Topological Vortex Logic: Generation and Colour Structure from T³/Z₃

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 structural identification connecting it to QCD. 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 single structural identification — the orbifold Z₃ with the centre of SU (3) — fixes B = ±1/3 on q₃ = 1, 2; the specific SU (3) representations and particle labels form a separate interpretive layer. 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. The correspondence concerns structure and symmetry, not dynamics: the mass magnitudes and the electroweak sector lie outside what the geometry fixes, and the exotic rep-6 corner states are a structural feature with no Standard Model counterpart. These boundaries are set out in the paper's "Scope of the Correspondence. " Note on this record. This record contains the 12-page Generation and Colour Structure paper (the technical derivation with full proofs). The companion foundational paper, "T³ as a Closed Information-Processing Environment, " is archived separately at 10. 5281/zenodo. 20806554. The standalone TVL. py library — an importable classification framework that derives the full physical classification of any winding vector and labels every derived quantity by epistemic status (fixed by the geometry alone, imported, or open) — is archived separately via software DOI 10. 5281/zenodo. 19683377 and serves as independent confirmation of the paper’s results. A full-paper audit found five defects, two of independent severity, none touching the five theorems; all are corrected in the prepared v1. 0. 6, which also reframes the companion software as confirmation rather than dependency. Changelog v1. 0. 6 (July 5, 2026) — 10. 5281/zenodo. 21206686 — five fixes, no theorem weakened. The SU (3) -representation table’s 2/3-shell breakdown is corrected from 9/9 to 6/6 (the overall 8-singlet/9-quark/9-antiquark split across all 26 states was and remains correct). Theorem 5’s hypothesis is corrected to require both ρ₀ multiplicity and fixed-point count jointly, not multiplicity alone, which gives a degenerate 2: 4: 4 coupling pattern; the conditional-assumptions list is aligned to match, closing a residual inconsistency where it had still named the insufficient ρ₀-alone condition. Axiom 3’s inner-product value set is corrected to −2, −1, +1, matching Axiom 2. The residual bound is rewritten via the general identity (|wi|−1) ² ≥ 1. The ρ₀-subspace’s two-dimensionality is clarified as a statement about its coefficient field, distinct from the Z₃-module action. Presentation: the subtitle is simplified, computational verification is reframed as confirmation, and TVL. py’s description is consolidated to a single entry in the implementation appendix. v1. 0. 5 (June 22, 2026) — 10. 5281/zenodo. 20806212 — record reorganization and reframing; the five theorems and proofs are unchanged. The companion paper "T³ as a Closed Information-Processing Environment" is moved to its own Zenodo record (10. 5281/zenodo. 20806554) ; this record now carries the Generation and Colour Structure paper alone (v1. 0. 0–v1. 0. 4 bundled both). The paper is retitled "Topological Vortex Logic: Generation and Colour Structure from T³/Z₃" and reframed as a structural correspondence rather than a complete derivation, with §8 recast as "Scope of the Correspondence. " The external-input accounting is corrected (the sole structural identification is the orbifold Z₃ with the centre of SU (3) ), a colour-grading consistency remark is added, and the bridge to the B₃ ≅ so (7) root system is noted. The software filename is standardized to TVL. py. v1. 0. 4 (April 26, 2026) — 10. 5281/zenodo. 19779092 — corrections and refinements (nine areas) ; mathematical content unchanged. Theorem numbering is corrected so the five theorems number 1–5 with their headings, Tc is defined at first use, and a §2. 4 sign typo is fixed. References are cited inline and reordered by first citation, and removed from section titles and displayed equations. The end-of-proof symbol is unified throughout (both papers) and an AI-audit attribution is dropped from the §4. 2 scope note. v1. 0. 3 (April 25, 2026) — 10. 5281/zenodo. 19752456 — GitHub URL corrected in the T³ paper; all other files unchanged. v1. 0. 2 (April 24, 2026) — 10. 5281/zenodo. 19750357 — presentation fixes; mathematical content unchanged. References are reordered by first citation and Appendices A and B are formally titled. The concept DOI is added to both title pages, and the derivation is streamlined from 23 to 12 pages (the Appendix C code listing becomes a pointer to the software record). v1. 0. 1 (April 21, 2026) — 10. 5281/zenodo. 19688501 — notation and one correction; mathematical content unchanged. ℏ, h, and c (with m and ξ) are defined at first use and all key display equations are numbered. §4. 4 is corrected (f₁, f₂, f₃ mutually incommensurate, f₁ the master clock) and acknowledgements are added. v1. 0. 0 (April 21, 2026) — 10. 5281/zenodo. 19682634 — initial release.