KNAPSACK, in the subset-sum form of Karp: given weights and a target, is there a set of items whose weights sum to the target? Numbers are written in binary, as the last four problems of Karp's list require: under the unary encoding they are solvable in polynomial time by dynamic programming, so the representation is part of the statement. The vocabulary carries the items, the bit positions, the bits of each weight, the bits of the target, and a linear order fixing the place values; being a linear order is folded into the yes-instances. The decoding (binNum) sums $2^{\mathrm{rank}}$ over a set of positions, and is defined for an arbitrary relation so that invariance is a plain transport. Membership is by an existential second-order definition that walks the order on items to verify the arithmetic; hardness is an ordered first-order reduction from Exact Cover.