A trace-hierarchy theorem for shared splittings in coupled spin-½ NMR spectra

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Shared splittings are equal signed frequency differences in the labelled transition families of two nuclei.We prove a trace-hierarchy theorem for 𝑁 ≥ 2 coupled spin-½ nuclei: the traces of the total-𝐼 𝑧 blocks determine exactly 𝑁 -1 linearly independent shared-splitting functionals of the energy levels.Their energy-level weights are the Krawtchouk polynomials 𝐾 𝑟 (𝑘; 𝑁) , 𝑟 = 2, … , 𝑁 .For the pairwise scalar-coupling Hamiltonian, the member 𝑟 = 2 gives the spectator-averaged signed sum rule ∑ 𝑆 𝑖𝑗 av 𝑖<𝑗 = ∑ 𝐽 𝑖𝑗 𝑖<𝑗 , and every member 𝑟 ≥ 3 vanishes.The highest member gives complete spectator independence for three spins; for 𝑁 ≥ 4, the hierarchy includes 𝑁 -3 intermediate relations implicit in the classical block-trace formula.Completeness follows by intersecting the shared-splitting weight space with the space of weights constant on magnetization blocks.The underlying four-level shared-splitting equality is algebraic at any coupling strength.Its observation requires non-zero transition moments, sufficient signal and resolved line positions.Its value equals the coupling constant for an isolated pair, but need not do so in larger systems.The trace functionals are invariant under relabelling within each block and respond selectively to added longitudinal product terms of rank three or higher.We specify a continuation-based convention for individual labels, subject to non-degeneracy, and give worked AB, ABC and ABCD examples.We provide high-precision numerical evidence of global non-uniqueness in parameter recovery from shared splittings alone.On noisy simulated spectra with interleaved multiplets, we test candidate assignments using fitted frequencies and their covariance.Completeness is relative to the block traces; stronger affine-relation and generic-rank claims remain open.

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