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 (\varphi^2 r(v)<L(v)), where (\varphi=(1+\sqrt5)/2). More generally, for sequential rules whose shells contain only previously decided points, a rule (\alpha c(v)\le L(v)) induces the complementary rule (\frac{\alpha}{\alpha-1}r(v)<L(v)). In the golden case this produces the complementary threshold pair (\varphi) and (\varphi^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.