Mac Pro shows black screen with white stripes, stops at 40%

that does appear to follow the computer, not the display.

try safe mode, as I described above.

Because it does not load the accelerated display drivers, it MIGHT complete startup, and allow you to enable File Sharing or Screen Sharing to another Mac.

if safe mode does not complete startup, on an older machine like the 2013 Mac Pro, you can still invoke Target Disk Mode, which is implemented in the older Mac's ROM, and does not require MacOS to boot up.

Transfer files between two Mac computers using target disk mode - Apple Support

Those pictures look like either a bad GPU/video card or bad memory.

Try running the Apple Diagnostics to see if any hardware issues are detected.

FYI, the GPU/video card on the Mac Pro (Late 2013) "trashcan" model had a very high rate of failure. If it is the GPU, then don't bother repairing it. Personally I don't think it is worth spending any money on it at all for multiple reasons.

<< So plugging into a working newer Mac might work, but I still have data on the internal Mac HD. Which could come from Tine-machine? >>

The Internal drive in a 2013 Mac Pro is a proprietary SSD, which will be difficult to move to another enclosure. but if you have Time Machine backups, you don't need the original.

if you have been backing up, a copy of your data is on the Time Machine drive, and could be restored all-at-once to a different Mac, or you could restore individual files or folders.

"Tried Command + Fn1 on startup- nothing changes :( get to same progress stage and halts"

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Thanks for the photo.

Power Issue:

Is this plugged into a surge protector? If not, do so, as power surges and power spurs can result in this. These can occur from transformers failing to power outages.

"Mac Pro shows black screen with white stripes, stops at 40%: I have a 2012/3 MacPro (cylinder shape with the latest possible system) which is now suddenly showing a black screen with some evenly spaced white stripes down the screen. Progress bar gets to about 40% and stops there. Nothing else. Suggestions or prognosis? Without yet a service, any idea if the data is retrievable?"

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Retrieving Data from a Mac Pro:

Seems your display may be failing. So, do you have access to another external screen, from a monitor to a projector? If so, connect to it, and mirror your screen to it, through use of you keyboard. Try Either of the Following:

• [Command + F1]
• [Command + FN + F1]

"So plugging into a working newer Mac might work, but I still have data on the internal Mac HD. Which could come from Tine-machine?"

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Thanks for clarifying all of that.

Recovering this:

I'm not so certain you could remove the hard drive, but, if you could, purchasing a Hard Drive Toaster may allow you to get this data --if not erased nor failed. From Amazon: hard drive toaster

The prize in this case is your files, and I thought you said you had a Time Machine backup drive. If true, your files can be recovered from the Time machine drive or imported to a different Mac.

Apple does not provide any real service for the 2103 Mac Pro any longer, because they have no parts.

<< Then just stops like before at the 30% mark… for hours…. >>

That MIGHT be what is actually happening.

But it is just as likely that the software changes to a different screen buffer, but the display hardware never switches. so you see the same screen, but the software might have finished loading and you have no way to tell and no way to tell it to straighten up and refresh the display.

The classical way to debug screen issues is to take a screenshot, and examine the screenshot (such as with Preview) where you can scroll around a bit.

Is the defect shown in the screenshot?

-- Yes, then the screen buffer is bad, and the problem is inside the Mac

-- No, then the problem is in the display itself.

"Sorry about all the questions.. been a while since I had to get into the nitty gritty or restore and recover."

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No worries. But, if I may ask, why the need to perform this restore and recovery? Perhaps something was occurring prior to this, such as a bug or misconfiguration?