Paper B: Pythagorean-Golden Quantile Duality in the Prime Gap Ratio Distribution

Let r_n = gₙ₊₁/g_n be the ratio of consecutive prime gaps, and let F be the limiting distribution of {r_n}. We establish, conditionally on the Hardy-Littlewood prime tuples conjecture, that the two most prominent Pythagorean mass points of F — at ratios 2/3 and 3/2 — straddle the golden probability levels phi⁻² approximately 0.382 and phi⁻¹ approximately 0.618 respectively: F(2/3^-) < phi⁻² < F(2/3) and F(3/2^-) < phi⁻¹ < F(3/2). We call this the Pythagorean-Golden Quantile Duality. The inversion symmetry F(r) + F(1/r^-) = 1 and median exactly 1 are proved conditionally on Hardy-Littlewood. We identify a three-step conditional proof structure for the main inequality: Step 1 (proved) expresses F(2/3^-) as a Hardy-Littlewood weighted sum; Step 2 (proved) shows this sum is dominated by 6-smooth gap pairs; Step 3 (open) requires showing the algebraic structure of the Hardy-Littlewood singular series forces phi⁻² as the exact upper bound. An unconditional analytic proof remains an open problem (OQ-B-1). All results are verified over 1,270,605 consecutive prime gap ratios (primes up to 5 times 10^7), with convergence confirmed across multiple sample sizes. We prove uniqueness: among all Pythagorean ratios 2^a times 3^b carrying significant probability mass, 2/3 and 3/2 are the unique ratios straddling the golden levels. The Cramér model gives F(2/3^-) = 2/5 = 0.400 > phi⁻², confirming that the Hardy-Littlewood singular series structure — not the Cramér approximation — is essential to the result.