Sidorenko's property and edge retention in deep lower tails

When a random network contains very few copies of a given pattern, how many connections can its most likely configuration retain? For bipartite patterns, we prove that homogeneous minimization at every lower-tail threshold is equivalent to Sidorenko's property, for both Bernoulli graphon entropy and its sparse-limit functional. We then determine the deep-tail power of optimal edge retention. If the pattern has m edges and density-domination exponent CH = C(H,K₂), the normalized entropy saving from the zero graphon and the edge density of every normalized minimizer are θm/CH+o(1) as the relative threshold θ tends to zero, uniformly over all Bernoulli baselines. For a non-Sidorenko pattern, every optimizer therefore retains a diverging multiple of the homogeneous edge density. Applied to the counterexample reported by OpenAI, the characterization gives negative answers to two conjectures of Zhao. The variational comparisons also yield conditional edge-retention bounds for dense and suitably sparse random graphs.