Emergent Lorentz Invariance in Absolute Frame Theory: Substratum Isotropy and the Cosmological Frame

Absolute Frame Theory (AFT) embeds the observable four-dimensional manifold $\mathcal{M}$ in an $N$-dimensional Euclidean substratum $\mathcal{A}$ through a map $X:\mathcal{M}\to\mathcal{A}$, and reads the flow of time as a sliding of the embedding interface through $\mathcal{A}$ with a velocity $V$ defined relative to the substratum. A substratum velocity is, on its face, a preferred rest frame, and the name of the theory invites the objection that it conflicts with Lorentz invariance---the most precisely tested symmetry in physics. We show that the objection does not hold, and we locate exactly why. Effective \emph{local} Lorentz invariance emerges, for free fields, from a single structural fact: the \emph{isotropy} of the Euclidean substratum. The rotational freedom of the four image directions is the group $\mathrm{SO}(4)$, which the signature-fixing Wick continuation carries into the Lorentz group $\mathrm{SO}(3,1)$; the physical invariants---the substratum wavenumber $|k|^2=m_{\mathcal{A}}^2$ and the transverse eigenvalue---are scalars independent of the sliding velocity, whence the on-shell mass is a Lorentz scalar and $V$ is unobservable. The absolute frame survives only as the \emph{global} cosmological rest frame, spontaneously selected by our embedding solution exactly as the cosmic-microwave-background frame is in general relativity, and not as an explicit local violation. Because $\mathcal{A}$ and $\mathcal{M}$ are \emph{continuous} and the only discreteness in the theory resides in the action (a Lorentz scalar) and in the information bottleneck of the interface (a reparametrization invariant), the theory carries no spacetime granularity and therefore predicts none of the dispersion-relation Lorentz violation of lattice-like quantum-gravity scenarios. The interactions inherit the covariance, and we show it vertex by vertex: by the no-forces theorem of the programme they are geometric, built from the induced metric, the normal connection, and substratum scalars, and they admit no bare coupling to the sliding velocity, the sole avenue to Lorentz violation. What is established is therefore free-field local Lorentz invariance, the cosmological-frame identification, and the Lorentz covariance of the interactions to all orders in perturbation theory---no non-invariant regulator and no gravitational anomaly in four dimensions---leaving only the constructive, nonperturbative caveat common to every interacting four-dimensional quantum field theory.