A Generating Formula for Prime Candidates: the Mod-100 Wheel G(n,k) = 100n+k

We study the closed-form expression $G(n,k)=100n+k$, where n≥ 0 and k runs over the 40 residues in $[1,99]$ coprime to 10. We prove that G is a bijection onto the set W of positive integers coprime to 10, with explicit inverse x↦( x/100,\ x 100) computable in constant time. Every prime other than 2 and 5 is therefore generated exactly once, and the non-prime outputs are exactly the integer 1 together with the composites coprime to 10. We verify the classification of all outputs below 10⁹ (4·10⁸ values, of which 50,847,532 are prime), show that the outputs form a rooted tree of depth Ω(x), and quantify how the share of primes is replaced by higher levels as N grows: the mean of Ω over the outputs is lnln N + B - 5/4 + O(1/ln N) with B≈ 1.0347, while the count of primes grows without bound. The primes are distributed among the 40 classes with a relative spread of 0.085% at 10⁹. We introduce the divisibility filter x k ≠ 0, k∈ Kₘₐₓ(N)=≤√N:(k,10)=1\, and prove that it yields a generator with precision and recall 1 on $[1,N]$; with the fixed set k≤ 99 the precision is 1 for N≤ 10,200 and above 99% for N≤ 12,826. Using precision, recall and F₁ we compare the variants with wheels of other moduli and with the formulas of Mills and Willans, and we report measured running times, per-output costs and memory footprints.