Good question and the answers so far show how little folks understand about tyres.
Tyre grip is generated in two ways. One is pure mechanical 'interlocking' with the road surface. The second is at a molecular level - bonding to the road and having the bond broken.
Tyres are designed to work in a heat range. Within that range the compound softens to allow the rubber to 'flow' and mould to the road (mechanical grip) and the molecular action is enhanced.
If there is dust/pollen etc on the road the tyre happily sticks to the dust and relies solely on mechanical grip to get the job done resulting in a reduction of overall grip.
Water works against both mechanical and molecular grip
The rubber contains aromatics and overheating 'burns' excessive amounts of the solvents etc out. Evidence of this is the 'bluing' you see of a tyre that has had a beat up.
Leaving a tyre sitting for too long, esp exposed to UV evaporates all the goodies.
Green (new) rubber is in its optimum state. Each time the rubber heats and cools it hardens - hence the term 'heat cycles'
Tyres are load sensitive. The more load, the more grip.
Except the trend isn't linear. Double the load doesn't equal double the grip unfortunately and extra mass has to be accelerated and decelarated.
More grip = more heat and the total loading is the limitation on how soft the compound can be. The lighter the loading the softer it can be. Since water cools the rubber, rain tyres are much softer than dry.
So the short answer is no - if there were two riders of identical ability riding the same machine, the lighter one would have a performance advantage.
However, for all the theory to make a scrap of difference the riders would have to load and unload the tyres appropriately at the maximum availible traction (ie be a perfect rider)
For blokes at clubby level the truth is a blurred. New tyres feel great. Do they make you faster?
Often your lap time is dictated by your head not your arse.
Does the new tyre give you a very short peak and then back to same old same o?
In that case your bike settings/riding syle/technique etc aren't in sync and you are burning the tyre to get a very short lived gain.
The trick is going as fast as you can and keeping the machinery good
"I don't like it, and I'm sorry I ever had anything to do with it." -- Erwin Schrodinger talking about quantum mechanics.
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