Periodic characterization of the meta-family S_{1,g,g+d}

Prove that for every integer d≥2 and every integer g≥d+1, the greedy 3-subsumfree sequence S_{1,g,g+d} consists exactly of 1, g, 2g+d−1, and 2g+d together with the integers z>g+d whose residues modulo 5g+2d lie in {g+d−2,g+d−1,…,2g+d−2}, and consequently is eventually periodic with period g+1 after its first g+3 entries.

Background

The authors formulate a broad meta-conjecture for greedy 3-subsumfree sequences with initial values 1, g, and g+d. It predicts an explicit finite exceptional set and a modular description of all later terms.

The conjecture is supported by computational evidence from an implementation in Magma and by a Walnut verification for the particular case d=4 and g=4. The paper notes that exceptional small-g cases can exhibit different moduli and periods, but does not establish the stated result in general.

References

As a matter of fact we have firm computational evidence, from an implementation in Magma , for the following meta-Conjecture. \begin{conjecture}\label{conj_d_g} Let $d\geq 2$. For every $g\geq d+1$ the greedy 3-subsumfree sequence $S_{1,g,g+d}$ is characterized as follows:

— Using Walnut to solve problems from the OEIS  (2503.04122 - Bosma et al., 6 Mar 2025) in Section 4, Conjecture \ref{conj_d_g}