Paper L: The Odd Symmetry of the Logarithmic Derivative of the Riemann Xi Function and Variational Structure of the Gap Strip

We prove that the logarithmic derivative of the completed Riemann xi function satisfies an exact anti-symmetry about the critical line: Re[xi'/xi(sigma+it)] = -Re[xi'/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) = Re[xi'/xi(sigma+it)] for each fixed t; the problem of proving the Riemann Hypothesis is equivalent to showing f < 0 for sigma in (5/12, 1/2), a left-side reformulation accessible to different analytic tools; and the gap strip (1/2, 7/12) has a self-symmetric structure. The monotonicity equivalence Re[xi'/xi] > 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.