Topological Vortex Logic: Stability, Root Systems, and Module Structure of Winding States on T³ with a Selected Z₃ Grading

We present a body of exact, elementary results about winding (vortex) states on the three-dimensional flat torus T³ = R³/LZ³ equipped with a selected Z₃ grading, together with an explicitly conjectural analogy between those results and a few features of the Standard Model. The framework, called Topological Vortex Logic (TVL), establishes five proved theorems and one conditional proposition about a precisely specified discrete model; the Standard-Model reading is a separate, labelled interpretive layer, matched to the mathematics rather than derived from it. Theorem 1 (Stability): In the stipulated quadratic split model E (w) = ε₀|w|², a winding state w ∈ Z³ is stable if and only if |w|² ≤ 3. The proof is analytic in both directions: a coordinatewise inequality — each split changes the energy by 2ε₀·Σᵢ (uᵢ² − wᵢuᵢ), and every summand is non-negative when wᵢ ∈ −1, 0, 1 — excludes every energy-lowering split, while an explicit unit split witnesses the unstable direction. The stable vocabulary is exactly 3³ − 1 = 26 states in three shells: 6 face (|w|²=1), 12 edge (|w|²=2), 8 corner (|w|²=3). Theorem 2 (Z₃ grading): The charge q₃ = (w₁+w₂+w₃) mod 3 is a topological grading — a chosen diagonal homomorphism H¹ (T³, Z) → Z₃ — partitioning the 26 states 8/9/9. The 8/9/9 distribution is basis-independent; the per-state charges depend on the chosen functional. Identifying this Z₃ with the centre of SU (3) is an imported, conjectural reading that underlies the conventional baryon-number representatives B = 0, ±1/3. Theorem 3 (A₂ root system): The six traceless edge states form a root system of type A₂ (the root system associated with su (3) ), verified by the root-system axioms, the Cartan matrix, and an analytic reflection-closure argument. What is established is the root geometry, not the full Lie algebra. Theorem 4 (B₃ root system): The eighteen face-and-edge states form the root system of type B₃ (associated with so (7) ) — the six face states as short roots ±eᵢ, the twelve edge states as long roots ±eᵢ±eⱼ — with the A₂ system of Theorem 3 as its traceless sub-system. Proved analytically via the root-system axioms, the B₃ Cartan matrix, and generation of the order-48 Weyl group from the simple reflections. Theorem 5 (Non-isomorphism): For each shell F, the rational permutation module MF = QF under the coordinate cycle C₃ is pairwise non-isomorphic to the other two. The dimensions (6, 12, 8) already separate them; the finer invariants — the character trace (0, 0, 2) and invariant-subspace dimensions (2, 4, 4) — record how they differ. A sixth statement is a conditional proposition: a coupling injective on the invariant pairs (invariant-subspace dimension, character trace) would assign the three shells distinct values — but no distinct masses, and no ordering, follow from the non-isomorphism alone. The Standard-Model analogy reaches a few isolated features — the generation count, colour triality, and the SU (3) root system — and not the structure of the Standard Model: there are no fermion fields, no gauge dynamics, no electroweak sector, and no mass spectrum. The one imported ingredient is the identification of the grading Z₃ with the centre of SU (3). The corner ±2μ weight shell is a structural feature that is not the weight system of any single SU (3) irreducible representation. These boundaries are set out in the paper's “Scope of the Correspondence. ” Note on this record. The companion foundational paper, “T³ as a Closed Information-Processing Environment, ” is archived separately at 10. 5281/zenodo. 20806554. The standalone TVL. py library implements and tests the classification; it is split into a mathematical core (returning only the geometric quantities) and a separate, optional adapter that supplies the conjectural physical reading, and is archived via software DOI 10. 5281/zenodo. 19683376. It is a verification tool, not an independent confirmation of the mathematics. Changelog v1. 0. 7 (July 23, 2026) — 10. 5281/zenodo. 21501526 — revisions to the proofs, the scope statements, and the topological setup. The mathematical core is unchanged and strengthened, and no proved result is weakened. The stable-direction proof is replaced by a complete coordinatewise inequality (the previous passage argued from three example decompositions and “by symmetry, ” which is not a proof over all integer splits). The B₃ result is promoted to a numbered analytic theorem (Theorem 4) with a full proof; the non-isomorphism becomes Theorem 5, with a remark that the shells already differ in dimension. The false “6” (sextet) assignment is corrected throughout: the six corner non-diagonal states project to ±2μᵢ, which is not the weight system of any single irreducible representation. The non-isomorphism proof is corrected to use the character trace χ (g) = |Fix (g) | (an isomorphism invariant) rather than the incorrect claim that an isomorphism maps fixed basis points bijectively. The former “mandatory hierarchy” is demoted to a conditional proposition requiring an injective coupling law; no mass ordering is derived, and the surviving “proves the ordering is forced” sentence is removed. The orbifold framing is corrected to a selected Z₃ grading (no orbifold quotient is constructed) ; the SU (3) -centre identification and baryon-number representatives are confined to the imported/conjectural layer. The Standard-Model “interpretation” is reframed as a conjectural analogy reaching only a few isolated features; root-system/Lie-algebra terminology is made precise (“type B₃, associated with so (7) ”). Presentation: the title becomes “…Stability, Root Systems, and Module Structure of Winding States on T³ with a Selected Z₃ Grading”; TVL. py is described as a math-core-plus-adapter verification tool, not independent confirmation. The winding class is correctly typed as an element of H¹ (T³;Z) (equivalently a homomorphism H₁→Z), rather than H₁ itself; the complex order parameter and its normalized U (1) phase are separated so that a phase slip can pass through zero without asking a U (1) -valued map to equal zero. The localized-vortex logarithmic energy formula is removed from the proof setup. The quadratic norm is instead motivated by the phase-only Dirichlet energy of the harmonic representative, while split additivity remains explicitly a stipulated non-interacting model assumption. The superfluid/Ginzburg–Landau references that supported the removed derivation are retired with it, and a standard algebraic-topology reference is added for the homotopy classification; the shorter bibliography reflects the withdrawal of the physical derivation, not reduced sourcing. The coordinate order-three action is denoted C₃ and its permutation modules are stated over Q; the B₃ Cartan matrix is put in the standard row-coroot convention. The software documentation and verifier are synchronized to these conventions. The TVL. py reference in the Computational Implementation section (and the record's companion-software lines) now cites the software's concept DOI, 10. 5281/zenodo. 19683376, which always resolves to the latest version, in place of the version-specific DOI. The Pati—Salam 1974 reference is corrected to the published title (“Lepton number as the fourth ‘color’”) and now notes the published Erratum, Phys. Rev. D 11 (1975) 703. 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 u