We prove that the logarithmic derivative of the completed Riemann xi function satisfies an exact anti-symmetry about the critical line: Rexi'/xi(sigma+it) = -Rexi'/xi((1-sigma)+it) for all sigma, t in R. The proof is one line from the functional equation. This odd symmetry theorem has three immediate consequences: the critical line sigma = 1/2 is the unique zero of the function f(sigma) = Rexi'/xi(sigma+it) for each fixed t; the problem of proving the Riemann Hypothesis is equivalent to showing f 0 iff RH was proved by Sondow and Dumitrescu (2010); the odd symmetry is new and provides its organising principle. We prove two unconditional results: the outer boundary Delta E > 0 for sigma > 7/12 (from the Heath-Brown density theorem), and a symmetric pair reinforcement theorem showing that hypothetical off-critical zeros within the density-allowed band preserve rather than undermine the sign. Three paths to close the remaining strip (1/2, 7/12) are identified as open problems.
Paul Buchanan (Sun,) studied this question.