The Battle for a Cross-Representation Invariant: After the Existential Quantifier Collapses, What Must Still Remain?

Team Not-Equal proposes its first candidate — the residual distinguishable load H_res: classify the residual functions of a SAT formula after a partial variable assignment by semantic equivalence; if some cut has many pairwise-distinguishable residual functions, then any system that summarizes what it has read using only finite state must have enough states — |Q_S|≥N_res(φ,S) is a genuine semantic lower bound. But Team Equal immediately points out that this holds only under a fixed variable order and a fixed-cut model — a general algorithm can reorder variables, revisit the input, use global algebraic summaries, or introduce auxiliary variables to escape. Establishes a six-part qualification test for candidate invariants, and uses the parity function PARITY (2^n candidates, yet only 1 bit of state is needed) to break the intuition that “a large candidate space requires exponential state.”