A couple of questions here from Specialist290, which I happen to already have the answer to, so let’s knock those off straight away:

A pair of questions whose answers might overlap just a bit:

1. How does the Imperial law handle cases of lost, mislaid, and abandoned property?

Down in the Imperial Service, strictly speaking within the Central Office of Records and Archives rather than any of the Ministries, is the Office of Title and Estate. Their job, as their name implies, is the proper recording of property titles and other property rights, which drops this firmly and inconveniently into their bailiwick.

(I take a moment here to note that you don’t need a title on file for personalty which you hold all the property rights over, which is the normal case, or where the ones you don’t are any sort of standard-form financial contract – but if you’re getting fancy, it saves time and trouble later to write these things down and send in a copy before anything ends up in dispute.)

Dealing with property of unclear title, the above three cases, is the job of a small and irked sub-bureau of theirs. Not initially: initially, law and custom gives the finder of such property a reasonable amount of time to return it to the owner themselves or to let the owner pick it up from them, because that saves everyone time and trouble. In the modern era, it’s especially trivial, because everything knows who it belongs to and will tell you, if it hasn’t already called them itself and asked to be collected.

If that doesn’t work, they turn it in to said small sub-bureau, or more accurately the staffer representing it ad-hoc, at the lost property desk at the local Imperial Services office. They can engage in more thorough searches of their records and other generalized detective work to figure out exactly who it belonged to and get it back to them.

(And, in the case of intentionally abandoned property, indict them for “trespass by chattel”.)

Ultimately, if it proves impossible to determine who the rightful owner is, it ultimately escheats, and then becomes available for homesteading by anyone who cares to pick it up at the warehouse, as other res nullia.

2. How does Imperial law treat intestate estates? (Part of me thinks that this is probably more of a historical footnote in this day and age, but I’d imagine there might still be that occasional one-in-umptillion case where someone dies without a will just before they remember that they’re overdue to renew their reinstantiation insurance…)

It’s also the job of the Office of Title and Estate, who process all the transfers of title during administration of the estate with some rubber-stamping help from the local Court of the District or, if it’s big enough, the Court of Claims.

In the event of intestacy when they don’t have a will to follow, there are sets of default rules. (This was the subject of controversy very briefly during the Great Conclave, when the question was asked as to exactly what right they had to distribute someone’s property in ways that they might not have approved of, shortly thereafter answered by pointing out that if they had something specific in mind they should have bloody well told someone, and if they were inconsiderate enough to die without filing the proper paperwork they can’t really complain when someone has to pick up their mess for ’em.)

The original set of rules for the intestate starts out with the general principle that for participants in marriage contracts, all non-entailed or otherwise held-in-manners-with-their-own-rules property becomes property of the marital coadunation. After that, for eldrae, the search first skips down the generations, first to children (who are expected to reach an equitable settlement between themselves, or else the court and Office will make a less preferable one for them), then to siblings if there are no children (likewise), then back to parents (if there are neither). If there are none of those, then it becomes the trust of the head of your lineage and/or the genarch of your House to resolve the issue and decide where it goes, including potentially into the familial properties.

If you are the genarch, die intestate, and the process gets this far because you have no direct heirs, it goes to the next genarch, once the appropriate processes find one. If there’s no-one of the blood who can be the next genarch, then technically it would escheat, but the process has never actually got this far ever. I don’t think any genarch has ever either died intestate or lacked heirs, in fact, so.

There are alternate sets of rules, the Empire being polyspecific, for species with very different traditions and biologies, but they generally run along similar lines.