Deterministic Certification of Approximation Gaps via Structural Lifting

We present a deterministic framework for extracting algebraic certificates from approximation gaps in bounded-query constraint systems. For verifiers satisfying a structural loopability condition, we give explicit algorithms that convert witnesses with robust gaps into low-degree polynomial obstructions, revealing deterministic algebraic structure underlying hardness of approximation. Our approach consists of three components: (i) structural lifting, which reorganizes overlapping constraints into consistency packets; (ii) a noise-to-structure theorem, showing that persistent gaps force repeated local failure patterns; and (iii) an algebraic rigidity procedure that deterministically constructs bounded-degree polynomial certificates from these patterns. Together these yield a deterministic gap-certification algorithm running in time 𝑂(∣ℒ∣⋅poly(𝑞,𝑡)). We instantiate the framework for Unique Games on dense graphs and algebraic CSPs, showing that randomized gap amplification can be replaced by deterministic structural extraction. Finally, we formulate the Intrinsic Error Ideal Conjecture, positing that all robust approximation gaps arise from low-degree algebraic obstructions invariant under structural lifting.