In the town of Trebon Kvothe hisself links all the fire in the town to one fire in his hand and drowns that one fire as a slapdash heatsink.

Nope - he grabs a roof tile, links it to all the other roof tiles (which are on fire) and hammers it under the waterline in the water tower. This dumps all the heat from the burning roofs into the water tower, and dampens the flames.

That heatsink is 'roof tiles' to 'roof tile', and then cooling the roof tile. It is not flame to flame.

An unlit box is full of shadow, it matters not what else is in it.

Uh, sure, I guess, but again, shadow isn't a thing, it's the absence of light.

Sympathy works by your alar deciding two unconnected bodys are actually connected magically. You have to supply the effort to enact change between them.

Yes, exactly - you connect two unconnected bodies, and through your will, they become connected. The 'supplied effort' you describe is the energy being transferred, i.e., the heat sources or movement sources.

I meant to imply your understaning of the universes parameters was incomplete.

I disagree.

I'm not convinced. Just becasue they aren't the same doesn't mean they cannot act in unison. I would actually speculate that Naming and Alchemy are more similar in that they are deduced from the natural world of Temerant, but those would be different to Sympathy and Faen Magic because those two are artificial and may be more closely related than the naturaly occuring magic.

Are you implying they're the same magics? Rothfuss has stated they are not. I wager there are similarities or overlaps, but Sympathy is not Naming is not Fae Magic is not Alchemy is not Yllish Knots. I would agree that Naming and Alchemy have more in common - indeed, they appear to be older magics, while Sympathy appears to be more modern.

The point here is that using Fae Magic to craft a cloak of shadow does not mean that a Sympathist could also link to shadow, for the same reason that a Namer knowing the Name of Fire and creating flame from nothing doesn't mean that a sympathist could point at a candle, and without anything else going on, light the candle.