A Seeded Automaton View of Goldbach Representations and Incremental Prime Emergence

We propose a structural reformulation of even Goldbach representations organised aroundtwo canonical geometries. For each even integer N = 2c, we distinguish the left-anchorgeometry, given by the Goldbach representation with smallest left prime, and the centralgeometry, given by the representation with minimal symmetric deviation from the centre c.Under this separation every odd prime p ≥ 7 has a canonical birth at N = p + 3 via thepair (3, p), and a canonical reflection at N = 2p via the pair (p, p). These two events delimita generative intervalI(p) = [p + 3, 2p].Inside this interval, every Goldbach representation N = a + b with a ≤ b has its left memberbounded by a ≤ p. Thus, at the birth of p, all possible left candidates needed to testGoldbach locally up to 2p are already available.We prove an internal self-generation property: every prime q with p < q ≤ 2p − 3 has itsbirth node q + 3 inside I(p). The only possible missing boundary case is q = 2p − 1, whosebirth node is the next even point after the reflection node 2p. This gives a natural transitionbetween consecutive generative zones.We also exhibit a seeded prime generator which starts from {2, 3, 5} and recovers allsubsequent primes through the birth rule q = N − 3, rejecting composites by divisibilityagainst previously generated primes. The seed {2, 3, 5} corresponds precisely to the wheelmodulo 30 = 2 · 3 · 5.We do not claim a proof of Goldbach’s conjecture. The contribution is a self-containedstructural viewpoint linking Goldbach representations, prime birth, reflection, local closure,wheel-sieve geometry, and a birth-indexed order of prime generation.