How much power output can an 8-pin to 2x 6+2-pin connector output?

I did a significant amount of research on my own, as well as learning a lot about basic electrical concepts to figure out the answer to my question. First off, I am not an electrician and my answer here certainly doesn't meet any professional or legal standards.

This question can be broadly broken down into 3 parts: the power source (PSU), the power transfer (the wires connecting the PSU and the GPU), and the GPU itself.

First, the PSU. The ATX standard determines a lot of the numbers thrown around here: 6 pins being restricted to 75W, and 8 pins being restricted to 150W. These standards are reasonable, safe, and followed by virtually all consumer desktop hardware. However, they are only standards, and electricity listens to physics, not standards. All power in the computer comes filtered through the PSU, subdivided into different voltages for various pieces of hardware. Broadly speaking, these voltages are divided into "rails" - so everything of a single voltage (12V, 5V, 3.3V, etc.) comes out of the same power source*. Each of these rails is limited to a certain share of the total power.

The relevant rail for this question is the 12V rail, where by far most of the power is drawn from in a modern desktop. PSUs differ in their specifications, but in my example 850W PSU the 12V rail can output the full 850W**, further limited by a max amperage of ~70A. Basic electrical formulas indicate this makes sense: Watts = Volts x Amps. This tells us that the 12V rail - where the 8 pin connectors on the PSU connect to - can supply plenty of power, far more than the GPU in question needs.

The power must still be carried from the PSU to the GPU, and it's here where the ATX standards are the most restrictive, and for good reason: the PSU supplies power and the GPU will take as much power as it needs, but neither does so with much regard to what's between them. If the PSU can't supply enough power, it will shut itself off*. If the GPU can't get enough power, it will shut itself off. If the wires get too much power, they will melt - or worse, catch fire. Accordingly, the ATX standards set the safety margins (6 pins for 75W, 8 pins for 150W) far below the point where this will occur. But to further confound the issue, both 6 pin and 8 pin connectors have the same number of current-carrying wires in them: 3. While there's some debate as to the purposes of the extra pins, the 3 hot wires within both cables indicate that both can carry the same amount of current. This is why a 6+2 pin is effectively equivalent to an 8 pin cable - there's no increased electrical risk. At the end of the day, the only real difference between the 6 and 8 pin cables in general is to inform whatever is using them of how much current is intended to be provided. It does nothing to stop anything from drawing more.

This brings us to our hypothetical GPU - it asks for three 8 pin connectors. Assuming it is responsible (I have not tested it), it will only function if it it senses three 8 pin connectors plugged into it. At that point, it assumes it can safely draw all the power that the ATX spec says it can: 450W (ignoring the PCI slot itself). Importantly, it will evenly draw this power across all 9 (3 for each 8-pin) 12V wires connected to it. This is pretty straightforward and safe - but alas, we have our goofy 8 pin to 2x 6+2 pin cable. If you've been following along, you will realize that the GPU will still pull all of the power it needs across the wires - because it assumes everything else is following ATX spec. It detects 3 8-pins connected to it (6+2 pins, but again they're the same) and will try to pull the 350W it needs across the 9 12V wires. (Here we are using the max estimated TDP of the theoretical graphics card).

But there are not 9, there are 6.

Regardless of however the 8-pin to dual 6+2-pin cables split themselves, at one point during the transfer they are limited to a single 8-pin cable. The PSU doesn't care - the power is coming off the same 12V rail no matter how many cables it's traveling through. The GPU doesn't care - it's getting its power one way or another. But remember the detail about the wires catching fire? It's a risk, and one we need to be wary of. The safe ATX spec says we can do 450W across 9 wires: that's 50W per wire. More usefully, it's ~4.2 amps per wire (50W / 12V = 4.17A). Our scenario? 350W over 6 wires: ~4.9A per wire. So if we connected the 3 8-pin requirement over only 2 8-pin cables (2 8-pin to dual 6+2-pins), we are breaking the ATX spec.

Through online calculators and speaking with people more familiar with electricity than myself, this does not appear to be a fire hazard: the 18 gauge wiring that is typically the standard for these cables isn't a fire hazard until around 15-20A per wire (over a 1 to 2-foot run). There's lots of variables in this calculation though, so it's best to stick to the ATX standard and never run splitters like the 8-pin to dual 6+2 pin.

Notes:

{1} Some PSUs subdivide voltages into 1 or more rails that split the available total power (wattage) between them.

{2} Realistically a PSU would shut itself down long before this, as the other rails are also drawing power and the maximum output of the entire power supply is also 850W.

{3} Good power supplies have safety features to detect unsafe power variations. Not all do, however.