Ramification Portraits of Rigid Lattès Maps

We classify the abstract weighted ramification portraits of rigid complex Lattès maps. For a Lattès map induced by an affine torus endomorphism of degree \(d\) , the complete portrait is determined by its action \(\phi\) on the finite branch-value set and by a uniform fiber formula. Reducing the affine map on the four possible Euclidean orbifolds yields nine affine functional graphs for signature \((2,2,2,2)\) , four graphs for \((3,3,3)\) , four Gaussian parity cases for \((2,4,4)\) , and four Eisenstein divisibility cases for \((2,3,6)\) . We give the critical-leaf multiplicities in every case and prove realizability. We also answer the overlap question from AIM Problem 6.4: every flexible Lattès portrait, in every possible square degree, occurs for a rigid Lattès map of the same degree. The result classifies weighted directed graphs; it does not classify maps up to Möbius conjugacy or retain cross-ratios and multiplier data.