Interesting concept, but I don't see the relation between collatz conjecture and incremental mechanisms.
Currently, the only incremental mechanism in the game that makes use of the conjecture, is how many steps each number will take to go back to 1. This determines how many producer ticks can be made per verifier tick. This is just 'idle clockwork', not 'collatz clockwork'.
I have some ideas for improvement:
1. Concept:
This is just a game, and players' computers are nowhere near enough to find nontrivial results to collatz conjecture.
Even if a number diverges, the player's computer has no way to determine it's actually a counterexample. This process can't prove the conjecture either.
Limiting the money gain by the player's computer power will also severely limit the incremental growth of the game, and therefore limiting game design.
So you can forget about proving the lower bound, and instead make use of the sequence produced from the numbers. This will allow for more interesting incremental mechanisms, and stays close to the conjecture.
2. 'Level' skips:
Instead of trying to check every number to 1, you can instead skip some numbers that will trivially reduce to some previously checked numbers, like even numbers.
3. Sequence based mechanisms:
It's very simple to design, like 'multiply money gained based on current sequence length', or even 'multiply money by log of highest number reached in the current sequence'. These are much more interesting and also more connected to the conjecture.
These upgrades will benefit from finding a longer sequence that oscillates wildly before eventually converges (or maybe doesn't converge within the available steps). The level skip could completely discard the current verification scheme, and instead focus on searching numbers that leads to such interesting sequences.