The Golden Assay Sieve: Complementary Threshold Duality for the Golden Exclusion Sieve

A companion note to The Golden Exclusion Sieve (Zenodo DOI 10.5281/zenodo.21660281). The complement of the published sieve is shown to generate itself autonomously from its own previous shell counts using the strict threshold (φ^2 r(v)<L(v)), where (φ=(1+√5)/2). More generally, for sequential rules whose shells contain only previously decided points, a rule (α c(v)≤ L(v)) induces the complementary rule (α/α-1r(v)<L(v)). In the golden case this produces the complementary threshold pair (φ) and (φ^2). The paper derives an exact integer-only membership test, a dual two-state shell-occupancy law, an (O(log n)) counting recursion, and the reflected discrepancy identity, together with inherited substitution, Zeckendorf-automaton, run/gap, Fibonacci, and Lucas structure. The reciprocal threshold relation is classical in complementary Beatty theory; the contribution here is its exact autonomous realization on the same self-referential shells as the Golden Exclusion Sieve. The complementary sequence is called the Golden Assay Sieve.