The Ram vs Physical memory is confusing, since they should be the same...

My gaming system has 32Gb Ram, so 20Gb isn't too hard to believe. Back in 2017 i had 16-20Gb on my computer.

Course laptops and integrated systems enjoy 'shared' memory, where a block of your memory can be diverted for video memory. But hard to say.

Looking at the specs listed it is a bit confusing, especially when free drive space isn't listed.

Was referring to "over 800 MB's physical memory but I'm about to del some unwated stuff to create more memory" when the 837 Mb from the system report refers to physical RAM unused at the moment the report was generated while deleting things frees up disk space.

You won't need to delete anything. That's not gonna help (or, rather make it worse with machine essential files. Do not do it!). The RAM requirements are a hardware requirement, which you have plenty, and there's nothing you can really do about that if you don't have enough without upgrading the chips (which for HoI's case, you don't need to do that either). Your machine well exceeds the requirements, and for that matter there's no such thing as a "laptop version" or a "desktop version" or any such silly notion you're thinking. Any game you've downloaded on GOG is essentially the same game made for the operating system installed. That is it. One singular copy of the same game. Meaning the one that works on your desktop is the same copy you've installed on your laptop. You're having an issue with running the game on that laptop because it's an issue with the laptop, not installing (which is a mistatement in your title of this thread).

I would note the GOG page requirements are rather exceedingly overstated compared to the original retail version I'd presume is being run. I don't know if it was because it's running third party emulation or that it was patched up. Regardless, the specs you've thus far shown basically can run it, but you've left out the 3D card info which I have very strong suspicion is part of the problem.

I'm pretty sure those are the (probably still somewhat inflated) requirements for GOG's offline installer, which GOG substitutes (in most cases) whenever the requirements for the game itself are so minimal as to present very little chance that any remotely "modern" PC or proper desktop-OS laptop would fall below them.

Didn't even consider that. The actual retail copy barely makes a dent on consumption. So ostensibly I can only think it would be a compatibility issue. There was no need to compensate for resources. It would barely use much in the first place.

What is the strangest part of all is I can Play Hearts of Iron II which takes up 758 mbs compared to HOI 1 which only takes up 300 plus mbs. I just don't get it
So how is it that I can run HoI II but not HoI 1. I have both games installed on laptop, Hoi II works but not HoI 1 they both use the same 3D cards. So explain that

That's not a critical comparison. Sequels and prequels of games can be rapidly different from one another. It might have even been made with a different engine and have different components. Because games made after the first one were made to compete with it's predecessor and even some in the current market at the time, it's bound to have some differences. There's at least a year difference at a glance between these games (which makes sense given the time they had to produce number 2). I wouldn't use the second game as a goalpost for anything. Each game has its own quirks. The speed a game runs has variables. Your hardware as well as the way the game is coded are all dependencies. To the point a game stops running is another variable.

That is the mystery we are trying to solve. Again, a newer game wouldn't be much to glean over as a basis of comparison. There's just something very specific about the first game.