Sorry I was away for a while, Dale.
Anaerobic bacterial decomposition basically turns large organic molecules into equal parts carbon dioxide and methane. This is what goes on in septic tanks and biogas digesters. If it goes on for a long time, there's not much carbon left, it's all gone up in the air. You don't want this going on in your soil.
With aerobic bacterial decomposition, the bacteria are still using large molecules for
energy and give off small molecules in the process, they just don't put out methane. Instead they use it for energy and oxidize it to CO2. This is good to have going on in your soil, because the small molecules they put out can be taken up as nutrients by plant
roots: CO2, NH3, PO4, etc. But this is no way to build up stores of soil carbon. Bacteria are too small to sequester any appreciable amount of mass in the soil. For that you need fungi.
Fungi have a lot in common with aerobic bacteria, they breathe in O2 and give off CO2. But while they do that, they are building structures in the soil, chains of hyphae, which if unraveled and laid end to end give ridiculously large numbers. Like miles and miles of hyphae per cubic foot of soil. This is what ties up carbon and forms the base of the soil food chain. There are lots of soil critters that live off of soil fungi, so they eat it and excrete nutrients. And because of being a constant buffet for all these critters, fungi have to stay two steps ahead by growing very fast.
Green mulches contain nitrogen which goes away fairly quickly as they become browns in the decomposition process. It's nice when you want to give a nitrogen
boost to a plant that needs to put on some green, but browns are where the long term nutrients are held. Fungi that break down brown materials are constantly providing those nutrients to plant roots, even actively transporting them over distances -- something that bacteria just can't do.