Speed of groove.

For sake of simplicity, let's consider the vertical movement only and 1kHz sine wave shaped groove for a constant amplitude of the signal. And let's take only the positive semi-period, thus a nice rounded "bump".
At the outer edge of the record, the bump will be "elongated" whereas towards the center of the record it will be "shortened". But in both cases the time the stylus "climbs" from the bottom to the peak of the bump stays the same. And given the fact the the bump has the same height at its peak in both cases, so does the "climbing speed".
This is the speed I'm referring to and I still fail to understand how it could be correlated to the linear velocity of the spinning record.
The linear velocity was "abstracted away" by the "length of the bump".
I'm only trying to understand here, by no means my knowledge of electronics could even come close to Al's...

You can us the math to help understand it. Take a 1kHz waveform again for an example. At the outer groove, the velocity is 20.1"/s. So 1000 (cycles/sec)/20.1 (inches/sec)= 49.8 cycles/inch. Now at the inner groove the velocity is 8.7"/s so the same 1kHz waveform is generated by 114.5 cycles/inch. The amplitude of the cycles is the same at the two locations of the record for the same volume level because the excursion of the stylus must be the same. That makes the inner grooves more demanding because the stylus has to track these 2.3 times more dense undulations in the grooves.

The linear velocity was "abstracted away" by the "length of the bump".

Precisely! That's all we are saying. As you correctly put it, "at the outer edge of the record, the bump will be 'elongated' whereas towards the center of the record it will be 'shortened.'" That shortening or lengthening results in the velocity of the stylus deflection being the same for a given signal at different points on the record, despite the fact that tangential ("linear") velocity is not the same.

Regards,
-- Al

Thanks for joining this discussion!
Would you consider a decreased performance on the inner groves normal in some of records especially fully "packed"?

Precisely! That's all we are saying

Well, I understood this all along, but the original poster was talking about correlating the cartridge's output voltage with the linear velocity of the record.
This is the correlation I said I didn't understand and I don't believe exists.
Everything else is clear to me. Maybe I misunderstood the post itself, wouldn't be the first time... ;-)
There's variation in sound quality between outer tracks and inner ones, due to the in formation being more densely packed together on the inner grooves, but not "output level" differences. Nobody has to turn the volume up as the inner tracks are played.