Adversarial Transport for Extremal ABC Behaviour

We formulate an adversarial transport framework for extremal behaviour in the ABC setting, focusing on a parametrised family of highly imbalanced near-collision triples. Within this family we define a cobordism-valence move system and construct discrete spines in the configuration graph. By interpolating spines to Lipschitz splines in the height–radical plane, we identify geometric barriers: any spline connecting a safe configuration to a “monster’’ configuration must traverse a wedge region and incur a definite geometric path length. We then introduce a simple Heston-type stochastic-volatility model for the evolution of height along a spline. Coupling volatility to geometric speed yields a quantitative “no cheap monster’’ inequality: any adversarial attempt to reach the monster region must pay a minimal integrated volatility cost. Although presented as a toy model, this framework suggests that deterministic analogues of volatility constraints—derived from arithmetic height inequalities or the appearance of new prime factors—may produce genuine obstructions to extremal ABC configurations. We conclude with potential extensions to deterministic volatility bounds, other Diophantine families, and speculative analogues in complexity theory.