A rational quantum channel beyond finite tracial baths

A quantum channel on a finite system always has a finite realization with a pure environment. With a maximally mixed environment this can fail even approximately: since MIP* = RE, some factorizable channels lie outside the closure of the finite matrix-tracial factorizations (Haagerup and Musat). This paper constructs an explicit one: a unital channel on fifteen qubits with 11,230 Kraus operators, whose entries are integers divided by five, that factorizes exactly through a type II_1 factor but cannot be approximated with any finite maximally mixed environment. The construction starts from an explicit finite presentation, with 43 generators and 4,417 relators, of a group that maps into the amalgamated double of a Kun–Thom pair over the field with five elements. In it, Thom's witness commutator becomes a word of length six, which every tracial matrix-ultraproduct representation kills by the centralizer-rigidity theorems of Alekseev, Liu and Thom and of Thom (recent preprints). Rational two-level blocks enforce the relations, and the Choi matrix gives the word a trace the identity cannot have. Against finite tracial channels whose Choi support lies inside the target's, the channel is at normalized Choi distance more than 1/316 and half-diamond distance at least 1/2. Its positive distance to the closure forces memory linear in the number of uses for causal repeated use at small error and bounded exchange. For a larger, fully referenced version of the channel, a uniform relator tolerance, which exists but is not evaluated, would give explicit lower bounds on the distance and on the memory constant. This archive includes the paper, whose appendices give the full presentation and all proofs, the exact channel data, programs that rebuild every channel from the presentation, and independent finite checks.