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| Format: | Recurso digital |
| Language: | English |
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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19478368 |
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| _version_ | 1866902257624154112 |
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| author | Lessard, Guillaume |
| author_facet | Lessard, Guillaume |
| contents | <p>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 (<span>$k=10$</span>, <span>$dim(H_C)=1024$</span>). 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 (<span>$d \ge 4$</span>) through a graph-girth proof derived from quadrilateral and pentagonal face-degree distributions.</p> <p>The work formally proves the complete positivity and trace preservation (CPTP) of the logical channel <span>$\Lambda_L$</span> under composition and derives the Liouvillian eigenvalue spectrum <span>$\lambda_1 = -\alpha \pm i(g-1)\omega$</span>, confirming the frequency-to-decay ratio of <span>$g-1=4$</span>. Technical utilities include the derivation of the machine-precision stopping law (<span>$T_{min}=18$</span>) for IEEE 754 double-precision compliance and a first-principles <span>$SU(2)_3$</span> 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 <span>$Z_{12} \times Z_{12}$</span> logical governance.</p> <p><strong>Keywords:</strong></p> <p>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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19478368 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | SATI CODEX / LCL-832 / LCL-833: Unified Scientific Publication on Genus-5 Topological Error Correction and Liouvillian Spectral Dynamics (April 2026 Final Edition) Lessard, Guillaume 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 <p>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 (<span>$k=10$</span>, <span>$dim(H_C)=1024$</span>). 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 (<span>$d \ge 4$</span>) through a graph-girth proof derived from quadrilateral and pentagonal face-degree distributions.</p> <p>The work formally proves the complete positivity and trace preservation (CPTP) of the logical channel <span>$\Lambda_L$</span> under composition and derives the Liouvillian eigenvalue spectrum <span>$\lambda_1 = -\alpha \pm i(g-1)\omega$</span>, confirming the frequency-to-decay ratio of <span>$g-1=4$</span>. Technical utilities include the derivation of the machine-precision stopping law (<span>$T_{min}=18$</span>) for IEEE 754 double-precision compliance and a first-principles <span>$SU(2)_3$</span> 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 <span>$Z_{12} \times Z_{12}$</span> logical governance.</p> <p><strong>Keywords:</strong></p> <p>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.</p> |
| title | SATI CODEX / LCL-832 / LCL-833: Unified Scientific Publication on Genus-5 Topological Error Correction and Liouvillian Spectral Dynamics (April 2026 Final Edition) |
| topic | 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 |
| url | https://doi.org/10.5281/zenodo.19478368 |