Exact measure-and-prepare copying of qudit phase states for every number of copies

We determine exactly how well a measure-and-prepare ("measure-first") forger can copy qudit phase states. For one input copy and every number m of output copies, in every dimension d, we prove that the flat seed is the optimal measure-first strategy for the continuous family of phase states, the optimum being the top eigenvalue of an explicit matrix on the symmetric subspace (for example (7d − 8 + √(25d² − 64d + 48))/(2d³) at 1→2). The proof reduces the question to a negative-correlation inequality for closed lattice walks in symmetric time-dependent windows, which we prove by summation by parts; this combinatorial core is verified in the Lean 4 proof assistant. Consequently, for cyclic phase-state codebooks ψ_x = d^(−1/2) Σ_j ω^(x·k_j) |j⟩, the states used in quantum fingerprinting and quantum hashing, in every dimension and every 1→m task the measure-first optimum equals the continuous one if and only if the keys form a Bₘ₊₁ set, whereas at 1→2 the coherent optimum does if and only if they form a Sidon set. Around this law we compare, for every copying task T→T+k, the optimal coherent copier (any quantum channel) and the optimal measure-first copier on a finite codebook with the continuous family: both finite-codebook optima are at least the continuous ones; if the keys have no additive collisions among multisets of size 2T+k, both are equal for every dimension; and for codebooks that are collision-free at size T, every collision at size 2T+k strictly increases the value of the natural measure-first strategy. For cyclic qubit codebooks of q states and every 1→m task (m ≥ 2) we prove an exact leak-order theorem: the measure-first optimum equals the continuous one if and only if q ≥ m+2, while the coherent optimum already equals it when q ≥ m+1, so q = m+1 separates the two attackers for every m. Pre-registered experiments on IBM superconducting processors, in which the strategies are emulated by classically recombining two circuit stages, confirm the predicted qubit gaps at 1→2 and 1→3. We give an explicit polynomial-phase hash with a guaranteed (sufficient) copy budget and prove a separation for it. Whether measure-first attackers leak no later than coherent ones for every codebook remains open; the stronger statement that the measure-first gap always exceeds the coherent gap is false for general finite phase codebooks.