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 L under composition and derives the Liouvillian eigenvalue spectrum ₁ = - i (g-1), confirming the frequency-to-decay ratio of g-1=4. Technical utilities include the derivation of the machine-precision stopping law (T₌₈₍=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.