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On MEZO veBTC boosting

The formula as given in the whitepaper reads:

votingPower = \|veBTC\|\cdot min(5, 1 + BOOST)

Where

BOOST = 4\frac{\|veBTC\|_{total}}{\|veBTC\|}\frac{\|veMEZO\|}{\|veMEZO_{total}\|}

Where $\|\cdot\|$ is taken to mean the size of a user's position, and $\|\cdot\|_{total}$ is taken to mean the size of the totality.

It immediately follows that a quantity of veBTC is strictly necessary to obtain franchise, and that the additive boost in the unsaturated regime equals $4\|veBTC\|_{total}\frac{\|veMEZO\|}{\|veMEZO\|_{total}}$ . If one were to rearrange the terms, he would have arrived at $4\|veMEZO\|\frac{|veBTC\|_{total}}{\|veMEZO\|_{total}}$ . Since the second multiplicand serves to proportionally rescale veMEZO into veBTC, it follows that the boost grows until the user possesses equal shares (of totality) of veMEZO and veBTC; when the share of veMEZO then exceeds that of veBTC, the boost saturates at 5. Mezo arrive at the same conclusion in their whitepaper's section 3.6.

On MEZO rebases

Each epoch, a quantity $E_t$ of MEZO is minted; this quantity is split between Rebases and Rewards. Rewards are not impacted by MEZO OoVs directly and are therefore out of scope of this token page. Define now the stake rate (the notation differs from that employed by Mezo, who use stake rate's additive complement, in the interest of preserving internal consistency between our descriptions of different veTokenomics; cf. YB):

s = \frac{\|veMEZO\|_{total}}{\|MEZO\|_{total}}

Then the Rebase/Reward split is calculated thus:

E^{rebase}_t = 0.5E_t(1-s)^2

Observe that the rebase portion grows quadratically with the portion of unlocked MEZO. It is easy to show by explicitly writing out

E^{rebase}_t = 0.5E_t(1 - \frac{\|veMEZO\|_{total}}{\|MEZO\|_{total}})^2

If one then differentiates w.r.t the veMEZO supply, one obtains that the derivative is of the form

E_t\frac{\|veMEZO\|_{total} - \|MEZO\|_{total}}{\|MEZO\|_{total}^2}

Evidently, this derivative is nonpositive for all admissible values of $\|veMEZO\|_{total}$ , with its only zero being reached when all the supply is staked. This implies that the system is unstable in the sense of lacking a global stable equilibrium, and it therefore can be expected that it will engender a dynamic pattern of MEZO locking/unlocking by profit-maximising holders (again, cf. YB's stake rate dynamics).

Once the rebase rate is computed, the distribution between veMEZO holders is proportional.