Not in general. If every route to the exit crosses a wall, the two answers differ. A possible state is (row, col, tokens_left). The test is not “do these journeys look similar?” It is “can anything about their different histories change their legal continuations or remaining answer?”

Imagine handing the unfinished problem to a friend. Give them enough information to finish correctly, without making them replay your whole journey. That handoff is your state.

For a fixed board with one removable wall:

def can_enter(cell, tokens_left):
    return cell == "." or tokens_left > 0

assert can_enter("#", 1)
assert not can_enter("#", 0)

If you are minimizing total cost, distinguish the best remaining cost from the cost already paid. Two histories may share a remaining-cost calculation even if their accumulated costs differ. The caller adds its own past cost.

Conversely, if the rule forbids revisiting any previously visited cell, that history affects future legality. Coordinates and token count alone may no longer suffice; a visited set can be part of the state, often making the problem much larger.

Recognition test: deliberately construct two histories that your proposed key merges. If their remaining answers can differ, your key has forgotten something important. If a coordinate is completely determined by the others, it may be redundant.

Cost connection: adding a budget 0..K can multiply the state count by K + 1. More information fixes correctness but may increase work.

Analogy limit: “the future” means the mathematical subproblem under the stated rules, not every real-world consequence.

If the state is measured just after entering and resetting, the token value is always known and need not be stored. A state measured before entry may still need it to decide whether entry is legal.