TickBitmap: Finding the Next Tick

Key takeaway: The swap loop needs "next initialized tick from here, in this direction" to be cheap, so V3 packs tick initialization state into a bitmap: one bit per usable tick (compressed = tick / tickSpacing, floored toward negative infinity for negative ticks), 256 bits per storage word, keyed by wordPos = compressed >> 8 with bitPos = compressed % 256. flipTick toggles a bit with XOR whenever a tick's liquidityGross crosses zero, and nextInitializedTickWithinOneWord finds the nearest set bit in one word with a single storage read: build a mask of bits at-or-below (searching down) or strictly-above (searching up) the current position, AND it with the word, then jump straight to the most/least significant set bit.

What You Are Building

The TickBitmap library: position, flipTick, and nextInitializedTickWithinOneWord. BitMath (branchless-ish MSB/LSB scans) is provided. This is the data structure that makes V3 swaps affordable: without it, every swap step would scan tick space one tick at a time; 60,000 ticks of empty space between two positions would cost 60,000 storage reads. With it, the swap loop covers 256 usable ticks per storage read.