A Hybrid No-Sum Sequences (HNS)

This paper presents a family of No-Sum (NS) sequences defined by read-once arithmetic derivations, as introduced in a previous paper, and introduces the scalable Hybrid NS algorithm, which maintains strong combinatorial hardness while enabling long-range generation. In Phase 1, we construct a strict NS sequence under (+,−,×) to establish the existence, uniqueness, and finiteness of a governing derivable set for greedy progression. The paper then introduces a prefix-lock, the positive derivability closure of the strict prefix, which is employed in subsequent phases to avoid collisions with previous exclusions. Phase 2 applies a relaxed NS rule (e.g., (+,−) or bounded read-once derivations), and Phase 3 applies an efficient sum-free rule (+), with all phases prefix-locked to preserve the definitional integrity of the phases and the greedy-minimality of the entire sequence. In this paper, we present a formal sequence construction with provable uniqueness and a scalable hybrid extension, and discuss the complexity of the sequence motivated by cryptographic hardness.