A λ-Resonance Theorem for Constrained QAOA: Spectral Structure and Recurrence Bounds

Training the variational angles of the Quantum Approximate Optimization Algorithm once on a small instance and reusing them on larger ones, known as parameter transfer, is the standard route past the exact-simulation wall. Existing explanations of parameter transfer focus primarily on structural similarity, but constrained encodings introduce a second quantity that must also be tracked: the penalty scale. We characterize the dependence on the deployment penalty scale when the circuit angles are held fixed. We show that for a scalarized cost C_λ = C + λv with integer-valued violation count v, the feasible probability F(λ) at arbitrary fixed depth p is a finite real trigonometric polynomial whose frequencies are integer linear combinations of the fixed γ angles. We call this dependence λ-resonance. It recasts penalty mismatch as a phase-interference problem and yields an explicit spectral-support formula, a Bernstein lower bound on the detuning required to change F by a specified amount, and exact recurrences when the layer phases share a common period. These statements hold for arbitrary fixed angles, including untrained angles; they establish spectral support and recurrence conditions without asserting that the training penalty is a preferred operating point. Exact-statevector experiments on a 20-qubit resource-allocation QUBO exhibit sharp and blunt response profiles and empirical recurrences consistent with the predicted spectral support. Independent penalty-phase sampling recovers the full Fourier coefficients and allows coefficient-based recurrence bounds to be evaluated. A trained two-qubit example shows that preserving the training phase scale need not maximize feasibility: a nearby penalty can perform better. Together, the theory and experiments motivate a conditional transfer protocol: treat (β, γ, λ_0) as a single calibrated parameter set, verify target validity, and bracket the matched penalty with a small fixed-angle sweep when feasible.