Golden Contraction: An Integer Recurrence with Golden-Ratio Modular Gating, Additive Growth, and Proportional Contraction

The Golden Contraction is an integer recurrence that alternates between additive growth and proportional contraction. Let (φ=(1+√5)/2), set (a(1)=1), and for (n≥ 2) let (m_n= φ n). If the least nonnegative remainder of (a(n-1)) modulo (m_n) is less than (n), the recurrence adds the current index, giving (a(n)=a(n-1)+n); otherwise it contracts the previous value, giving (a(n)= a(n-1)/φ). The resulting orbit displays a rising sawtooth structure produced by the interaction of a golden-ratio Beatty gate, modular branch selection, linear pumping, and nonlinear release. This deposit introduces the recurrence, gives exact integer implementations, develops its full-state formulation on clock-value pairs, and studies the symbolic sequence of ADD and CONTRACT decisions. The universal claim that no four identical decisions can occur is false: four consecutive ADD decisions and four consecutive CONTRACT decisions both occur for suitable noncanonical states. The manuscript replaces that false universal pattern with proved regional and conditional results, including a theorem showing that any CONTRACT run beginning below three golden moduli has length at most three, together with an exact classification of two routes through which four consecutive ADD decisions can occur under the stated hypotheses. These results reduce the unresolved canonical run-length problem to the study of bounds on the normalized orbit. The canonical orbit beginning at (a(1)=1) was computed using exact branch arithmetic through (n=100067704832), where (a(n)=462116062084). No ADD or CONTRACT run longer than three was observed on that finite prefix. An independent census through (10^8) found only one CONTRACT run of length three, beginning at gate 37, and no canonical run of length four. These computations are reported as finite evidence and not as proofs of permanent behavior. Open problems include proving an eventual normalized corridor, determining whether the canonical orbit permanently excludes four-runs, explaining missing symbolic words, studying attraction and merger between different initial states, and analyzing the branching predecessor geometry of the autonomous clock-value map. The accompanying reproducibility package contains the manuscript source, figures, exact Python and C implementations, machine-readable audit results, the first 10,000 sequence terms, the first 1,000 annotated branch decisions, and the checkpoint from the computation beyond 100 billion steps.