  The synthesis of quantum and gravitational physics is sought through a
finite, realistic, locally causal theory where gravity plays a vital role not
only during decoherent measurement but also during non-decoherent unitary
evolution. Invariant set theory is built on geometric properties of a compact
fractal-like subset $I_U$ of cosmological state space on which the universe is
assumed to evolve and from which the laws of physics are assumed to derive.
Consistent with the primacy of $I_U$, a non-Euclidean (and hence non-classical)
state-space metric $g_p$ is defined, related to the $p$-adic metric of number
theory where $p$ is a large but finite Pythagorean prime. Uncertain states on
$I_U$ are described using complex Hilbert states, but only if their squared
amplitudes are rational and corresponding complex phase angles are rational
multiples of $2 \pi$. Such Hilbert states are necessarily $g_p$-distant from
states with either irrational squared amplitudes or irrational phase angles.
The gappy fractal nature of $I_U$ accounts for quantum complementarity and is
characterised numerically by a generic number-theoretic incommensurateness
between rational angles and rational cosines of angles. The Bell inequality,
whose violation would be inconsistent with local realism, is shown to be
$g_p$-distant from all forms of the inequality that are violated in any
finite-precision experiment. The delayed-choice paradox is resolved through the
computational irreducibility of $I_U$. The Schr\"odinger and Dirac equations
describe evolution on $I_U$ in the singular limit at $p=\infty$. By contrast,
an extension of the Einstein field equations on $I_U$ is proposed which reduces
smoothly to general relativity as $p \rightarrow \infty$. Novel proposals for
the dark universe and the elimination of classical space-time singularities are
given and experimental implications outlined.
