What if we...

Let a Rational number stand for a real particle -- like a quark or an electron -- defined minimumly as a detectable excitation of some underlying field, a quantum fluctuation that brings with it the property of permanence and continuity of identity.

The Irrationals can then stand for virtual particles. Virtual particles also considered as excitations of this underlying field from which the real particles emerge, but manifested as the forces operating between real particles; that is, acting as intermediaries -- the mesesenger particles.

Here is where I have a problem: In the literature, virtual particles play two different roles. One: force-particles and the field supporting their existence. And two: emerging as matter/antimatter pairs from the vacuum [e.g., quark-antiquark pairs], borrowing energy for a Planck time, then dissolving back into the morass of potential quantum energy. In any event, each virtual particle model shares the concept of virtual particles emerging from the vacuum of pre-space.

So, we have quantum fluctuations, both real and virtual, underlying the reality of and giving substance to quantum space.

The real particle fluctuations -- corresponding to the Rationals -- are discrete (quanta) and may be considered as nodes of a network. Topologically, each node is compact in Hilbert space [infinite dimensional] and surrounded by a neighborhood of energy.

Interaction of the Rationals through virtual entanglement brings into being, or precipitates, the Irrationals, which may be thought of as dynamic interconnections -- bosons, force particles -- between nodal neighborhoods. Their intermingling, in turn, generates and defines the Real Numbers -- the Continuum.

One could argue that the particles and their interactions arise simultaneously, thereby occluding the logic of cause and effect. Nonetheless, the point is that their joining as continuous process produces the space we're familiar with.

Gravity, residing as potential only in quantum space, is an effect of this interaction and as such is not fundamentally a quantum phenomenon and cannot therefore be a gradient of the underlying virtual quantum field. With time it forms a hypersurface or multi-dimensional manifold -- the creation of the continuum of spacetime -- comprised of families of curves -- pathways, trajectories, histories -- in space. It acts to cohere and maintain the integrity of the quantum world, to give it sense and definition, as does an envelope of a tangential family of curves on an ordinary surface.

But for the sake of argument, if virtual gravitons -- the force-particles of a gravitational field -- can be construed as points (quanta) of those curves, then the manifold/field thus generated would be of the second order, the order of acceleration.

But is that necessary?