This page derives the exact finite object built by constructor.py. It does not use the instantaneous toric-code equivalence as a substitute for the schedule algebra.
Index the two vertices of cell (i,j) by
A(i,j) = 2(i L2 + j) and B(i,j) = A(i,j) + 1,
with both coordinates reduced periodically. The constructor uses
z: A(i,j)--B(i,j),x: B(i,j)--A(i+1,j),y: B(i,j)--A(i,j+1).Direction fixes the measured operator: ZZ, XX, or YY. This is the first edge label.
The hexagons are properly colored by 0,1,2. If an edge separates faces with colors a and b, its round label is the missing color
r = 3 - a - b.
This is the second edge label. It is transverse to direction: on the shipped 3 x 3 torus, each round contains three XX, three YY, and three ZZ measurements.
At a honeycomb vertex, the three incident edges separate the three pairs of neighboring face colors: {0,1}, {1,2}, and {2,0}. Their missing colors are respectively 2, 0, and 1. Exactly one incident edge therefore belongs to each round.
Consequently, edges with one round label are disjoint and cover every vertex. Their two-qubit Pauli measurements commute because they have disjoint support. For n = 2 L1 L2, every round contains n/2 = L1 L2 measurements. The constructor turns this proof obligation into a hard check by sorting all round supports and requiring the list 0,...,n-1 exactly once.
For a face p, multiply the directional Pauli operator on each of its six boundary edges. Two different letters meet at every vertex. Their product, up to a phase omitted by the Atlas sign-free representation, is the third Pauli letter. The resulting W_p has weight six.
There are two commutation cases:
W_p at two endpoints. Its Pauli letter differs from the plaquette letter at each endpoint, so two anticommutations cancel.W_p at that vertex, so the local operators commute.Thus W_p commutes with every edge measurement in all rounds.
For the 3 x 3 convention, the first three plaquette words are
XY___ZZ___YX______
_ZXY__YXZ_________
___ZXY__YXZ_______The reproduction script constructs all nine words from the six-edge face walks. Every word has weight six, all lie in the steady snapshot row space, and their binary rank is eight. The one dependency is the closed-torus global product relation.
Let the current commuting ISG generators be rows g_i and let p be the next measured Pauli.
p commutes with every g_i and is outside their span, append it.p is already in the span, only its outcome sign changes; the sign-free row space stays fixed.p anticommutes with generators indexed by a set A, choose one pivot g_a. Replace every other g_b, b in A, by g_b g_a, which now commutes with p, and replace g_a by p.The third rule trades one generator for one measurement rather than increasing rank. Binary symplectic addition implements multiplication and the symplectic inner product selects A. This is the exact ISGTracker.measure algorithm used by the constructor.
Starting from no generators on the 3 x 3 instance, the ranks after each complete round are
9, 12, 14, 16, 16, 16,
16, 16, 16, ...Once the rank reaches sixteen, the phase-specific row spaces repeat after each three-round period. Nine current-round measurements plus eight plaquette generators have combined rank sixteen, so their spans share one relation. On eighteen qubits, dynamic_k = 18 - 16 = 2.
The face-adjacency graph is triangular. In the constructor coordinates, a valid closed formula is
c(i,j) = i - j mod 3.
Transport around the first periodic handle shifts i by L1; transport around the second shifts j by L2. The color returns to itself only if both lengths vanish modulo three. The constructor also tries the opposite orientation and constraint propagation, then rejects a torus when no proper coloring is found.
The reproduction script checks all 2 <= L1,L2 <= 9 face graphs and finds a coloring exactly when both dimensions are divisible by three. This finite scan is a regression certificate for the shipped constructor. The winding argument supplies the reason for its accepted family.
The most dangerous wrong constructor measures all XX direction edges, then all YY, then all ZZ. Every round is still a perfect matching. Plaquette products still commute with every measurement. Those two local checks therefore cannot distinguish the protocol.
The long-time ISG does. On the 6 x 6 torus:
| grouping | physical qubits | steady ISG rank | dynamic_k |
|---|---|---|---|
| plaquette missing color | 72 | 70 | 2 |
| edge direction / Pauli letter | 72 | 52 | 20 |
Every entry in this table is recomputed by reproduce.py. The wrong sequence is not a small convention change; it preserves an extensive collection of logical degrees of freedom instead of the two toric logicals.
The schedule's correctness is certified by a chain of independent invariants: proper face coloring, perfect matchings, mixed Pauli content, commuting plaquette products, startup rank profile, periodic rank, and a counterexample that passes the local checks but fails the dynamical count. The literature is needed to interpret the steady ISG as topological order; the finite algebra above is executable and recomputed.