Topological Vortex Logic: Stable Winding States on the Three-Torus and Their Classification

We derive the complete classification of stable winding states on the three-dimensional flat torus T³ = R³/LZ³ equipped with Z₃ = center (SU (3) ) orbifold symmetry. The framework, called Topological Vortex Logic (TVL), establishes five theorems from the T³/Z₃ geometry with zero free parameters and a single external identification. Theorem 1 (Stability): A vortex state w ∈ Z³ is stable against spontaneous splitting if and only if |w|² ≤ 3. The proof combines a closed-form algebraic argument for the unstable direction with exhaustive verification of the stable shells. The stable vocabulary consists of exactly 3³ − 1 = 26 states in three families: 6 face (|w|²=1), 12 edge (|w|²=2), and 8 corner (|w|²=3). Theorem 2 (Z₃ Invariant): The charge q₃ = (w₁+w₂+w₃) mod 3 is a topological invariant from the orbifold holonomy, partitioning the 26 states into 8 singlets, 9 unit-charge, and 9 anti-unit-charge states. The assignment B = ±1/3 to q₃ = 1, 2 is the single external identification, imported from QCD. Theorem 3 (A₂ Root Embedding): The six traceless edge vortices form the A₂ root system of su (3), verified by all four root system axioms including the Cartan matrix and reflection closure. What is derived is the root geometry — the kinematic skeleton — not the full Lie algebra. Theorem 4 (Non-Isomorphism): The three vortex families are pairwise non-isomorphic as Z₃-modules, distinguished by fixed-point counts (0, 0, 2) and ρ₀ multiplicities (2, 4, 4). Theorem 5 (Mandatory Hierarchy, conditional): The non-isomorphism forces any coupling sensitive to the Z₃-module structure to treat the three families unequally. The non-isomorphism is unconditional; the mass ordering consequence requires a coupling assumption. The energy model E (w) = |w|²ε₀ operates in the non-interacting limit. Three open walls remain: mass magnitudes, the electroweak sector (requiring an S³ Kaluza-Klein sector), and the exotic rep-6 corner states. This upload contains: the main 18-page paper (T³ as a closed information-processing environment), the 12-page technical derivation with full proofs (code available separately via software DOI 10. 5281/zenodo. 19683377), the standalone TVL. py framework (44 self-test assertions, B₃ root system verification, Z₃-module invariant table), and a README with usage instructions. ─────────────────────────────────────────────────────────────────────────────────────────────────Version History───────────────────────────────────────────────────────────────────────────────────────────────── v1. 0. 4 (April 26, 2026) — DOI: 10. 5281/zenodo. 19779092 TVLCompleteDerivation. pdf — The technical derivation document has been corrected and refined in nine areas. Theorem numbering has been corrected by giving the Definition environment an independent counter, so that the five theorems now number 1 through 5 in exact correspondence with their section headings. The superfluid critical temperature Tc is now explicitly defined at its first use in Theorem 2. A double minus sign typo in the §2. 4 state table has been corrected: (--1, 1, 0) and (--1, 0, 1) now correctly read (-1, 1, 0) and (-1, 0, 1). The AI audit attribution has been removed from the scope note in §4. 2; the mathematical content of the note is unchanged. All seven references are now cited inline in the body text; previously they appeared in the bibliography but were never called out in the text. The references have been reordered 1–7 to match the order of first citation in the body. Citations have been removed from section titles and from inside displayed equations; all citations now appear in the surrounding prose before the relevant formula or statement. A duplicate end-of-proof symbol has been removed from the §4. 2 A2 root system proof, where both a manual blacksquare and the proof environment symbol were appearing simultaneously. The end-of-proof symbol has been unified to the filled black square throughout, replacing the previous inconsistent mix of filled and outline squares. T3InformationProcessingEnvironment. pdf — The end-of-proof symbol has been updated from the outline square to the filled black square to match the derivation paper. Both papers now use the same symbol consistently throughout. The mathematical content, all five theorems, and the SAP epistemic labelling are unchanged from v1. 0. 3. v1. 0. 3 (April 25, 2026) — DOI: 10. 5281/zenodo. 19752456GitHub URL corrected to github. com/MerebashviliV/topological-vortex-logic in T3InformationProcessingEnvironment. pdf. All other files unchanged from v1. 0. 2. v1. 0. 2 (April 24, 2026) — DOI: 10. 5281/zenodo. 19750357 (1) References 1–8 reordered to match order of first citation in text in T3InformationProcessingEnvironment. pdf. (2) Appendix A and Appendix B formally titled "Verification of the B3 Root System" and "Legendre's Three-Square Theorem" respectively. (3) Concept DOI (10. 5281/zenodo. 19682633) added to title page of both papers. (4) TVLCompleteDerivation. pdf streamlined from 23 to 12 pages. Appendix C code listing removed; replaced with pointer to standalone software record (10. 5281/zenodo. 19683377). Mathematical content and all five theorems unchanged from v1. 0. 1. v1. 0. 1 (April 21, 2026) — DOI: 10. 5281/zenodo. 19688501 (1) hbar defined as reduced Planck constant at first use (§3. 2), along with m as particle mass and xi as coherence length. (2) h defined as Planck's constant at first use (§5. 1). (3) c defined as characteristic signal propagation speed at first use (§4. 2). (4) §4. 4 corrected: f1, f2, f3 are mutually incommensurate; f1 is the master clock; edge and corner frequencies are not harmonics of f1. (5) All key display equations numbered: 10 in main paper, 13 in technical report. (6) Acknowledgements added thanking Dr. Z. Merebashvili for proofreading and identifying the Section 4 issue. v1. 0. 0 (April 21, 2026) — DOI: 10. 5281/zenodo. 19682634Initial release. Concept DOI (always resolves to latest version): 10. 5281/zenodo. 19682633─────────────────────────────────────────────────────────────────────────────────────────────────