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 M_F = Q[F] 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, T_c 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