The Rank Price: the interference dimension of Clifford+T circuits

Representations of a Clifford+T phase polynomial, modulo Clifford operations, form a coset of the punctured Reed-Muller code RM(h-4, h). The optimization literature decodes this coset for weight: the number of odd-coefficient parities is the T-count, and its exact minimization is NP-hard. We decode the same coset for rank: the F2-dimension d of the span of the odd-parity support, minimized over the coset, which we call the interference dimension, because exact path-sum simulation costs poly x 2^d rather than 2^(T-count). We prove that the subspaces supporting some coset element form an intersection-closed family, so a unique minimal core subspace V0 exists with d = dim V0; that the Clifford-equivalence class is computed by a syndrome map into the shortened code RM(3, h); and that d equals the F2-rank of an explicit h x O(h^2) matrix read off the circuit: polynomial time, in sharp contrast with the NP-hardness of the weight objective. For the standard GF(2^k) multiplier benchmarks the invariant is a convolution tensor and d = 4k-1 exactly; combined with the hierarchy log2(t+1) <= d <= t, where t is the optimal T-count, this yields a linear lower bound on multiplier T-count in the coset model, which meets Karatsuba-type upper bounds at t = Theta(k). A counting argument gives families on which the rank price beats the optimal weight price superpolynomially. Every theorem is machine-verified (exhaustive small-case checks; 142/142 agreement with brute-force coset scans), and the method is benchmarked against the feynman suite and the quizx stabilizer decomposer, with both engines agreeing on every exact amplitude. Optimal T-count is NP-hard; the interference dimension, which is what exact simulation actually pays for, is a matrix rank. Code and machine verification of every proof step: https://github.com/0verflowme/rankprice