The Balanced Tribonacci Decomposition: Extending Optimization-Based Signed Representations to Higher-Order Recurrences

In a companion study, we introduced the Balanced Zeckendorf Decomposition (BZD), replacing rigid syntactic gap rules with a classical non-adjacency condition and a three-tier lexicographic optimization criterion to achieve globally minimal weight and reduced maximum indices for signed Fibonacci representations. In this paper, we extend this framework to higher-order linear recurrences, specifically the Tribonacci sequence. We define the Balanced Tribonacci Decomposition (BTD), retaining the non-adjacency rule (index gap ≥ 2) and applying the identical three-tier optimization criterion (minimizing weight, maximum index, and total positional cost). Unlike the Fibonacci sequence, where the standard greedy algorithm naturally enforces non-adjacency, the classical greedy Tribonacci representation may contain adjacent indices; BTD strictly enforces the gap ≥ 2 rule. We demonstrate that BTD consistently achieves optimal weight while systematically minimizing the maximum Tribonacci index. Extensive computation shows that integers of the form T_k − 1 collapse from Θ(k) terms in the classical representation to exactly two terms in BTD (T_k − T_1). A simple recursive algorithm with lookahead computes BTD in O(log n) arithmetic operations. This work resolves one of the primary open questions posed in the original BZD framework and opens the door to generalized signed numeration systems for arbitrary linear recurrences.Key Findings & Empirical ResultsWorst-Case Compression: Integers of the form T_k - 1 collapse from Θ(k) terms in the classical greedy representation to exactly 2 terms (T_k - T_1) in BTD.Weight Reduction: Averaging over a deliberately hard test set yields a ~59.88% reduction in average weight compared to the classical Tribonacci representation. Over all integers from 1 to 3000, BTD achieves an 18.7% average weight reduction.Algorithmic Approach: The manuscript details an O(log n) recursive lookahead heuristic, alongside a rigorous mathematical proof of uniqueness.Supplementary Materials Included in this UploadThis repository/upload includes the final preprint manuscript (BTD3.pdf) alongside the supplementary Python implementation (BTDtreb.py) and its empirical output log (BTDtreboutput.txt). While the paper outlines an O(log n) heuristic, the provided Python script implements an exact, exhaustive signed Depth-First Search (DFS) with aggressive mathematical pruning. This exact solver was used to verify the absolute global minimum across all valid representations for the empirical results presented in Section 4 of the paper, guaranteeing the strict optimality of the reported data.