LCL-833: A Jones,Khovanov Augmented Liouvillian Framework for Logical Protection in Genus-5 Surface Codes

LCL-833: A Jones,Khovanov Augmented Liouvillian Framework for Logical Protection in Genus-5 Surface Codes

Authors/Creators: Lessard, Guillaume (El'Nox Rah), iD01t Productions, Quebec, Canada

ORCID: 0009-0000-3465-3753

Description:

We present LCL-833, a mathematical framework for topological quantum error correction on the genus-5 surface code [[832, 10, 4]]. Logical protection is achieved by weighting homology cycles with the Jones polynomial evaluated at the fifth root of unity q = e^(2πi/5), augmented by a perturbative correction derived from the total rank of the reduced Khovanov homology. The resulting effective protection parameter δeff(K) = vJ(K) + 0.05 RKh(K) is calibrated to control the operational coherence weight αop and the spectral gap Ggap of an explicit Gorini,Kossakowski,Sudarshan,Lindblad (GKSL) Liouvillian generator for the logical channel ΛL = D ∘ N(n) ∘ E.

All quantities are anchored to the trefoil knot 3₁ using the LCL-832 calibration (αop = 0.8783, Ggap ≈ 0.1217). A stratified mapping based on Johnson's Stratified Axiomatics (2025) enforces acyclic dependency between topological invariants and operational parameters, preventing semantic drift. The construction yields a CPTP channel with a unique fixed point. Numerical validation on 31 knots confirms consistency with an overall improvement in logical protection as knot complexity increases.

The framework analyzes a topological,Liouvillian relation ω/α = g, 1, a heuristic universal genus-g scaling law, and compares the protection hierarchy with standard planar and toric surface codes. An empirical anchor from LCL-832 provides a minimum-time prediction Tmin = ⌈53 ln 2 / |ln Ggap|⌉, machine-verified for all 31 knots.

Keywords: topological quantum error correction, surface codes, logical channels, Liouvillian dynamics, Jones polynomial, Khovanov homology, stratified axiomatics, genus-5, GKSL, stabilizer codes.

Resource type: Preprint

License: CC BY-NC-ND 4.0

Publisher: Zenodo

Language: English

Community: LCL-832: Quantum-Computational Framework for Self-Referential Systems (id01t)

Related identifier: Preceded by DOI 10.5281/zenodo.18743234 (LCL-832)