integrated amp with phone for graham ls6 under $4000 new

What do you mean “the ones I’ve seen?”  How long do you need?  Personally, I would just start with Belden 12 from Blue Jeans cables and listen for a few months until everything breaks in. Then you can try something else to see if you like it better. 

I’ve used those speakers with several different integrated and they always sounded good, but they will step up their game when you move up the amp/source chain. I don’t think you need to agonize over your choices, but if you like classic rock, you might want the amp with a little more power. I really liked the Heed Obelisk myself. 50 watts and minimalist, but sounds really good.

I was thinking about this some more and if you’re in the US, I would recommend you contact Gene Rubin. He carries Heed, Belles, LFD and others and he can really help you set up something good with the Chartwells. He knows what works and he is excellent to deal with.  

I recommend Belden 12 gauge from Blue Jeans with locking bananas. If you’re using 6 feet now, order 8 feet to get a little positioning flexibility. Later on you can fool around with more expensive cables. 

Morrow speaker cables are good, but I would highly recommend the break in service. I’m not sure how many days he offers now. I would take at least 5 days. Whatever you buy, there will always be something else and another opinion. Personally, right now I’m listening to a pair of Worlds Best Canare Bi-wire cables I bought on Amazon for $120 and they sound damn good, even compared to cables costing 10x as much. But Morrows are fine.