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. The Relational Coherence framework (RCF) responds to this constraint within the relational reading of Rovelli: quantum properties are encoded in interaction histories, not assigned to systems in isolation. The RCF packages one technically explicit aspect of this reading into a normalized Hilbert–Schmidt coherence quantifier ᾱ (S, E) ∈ 0, 1 — the scalar affinity — grounded by einselection in the pointer basis fixed by the system–environment Hamiltonian. Four standard identities (property indefiniteness, exponential decay under Markovian dephasing, range and extremal characterization, and coherence redistribution under unitary S–E evolution) are collected in ᾱ-language, suitable as a per-layer coherence-budget diagnostic in two-qubit circuits. The technical contribution of this paper is the Coherence-Budget Allocation Theorem: under Markovian gate-time-proportional dephasing with effective per-layer rates γ̃_ℓ and an entanglement target Eₜarget = Σ_ℓ |sin 2θ_ℓ| ∈ (0, L), the unique cost-minimizing exchange-angle schedule for the isotropic Heisenberg-exchange unitary URA (θ) = exp−iθ (σₓ⊗σₓ + σᵧ⊗σᵧ + σᵦ⊗σᵦ) /2 is θ*_ℓ = ½ arccos (min (1, γ̃_ℓ/ (2λ*) ) ), with shadow price λ* fixed by the entanglement budget. The theorem yields three falsifiable predictions: (P1) distributing entanglement across layers beats concentration in the uniform-noise case; (P2) quieter layers receive heavier entangling weight in the heterogeneous case; (P3) layers whose effective rate exceeds the global shadow price are skipped entirely. A reference scheduler implementing the theorem (figures/rcbₒptimizer. py) beats both fixed-gate and uniform-θ baselines in multi-layer density-matrix simulation, with fidelity advantages of ΔF = +0. 040 over concentrated full-entanglers at γ = 0. 15, Eₜarget = 1, L = 8, and ΔF = +0. 025 over uniform-θ in the heterogeneous regime. Qualitatively similar noise-aware angle-routing and gate-removal behaviours have been demonstrated empirically through compiler optimizations for continuously parameterized ZZ gates on trapped-ion hardware (Yale et al. , Phys. Rev. Appl. 24, 024057, 2025). The present work differs by deriving these behaviours as the unique analytic solution of a convex coherence-budget program grounded in einselection, rather than through hardware-calibrated heuristic search. The fidelity advantage of partial-exchange gates is contingent on direct exchange calibration; on platforms where URA (θ) must be decomposed into fixed native primitives the advantage may shrink or vanish.