This record presents exact results about a stipulated quadratic split model on the winding lattice of the flat three-torus T³ = R³/LZ³, together with a separate, explicitly model-dependent information-processing analogy. The exact results and the analogy are kept distinct: the geometry and split relation do not by themselves define a physical clock, transition dynamics, thermal barrier, error-correction mechanism, or closed system. Stable winding labels: in the stipulated cost E (w) =ε₀|w|², a label is stable against splitting if and only if |w|²≤3. The nonzero stable set is exactly −1, 0, 1³ ∖ 0, containing 26 labels in shells of 6, 12, and 8. Cascade termination: every nonzero unstable winding label admits a finite favourable-split cascade ending in stable pieces; every maximal favourable cascade terminates because the non-negative integer multiset cost Φ (M) =Σv∈M|v|² strictly decreases at each favourable split. This is a theorem about the split relation, not a physical decay law or rate. Root and module structure: the eighteen face-and-edge labels form the root system of type B₃, associated with so (7), and the three shells are pairwise non-isomorphic as rational permutation modules under cyclic coordinate permutation. No physical coupling or mass ordering follows from that non-isomorphism. Scalar-Laplacian spectrum: the free scalar-Laplacian eigenvalues are λn= (2π/L) ²|n|². Frequencies such as fn=c|n|/L arise only after assuming a wave equation with propagation speed c. Harmonic fields: a vector field on the flat T³ that is both divergence-free and curl-free is harmonic and hence constant. This classifies a restricted class of fields; it supplies no turbulence or transport dynamics. Arithmetic shell gaps: by Legendre’s three-square theorem, no free scalar-Laplacian eigenmode exists when |n|²=4a (8b+7). This does not exclude driven, interacting, nonlinear, or different-field responses at the corresponding numerical frequencies. Winding conservation: for a specified nonvanishing complex order parameter with normalized phase ψ: T³→U (1), the winding class lies in H¹ (T³;Z) ≅Z³ and is invariant under nonsingular homotopy. The bare manifold alone does not supply those field labels. Part II explores a possible information-processing reading after adding explicit assumptions. It may use the 26-element set as an alphabet, adopt the lowest wave frequency as a reference, track three winding labels together with Noether charges of a chosen translation-invariant action, and introduce a connected graph whose face/edge/corner degrees are 16/10/6. The graph is a chosen finite state-transition graph, not a physical transition law or a finite-state automaton without further input and output data. The winding lattice w and the Laplacian mode lattice n are isomorphic copies of Z³ but are not identified by the exact results. Note on this record. The companion paper — “Topological Vortex Logic: Stability, Root Systems, and Module Structure of Winding States on T³ with a Selected Z₃ Grading” — is archived separately at 10. 5281/zenodo. 19682633. The standalone TVL. py framework is archived via software concept DOI 10. 5281/zenodo. 19683376. Changelog v1. 0. 7 (July 24, 2026) — 10. 5281/zenodo. 21522981 — recast from a claimed closure theorem to exact lattice-model and flat-torus results with a separately labelled information-processing analogy. The central claim that T³ is automatically a closed information system is withdrawn. Boundarylessness removes a boundary but does not establish physical isolation, conservation of every field, or exclusion of external coupling. The false claim that every immediate split product is stable is replaced by the stronger cascade-termination theorem for the multiset cost Φ. The winding-cost lattice w and the scalar-Laplacian mode lattice n are separated throughout; neither is treated as the physical trajectory of the other. The frequency formula is made conditional on an assumed wave equation and speed c. Clocking, gating, damping, Q-factor, latency, and sampling language are confined to the model-dependent reading. Phase-slip suppression is no longer inferred from ε₀. A physical barrier and a thermal equilibrium model are stated as separate assumptions. Noether charges are attributed to symmetries of a specified action, while the three winding labels are attributed to the homotopy class of a specified phase field. “Seven” is conditional bookkeeping, not an exhaustive topological count. The harmonic-flow result is narrowed to divergence-free and curl-free fields; curvature alone is not claimed to create compression or disorder, and no causal link from laminarity to splitting is asserted. The former mixed “noise-filter” table is replaced by a side-by-side statement of two independent mechanisms: favourable splitting on w and arithmetic absence in the free scalar-Laplacian spectrum on n. The transition construction is defined as a chosen connected graph on the 26 stable labels with degrees 16/10/6. Pair creation, annihilation, rates, automaton inputs, and computational universality are not derived. The Summary and Conclusions are rewritten so exact results and assumptions remain separated; the appendices are synchronized to the narrow free-Laplacian claim. Finalization: the bibliography is placed in first-citation order; the Noether and Shannon references are explicitly cited; the Legendre, Schumann, and Shannon metadata are corrected or completed; PDF metadata, the July 24, 2026 date, the reserved version DOI, and the concept DOI are included. v1. 0. 6 (July 5, 2026) — 10. 5281/zenodo. 21200822 — erratum from a full-paper audit; 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. §8. 6–8. 7: the transition graph is corrected from a regular Cayley graph to an induced graph with degrees 16/10/6. §6. 2: the maximum-principle argument is corrected for a compact boundaryless manifold. §8. 6: vocabulary closure is restricted to a chosen transition rule rather than general addition. §2. 3: the selected charge grading and coordinate-permutation Z₃ are distinguished. §6. 5: the flat-metric distance bound is corrected to √3/2·L/c under an assumed propagation law. The conservation and external-parameter statements are qualified. A non-interacting caveat is added to the quadratic split cost. v1. 0. 5 (June 23, 2026) — 10. 5281/zenodo. 20806555 — separated into its own Zenodo record; mathematical content unchanged. This paper moved to its own record, split from the companion TVL derivation. The title-page DOI, companion reference, and software citation were updated. Pre-split history (v1. 0. 0–v1. 0. 4): bundled with Paper A under concept DOI 10. 5281/zenodo. 19682633; those versions are recorded in Paper A’s changelog above.
Vladimer Merebashvili (Fri,) studied this question.
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