Exact discrepancy of letter positions in fixed points of binary morphisms, with an application to conjectures of Kimberling

In 2017 Kimberling added to the On-Line Encyclopedia of Integer Sequences a large family of conjectures of the form L < nr − a(n) < U for all n ≥ 1. Here a(n) is the position of the n-th 0 (or 1) in the fixed point, or limiting word, of a morphism on {0,1}, and 1/r is the frequency of that letter. Using the Dumont–Thomas numeration we show that, when the second eigenvalue λ₂ of the incidence matrix satisfies |λ₂| < 1, the exact supremum and infimum of nr − a(n) are affine functions of the unique solution of a four-variable max–min linear system over Q(√d). The solution comes with an exact certificate. We compute the exact supremum and infimum for all 78 position sequences in this family. Sixty of the conjectures hold (four of them after correcting an evident misprint), and 57 of these had not been proved before. Eighteen fail. Most failures occur at very small n, but two do not: the conjectured bounds for A284365 hold for n < 8 933 313 and fail at n = 8 933 313, and those for A284366 hold for n < 2 977 771 and fail at n = 2 977 771. The optimal bounds are −(12 + 2√21)/15 and (6 + √21)/5 for A284365, and (√21 − 9)/5 and (6 + √21)/5 for A284366. For four sequences with |λ₂| > 1 the discrepancy is unbounded on one side, and we determine the other side. We also determine the exact ranges for six older entries defined by other recursions (A026363, A026364, A026367, A026368, A045671, A045672), after showing that they are position sequences of such fixed points; this confirms Kimberling's conjectures for them. We supply the certificates together with an independent checker.