Sharp colour bounds and structural obstructions for three-direction sums–differences constructions

We prove the sharp colour bound |S_0|,|S_1| ≥ 3D for every complete literal ladder in the three-direction sums–differences framework, at every length and initial kernel dimension D. For either coordinate rung the proof is entirely by hand; the sum-rung case uses the preceding ladder score theorem. Equality is attained in every positive dimension. Combined with shape-independent bounds in dimensions one and two and further infinite-family theorems, this gives a structural obstruction: any construction improving the Katz–Tao exponent 11/6 must have dimension at least three and escape the ladder, double-path, unpinned three-layer and wide constant-rung families. Computer-assisted extensions cover constant coordinate-rung width-three ladders in dimension three, width-four ladders through dimension five, dimension-three width-four boxes with at most six rung positions, and block-free sum-rung boxes in dimension three. A rectangular copy lemma and explicit nonliteral counterexamples identify essential structural restrictions. Three hexagon-bundle theorems preserve the sharp colour bound under specified links, and exact checks of 31,064 hexagon-join constructions confirm the dimension-two colour bound with margin. The proofs include a self-contained entropy translation, reproducible finite verification and a Lean formalization of the double-path algebra. The results were discovered and developed using CASPAC, the computational mathematics engine of the author's CASP research system.