TrueSurfing is not as good as GotoSurfing, period.

Here's the prove in my mind previously, refined with Claude

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Proof: Equivalence of Surfing Algorithms

Assumptions

1. All waves W₁, W₂, ..., Wₙ are known in advance

2. Robot uses orbital movement (heading determined by position relative to opponent)

3. Danger is a function only of (position, velocity) at wave intersection

Definitions

State: S = (p, v) where p is position, v is velocity vector

Reachable set: R(S₀, t) = all states reachable from S₀ in time t under physics constraints

Intersection state set: I(S₀, Wᵢ) = all possible (position, velocity) states at which Wᵢ can intersect the robot, starting from S₀

Lemma: Reachable State Coverage

All three algorithms can evaluate every state in I(S₀, Wᵢ).

Proof:

• True Surfing: Direction choice determines trajectory → intersection state. Evaluates both directions.
• GoTo Surfing: Destination choice determines path → intersection state. Sufficient destinations cover I(S₀, Wᵢ).
• Path Surfing: Action sequences cover all trajectories → all intersection states.

Since I(S₀, Wᵢ) is determined by physics, not algorithm, all three cover the same set. ∎

Theorem

All three algorithms find the same optimal intersection states (S₁*, ..., Sₙ*) minimizing Σᵢ D(Wᵢ, Sᵢ).

Proof: By Lemma, all evaluate the same state space. Optimal selection over identical sets yields identical result. ∎