Algorithmic-Trajectory Cuts and the Causal Bottleneck: Must Every Exact Solver Expose a Distinguishable Bottleneck?

Round 2's residual distinguishability holds only in a “read-once, never look back” model. This round hands the power to choose the cut back to the algorithm itself, observing its actual run trajectory C₀→C₁→⋯→C_T directly. Team Equal immediately counterpunches: a general Turing machine can re-read its input, so even if two inputs reach the same configuration at some moment, their futures can still diverge because the read-only input differs; moreover, the input has only n bits and the decision output only 1 bit, so there is no simple information-conservation law of the form “exponentially many candidates → exponentially many bits must be transmitted.” This is the round's most important reversal: the real difficulty is not “how much information is needed,” but “what structural transformation this information must undergo to yield the correct answer.” Team Not-Equal therefore proposes causal reconstruction complexity (CRC) as a new object of study, while explicitly conceding: defining CRC directly as the minimum solving time would circle straight back to the original problem itself.