the scalar vacuum. Building upon the UV-finite framework established in Parts I-VII, we derive the following key results:Algebraic Origin of Lorentz Symmetry: We demonstrate that the Lorentz algebra $SO(1,3)$ arises naturally as the automorphism group of the internal Split-Quaternionic state space, with the speed of light $c$ emerging from the Berry connection.Optical Gravity: We propose a model where the gravitational field is described as the refractive index of the vacuum, $n(\mathbf{r}) = \exp(-\Phi/c^2)$. We show that this scalar-optical approach reproduces the Schwarzschild metric in isotropic coordinates in the weak-field limit.Classical Tests of GR: The optical metric correctly predicts light deflection and the perihelion precession of Mercury, resolving the historical difficulty of scalar gravity theories in matching observation.Resolution of Singularities: We suggest that the central singularity of black holes is resolved by a ghost-condensation mechanism at the fundamental Lee-Wick scale ($M \approx 11.3$ TeV), forming a stable "Black Droplet" instead of a singularity.Thermodynamics: We provide a new interpretation of the Third Law of Thermodynamics, where zero temperature corresponds to the perfect cancellation of observable degrees of freedom by ghost modes.This work suggests that gravity is not a distinct fundamental interaction but a macroscopic manifestation of the same scalar field that constitutes matter and gauge forces, completing the unification program of the HDLW theory.