The Blass-Gurevich-Shelah conjecture: choiceless polynomial time with counting does not capture polynomial time

Main theorem: over a fixed vocabulary of eight binary relations, consistency over $\mathbb F_3$ of an explicitly defined linear system (normalisation equations $\sum_{a\in X_t}\lambda_a=1$ and consistency equations $\mu_y=\sum_{a\in X_t}\lambda_aC(y,a)$) is a polynomial-time query on all finite structures, with no promise, that no CPT program with counting defines, allowing hereditarily finite sets of unbounded rank. Priority: Shelah (2000) claimed noncapture for counting extensions of choiceless computation; later accounts kept the problem open, and the manuscript gives a separate proof not relying on that claim. Not shown: Gurevich's conjecture that no logic captures polynomial time, or anything about rank-based or other extensions.

Main theorem: over a fixed vocabulary of eight binary relations, consistency over $\mathbb F_3$ of an explicitly defined linear system (normalisation equations $\sum_{a\in X_t}\lambda_a=1$ and consistency equations $\mu_y=\sum_{a\in X_t}\lambda_aC(y,a)$) is a polynomial-time query on all finite structures, with no promise, that no CPT program with counting defines, allowing hereditarily finite sets of unbounded rank. Priority: Shelah (2000) claimed noncapture for counting extensions of choiceless computation; later accounts kept the problem open, and the manuscript gives a separate proof not relying on that claim. Not shown: Gurevich's conjecture that no logic captures polynomial time, or anything about rank-based or other extensions.