Debugging an FD problem through three visuals The front derailleur cable is as taut



as I can get it.

To attach the cable with as much tension, it's first necessary to understand the inside of the barrel adjuster.

And the H limit screw on a Sora FD is so undone (top red arrow)



it does not even touch its footing (bottom red arrow).  (The yellow ball is removed.)

Yet when the chain is on the larger chainring and on any of the three smallest cogs, the chain rubs (red arrow)



against the FD cage on the outside.

What else should I try?

If at this point I insert a tougher rubber ball to tighten the FD cable even further, the cage does get off the chain. Is that an indication that my best effort to pull the cable hard before tightening it was not good enough, and I should try pulling harder?

Components: Sora Sora R3000 front derailleur and ST-R3000-L shifter.

Update (question re "judgment" tool, following Nathan Knutson's answer):

The cable goes from one geometric point (bottom red arrow) on the bottom bracket to another geometric point (top red arrow) on the derailleur. There is no tweaking possible to the location of these two geometric points. What is, then, the purpose of the judgment tool?
I'm going to assume you've got FD-R3000 and are pairing with a compatible road double shifter. 

The anchor position is adjustible. Follow the procedure in the Shimano dealer manual for your model. It is available at si.shimano.com. There are only two positions so if you don't have the special angle-finding tool you can use process of elimination. Failure to use the right position can cause what you're experiencing. The two different positions slightly alter the lateral point on the arm where the cable is anchored, which tunes the total throw.

All the "long-armed" Shimano road front derailleurs have the quirk of needing a ton of cable tension. The axiom with them is that once the right anchor position is chosen and assuming compatible parts, setup problems are almost always due to inadequate tension. They don't conform to generic instructions for FD setup. A good procedure is start with the shifter and barrel adjuster in the slackest position, then use pliers, a fourth hand tool, or the inner limit to pre-actuate the heck out of the FD as you pull the cable tight while anchoring the cable, then add some more tension as needed with the barrel adjuster after that.

You usually must accept essentially zero clearance between the chain and inner cage in the low gear with these derailleurs. 

Another compounding factor to difficulty adjusting the "long arm" FDs is that the cable is anchored in a rather bent position with them, which can make it difficult to simply loosen the anchor bolt and pull out more slack - the memory of the material makes it want to return to how it was. You can try making a marking on the cable where it was exiting the anchor previously in order to be more sure you've actually added tension.