Could "distance" be an emergent property like temperature?

If you think about a pre-geometric theory of particles which might consist for example of only fermions, bosons and gravitons but no notion of space.

Then the initial state and final state of an experiment would just be a collection of particle types and we would have no other information about the positions of these particles.

But, if we are only able to detect the high energy particles such as the electrons and photons, but not the gravitons then… the gravitons would be like particles in a hot fluid in which we can’t measure the individual positions of the atoms but we can have some general measurement of temperature. Except in this case instead of "temperature" the information would be encoded in the distances between the particles in the initial or final state.

In this way, for the initial and final states of an experiment the distances between particles might be an emergent property of certain collections of gravitons.

We already have an idea that "time" is an emergent property which arrises because of thermodynamics and the fact that sets of particles are grouped together and given a property called "entropy" or temperature. States of the universe with higher entropy are considered at later times than those of lower entropy. I would think that since space and time are similar concepts that "space" too should be an emergent property similary of a kind of information averaging.

In other words, a certain set of states in the pre-geometric theory where none of the particles have any idea of position:

$$left{|text{photon+electron+ (set of gravitons A)}rangle, |text{photon+electron+ (set of gravitons B)}rangle, |text{photon+electron+ (set of gravitons C)}rangle,….right}$$

may well correspond to a percieved state in a geometric theory:

$$approx |text{photon+electron 3 nanometres apart}rangle$$

But I wonder, would there be enough information in those set of states consisting only of particles without positions, which could encode distances between particles. I’m not sure how the gravitons would be able to tell you which particles are spaced apart.

For example if there were N photons in the output state, the sets of gravitons would have to encode somehow 3(N-2) distances I believe. (Since photons are identical the ordering of the photons would not be important).

So the question is, would there be enough information to make this work?

Edit: I suppose the distance might also be encoded somehow in the amplitudes as such:

$$psi(e,gamma,L) approx sumlimits_{G} f(G,L) tilde{psi}(e,gamma,G)$$

where $L$ is the distance separating the electron and photon and $G$ represents a set of gravitons. As an example let $alpha_n$ be the amplitude for there being $n$ gravitons. The distance between photon $N$ and photom $M$ might be encoded something like: $sumlimits_i |alpha_{iN}|^2 |alpha_{iM}|^2$ [as a very crude example which won’t work in practice], so yes I think there is enough information if we allow the amplitudes to encode the geometry. Although, I’m not sure this is physically allowed.