Self-Referential Fibration Snake 2: Higher-Order Snake Equations and Cascaded Error Spectra of Topological Codes
Abstract
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.
Related
- Self-Referential Fibration Figure 3: Order Effects of Cosmological Comoving Truncation—Window-Operator Commutator and Fisher Adjudication
- Self-Referential Fibration Figure 1: Naturality Failure of Coarse-Graining—Explicit 2-Morphism Coordinates of the Commutation Obstruction
- Self-Referential Fibration Figure 4: Full-Order Combinatorial Enumeration Formula for Gauge Obstructions on Coarse-Graining Fibers
- Yuanxian Theory: Explicit Evolution Differential Equation of the Self-Referential Mind Field — A Unified Field Framework Based on the 𝒯⁶⁴ Topological Ontology
- Predicting Entanglement Spectra of Gapless Symmetry-Protected Topological States in One Dimension
BibTeX
@misc{zhou2026selfreferential,
title = {{Self-Referential Fibration Snake 2: Higher-Order Snake Equations and Cascaded Error Spectra of Topological Codes}},
author = {changzheng zhou and ziqing zhou},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23493971},
url = {https://zenodo.org/records/23493971},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Self-Referential Fibration Snake 2: Higher-Order Snake Equations and Cascaded Error Spectra of Topological Codes
AU - changzheng zhou
AU - ziqing zhou
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23493971
UR - https://zenodo.org/records/23493971
AB - 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.
ER -