Nonexistence of non-degenerate third-order sum-free functions from F_2^6 to F_2^5

Abstract: A function F₂ⁿ → F₂ᵐ is third-order sum-free if the sum of its values on every affine 3-flat is nonzero. Heering, Kaspers and Taranchuk related non-degenerate higher-order sum-free functions to high-distance subcodes of Reed–Muller codes. We settle the first parameter with m < n permitted by their general lower bound and prove that no non-degenerate third-order sum-free function F₂⁶ → F₂⁵ exists. Equivalently, RM(3,6) has no codimension-5 binary linear subcode of minimum distance at least 12. The proof works in the 42-dimensional quotient of Boolean functions modulo RM(2,6). A first-moment identity forces every nonzero element of a hypothetical 5-space to have maximal affine-3-flat weight 5760. An intrinsic eleven-type analysis of the high-degree quotient shows that only the zero type and a decomposable quartic type can be extremal; the last sextic type is controlled by hyperplanes of the dual polar space DW(5,2). Projecting the hypothetical 5-space to the high-degree quotient then leaves three cases. Positive projection of dimension at least two is excluded by an alternating-form obstruction, dimension one by Pfaffian geometry, and zero projection by a Plücker-parity theorem proved using Chevalley–Warning.