  Elements of a novel theory of quantum physics are developed, synthesising the
role of symbolism in describing quantum measurement and in the topological
representation of fractal invariant sets in nonlinear dynamical systems theory.
In this synthesis, the universe $U$ is treated as an isolated deterministic
dynamical system evolving precisely on a measure-zero fractal invariant subset
$I_U$ of its state space. A non-classical approach to the physics of $U$ is
developed by treating the geometry of $I_U$ as more primitive than dynamical
evolution equations on $I_U$. A specific symbolic representation of $I_U$ is
constructed which encodes quaternionic multiplication and from which the
statistical properties of complex Hilbert Space vectors are emergent. The
Hilbert Space itself arises as the singular limit of Invariant Set Theory as a
fractal parameter $N \rightarrow \infty$. Although the Hilbert Space of quantum
theory is counterfactually complete, the measure-zero set $I_U$ is
counterfactually incomplete, no matter how large is $N$. Such incompleteness
allows reinterpretations of familiar quantum phenomena, consistent with realism
and local causality. The non-computable nature of $I_U$ ensures that these
reinterpretations are neither conspiratorial nor retrocausal and, through a
homeomorphism with the ring of $2^N$-adic integers, are robust to noise and
hence not fine tuned. The non-commutativity of Hilbert Space observables
emerges from the symbolic representation of $I_U$ through the generic
number-theoretic incommensurateness of $\phi/\pi$ and $\cos \phi$. Invariant
Set Theory implies a much stronger synergy between cosmology and quantum
physics than exists in contemporary theory, suggesting a novel approach to
synthesising gravitational and quantum physics and providing new perspectives
on the dark universe and information loss in black holes.
