SKARDAVNELNATE: As expected the "Support Stuff" didn't have an answer.

They did identify an out dated driver on my computer but of course that didn't provide the missing file.

Now I have two installers stored on my computer.
setup_ashes_of_the_singularity_escalation_3.10_(64bit)_(49861).exe
setup_ashes_of_the_singularity_escalation_3.11.2_(64bit)_(58286).exe
Between these versions something was updated in the game to now require XAudio2_9.dll.

The store page lists Windows 7 in the minimum system requirements. However XAudio2_9.dll is not normally found on Windows 7 systems.

Windows 7 has XAudio2_0.dll through XAudio2_7.dll. XAudio2_8.dll is a Windows 8 native library. XAudio2_9.dll is a Windows 10 native library.

It is not clear what redistributeable package will provide XAudio2_9.dll for Windows 7. The installer does not include this as a dependency the way it does with Visual C++ packages during installation.

I found this reference:
https://learn.microsoft.com/en-us/windows/win32/xaudio2/xaudio2-redistributable

According to that page it's possible for developers to distribute XAudio2_9.dll to Windows 7 users. However, it seems that if the developer doesn't do that then the only way to get the library is to be a developer yourself in order to have access to the NuGet Package Manager in Microsoft Visual Studio.

SKARDAVNELNATE: I never installed them in the first place seeing as how I'm on Windows 7 and there is no DX12 package for it.

The minimum system requirements for the game still say Windows 7 (64-bit), DirectX 11.
If it no longer works with Windows 7, or it no longer works with DirectX 11, then they should make that known to avoid false advertising.

SKARDAVNELNATE: Despite the lack of assistance I did find one way to get the missing file.

Go here:
https://www.nuget.org/packages/Microsoft.XAudio2.Redist/

Download microsoft.xaudio2.redist.1.2.10.nupkg
Use 7Zip to extract it.

Find xaudio2_9redist.dll in "build\native\release\bin\x64"

At this point I renamed it to XAudio2_9.dll and placed it in "C:\Windows\System32\"

hhtfuh: Folks, let's face it, this isn't the 1st nor it'll be the last time.

Games like "Saints Row IV" or "Lost Ember" or many others were similarity made not working on Windows 7 out of the box, while "No Man's Sky" has phased out Windows 7 overtime without changing the GOG page. I won't even longmention in-depth the fact that many games like "Ashes of the Singularity: Escalation" have not received the latest update as it is on Steam, are basically abandoned, or that games like "Mechwarrior 5" scream about a missing Windows 10 DLL (that doesn't prevent launch, it is used for debug, not important but quite annoying). It does not help that games like "Bite the Bullet" and "Comanche" do freeze to death on having LAN plugged-in without the Internet access, or how "Beat Hazard 2" is under Online DRM and won't work offline at all, this all is so wrong, the store doesn't do a thing to prevent it. Too many issues that stand unresolved forever, folks!

The truth is GOG doesn't care and likely wouldn't care in the future, it certainly doesn't honour you the customers, it won't do as little as to just autodownload a product from its own GOG Galaxy and automatically launch on 7/8.1/10 (or whatever the supported systems are) to see an error popup that demonstrating that the page must be corrected or the developer shall fix the compatibility issue instantly: Not full testing, mind you, just launch and quit, it won't take more than 8–16 workdays of 1 employee per OS for all the games, assuming it's pre-downloaded. In an ideal world, GOG could care, do basic quality control, but, we are in the realm of negligence, it very sadly seems so, where GOG has grown numb. I truly do wish that wasn't the case.

What can you do as a mere customer like myself? Do not trust GOG Galaxy for storage, do keep tested offline installers, never expect GOG/devs to care about your compatibility, do not pay for games whose developers do misbehave such rotten ways deliberately. Not much can be done beyond that for now.