How Goku knew he surpassed Vegeta

[Piramid]

In the Cell Saga, how did Goku know that Vegeta and Trunks still had levels to surpass, and how was he so sure that he and Gohan had already surpassed them?

When Goku and Gohan came out of the Hyperbolic Time Chamber, Trunks explains everything that happened while they were inside. This is when Goku discovers that Vegeta and Trunks used SSJ Grades 2 and 3, and he deduces that they hadn’t tried the same approach as him and Gohan—getting used to the standard SSJ form.

But if we think about it, this alone isn’t enough to be sure that he and Gohan were stronger, because in theory, their SSJ form has a lower multiplier than SSJ Grade 2. So, unless Goku somehow assumes that their base forms are way stronger than Vegeta’s and Trunks’, it doesn’t make sense for him to be so sure they had surpassed them.

Unless Goku and Gohan’s SSJ form has a higher multiplier than SSJ Grade 2.

[LightBing]

The point of the arc was surpassing SSJ. Trunks and Vegeta failed in the task, while Goku choose the correct option. From that he could easily ascertain his superiority.

Base's effect should be limited. As we know, Goku never grew enough to face Freeza in base.
The perspective of the characters, when choosing to surpass SSJ instead of growing in base which has multiplicative effects, points in that direction as well.

Another point is, SSJ grade 2 and 3 even if stronger are unbalanced. I wouldn't be surprised that Trunks and Vegeta could output more power than Goku in Grade 3 but that's useless if your other stats drop.

To answer your question, if SSJ then had an higher multiplier I'll say yes and no.

Yes, because obviously Goku had to grow stronger and something had to change.
No, because simply having an higher multiplier seems opposite to the point of the story, grade 2 and 3 are already the higher multiplier wrong transformations.

My conclusion is that SSJ become like a new base. The way Piccolo and Vegeta describe it is what makes most sense.

Basically (Base x 50) + mastery would be the formula.
Some random numbers:

Goku when he returned to Earth:

100 x 50 = 5000 + 500 for a power level of 5500.

After RoSaT.

170 x 50 = 8500 + 9000 for a power level of 17500.

Of course you could reach the same result by increasing the multiplier. Personally it seems too rigid and like I said before feels opposite of Toriyama's intention. After all, at the end surpassing SSJ was the wrong path.

[Piramid]

I see Grade 1 and 2 as corrupted forms of SSJ1, due to not mastering the transformation. When you master the transformation, you're able to access more power without corrupting it.

Moreover, if we assume that Goku has the same multiplier in Namek as in the Cell Games, that means he didn't need to surpass the power limit of Super Saiyan, something that was always his goal.

It actually explains what happens better without the need for Goku and Gohan to become several times more powerful than Vegeta and Trunks in base form, and only by spending less than a day in the Hyperbolic Time Chamber, while Vegeta and Trunks spent two full days.

In addition, we have Goku’s confirmation when he says that training in the Room only results in torture, not actual training. This means that Goku didn’t use the Room to become more powerful through mere training but to properly improve a transformation, unlike Vegeta.

[PerhapsTheOtherOne]

I've always taken this part of the series as Goku and Gohan just becoming stronger as a whole by focusing on utilizing Super Saiyan with complete control, rather than forcibly empowering it.

Like the difference between working out your upper body exclusively vs. doing all-around muscular workouts.

That, and Goku and Gohan explicitly trained together rather than separately like Future Trunks and Vegeta did. This helped magnify their training gains, as they could push each other and help grow stronger more efficiently by having a partner to spar with and actually practise what they were trying to accomplish in action.

It's continuing the tradition of working more efficiently being a better method than working harder. By training together and focusing on mastering their usage of Super Saiyan, they became far stronger simply due to better acclimating to controlling their immense power in a focused manner.

[Godo]

Goku had knowledge of SSJ grade 2 and grade 3's power first hand and knew that his mastered form was stronger.
Just by Vegeta and Trunks being in base should have been enough to know that they hadn't reached the same level as Goku and Gohan.

[Piramid]

I've always taken this part of the series as Goku and Gohan just becoming stronger as a whole by focusing on utilizing Super Saiyan with complete control, rather than forcibly empowering it.

Like the difference between working out your upper body exclusively vs. doing all-around muscular workouts.

That, and Goku and Gohan explicitly trained together rather than separately like Future Trunks and Vegeta did. This helped magnify their training gains, as they could push each other and help grow stronger more efficiently by having a partner to spar with and actually practise what they were trying to accomplish in action.

It's continuing the tradition of working more efficiently being a better method than working harder. By training together and focusing on mastering their usage of Super Saiyan, they became far stronger simply due to better acclimating to controlling their immense power in a focused manner.

Do you think then that Grade 2 decreases speed compared to the regular SSJ?

If not, then why didn’t Goku, Vegeta, and Trunks use Grade 2 during the Cell Games?

Goku had knowledge of SSJ grade 2 and grade 3's power first hand and knew that his mastered form was stronger.
Just by Vegeta and Trunks being in base should have been enough to know that they hadn't reached the same level as Goku and Gohan.

So you believe that Grade 4 exists.

[GatoF]

Grade 2 doesn't decrease speed like grade 3 but still has a high energy consumption and stamina drain and these are minimal in full power ssj (or grade 4)

[Hugo Boss]

He just knows it. It’s not exactly explained how he got that much accuracy in reading his opponent’s strength, but this is probably attributed to his special training, as he focuses mostly on the spiritual aspect of fighting improvement. And he knows Vegeta isn’t the kind that goes deep in that department. Goku even knows he is not as strong as Cell, despite not even having a clue of his true strength.

In resume, this version of Goku knew stuff that would make (Super) Goku look bad in comparison.

[Godo]

I've always taken this part of the series as Goku and Gohan just becoming stronger as a whole by focusing on utilizing Super Saiyan with complete control, rather than forcibly empowering it.

Like the difference between working out your upper body exclusively vs. doing all-around muscular workouts.

That, and Goku and Gohan explicitly trained together rather than separately like Future Trunks and Vegeta did. This helped magnify their training gains, as they could push each other and help grow stronger more efficiently by having a partner to spar with and actually practise what they were trying to accomplish in action.

It's continuing the tradition of working more efficiently being a better method than working harder. By training together and focusing on mastering their usage of Super Saiyan, they became far stronger simply due to better acclimating to controlling their immense power in a focused manner.

Do you think then that Grade 2 decreases speed compared to the regular SSJ?

If not, then why didn’t Goku, Vegeta, and Trunks use Grade 2 during the Cell Games?

Goku had knowledge of SSJ grade 2 and grade 3's power first hand and knew that his mastered form was stronger.
Just by Vegeta and Trunks being in base should have been enough to know that they hadn't reached the same level as Goku and Gohan.

So you believe that Grade 4 exists.

Nope, never mentioned "Grade 4", did I? I even mentioned "mastered form", as in mastered SSJ, which is good old SSJ without the stamina drawbacks.