Where can I find 2 of the steps/proofs described in Dan Boneh's video on PLONK in the PLONK Paper? The 2 don't seem to match

This is Dan Boneh's video on PLONK - https://www.youtube.com/watch?v=vxyoPM2m7Yg

I went through the video multiple times & also tried to go through the original PLONK paper - https://eprint.iacr.org/2019/953.pdf

Boneh's explanation of PLONK involves the steps

1) Boneh consider's the trace of the equation as the inputs (public & private) & the gates. Let's say there are 3 gates & 3 inputs, each gate can be considered as

$Left$ $o$ $Right$ = $Output$

So 3 gates become 9 points. Adding to this, the 3 points from the 3 inputs. We get a total of 12 points.

So he interpolates a polynomial $P(X)$ of degree 11 using the 12 points.

PLONK Paper:. I don't see the PLONK paper doing this step at all.

2) Boneh proves that this polynomial is the input correctly by interpolating a polynomial $v(X)$ for the 3 inputs & verifies if $P = v$ at the 3 points.

PLONK Paper: Since the PLONK Paper never creates $P(X)$, it doesn't do this step either.

3) Boneh proves that every gate is evaluated correctly

He creates a selector Polynomial $S(x)$ which represents whether a gate is an addition or a multiplication gate & then checks whether the below is true

$S(y)\cdot [P(y) + P(\omega Y)] + (1-S(y))\cdot P(y)\cdot P(\omega Y) = P(\omega^2 Y)$

PLONK Paper: The PLONK Paper does do this though they expresses this polynomial in a slightly different way

$q_L \cdot f_L + q_R \cdot f_R + q_O \cdot f_O + q_M \cdot f_L \cdot f_R + q_C = 0$

Here $q_L$, $q_R$ & $q_O$ act like the selector polynomials & I believe do the same thing which Boneh does with his $S(x)$ selector polynomial.

So I believe this step would match in both cases.

4) Boneh proves that the wiring has been done correctly using a prescribed permuation check.

PLONK Paper: This is also done in the PLONK Paper, they call it "copy constraints" checking using a permutation done by grand product argument.

So this also matches.

So I am confused as to why the first 2 steps aren't covered in the PLONK Paper. And I am just unable to recognize them in the paper? The PAPER covers 10x as much as Boneh's video - in terms of details, proofs, optimisations etc so I may be not recognising it in the paper - which is what i want to find out.

I am looking to find out if those steps get implicitly performed in some other step in the paper - if so, which steps are those?

Or if the same check/purpose is done in a different way (like for e.g. the selector polynomial looks quite different between the two) - if so, I want to know which step is it in the paper?

It is very common for a talk/presentation to not match all the details of a paper. There can be many valid reasons for this.

1. Illustrating crucial concepts without too much detail that won't fit into a regular length talk/lecture
2. There may be multiple versions of the same paper, especially in applied fields, as details get updated/improved

Maybe try to see if an earlier preprint version of the paper is available. If anyone else here has noticed this they may respond.