The uncertainty principle in the distribution of primes - two structural obstacles to the Riemann Hypothesis

Twenty attempts to make progress on the Riemann Hypothesis were carried out with numerical verification and pushed to the point where the obstacle became explicit. In every case the difficulty reappeared at comparable magnitude, migrating from one object to another without diminishing. Two distinct mechanisms account for this pattern. The first is the Fourier duality between primes and zeros established by the explicit formula; from it follows an uncertainty principle — a particular case of the Heisenberg–Weyl theorem for conjugate pairs — under which reformulation redistributes information without creating any. The second lies in the structure of the target: the critical line has measure zero within the strip, and no finite verification covers a positive fraction of the infinitely many zeros — the 1013 already confirmed represent a fraction of exactly zero. One accounts for the sterility of reformulating; the other, for that of computing. This work does not prove the Riemann Hypothesis, nor does it propose a route to proving it; it documents why two natural modes of attack cannot, on their own, succeed.