Polylogarithmic List Replicability of Sign Matrices: Entropy Tilting, Hanner Lower Bounds, Combinatorial Bridges, and the Remaining Logarithm

BHHLT26 conjectured that every N x M sign matrix has list replicability number O(log N log M). For N, M >= 2, we prove lr(B) <= 2^31 ln^2(2N) ln(2M). The proof combines a nonlinear power-map encoding with a relative-entropy tilt that transports bounded-overlap covers without a boosting factor. The corresponding topological bound is Ind S(B) + 1 <= 2^31 ln(2N) ln(2M) min{ln(2N), ln(2M)}. Hanner matrices give Omega(log N log M) lower bounds, a quantitative consequence of existing Hanner geometry and the index-list-replicability bridge. We also establish an unbounded restricted-output/improper separation, exact values for anchored cubes and Hadamard matrices, and exact learning-theoretic values for projective-plane classes. The binary-cube value is due to BHH+26c; the threshold value is due to CMY23. A combinatorial bridge gives lr(C_B, eps) <= chi*(B) <= h_+*(B) <= h(B) <= r_2(B) + 1, while a two-piece equivariant join construction gives the sharp bound Ind S(C) <= 2s + 2 under bounded error intersections. Power-map, affine-sketch, static-label, and average-stability obstructions identify limitations of the available methods. For Paley matrices, inversion mixing reduces the residual-emptiness approach to bounding switchable monochromatic squares. The conjecture remains open, with one logarithmic factor separating the general upper bound from the Hanner lower bound in the square case.