Unifying Goldbach's Conjecture through Geometric Suppression and Distributional Concentration

This work presents an interdisciplinary synthesis addressing Goldbach's conjecture from the perspective of theoretical physics. We propose that the distribution of prime numbers can be modeled under the Feynman path integral formalism. Using a test family of paths xn(t)=tn+1xn(t)=tn+1, we demonstrate that phase cancellation in the minor arcs of Goldbach is analogous to the suppression of high-energy paths in quantum systems. We show that the classical action Sn∼n/4Sn∼n/4 leads to rapid oscillations, rendering all but the classical path negligible in the semiclassical limit. Furthermore, we prove that the normalized curvature converges weakly to a Dirac delta: x¨n/(n+1)⇀δ(t−1)x¨n/(n+1)⇀δ(t−1). This "mass concentration" is then mapped to the Goldbach problem, where the exponential sum S(α)=∑p≤ne2πipαS(α)=∑p≤ne2πipα exhibits a similar concentration in the major arcs. We introduce a heuristic rarity function F(n)=exp⁡(−S(n)n/(log⁡n)2)F(n)=exp(−S(n)n/(logn)2) and show that its integral converges, with its mass concentrated on numbers already verified up to 4×10184×1018. The parity barrier, which stymies classical sieves, is reinterpreted as a smoothness condition that the Dirac delta (via its singular nature) can potentially bypass. The work is an expository synthesis, offering a new geometric lens on an old arithmetic problem.