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Scoring the fit with Kullback–Leibler divergence, $D_{\mathrm{KL}}(p \,\|\, q)$ = $\sum_i$ $p_i$ $\log_2(p_i/q_i)$ , from the observed August distribution to each candidate: the solved game scores 10.15 bits, the uniform null scores 0.045 bits, and July’s raw persistence scores 0.00068 bits. On a chi-square test against the eight species’ actual August battle counts (summing to 2,137,239 team-appearances, treating each candidate distribution as the expected shares), the solved game scores approximately 2.3$\times$10^{13} — a figure that large only because the model assigns four real, frequently-played species an expected share of essentially zero, and squaring a large real count against a near-zero…

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