Deriving CSS codes from a single zero-sum axiom

We show that the algebraic skeleton of **CSS (Calderbank–Shor–Steane) quantum error-correcting codes** follows from a single axiom rather than being postulated. The axiom — *zero-sum*: the weights of a closed sequence of moves sum to zero — is the content of Z0 in the Zero programme, a discrete-physics derivation of spacetime and matter from one primitive. From it (together with local compensating moves) we obtain, in order: 1. a **cycle space** over $\mathbb F_2$, i.e. a classical linear code with parameters $[E,\ \beta_1=E-V+c,\ \mathrm{girth}]$; 2. the **commutation criterion** for vertex/face stabilisers, proved unconditionally when faces are closed words (Theorem A, §4); 3. the **closed form** for the number of logical qubits, $k=E-r_X-r_Z$ (Theorem B, §5); 4. the **distance** as an *essential girth*, and the observation that the naive $d=\mathrm{girth}$ is false (Theorem D, §6); 5. the **binary operator algebra on each edge**, including a *real* anticommutation $F_eZ_e=-Z_eF_e$ with no factor of $i$, and the tensor-product structure $(\mathbb C^2)^{\otimes E}$ as a consequence rather than an assumption (Theorems E and F, §7); 6. the **face-selection rule**: not determined by the four candidate primitives we tested (Theorem G, §8.1), but a necessary relation $\sum_f\partial f=0$ always holds, forcing $k=\beta_1-|F|+\mathrm{rel}$; when $K$ is a closed connected surface this collapses to $k=2g$, depending only on the genus (§8.2). The net result is a **reduction of borrowings**. Standard CSS constructions import three ingredients — a qubit register, Pauli anticommutation, and the tensor product. Here the first is *replaced* by a derived per-edge binary operator space (Theorem F), the second is *derived* (F2, or E1 in matrix form), and the third is *derived* (F4). That leaves exactly **one** genuine input — the choice of faces. We also report a structural negative result: the derived constraint is *weaker* than standard homology, so this route cannot **improve** on CSS codes (§9). ---