An Entanglement-Sensitive Return-Amplitude Residue and an Encoded Two-Boson Holonomic Gate

We define a basis-independent two-particle return-amplitude residue H_E = sin²(2γ) sin⁴(χ/2), constructed from a single-particle return angle χ ∈ [0, π] and a local dyad rotation γ. The residue is bounded in [0, 1], scales quartically for small χ, and isolates an off-diagonal cross term that appears in the symmetric entangled return amplitude but not in the corresponding product-state amplitude. It is invariant under common unitary changes of basis at the basepoint and depends only on operational overlap data, not on the mechanism that produces the loop. A minimal number-conserving two-boson hopping Hamiltonian is then shown to carry a nontrivial encoded holonomic gate. The encoded subspace span{|20⟩, |02⟩} satisfies the scalar-compression condition with vanishing compressed Hamiltonian, and at T = π/(2J) the cyclic return map is U_code = −σ_x, purely geometric in the projected-evolution sense. The associated one-particle obstruction is calibrated using higher-rank numerical ranges. For a Hermitian one-body Hamiltonian κ on m modes, the rank-2 scalar-compression condition is governed by Λ_2(κ) = [λ_2, λ_{m−1}]. The obstruction holds at m = 2 unless κ is scalar, collapses to a single value at m = 3, and becomes a generic interval at m ≥ 4. The two-boson construction is therefore a minimal mode-resource separation, not an absolute single-particle no-go theorem. The paper also gives an interferometric estimation protocol for H_E, the resulting shot-count scaling, and a comparison with Lusk's spacetime-specific entanglement holonomy in Kerr geometry.