A Proof of the Golomb Meta-structure Conjecture for Almost Golomb Sequences

This is a mathematical proof preprint studying the core chain of Golomb meta-structure conjectures proposed in Section 9 of Benoît Cloitre’s Almost Golomb Sequences. This conjectural chain describes how the maximum multiplicities of the family of almost Golomb sequences are controlled by the classical Golomb sequence. The paper proves two key boundary conjectures: the Prefix Conjecture and the Domination Lemma. First, it proves that for every order r≥3, the value of the almost Golomb sequence of order r at the r-th position is equal to the term of the classical Golomb sequence indexed by r−1. Second, it proves that for every order r≥5, the maximum run length occurs on the run of the boundary value r−1. This yields the formula for the maximum multiplicity and further derives the threshold law in Section 9: the threshold positions and their consecutive gaps are both determined by the classical Golomb sequence. The proof relies on two main ideas. First, it constructs an auxiliary sequence obtained by shifting the classical Golomb sequence and establishes prefix agreement between the almost Golomb sequence and this auxiliary sequence. Second, it uses a minimal counterexample argument to show that no later run can exceed the boundary run length. Together, these two parts give a complete derivation from the local boundary value to the global maximum multiplicity, and then to the threshold law. The result should be understood as a proof of the core Golomb meta-structure conjectural chain in Section 9 of Cloitre’s paper. It does not address the other open problems listed in Section 10, nor does it claim to solve all subsequent directions in the theory of almost Golomb sequences.