The pure Yang-Mills mass gap does not determine the physical hadron mass gap

The axiomatic quantum field theory formulation of Yang-Mills existence and mass gap asks whether, for every compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on four-dimensional Minkowski space and has a spectral gap above the vacuum. Constructive programs typically treat the pure-gauge case as a self-contained mathematical object and defer matter to a downstream step. The physical hadron mass gap, however, is the infimum of the positive spectrum of the full-QCD vacuum Hamiltonian, and full QCD contains quarks that pure Yang-Mills lacks. Deferring matter to a separate step can suggest that a pure-gauge solution determines the hadron gap. We prove otherwise. For any pure Yang-Mills record X with finite gap G(X), and any pair of real numbers μ₁ < μ₂ below G(X), the two decoupled matter extensions H₁, H₂ built from the same X with a C² spectator diagonal Hamiltonian of gap μ_i satisfy gap(H_i) = μ_i. Any function of the record alone (any source-only procedure) has, at the pure Yang-Mills input X, error at least (μ₂ - μ₁)/2 on at least one of the two extension gaps. The bound is sharp: the constant procedure returning the midpoint (μ₁ + μ₂)/2 attains it. The proof uses only the tensor-decoupling identity for positive commuting self-adjoint operators (Reed-Simon VIII.10 spectral calculus) and the triangle inequality. A companion proposition shows that the four attached-structure classes commonly invoked in constructive quantum field theory (isometric embedding to a target Hilbert space; Radon-Nikodym derivative against a source path measure; algebra inclusion; heavy-mass-limit family) are not exhaustive: the simple procedure c · G(X) attaches none of them. Nine analytic and numerical methods surveyed in Appendix A (lattice QCD, effective field theory, bootstrap, functional-integral, semiclassical, TQFT-embedding, large-N, holographic, renormalization-group flow) each supply matter data as external input. The theorem clarifies why: matter data cannot be recovered from the pure-gauge record alone.