Architecture & Education

1 · The mod-30 partition, N = 30·B + L

Every integer N is written as a block index B and a residue L with 0 ≤ L < 30. Because 2·3·5 = 30, the residue alone decides whether N can avoid the base primes. Exactly eight residues are coprime to 30:

The other 22 residues per block are excluded immediately by 2, 3 or 5 and never enter a lane. 2, 3 and 5 themselves are base primes that build the wheel, so they sit outside the eight lanes.

2 · Discover → absorb → preserve → observe → activate

• Discover — candidate admissible locations exist inside the eight lane nodes as soon as their block is materialised.
• Absorb — every observed prime p owns a composite wave which projects collisions onto future locations. A hit location is absorbed (composite).
• Preserve — an untouched location is a survivor. It waits in the lane reservoir as survivor-unobserved until the frontier reaches it.
• Observe — when the observation frontier arrives at a surviving location, it becomes an observed determinant prime.
• Activate — the new prime's wave switches on at p², because every smaller multiple of p was already absorbed by a smaller wave.

3 · Why there is no per-candidate trial division

A trial-division test asks, for a single N, whether any divisor up to √N divides it. The PCF engine never asks that question. It only records whether a location was hit by an already-active wave. Work is done once per wave, not once per candidate: primes are found by absence of collisions, not by testing divisors.

Mathematically this is an incremental wheel-sieve construction (a mod-30 wheel over a sieve of Eratosthenes with p² start). That is well-understood, correct, and efficient — but it is not a novel primality proof, and mod-30 location by itself proves nothing about primality. PCF outputs are therefore labelled “PCF survivor / observed prime”.

4 · State vocabulary

Absorbed locations carry a collision certificate: the absorber prime p, its activation p² and the collision position. The cofactor is reported from the known schedule, never used to make the decision.

5 · Eight independent lane nodes and reintegration

Each residue class runs as its own node with its own visible collision frontier, absorbed counter and survivor list. Because each wave projects onto the lanes on a fixed schedule, the nodes can be advanced independently and then reintegrated in numeric order to produce the combined prime stream.

block B      L=1   L=7   L=11  L=13  L=17  L=19  L=23  L=29
B = 3         91    97   101   103   107   109   113   119
              ×     ◆     ◆     ◆     ◆     ◆     ◆     ×
              7-wave                                    7-wave