The Slepian commutation is strictly archimedean: a rigidity theorem for semilocal time-band limiting

Time-band limiting rests on what Slepian called a lucky accident: a differential operator commutes with both the time and band cutoffs. Connes, Consani, and Moscovici transplanted the construction to the semilocal adele class space, whose Fourier transform carries the local zeta factors in its Mellin multiplier. Within the fixed K_S-invariant, fixed-cut algebraic setting, the accident is strictly archimedean. For every finite set S of places of Q containing ∞ and at least one finite prime, the semilocal time-band limiting pair (P, P̂) has trivial commutant inside the specified difference-differential algebra D_S^+ (whose archimedean specialization contains the classical prolate operator): in either of two unbounded-operator readings—a formally symmetric element admitting a self-adjoint realization that spectrally commutes with both cutoffs, or a general element satisfying the form-commutation identities with both cutoffs on smooth functions compactly supported off the cut—the element is a constant. At one prime the commutator defect resolves into lattice combs separated by exponential rate, and half-line uniqueness at mixed rates forces triviality. At several primes the combs are incommensurable, and the proof passes through an effective lower bound for linear forms in two logarithms and a Gevrey window class. No statement about the Riemann hypothesis is made or implied.