SATI CODEX / LCL-832 / LCL-833: Unified Scientific Publication on Genus-5 Topological Error Correction and Liouvillian Spectral Dynamics (April 2026 Final Edition)

This unified publication consolidates the SATI CODEX and the LCL-832/833 frameworks into a definitive scientific specification for topological quantum information processing. The manuscript provides the rigorous theorem-grade core for the [[832, 10, 4]] CSS surface code constructed on a closed orientable genus-5 surface ($k=10$, dim(HC)=1024). Central to this edition is the T1 Restoration Patch for Proposition 7.4, which resolves prior logical inconsistencies by establishing a code distance lower bound (d ≥ 4) through a graph-girth proof derived from quadrilateral and pentagonal face-degree distributions. The work formally proves the complete positivity and trace preservation (CPTP) of the logical channel ΛL under composition and derives the Liouvillian eigenvalue spectrum λ₁ = -α ± i(g-1)ω, confirming the frequency-to-decay ratio of $g-1=4$. Technical utilities include the derivation of the machine-precision stopping law (Tₘᵢₙ=18) for IEEE 754 double-precision compliance and a first-principles SU(2)₃ calibration map linking the trefoil Jones polynomial to logical error rates. This edition officially closes the four historical mathematical gaps (G1–G4) regarding parity-check construction, spectral gap uniqueness, Kraus irreducibility, and knot-based calibration, providing a turnkey operator algebra for Z₁₂ × Z₁₂ logical governance. Keywords: Quantum Error Correction (QEC), Topological Surface Codes, Liouvillian Spectral Theory, Genus-5 Topology, [[832, 10, 4]] Code, SATI CODEX, Logical Channel Dynamics, CPTP Closure, Jones Polynomial, Khovanov Homology, Machine-Precision Stopping Law, Stabilizer Codes, Binary Symplectic Geometry, Open Quantum Systems.