Running least common multiples: distinct values and multiplicities

The unresolved repeated-sum question

The distinct sum at $\{2,3,5\}$ is settled by Theorem 1.1. The general and subseries arguments remain ordinary proofs pending formalisation (Remarks 5.1 and 5.10). The finite-state calculations of Remark 5.8 have their own scope; the $\{2,3,5,7\}$ automaton collision limits that interval criterion only.

For ${\mathcal{R}}_{\{2,3,5\}}$, the missing ingredient is not just another normalisation or one sharper finite enclosure. The coefficients, recurrence, quadratic bounds and denominator-clearing formula are explicit. We must exclude eventual integral reduced tails for every possible denominator, or equivalently prove the residue inequalities beyond every prescribed onset. The fixed-start formula explains why an early success does not suffice; the strip identity explains why avoiding floor crossings does not supply the proposed cancellation.

The rank and uniform-norm results concern the kernel, not the arithmetic of its sum. Finite tests, countability, radix recoding and denominator support do not close the remaining argument. The settled two-prime case uses Fan’s earlier reduction to the cited Hecke–Mahler value theorem.