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

Abstract

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 paper, “Exact Results on the Flat Three-Torus,” 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 v1.0.8 (August 3, 2026) — 10.5281/zenodo.21765104 — four corrections and seven added remarks; no existing theorem, proof or numerical value changes. The exotic-sector attribution is corrected. Under the paper's own conventions the three non-diagonal corner states of unit charge project to the negatives of twice the fundamental weights, and those of anti-unit charge to the positives; the previous text stated the reverse. The conclusion is unaffected, the weight set being symmetric under negation either way. An abstract cross-reference to an appendix is corrected to a section reference, the target being an ordinary numbered section. A symbol collision is removed: the classification map and the hypothetical coupling law were both written as the same letter, and the classification map is renamed. Three descriptions are made precise: the two-dimensional module is identified as a rational module of the cyclic group of order three, the shell ratios are called bare shell-energy ratios rather than mass magnitudes, and an uncited and scheme-dependent quark mass ratio is replaced by the cited statement that quark masses span several orders of magnitude. A characteristic-free refinement remark is added after Theorem 5: over F₃ the coordinate cycle acts with Jordan types J₃², J₃⁴, and J₁²⊕J₃² on the three shells, so by Krull–Schmidt the permutation modules are classified completely in the modular case, and the classification holds over any commutative ring; the rational hypothesis is a convenience. A joint-symmetry remark is added after Theorem 4: since the charge is the mod-3 reduction of the pairing with (1,1,1), the part of the Weyl group W(B₃) preserving the grading pointwise is exactly W(A₂) — the Weyl group of the sub-system of long roots orthogonal to (1,1,1) — at index 8, enlarging to S₃×{±I} of order 12 when the two nonzero classes may exchange; elementary facts recorded for the connection they draw. A Weyl-orbit remark is added: with the fundamental weights of B₃ written ω₁, ω₂, ω₃, the face, edge and corner shells are exactly the Weyl orbits of ω₁, ω₂ and 2ω₃, of sizes 6, 12 and 8 and squared norms 1, 2 and 3, so the shell trichotomy, the ratio of shell energies and the root identification are one statement; the doubling in the third is essential, ω₃ not being an integer vector. A projective remark is added: the stable states are canonical representatives of the nonzero vectors over the field of three elements, and modulo conjugation they are the thirteen points of the projective plane of order three, under which the shells become the strata of the coordinate triangle, the neutral states descend to a line, and the hyperoctahedral symmetry acts as the setwise stabiliser of that triangle. A dictionary remark is added: every family named in the paper receives an exact projective identity, the two edge families being separated inside one stratum by the grading alone; the exotic sextet is identified positively as the frame quadrangle with the image of the corner diagonals removed; and the shell sizes and the charge distribution are derived, the first as the coefficients of the cube of one plus twice a variable, the second as a two-dimensional subspace with its two cosets. A conic remark is added: the sum of squares is a nondegenerate quadratic form over the field of three elements whose conic is exactly the frame quadrangle, so the shells are its level sets; the distinguished line is the tangent at the image of the corner diagonals; the projective image of the Weyl group is the full stabiliser of the conic, acting on its four points as the full symmetric group; and the orbit of the grading form is exactly the four tangent forms, so the selected grading is a choice among four equivalent ones. A polarity remark is added: the interior, exterior and on-conic trichotomy of the thirteen points is the shell trichotomy; the coordinate triangle is a self-polar triangle whose sides are the three external lines; each edge class corresponds to an unordered pair of conic points, hence to a pair of grading choices; and the polarity carries each family of points to a class of lines, with uniform incidence profiles. 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