Self-Referential Fixed Points and the Physical Horizon of NP-Complete Computation: A Structural Analog of Gödel–Löb Incompleteness in Resource-Bounded Computation

A machine is considered whose task size is set by the resources its own computation will leave behind: decide the largest satisfiability instance that the residual budget permits by exhaustive search. Self-consistency is a fixed point of a self-consumption operator on budgets. In the continuous model the fixed point is half the initial budget for every specification matched to its cost function, with an exact stability criterion. With integer problem sizes a fixed point exists only under an arithmetic condition on the budget; every budget that is not a power of two has exactly one two-cycle, into which re-planning from the full budget always falls. Resource counterparts of the diagonal lemma and of the theorems of Gödel, Löb and Tarski are proved for a feasibility predicate. For the observable universe the fixed-point size is about 305 variables, a value specific to the algorithm and to the physical accounting.