The common defense of high-entropy football is that direct play is the high-variance gamble that gives an outmatched team a chance to compete. This experiment tests that defense empirically. Each chain is run thousands of times, and the resulting distribution of expected goals per possession is plotted. The structured chain has both a higher mean and a lower variance than the direct chain. Low-entropy football is not the high-risk choice; it is the opposite. The statistical case for structure that the public discourse never makes is contained in the histogram below.
Simulated possessions are aggregated into matches and the per-match expected goals are histogrammed. Bin width is automatic. The structured distribution sits to the right of the direct distribution (higher mean) and is narrower (lower variance). This is the central empirical finding of the experiment, and it inverts the common defense of high-entropy play.
The three claims the experiment is designed to test, with the empirical answer from the current simulation run.
The public defense of direct play rests on a variance argument: when you are outmatched, embrace chaos because chaos gives you a chance. The experiment shows that the variance argument is empirically wrong on its own terms. The high-entropy chain produces a wider distribution, but the increased dispersion does not translate into a higher ceiling for the underdog. It produces a lower floor without producing a higher peak. The team running the high-entropy chain is more likely to underperform its own mean than the team running the low-entropy chain.
The reason is mechanical. The structured chain's mean is higher because the chain reaches better shot origins more often. The structured chain's variance is lower because the sequence is more deterministic; the same starting state produces a more concentrated distribution of outcomes. Combining these properties, the structured chain dominates the direct chain on first-order and second-order statistics simultaneously. There is no statistical case for choosing the direct chain when the substrate supports the structured one.