Holes in Valid-Extension Sets of Finite Gilbreath Sequences

Abstract

Given a finite sequence of integers, form its difference triangle by repeatedly taking absolute differences of consecutive entries. We call the sequence Gilbreath if the leftmost entry of every row below the top is 1. The Gilbreath conjecture, which remains open, asserts that every initial segment of the primes is a Gilbreath sequence. This paper studies the local extension problem: given a Gilbreath sequence, which integers can be appended to it while preserving the Gilbreath property? We call the set of such admissible values the valid-extension set of the sequence. A previously proposed characterization in the literature predicts that this set always fills a natural parity interval around the last term. We show that this fails in general: the valid-extension set can have interior holes, with the smallest failure occurring at length 5 for the sequence (2,3,5,9,15). The paper develops a corrected theory of valid extensions. We give an exact criterion for membership in the valid-extension set, an algorithm that computes it, and a sharp condition determining exactly when the set fills the candidate interval. This last condition is an order-sensitive analogue of the classical Brown completeness criterion for subset sums. We also establish endpoint validity and reflection symmetry, determine the exact minimum size of the valid-extension set together with its unique minimizer, exhibit a family whose valid-extension set has exponentially many components, and provide enumeration data through length 11.