The shortest clean account keeps three labels distinct:
A green face supports both X and Z checks. A green boundary is not therefore an "X boundary." The anyon-condensation classification of Kesselring et al. supplies the invariant meaning: name a boundary by the excitation sectors it condenses, then use color or Pauli words only after that dictionary is fixed.
The constructor chooses every data qubit whose integer-grid column is zero. For odd distance d, this left-side support L contains exactly d qubits. Create two Pauli operators on the same binary support:
X_L = product_(q in L) X_q,
Z_L = product_(q in L) Z_q.
Every retained face meets the left side on zero or two vertices. Hence the binary incidence vector ell satisfies H ell^T=0: X_L commutes with all Z checks and Z_L commutes with all X checks. Their mutual symplectic overlap is |L| mod 2=d mod 2=1, so they anticommute.
The patch has k=1. An anticommuting normalizer pair must therefore represent its one logical qubit rather than a spare stabilizer pair. Multiplying either operator by face stabilizers deforms the support without changing the protected action. The colored-homology framework of Bombín and Martín-Delgado explains which deformations and endpoint condensations are topologically equivalent.
The declared support proves
d_code <= weight(X_L)=weight(Z_L)=d.
It does not prove equality. Distance minimizes weight over every Pauli in the normalizer but outside the stabilizer group. A different colored string, a branched string-net, or a mixed representative could in principle be lighter. The lower-bound step must exclude all of them.
The Atlas uses two evidence routes:
| instance | upper witness | lower-bound route | structured origin |
|---|---|---|---|
d3_triangle | left side, weight 3 | exact CSS normalizer enumeration | verified |
d5_triangle | left side, weight 5 | exact CSS normalizer enumeration | verified |
d7_triangle | left side, weight 7 | triangular-family literature theorem | claimed, chamberland2020triangular |
For the first two instances, qeclib.distance.exact_distance enumerates the CSS normalizer quotients and returns no lighter nontrivial logical. For the thirty-seven-qubit instance, the constructor still recomputes the support, zero syndrome, anticommutation, and weight seven. The missing machine-owned fact is the exclusion of weights below seven. The card therefore routes the family claim to wanted:certified-lower-bounds and does not relabel the cited lower bound as verified.
At d=3, the three clipped faces have supports
{0,1,2,3}, {1,2,4,5}, and {2,3,5,6}.
Using them in both CSS sectors gives the seven-qubit self-dual Hamming/Steane stabilizer code up to qubit and generator relabeling. This is an intersection of descriptions: the Steane entry emphasizes a fixed algebraic block, while this entry emphasizes the first patch in a growing local topological family. It would be wrong to infer that every Steane-code protocol scales with this geometry, or that evidence on one seven-qubit block demonstrates family suppression.
On a closed color-code surface, global face relations alter stabilizer rank and handles supply different logical classes. A triangular patch with Pauli boundaries changes which sectors condense. Another finite staircase can keep the honeycomb bulk but change the qubit formula or boundary check weights. This is why the constructor rejects any boundary string other than three-color-triangle instead of treating it as display metadata.
The practical test is algebraic: after changing a boundary, rebuild H, recompute its rank and normalizer quotient, and re-establish the logical representatives. Neither 6.6.6 nor a triangular outline is sufficient evidence by itself.