Exact Results on the Flat Three-Torus: Stable Winding States, Cascade Termination, Harmonic Flow, and Arithmetic Spectral Gaps — with an Information-Processing Analogy

Abstract

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 {−1,0,1}³ ∖ {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 v1.0.8 (August 3, 2026) — 10.5281/zenodo.21764815 — three added results and five precision corrections; the former open-questions remark on the achievable set is superseded by two of the new results; no existing theorem or numerical value changes. A subsection on where a cascade terminates is added after the termination theorem. The total quadratic cost of a terminal multiset is bounded below by the taxicab norm of the winding vector and is congruent to it modulo two, and the bound is attained by an explicit cascade; the split relation is shown not to be confluent. A statement that a transition within the face family costs about one stiffness unit is corrected: the face shell lies one unit above the vacuum, and distinct face states are mutually degenerate, so the model supplies no face-to-face energy difference. The winding phase is written as a circle-valued map with its globally defined one-form, in place of a real phase function that is single-valued only on the universal cover; the selected grading is described by its character rather than as an action by scalar multiplication. Terminology is tightened where it outran what is established: the reference period is described as adopted rather than fundamental, an unsupported aliasing claim is removed, the symbol count is called a maximum symbol entropy rather than an information capacity, and the neutral states are no longer called singlets. The achievable terminal-cost set is determined exactly: it is the full arithmetic progression from the taxicab norm to the maximal cost in steps of two, proved by a downward-closure induction riding on the terminal-cost proposition alone. The maximal terminal cost is given a sharp upper bound — half the squared norm plus a correction depending only on the number of odd coordinates, with the stable states and the class of (0,2,2) as the tabled values — and with no unconditional closed form for the maximal cost claimed; the upper bound is proved by a parity-coupling case analysis; the exceptional class is derived as the set of two-corner profit-one sums; attainment of the bound is verified exhaustively on a stated finite range, with the general case left open. The verification range for the maximal terminal cost is stated exactly. A clause reporting agreement “on all larger states sampled” is removed, the sample being unspecified and therefore not reproducible; the exhaustive claim now stands on the stated finite range alone. The summary table’s entry for the information-capacity bound is renamed from alphabet capacity to maximum symbol entropy, matching the term used where the quantity is defined; the paper previously named one quantity two ways. 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 under concept DOI 10.5281/zenodo.19682633, where those versions are recorded.