Exact trace-invariance relations for shared splittings in coupled spin-½ NMR spectra

<jats:p> A shared splitting is a frequency difference that occurs identically among the transitions of two scalar-coupled nuclei. The relation is exact in every coupling regime, but the common value coincides with the corresponding coupling constant only in the first-order limit. We show that the shared splittings of an <jats:italic toggle="yes">N</jats:italic> -spin-½ system obey a complete and finite ladder of exact linear identities, one for each rank r = 2, …, N, with a single source: the invariance of the traces of the blocks of the spin Hamiltonian. The rung r = 2 is an exact sum rule relating the spectator-averaged shared splittings to the sum of the signed coupling constants; the rungs vanish identically, the topmost reducing, for , to the complete spectator-independence of the shared splittings. The admissible weights are the Krawtchouk polynomials , and within its class (linear combinations of shared splittings determined by the block traces) the ladder is proved complete. Because is the rank of longitudinal spin order, the ladder is also a graded diagnostic: a longitudinal term of rank added to the assumed Hamiltonian moves the rung and no other, so a violated rung reports the longitudinal order at which the model and the data disagree. The identities reduce the splittings readable from a spectrum to linearly independent combinations (from 12 to 3 for ABC and from 48 to 9 for ABCD); most of the reduction is combinatorial, with the ladder’s homogeneous rungs the only constraints that carry information about the Hamiltonian. All results are verified symbolically and against exact diagonalization, with residuals at the level of floating-point round-off; the identities constrain transition frequencies, not intensities. </jats:p>