Structural Reduction of Two Twin-Prime Conjectures and Conditional Resolution under Hypothesis H

Abstract

I introduced the Twin-Prime Propagation Conjecture and the Twin-Gap Existence Conjecture in two earlier notes. Here I give a complete structural reduction of both statements.A covering argument shows that the second linear form Dn is never a lower twin prime after the first pair; consequently the propagation conjecture reduces to a single admissible 6-tuple of linear forms. That 6-tuple is a special case of Schinzel’s Hypothesis H, so the propagation conjecture follows from Hypothesis H. The gap-existence conjecture is strictly stronger: it is a uniform short-interval form of the Bateman–Horn law. Both statements imply the classical twin-prime conjecture. Neither is accessible to present sieve methods unconditionally. A companion Python certificate (prove section6.py) machine-checks everymodular lemma, the admissibility statements, the logical reductions, and the computational claims of Section 6.