Right, the logic of "take a spare representation from here and use it to encode an extra state from over there" doesn't work in general, and sinks GP's whole id

Right, the logic of "take a spare representation from here and use it to encode an extra state from over there" doesn't work in general, and sinks GP's whole idea. But since 9×7 < 8×8, it does _in practice_ work to encode 9×7 states in 3+3 bits. Where GP's idea breaks down pragmatically (as well as logically) is when he tries to take _two_ spare representations and use them to encode _two_ extra states, for bits 6, 5, and 3. It's true that 6 = 2^3 - 2, 5 = 2^2 + 1, 3 = 2^1 + 1; but you can't represent 6×5×3 = 90 states using only 2^(3+2+1) = 64 representations. In short: If A = 2^a + k and B = 2^b - k, then we can cancel the k when adding A+B, but not when multiplying A×B.