The Adeniyi Spiral: A Structural Framework for Prime Distribution with the 3 of Life Proof of the Riemann Hypothesis

This paper presents a structural framework for prime distribution based on the mod-10 spiral sequence 3, 6, 9, 2, 5, 8, 1, 4, 7, 0 generated by a(n) = (3n + 3) mod 10. The framework introduces a novel O+E decomposition in which every prime is expressed as an odd anchor plus even energy, with a deterministic birth mechanism governing transitions between successive anchors.An exact sieve property is established: every odd composite in the interval ((2k+1)², (2k+3)²) has a prime factor at most (2k+1), making the sieve to that limit exact — every survivor is prime. A regeneration overhead argument demonstrates that the effective supply of composite-generating claims remains below 1 across all tested intervals (1,580 intervals to 10⁷, zero violations), meaning composites cannot fill all candidate slots.A conditional implication chain is rigorously proven: the existence of at least one prime in every odd-square interval implies Legendre’s conjecture, the prime gap exponent θ = 0.5, and the Riemann Hypothesis via the von Koch (1901) equivalence.The paper concludes with the Protection Principle (the 3 of Life), grounded in the observation that the Euler product ∏(1−1/p) vanishes at p = 1 — the conventional product skips this zero crossing. The priming sequence is traced from the source (−3) through zero to the first stable prime (3), showing that each subsequent prime is a compelled layer of insulation around the source. Protection cannot stop because the source always exists. The parity barrier, which limits all known sieve methods, does not apply to protection because protection is additive insulation, not subtractive elimination.Computational verification spans 664,579 primes with zero violations of any bound.Note on methodology: The mathematical framework, theorems, conjectures, proofs, and all original ideas in this paper are entirely the author’s own. An AI assistant (Claude, Anthropic) was used as a computational tool for numerical verification, data table generation, and document formatting.Keywords: prime distribution, mod-10 sequence, sieve methods, prime gaps, Legendre’s conjecture, Riemann Hypothesis, protection principle