Self-Referential Fibration Snake 2: Higher-Order Snake Equations and Cascaded Error Spectra of Topological Codes

This paper establishes higher-order snake equations within the framework of
adjoint chains in tensor categories, and strictly restricts their physical carrier to
the encoding–recovery composite of concatenated quantum codes. The core con
clusions have a three-layer structure. First, the snake equations induced by duality
hold automatically by the coherence theorem for compact closed categories and
carry no error-correction information. Second, the encoding–recovery composite
holds in the exact-recovery case if and only if the Knill–Laflamme condition is sat
isfied; in the broken case, it is characterized by two-sided inequalities between the
logical error rate and the deviation of the snake diagram in diamond norm. Third,
taking the Steane [[7,1,3]] code as the component code, within the model of a
single bit-flip sector, ideal syndrome measurements, and per-syndrome minimum
weight decoding, we exactly enumerate the decoding failure set of all 128 errors,
obtain a degree-seven polynomial for the logical error rate and the complete er
ror spectra for concatenation depths one through six; the threshold fixed point is
0.0645962393 (single sector), and the leading coefficient 21 is determined by the
combinatorial fact that none of all two-qubit errors escapes. Under complement
symmetry, the fixed-point ledger is completed to five points (including the end
point 1), and the basin-of-attraction structure is closed accordingly. The recursion
structure of the cascaded error spectra is an exact equality under independent
and identically distributed Pauli noise, blockwise independent decoding, and per
syndrome minimum-weight strategies; the corresponding relations under full Pauli
noise, correlated noise, measurement noise, and non-Pauli effective channels are
explicitly registered as open boundaries, and partial results are not used to replace
general conclusions.