A Root-breaking Pointwise Superhypothesis and its Consequences

This paper investigates three central analytic challenges in number theory—the BinaryGoldbach Conjecture (BGC), the Riemann Hypothesis (HRH), and Legendre’s Conjecture(LC)—and formalizes their interrelations through two analytic postulates. We define theWeak Minor Arcs Hypothesis (HWMA) and the Root-breaking Pointwise Superhypothesis(HRBPSH), the latter surpassing the classical x1/2-barrier of HRH. By analyzing the circlemethod, short-interval estimates, and bilinear exponential sums, we show how HRBPSHwould imply both HWMA and HRH, while HRH alone cannot reach the pointwise precisionneeded for LC.