Edit based on Tim B's new hint:
The clues (including their answers) are:
1 Escorted by --> WITH
2 Grips --> HOLDS
3 Italian sequencer --> FIBONACCI
4a Decay --> ROT
4d Input --> READ
5 Not just a --> THE
6 Cipher reversed --> DECODE
7 DNA scrambled --> AND
8 Piano lever --> KEY
9 On a slant --> DIAGONALLY
The filled out puzzle then looks like:
And the message reads (based on the clue 6 1 4a 7 4d 9, 3 2 5 8):
Decode with rot and read diagonally, Fibonacci holds the key
The diagonals of the grids are:
So we have to:
Decode these diagonal parts (xifsf, ku, wkh, ywzym) with ROT(X), X being the corresponding numbers that are found in the grid (as found by Tim earlier on).
And it leads to:
xifsf (ROT 1) --> where
ku (ROT 2) --> is
wkh (ROT 3) --> the
ywzym (ROT 5) --> truth
So the solution is: Where is the truth
The latest hint suggests to use alphametics to solve the puzzle.
Since we're given the clue 3+4+5=6, we probably have to sum, the words found in the corresponding grids.
WHERE+IS+THE=TRUTH, which has the following solution: 69404 + 81 + 794 = 70279. So the value we place on truth is 70279.
Previous update: (going in the wrong direction)
Apparently this is not yet the solution to the puzzle, nor is the number 5 or 6 (the rotation used for the last (truth-)grid/the number found written in the last grid).
I'm not sure whether the hint 3+4+5=6 still needs to be used at this point..
Another way to go could be to start looking for the truth, since that's what the previous part actually suggests. Obviously the word truth was found in the fourth and last grid, which also has the word "six" in it, but we already discarded that possibility. Maybe we should go by the saying "the truth lies somewhere in the middle", leading us to the centre part of the grids/crossword, but that's just guessing and it has no logical basis.
The latest hint seems to suggest looking at the crossings of the crossword puzzle, which would be the letters H R T D O E A D I Y, but that doesn't really help either. Maybe we should focus just on the only real crossroads, the one where 3d and 6a cross around the letter O, but I have no idea what to do with that..
N.B., something that does add up nicely:
F(6)+F(7)+F(8) = 8 + 13 + 21 = 42 :P
I haven't made any real progress on this one, but I'll give you my two cents. Maybe someone else can make sense of it.
First of all:
The question is about "extracting the value" from the puzzle, implying that we're searching for a number here.
The clue 3+4+5=6 seems to be supporting that view. I've tried plugging in values into the grid to match this equation, but there's no solution, so that's not the way to go.
seventeen + two + one = twenty is the closest I could get, but seventeen and twenty don't line up correctly.
Since OP suggests to start with the image, I also tried changing the link (https://i.stack.imgur.com/K6ckx.png) to https://i.stack.imgur.com/K345ckx.png, but no luck there.
Then there's the numbers Tim found in the grids:
Maybe we should add those grids somehow, but I've tried this in lots of different ways and it doesn't work either. OP also changed one of the grids at some point to remove a red herring, so most likely the letters in these grids don't bear any further significance, other than the numbers inside of it.
About the five clues:
Are they even the clues??
They are capitalized weirdly, is EG IDI NC DNAP O perhaps a message? It's an anagram of ENCODING PAIN, but now I'm probably just seeing stuff that isn't really there...
Could the Italian sequencer be Fibonacci?
Second hint tells us that the clues to the puzzle can be found "in the end". So maybe in final letters of any of the words?