Fee calculation vs invariant in Uniswap V3

In v3, fee tokens are held/tracked separately from the liquidity pool. This means that, unlike in v2, the invariant product does not change after a trade, as we see below.

Consider a specific pool with a given range-position, and suppose it has virtual reserves

$x,y$ corresponding to assets X, Y respectively. Suppose these reserves initially satisfy the product-invariant

$xy=L^2$. When a trader wants to put in

$\mathrm{\Delta }x$ amount of X tokens and receive

$\mathrm{\Delta }y$ Y tokens in exchange, here is how we can calculate

$\mathrm{\Delta }y$ they will receive, taking fees into account. Suppose the fee rate is

$1-\gamma$, e.g.

$1-\gamma =0.003$.

1. Reduce the amount of X tokens converted, by the fee, so effectively they will convert
   $\gamma \mathrm{\Delta }x$ of X
2. Plug this reduced amount into the invariant formula to get the corresponding
   $\mathrm{\Delta }y$:
   
   $$(x+\gamma \mathrm{\Delta }x)(y-\mathrm{\Delta }y)=L^2$$
3. Track the fee paid as
   $(1-\gamma )\mathrm{\Delta }x$ amount of X tokens

Note that due to the non-fungibility of liquidity pools in v3, the fees are kept separate from the pool. Here are some relevant highlights from the v3 whitepaper:

As a consequence,

the exact same invariant continues to hold for the virtual reserves in a given liquidity pool.

This is in contrast to v2, where the

$(1-\gamma )\mathrm{\Delta }x$ is added back to the X reserves, so the new level of X tokens is

$x+\gamma \mathrm{\Delta }x+(1-\gamma )\mathrm{\Delta }x=x+\mathrm{\Delta }x$. But since

$\mathrm{\Delta }y$ was calculated to satisfy the above equation, the new product of reserves

$(x+\mathrm{\Delta }x)(y-\mathrm{\Delta }y)$ will exceed

$L^2$ slightly. This is in fact what the v1 whitepaper says (the fee mechanism is identical to v2):