d_code = 64

The framework assigns the boundary between branches a code distance of 64. This post explains where that number comes from, because it is one of the few that is both honest arithmetic and a cautionary tale about how I first used it.

What code distance means

In quantum error correction, the distance of a code is the number of local errors needed to turn one valid memory state into another indistinguishable one. Distance 64 means a chain of 64 elementary failures, lined up just so, before the protected information silently corrupts. Everything else is detected and corrected. Distance is the wall thickness of a storage medium, measured in failures rather than metres.

Where 64 comes from

The boundary category, from the earlier posts, has a large anyon content with gauge structure. Counting only the physical sectors, those that satisfy the Z6 admissibility condition from the congruence post, gives 50 anyon types. To label which of 50 possibilities you are holding, you need a register: 2 to the 6 is 64, and 64 is the smallest power of two that covers 50. So the natural encoding uses 6 qubits per anyon, and the code distance of the anyon register is 2 to the 6, which is 64.

Notice what this is. It is a statement about counting sectors and taking a ceiling. Two to the 6 exceeds 50, so the distance is 64. It is arithmetic, and the arithmetic is correct. Whether it deserves the name derivation is exactly the question the second half of this post is about.

The use that went wrong

Distance 64 entered the leakage model as the exponent in a suppression. The interface paper wrote the baseline leakage as an exponential whose action is the code distance times a logarithmic factor: about 147 in natural units, giving a suppression of roughly 1.07 times 10 to the minus 64. That is a small number, and for a boundary that is not supposed to let anything through, small is good.

The problem was not the arithmetic. It was the inference. A distance computed from a sector count is a property of the encoding chosen, not a measured thickness of anything. The paper treated the exponential of it as a physical prediction with 64 orders of authority behind it, when what stood behind it was: 50 sectors, rounded up to a power of two, exponentiated.

Two specific complaints, from the audit and from the later competing-avenues paper. First, the ratio inside the logarithm was reverse-engineered to hit a pre-committed target, which makes the 147 an echo of the target, not a prediction of it. Second, the whole construction is a proxy borrowed from quantum error correction for a quantity, branch leakage, for which no actual code is specified. An exponential of a proxy is a proxy with stage fright.

What survives

What survives is the structural claim, which is worth keeping and is all the papers now claim. If branch isolation is an error-correction problem at all, then leakage is suppressed exponentially in the code distance, and the distance is bounded above by the number of sectors the category can distinguish. Different sectors, thicker wall. That is a real constraint shape, even with the exponent demoted from prediction to illustration.

The numeric version, 64, and the suppression, 10 to the minus 64, are now labelled benchmark: consistent, reproducible, and carrying no derivational authority. The next post is the repair of the leakage formula itself, which was a worse error than this one and had a better ending.