Breadth-First Search

Breadth-First Search is the simplest correct answer to find a shortest route. It is Uninformed: it searches the full (x, y, tick) graph systematically, but it has no hint about which direction the exit lies, and no cost beyond one tick per step.

Uninformed does not mean dumb. It means the expansion order does not peek at the goal. Breadth-First Search expands every state one tick away, then two ticks away, and so on. That is exactly why the first time it reaches the exit, the path has the fewest ticks among legal states.

Planning happens before the walk. Put the entrance at tick 0 in a first-in-first-out queue. Pop the oldest state. If it is the exit, reconstruct the path and stop. For each orthogonal neighbour at tick + 1, skip walls, skip visited space-time keys, and skip any "x,y,phase" in the learned death set. Push surviving neighbours onto the queue. When planning finishes, walk the reconstructed path against real vision cones.

Soft patrol proximity is not part of the priority in Breadth-First Search. If the shortest legal route brushes a guard lane, Breadth-First Search takes it.

A catch at tick T adds every cell in that guard's cone as "x,y,T % period". The next Breadth-First Search run treats those as walls in time. Same square at a different phase can still be used. That is how a later attempt can cross a kill tile safely: it arrives when the calendar says the slot is free.

Breadth-First Search does not store confirmed-safe tiles. Anything not blacklisted is assumed walkable until a later death proves otherwise.