# Are Uniswap V3 liquidity units fungible across different price ranges?

If I provide liquidity to a Uniswap V3 pool with a set fee, and I provide `x` amount of token0 and `y` amount of token1 within a certain price range, let's call it `[pa, pb]`, and I receive a value of `L` (`positions[id].liquidity`). Would it be possible to get the same value of `L` by providing smaller amounts of both token0 and token1 (lower capital value) to a narrower price range? This would mean that liquidity units are not fungible across different price ranges as the same value for `L` could be obtained by providing a smaller capital value over a narrower price range.

In Uniswap V2, all liquidity provider (LP) positions fall in the price range `[pa, pb] = [0, Infinity]`. The value for liquidity is fungible across all positions and is calculated as follows:
This means that the value of liquidity is directly proportional to the values of `x` and `y` tokens provided, and the only way to get the same exact value of liquidity is to provide an amount of underlying tokens that falls on the curve described by the formula above.
In Uniswap V3, LP positions are more fine-tuned and have an additional level of depth. Uniswap V3 introduces the concept of concentrated liquidity, where positions are not only described by the amount of liquidity provided but also the price range `[pa, pb]` over which that liquidity is provided. Liquidity is described by the following formulas:
By examining the Uniswap V3 formulas, we see that the value of liquidity is directly proportional to the values of `x` and `y` and inversely proportional to the width of the price range. If we calculate the amount of liquidity that results from providing an amount of tokens `x1` and `y1` across a price range `[pa, pb]`, it should be possible to get the same exact value of liquidity by providing a smaller amount of both tokens `x2` and `y2` across a narrower price range `[pc, pd]`.