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FWIW I’ve been doing a really deep dive on 5e lately and I really think it may be the way to go here. I mean OSR isn’t some kind of pre-defined rules you can plug into or anything, but 5e is. While I have many pages of notes I think most of it really boils down to minor tweaks.
Classes in 5e do a lot of the heavy lifting for “Features” so something between Classes and Backgrounds would probably be where you would truly create a Codex Integrum “flavor”. That heavy lifting, though, may also be where you need to do a bit of tweaking for folks using the core Classes. In essence, though, I think the big thing would be to say something like “Use the Martial Pool mechanic in any situation in which you are using combat rounds”. Then modify the SRD definitions of Advantage to just be adding a Free Dice and Disadvantage as removing a die from the MP (or current MP roll? I wasn’t sure on that) unless doing so would bring the MP to 0, in which case roll 2 dice and take the lower result. Unlike the SRD you might add that you can have more than 1 case of Advantage or Disadvantage for a roll. Then for consistency sake you can replace using the phrases “add a Free Dice” or removing them with just saying you have Advantage or Disadvantage for the roll. Easy. MAYBE a review of how a 5e “Bonus Action” fits in may be helpful, but largely I think the above covers any situation where you have one that would affect the MP.
The term BaB in CM can probably be straight up replaced with Proficiency Bonus. (Proficiency Bonus is just level-based, so every Class just gets it the same way). But 2 things are based on that, MP size and when you get Martial Feats. So is a standard MP of 2 at 1st level, 3 at 5th and 4 at 9th good enough for that? Ditto for Martial Feats? Personally I think the former might be fine (depending on if you level cap or not, although 5e is a bit different here, too, but I’ll need to do some playing to definitively say if level capping is still needed to keep some of that feel to it or not), but I think the latter may be more problematic and may need to be linked in some other way. (If you linked it to Feats, of which the SRD only has 1 anyway, you would get those first at 4th, then 8th, 12th, etc.) I kind of think that will need to be addressed. Maybe one every other level? So 1, 3, 5, etc.?
You will need some slight revisions for any “negative hit points” statements you have, since you bottom out at 0 now (the only thing below 0 is if that last blow would be negative enough to equal your hit point total, in which case its insta-death. Otherwise you’re at 0 and either making saves to avoid death, or someone has to, well I guess we used to call it “bind your wounds”, basically.
Criticals should probably be reviewed compared to 5e, but changing the CM version may or may not be needed. I have some notes about it, but they may be worth reviewing, is all. Same with some of the equipment lists (mainly to either get rid of some fields no longer used in 5e or to just make sure you define them in Codex if they are still needed there anyway) and maybe a little review on Mounted Combat, but overall Codex Martialis should be a relatively easy plug-in to just use the SRD for 5e and you’ll be good to go (and have a game system a WHOLE lot easier to deal with.)
Less important, though still referenced in a few Martialis spots, would be “Skills”, in essence these are just Proficiencies in 5e and there are 3 types, Skill Proficiencies, Tool Proficiencies, and Languages. Most of them are granted by either Class or Background and having them just means you get to make the roll with your Proficiency Bonus (so just a level-based modifier, if you will). Tool Proficiencies just mean you have to have the “tool” kit for that Profciency to even be able to roll. Otherwise it’s just noteworthy that the Proficiencies are simply broader in scope now so you’re not drowned in a bazillion little “skills” to choose from. Considering that a common problem I’ve long had with skill-games like 3.5e is that historically people are a lot broader in skill than you could usually be by those rules I’d say this is another big advantage to broad proficiencies over focused skill lists.
Integrum would need a bit more of a deeper dive to get it jibed up, but that’s mostly just because of things like Spells being case from a level slot rather than always based on the caster’s level, so even that isn’t a huge deal. In other words Magic Missile is no longer automatically adding bolts because you’re a higher level caster, instead you have to use one of your higher level spell slots to do it, an that greatly reduces the damage some of those things did, too, since it’s not about caster level anymore but about spell level instead. That and cantrips are just always available to cast as an Action. There are a lot more little things including reviewing the new spell lists (remember you only need to address what is in the SRD, though, as nothing in the PHB would be something you’d bother with).
Still, while I have many pages of notes it took me only maybe 6 hours of solid work, if that, to essentially prove to myself that 5e is likely going to be the easiest way to go to fill in a game system for what CM doesn’t cover with really just some relatively minor tweaks to CM and, IMHO, fill in plenty enough of what OSR does anyway (besides just making sure to explicitly state that Rule #1 in CM is to “just wing it when you need it” practically covers the biggest thing OSR is supposedly trying to pull back into gaming). I’d be happy to sit down (or conference/phone call it if you’re still more COVID-averse right now, since I haven’t gotten the shot just yet) and go over these some day if you really want to see what I’ve got figured out with it.