I think they would rather continue pushing PS Now and selling remastered old titles than offer BC.

To my knowledge GPGPU compute, a completely new feature exclusive to current gen and modern PC gaming is by design allowing your GPU to handle some of the processing tasks normally reserved for a CPU. Sony went ahead and made some modifications to their pictairn based AMD graphics card which even improve upon this. You and the other guy are looking at the clock speed on the Cell, seeing it has a higher number and thinking its instantly impossible but theres more to it than that. While the Jaguar might have a lower clock speed, it still has modern day features and is far more efficient on a relative scale. If they were able to match the Cell performance with just the cpu performance on the ps4 then BC would seem far more possible. Truthfully though, this is entirely conjecture. Unfortunately no one on here has proven that I'm wrong. 

I really do think GPGPU would be key to PS3 emulation on PS4.

Given the current state of the PS4, Sony made a wise decision to NOT include backwards compatibility. As healthy as PS4 sales have proven right now, I don't think sales will have increased very much with backwards compatibility included. It's doing very well on its own by its own library of games. And the potential reality of PS4 users actually using backwards compatibility, I believe, is too small to justify Sony's efforts of including it and maintaining it through out the life span of the console. The far majority of users who buy a new console is to play those games for that console. And as inspiring as it is to have the convenience of playing an older game on a new console, it's very easy to prioritize the new games first while placing the older games to the back burner for who knows how long. I still have the original disc versions of PSOne games what were bought during the PSOne days that I have yet to play because I was distracted by the newer games of that generation. One still in shrink wrap.

No, at this point 99% of consumer do not care. There are almost no PS3/360 games on the horizon and there have hardly been any released in 2015. I can't imagine that PS4 sales in late 2015 are being limited by a lack of BC, people are just waiting for more games.

Fallout 4, Star Wars Battlefront and The fact that this years COD isn't a full release on last gen platforms will do signifcantly more for the PS4 then a mid gen BC update. It'd always be a good thing to have but from a marketing perspective its not worth the investment anymore. Would have been a huge edge if it happened in 2013/14 though

You don't understand how emulation works. 

Programs are usually built to utilize a specific processor type. The programs expect an IO request to occur within so many CPU cycles,  a certain memory address system, and specific drivers. 

The cell processor had a higher clock speed than the ps4, but worse multithreading. This causes asynchronous nightmare.  You will have threads finishing in unexpected orders...

Indeed. I'm looking forward to the day that I don't need to buy a console to be able to play the latest console games. Sony should be putting PS Now into all it's internet TVs and get deals with as many other TV brands as possible.

At this point, its completely unecessary. Would be a waste of resources.

Ummmm all 2014,2015 and 2016 Bravias, along with almost all 2015 and 2016 Samsungs come with PS Now.

BC is realy a minor feature that ahrdcore gamers (wich is a tiny fraction of the whole market of games) like to praise as the second comming of the messiah because we like to replay games we love over and over again. In reality it hardly even help console sales or affect the sales of new software its doesnt make a impact to be meaningfull either way.

Backward compatibility was never a system seller,PS Now would make more money for them