The Complexity Potential-Function Game: Amortized Tractability Certificates, Potential-Function Escape, and the Certificate-Completeness Trap

Formally imports the telescoping sum from standard amortized analysis: with amortized cost ĉ_t=c_t+Φ_(t+1)-Φ_t, as long as the initial potential, the per-step amortized cost, and the number of steps are all polynomially bounded in the input, the total actual computational cost is polynomial — this is a rigorous proof, not a metaphor. But this round catches a previously overlooked asymmetry: the P=NP side only needs a single SAT algorithm to exist, together with its own algorithm-specific potential function — it need not be solver-independent. The P≠NP side, if it wants to argue a lower bound from “no potential function of a given kind can be found,” must first prove that the certificate system is complete for all P algorithms; otherwise, a missing certificate only shows that the certificate system is too weak, not that no such algorithm exists — this is the “Potential-Function Certificate-Completeness Trap.” Upgrades ATC into a two-layer certificate (a Progress/Ranking layer governing termination, and an Amortized Potential layer governing total resource use), and distinguishes three layers of cost — object-level running time, certificate complexity, and discovery complexity — making clear that object-level running time is what the traditional P/NP classification actually targets.