Tunnell's Theorem and #P-Completeness

We propose a new hypothetical framework to explore the relationship between the Birch--Swinnerton-Dyer conjecture (BSD) and computational complexity theory. This paper introduces two central conjectures: a Reduction Conjecture, which posits the existence of a polynomial-time parsimonious reduction from \#P-complete problems to the counting of integer solutions for Tunnell's equations, and a Solution Density Conjecture, which posits that these solution counts are sufficiently well-distributed. We then formally prove that if these two conjectures are true, a surprising conditional result follows: the assumptions of P = NP and the truth of the BSD conjecture would imply the collapse of the counting complexity class \#P to FP. Our main conditional result is that if BSD is true, our conjectures hold, and the widely believed separation \#P $\neq$ FP holds, then P $\neq$ NP. This work reframes the P versus NP problem as a question about two deep, open problems in number theory: the existence of a "complexity-to-arithmetic" reduction and the statistical distribution of solution counts to Tunnell's specific quadratic forms.