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 (irrotational), 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, ρₛ, ξ). The closed T³ information system is a topological fact. Note on this record. This record contains the 18-page paper only. 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. A full-paper audit found nine peripheral-claim errors, two of independent severity, none touching the five central results (the 26-state vocabulary, the B₃ ≅ so (7) root system, the conservation count, the laminar-flow uniqueness, or the two-level noise rejection) ; all nine are corrected in the prepared v1. 0. 6. Changelog v1. 0. 6 (July 5, 2026) — 10. 5281/zenodo. 21200822 — nine peripheral-claim fixes, no change to the five central results. §6. 3: "any divergence-free flow is a uniform flow" is corrected to require the flow be irrotational, with the missing case (curl/co-exact fields, also divergence-free but not covered by the original argument) named and excluded by a Hodge-decomposition argument; swept through the summary table, the Conclusions, and the abstract. §8. 6–8. 7: the transition graph is corrected from "the Cayley graph of −1, 0, 1³" and "regular" to an induced subgraph of that Cayley graph (Ωstable is not closed under the group operation) with degree 16/10/6 by family (face/edge/corner), not constant. §6. 2: the maximum-principle argument is corrected — it no longer states a harmonic function "achieves its maximum on the boundary" of a manifold stated, in the same sentence, to have none; now the standard strong maximum principle for compact boundaryless manifolds. §8. 6 (transition rules): the vocabulary-closure claim is corrected from an unqualified "is closed" (false under general addition of stable vectors, e. g. (1, 0, 0) + (1, 0, 0) = (2, 0, 0) ) to closure under the restricted transition rule by construction. §2. 3: the holonomy ℤ₃ = center (SU (3) ) and a separate coordinate-permutation ℤ₃ are now named as distinct groups rather than conflated under the same symbol. §6. 5: the information-latency bound is tightened from the loose ≤√3·tq to the exact derived value √3/2·tq, consistently at every location it appears. §5. 1: a clarifying note distinguishes the parameter-independent existence and count of the seven conserved quantities from their parameter-dependent numerical values. §5. 2: the seven-conserved-quantity exhaustiveness claim is qualified to what follows from the topology and symmetry structure alone, since additional field content could carry further internal charges. A non-interacting-limit caveat is added for the energy formula E (w) =|w|²ε₀, matching the caveat already carried by the companion paper. v1. 0. 5 (June 2026) — 10. 5281/zenodo. 20806555 — separated into its own Zenodo record; mathematical content unchanged. This paper now has its own record, split from the companion TVL derivation with which it was bundled under concept DOI 10. 5281/zenodo. 19682633 (v1. 0. 0–v1. 0. 4) ; the companion, retitled "Topological Vortex Logic: Generation and Colour Structure from T³/Z₃, " continues at that DOI. The in-text companion reference, the title-page DOI, and the software citation are updated, and a provenance footnote records the former shared DOI. Pre-split history (v1. 0. 0–v1. 0. 4): bundled with the Derivation under concept DOI 10. 5281/zenodo. 19682633; those versions are recorded in Paper A's changelog above.
Vladimer Merebashvili (Sun,) studied this question.