Finite-time consensus in a compromise process

The paper proves: (i) finite-time consensus occurs with probability one iff N is a power of two (for non-powers, impossibility holds under a general-position hypothesis), and (ii) when N = 2^k, the exact minimal number of averaging steps is k·2^{k−1}. These are stated as Theorem 1 and Theorem 2 and proved via a constructive hypercube schedule (sufficiency) and an integer-coefficient/general-position argument (necessity and the sharp lower bound) . The candidate solution reaches the same conclusions. Its sufficiency argument uses the same hypercube pairing idea, and its necessity argument matches the paper’s integer-clearing/general-position approach. For the minimal-steps lower bound, the model uses an information-theoretic (entropy/Jensen–Shannon) potential to show a per-step increase of at most 2 bits in total row entropy, yielding the Nk/2 bound that the hypercube schedule attains; the paper instead uses a participation-count contradiction to derive the same bound . Thus results coincide; the proofs differ only in the lower-bound method.