Paper 13: leakage as a transmission problem

Paper 13 asks the same question as Paper 3, what is the leakage between branches, but in the place where the answer is forced to be clean: a slab of topological field theory. The result is the most stringently bounded claim in the framework, because in this setting the transmission is either zero or one, and the paper’s value is in locating which side of that dichotomy the framework’s hypotheses actually fall.

The setting

A SymTFT, a symmetry topological field theory, is the modern way to make symmetry structure geometric: a bulk topological theory whose boundary conditions encode a symmetry, with the boundary theory living on one face. The framework’s version is a slab: bulk category in the middle, branch theories on the two faces, and the question is whether an excitation on one face can propagate through the bulk and correlate with the other. In the topological setting this is not a fuzzy question. Operators either exist or do not, and any operator that exists has computable matrix elements.

The dichotomy

The paper’s theorem-shaped result: for the framework’s category, transmission through the slab is gated by whether the condensed anyon content supports a topological line connecting the two boundaries. If no such line exists, the transmission is exactly zero: the branches are absolutely isolated, stronger than any suppression, and the leakage hypothesis is dead in this setting, full stop. If such a line exists, the transmission is exactly one times the line’s algebra: a definite, structure-preserving channel, with matrix elements fixed by the category’s data, not tunable.

The interesting case, and the framework’s honest position, is that the answer depends on the condensation pattern, which the bulk theory does not currently fix. The paper computes both branches of the dichotomy: with the minimal condensation set, the connecting line does not exist and isolation is exact. With the enriched set that the bulk’s anomaly structure permits, a line exists, and its transmission is a specific unitary operation, patterned, with correlations that are the structure the lab protocols of Paper 14 look for.

Why a dichotomy is useful

A hypothesis that says some small amount leaks is hard to test, because every null result can be absorbed by making the amount smaller. A hypothesis that says the transmission is topologically gated, zero or a specific unitary, is different: it cannot be tuned. The framework’s models must commit to a condensation pattern, and each pattern has consequences that are exact. The paper’s contribution is making the commitment possible to audit: once the pattern is named, the transmission is arithmetic.

It also explains, structurally, why the leakage story and the noise story are different hypotheses. Thermal noise in a detector is local, featureless, and uninspiring as evidence. A topological transmission line’s correlations carry the category’s fingerprint: the same algebra, the same channels, every time. That is the patterned signature the whole test programme is built around, and this paper is where the pattern’s existence is either proven or refuted for the specific category, with no knobs in between. The next paper moves from the slab to the laboratory, where that difference has to survive contact with a real detector.

The next paper turns that signature into a laboratory protocol, written before anyone touches a detector: Paper 14, lab correlated errors.