The Golden Exclusion Sieve: Fibonacci Substitution and Zeckendorf Automata from a Self-Referential Density Rule

The Golden Exclusion Sieve is a deterministic subset of the positive integers defined by a self-referential density rule. To decide whether an integer (v) is admitted, the rule counts earlier admitted integers in the interval between ( v/φ) and (v), then compares that count with the interval’s golden-ratio capacity. Although the definition is nonlocal, the resulting system is governed by finite-state structure. The central theorem pins every shell count to one of two adjacent integers. This yields an (O(log n)) counting recursion and a primitive six-state substitution whose type projection is the Fibonacci substitution. The sieve has density (1/φ), sharp bounds on admitted and rejected runs, and exact gap and run distributions. Every Fibonacci number (F_k≥ 3) is excluded, every Fibonacci predecessor (F_k-1) is admitted, and every Lucas number from (29) onward is excluded. The discrepancy (D(n)=A(n)-n/φ) has an exact normal form. Its global maximum is (2/φ^2), its scale-wise maxima occur at Fibonacci predecessors, and its record minima are exactly ( F_m/2-1). Its lower envelope equals (-13log_φ n) plus a bounded asymptotically log-periodic correction of exact amplitude (1/2). Finally, every positive integer has a constrained Fibonacci-valued ancestry word. Reversed ancestry digits recover the full six-state substitution letter. Valid padded Zeckendorf representations are read by a minimal nine-state membership automaton, while simultaneous recognition of representation validity and sieve membership requires exactly eleven states. Thus a single self-referential crowding rule admits equivalent descriptions through arithmetic recursion, symbolic substitution, Fibonacci numeration, and finite automata.