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First, you will need a patched steam_api.dll for the game to work without steam. Make sure that you're running Windows Vista SP2 x64 (as its only tested to work on this version). The extended kernel is not required. If you happen to be running this on a Vista installation with an extended kernel version of 2023.03.06 or greater, it will crash. This tutorial only works for extended kernel v10192022 and below.
Step 1: Download the Universal C runtime.
Step 2: Install the Visual C++ 2015-2022 redistributables. if using extended kernel. Spoof the OS ver as 6.1 7601. If not, download only the 2015 visual c++ runtimes, x64 and x86.
Step 3: You're good to go! Try running SonicMania.exe. If you get an APPCRASH error or similar, try Step 4 and onwards.
Step 4: Enable automatic updates for Windows Vista, and update your system. The updates in question are feature updates, not security updates.
Step 5: Run the game again.
Step 6: Enjoy!

Mania mod manager does not work as it requires .NET 4.8, unsupported even with the extended kernel, so you cannot install mods . The game is known to run just fine in Vista, no lag or stuttering or some weird visual artifacts.

I forgot; It works even without the extended kernel, with the updates mentioned above.

Not sure if you've looked into this already but according to the comments on this site the game can also be made to run on XP with a few tweaks.
http://wp.xin.at/archives/4212