What Gets Extremized A cross-level audit of extremal principles, the inevitability of Maxwell points, and a counterexample family for maximum entropy production

Abstract A widely circulated reading unifies four levels — conservative systems, near-equilibrium steady states, far-from-equilibrium dissipative structures, and self-referential systems — with a single sentence: each level extremizes some quantity. This paper argues that the sentence names four objects of entirely different logical status: an identity rewriting, a theorem with extremely narrow conditions, an unproved conjecture, and a level on which no extremized object exists at all. It is therefore not a unification but an equivocation. Three contributions follow. First, an audit table filling in, for each level, the dimension, the reference zero, whether the quantity is maximized, minimized or stationary, its status, and the associated cost of maintenance. Second, a theorem: in any bistable coexistence window bounded by a pair of saddle-node bifurcations, the quasi-potential depth difference necessarily changes sign between the two ends, so at least one Maxwell point lies inside the window; if the entropy production rate is strictly monotone along the coordinate on which the bistability occurs, then the maximum entropy production conjecture (MEP) necessarily disagrees with the actual selection on a non-empty subinterval. The counterexample is therefore not model-specific: it holds for the entire class of codimension-one saddle-node bistabilities. Third, a fully reproducible worked example in a completely documented dissipative model, in which the region of disagreement occupies 36% to 39% of the coexistence window, a share robust to the choice of gain-factor family and to whether an adiabatic reduction is performed. The paper then argues that what actually runs across the levels above the first is not an extremum but a lower bound of the form "to maintain this structure, the flux may not fall below a given value," and states explicitly that this bound currently has an explicit form only on the light-tailed branch. Finally, a two-parameter self-referential model shows that on the fourth level a conserved quantity exists while an extremized quantity does not — a structure shared with the first level and absent from the second and third. All numerical results are deterministic algebra and root-finding, with no stochastic content.