Complete classification of weight-3 logical operators in the Face Centered Cubic lattice CSS code

The [[3L³, 2L³+2, 3]] CSS code on the Face Centered Cubic (FCC) lattice reaches a 67.7% encoding rate with weight-12 stabilizers. Its distance is 3. We classify every weight-3 logical operator of the code. The X-type ones are the 4L³ triangular faces of the lattice. The Z-type ones are the 4L³ claws, each formed by the three edges of a tetrahedron meeting at one of its vertices. The two families are exact duals under the exchange of vertices and octahedral voids, and each spans 2L³ - 1 independent logical classes. We prove this count analytically from the homology of the 3-torus, not only numerically: the cycle space has dimension 5/2L³ + 1 , the boundary space has dimension 5/2L³ - 2 , and subtracting the stabilizer rank leaves 2L³ - 1 for every even L. So k = (2L³ - 1) + 3 , and the excess rate is carried entirely by weight-3 operators. We then show that this sector cannot be protected. The three-body parity ZeₐZebZec at a face anticommutes with four octahedral stabilizers, so it is not measurable, and no measurement of any kind can distinguish |ψ from L̂T|ψ for arbitrary encoded |ψ . Removing the sector requires adjoining all 2L³ - 1 independent directions, which returns $$k = 3$$ . Verified exactly over GF(2) at $$L = 4, 6, 8, 10$$ .