Relational Coherence Budgeting for Tunable Exchange Gates: Pointer-Basis Affinity, Decoherence, and Exchange-Angle Coherence Cost

Bell's theorem and its loophole-free experimental verification rule out local non-contextual hidden-variable completions of quantum mechanics. A natural response, on which this work builds, is the relational reading of Rovelli, on which quantum properties are not assigned to systems in isolation but are encoded in their interaction histories. We package one technically explicit aspect of this reading into the Relational Coherence framework (RCF), centred on the Affinity Tensor α(S,E) — the off-diagonal part of the reduced state ρ_S in the einselected pointer basis — whose normalized Frobenius norm ᾱ(S,E) ∈ [0,1] is the Hilbert–Schmidt coherence quantifier of ρ_S in that basis. We collect four standard identities (property indefiniteness, exponential decay under Markovian dephasing, range and extremal characterization, and coherence redistribution under unitary S–E evolution) in the language of ᾱ, suitable for use as a per-layer coherence-budget diagnostic in two-qubit circuits. Building on existing parametric-gate work, we examine the isotropic-Heisenberg-exchange unitary U_RA(θ) = exp[−iθ(σ_x⊗σ_x + σ_y⊗σ_y + σ_z⊗σ_z)/2] with θ ∈ [0, π/2], which traces the diagonal (θ, θ, θ) of the SU(4) Weyl chamber and attains maximum entanglement capability at θ = π/4 (√SWAP). On platforms where the Heisenberg-exchange pulse duration scales linearly with θ, U_RA(π/8) accumulates half the coherence-limited error of a full θ = π/4 entangler. Numerical simulations in Qiskit and Cirq under gate-time-proportional depolarizing, dephasing, and amplitude-damping noise yield a per-gate fidelity advantage of ΔF ∈ [+0.038, +0.039] at p_base = 0.10 over CNOT, iSWAP, and CZ, consistent with the experimental continuous-fSim advantage reported by Foxen et al. on Sycamore. The advantage is contingent on direct exchange calibration; on platforms where U_RA(θ) must be decomposed into fixed native primitives, the advantage may shrink or vanish.