Semantic Hardness Conservation: An Operational Normal Form for Exact Reasoning After Compression

Abstract

Exact semantic computation can shrink dramatically when a task ignores distinctions preserved by reusable representations, yet smaller representations need not make reasoning easier. We develop Semantic Hardness Conservation, a normal-form theory that identifies where computational work goes after exact semantic compression. Building on Task-Semantic Images and Semantic Lifetime Width, we prove a canonical image-quotient bridge, a minimal exact-interface theorem, a natural same-image realizability separation between 2-CNF and 3-CNF, an audit-completeness theorem for encoding, realizability, separation, and transformation, and polynomial representation invariance. The Akhtar Semantic Convergence Theorem shows that uniform polynomial control of these four channels, their call counts, and stage count yields deterministic polynomial-time computation; the associated localization corollary identifies where hardness must survive if P differs from NP. We further define Akhtar Semantic Bottleneck Complexity, prove reduction monotonicity, normalize ten semantic strategies into the same operational form, and connect restricted representation hopping to known Tseitin lower bounds. Reproducible experiments generate 369 exact stage profiles across 45 controlled families, including Tseitin, pigeonhole, XOR, Horn, 2-SAT, random, and planted formulas. The results distinguish compression from realizability, certification, and conversion without claiming a universal polynomial algorithm for 3-SAT or resolving P versus NP and provide an auditable blueprint for future exact algorithms.