You ask such well-framed questions.
"Where if at all do fields arise in a paradigm of Matter-Time. Does the gaechron have a specific volume (?) or does it undergo a state change that would measure as a volumetric difference."
We have to proceed carefully lest you fall of the end of the world...space has a little different interpretation in matter time. Space is not where action occurs, rather space is the result of the action of matter in time.
Remember that fields are all a result of matter exchange between objects and fields are very convenient representations of matter exchange, especially for our minds. However, space is not really necessary to describe the matter exchange or forces or matter acceleration that is what we think of as fields.
The electron exchanges a Rydberg of matter with a proton to stabilize a hydrogen atom, 8e-21 me per period, and is what we call a charge field. The earth exchanges ~0.16 Mearths/yr with the sun to stabilize its orbit per period and that exchange matter is what we call a gravity field. Gravity seems a lot more expensive than charge...
The gaechron has a volume, kind of like a Planck volume, for every action. However, space is not where gaechron are because gaechron are the universe, not space. Space or volume is only a result of action.
One way to think about quantum gravity in matter time is entropy, since entropy has a lot to do with volume. However, entropy is just the log of the number of states and so is a counting game as well. Volume therefore is a result of entropy or the number of states and not the other way around.
Anyway, it is fun to think of a universe that does not exist in space, but rather produces space from its action. Understanding a universe that is not due to space, but rather creates space with action, is very challenging.