Instead of iterating over multiples and marking positions, GC-60 lets known divisors stamp their fixed cyclic pattern directly onto the structure, composites are never reached, they are already marked.
The foundation of GC-60 is a fixed repeating structure: the integers mod 60, shifted by 10. Within each block of 60 numbers, exactly 16 positions can possibly be prime (all others are divisible by 2, 3, or 5 by construction). These 16 offsets form the passive receiver — the binary that simply waits to receive projections.
Primes 2, 3, and 5 are excluded from the module by definition. The first useful prime is 7, whose first relevant multiple is 7×7 = 49. To locate 49 in the structure: (49 − 10) mod 60 = 39 → offset 39, list 0.
Every prime generates a cyclic pattern of offsets as its multiples land on the mod 60 structure. For prime 7, the full cycle across seven consecutive lists is:
This cycle is computed once. From that point on, there is no need to calculate 7×7, 7×11, 7×13, 7×17… individually. The cycle is a mask that tells exactly which offset in each list is hit by a multiple of 7.
The divisor does not scan. It projects. Given its cycle, it stamps its footprint onto the fixed structure by simple monotonic translation — sliding the same repeating pattern forward along the lists, indefinitely.
To reach any height on the number line, the prime's cycle is simply shifted forward by the prime itself. For prime 7, shifting the cycle by 7 lists gives:
The result is a double infinite cycle: the mod 60 structure repeats every 60 numbers; the prime's offset cycle repeats with its own period. To reach any target height — 10²¹, 10²⁶, 10²⁷ — all that is needed is to translate the prime's cycle along the mod 60 cycle until the target lists are reached. No precomputed archive. No lookup table. No limit imposed by register width.
The distinction that defines GC-60 is not about speed — it is about the direction of the operation.
The framework evolved in three steps, each preserving the passive projection principle while increasing efficiency and parallelism.
16 residue offsets repeating every 60 numbers. A single binary receives the projections of all known divisors. This is the original formulation — complete and deterministic from the first implementation.
The same geometry restated on modulo 30 — a more compact representation of the same 16 candidate positions, halving the period of the base cycle. Same determinism, smaller footprint per cycle.
The single binary becomes three: Sieve_0, Sieve_10, and Sieve_20 — each covering a disjoint partition of residue classes mod 30. No candidate is ever checked twice. This is not simply parallelism: it is a partition of the candidate space that eliminates redundancy at the root, delivering approximately 30% faster execution than the single-binary implementation at identical scale.