T³ as a Closed Information-Processing Environment

We derive the minimal mathematical structures required for the three-dimensional flat torus T³ = R³/LZ³ to function as a closed, stable, discrete information-processing environment. The question is not what external physics the geometry implies, but what the geometry itself requires: what conditions must hold for information to be stably encoded, reliably clocked, conserved under flow, and filtered against noise within a compact flat manifold with periodic boundary conditions. We identify five minimal requirements — stability, a well-defined clock, conservation, laminarity, and noise rejection — and show that each is satisfied by a unique mathematical object that T³ provides automatically, without external design. Stability: a vortex state is stable against Landau splitting if and only if |w|² ≤ 3, giving exactly 3³ − 1 = 26 discrete vortex states in three symmetry families, determined by the first Betti number b₁(T³) = 3. The 18 face-and-edge states form the root system B₃ ≅ so(7), verified through all four root-system axioms; its Weyl group is the symmetry group of the vocabulary. Clock: the natural clock is the lowest cavity resonance f₁ = c/L. Conservation: information is conserved by seven independent quantities — four continuous Noether charges and three discrete topological winding numbers. Laminarity: laminar, distortion-free flow is the unique consequence of the Laplace equation on a compact flat manifold. Noise rejection: operates at two complementary levels — the Landau energetic filter decays high-winding states within one clock period, and Legendre's three-square theorem geometrically forbids states at the shells |n|² ∈ {7, 15, 23, …}. These five mechanisms are not only each satisfied but jointly sufficient for closure, and the structure is independent of the four external scale parameters (L, c, ρ_s, ξ). The closed T³ information system is a topological fact. Contents of this record: the 18-page paper. The companion paper — “Topological Vortex Logic: Generation and Colour Structure from T³/Z₃,” which develops the particle-physics correspondence of the same T³/Z₃ vortex classification — is archived separately at 10.5281/zenodo.19682633. The standalone TVL.py framework, including the stability proof and the computational verification of the B₃ ≅ so(7) root system, is archived separately via software DOI 10.5281/zenodo.19683377. ─────────────────────────────────────────────────────────────────────────────────────────────────Version History───────────────────────────────────────────────────────────────────────────────────────────────── v1.0.5 (June 2026) — DOI: 10.5281/zenodo.20806555 This paper now has its own Zenodo record, separated from the companion TVL technical derivation with which it was bundled under concept DOI 10.5281/zenodo.19682633 (versions v1.0.0 through v1.0.4). The companion paper, retitled “Topological Vortex Logic: Generation and Colour Structure from T³/Z₃,” continues at 10.5281/zenodo.19682633, where the full pre-split bundle version history (v1.0.0–v1.0.4, with per-version DOIs) is recorded. In v1.0.5: the in-text reference to the companion report is updated to its separated record and new title, the title-page DOI is updated to this record's own concept DOI, the companion software is now cited by its Zenodo DOI, and a provenance footnote records the former shared DOI; the mathematical content of the paper is unchanged from v1.0.4. Concept DOI (always resolves to latest version): 10.5281/zenodo.20806554