A Restricted Jacobi Identity (Elementary Checked Boundary Case) for the Field-Level Tethered Bracket on the Divergence-Free Shear Class (Companion paper I)

Companion paper I, version 1.1 (finite-Fourier model, machine-checked). In a finite-Fourier model of vector fields on T^3 we consider the vector-valued tethered operation {a,b} = [a,b] + T(a,b), where [·,·] is the vector-field Lie bracket and T(a,b) = −(κw)⟨Π_u a, Π_u b⟩u. For every divergence-free axis u along e_x with ⟨u,u⟩ ≠ 0 (in particular u = (sin y, 0, 0)) the Jacobiator vanishes whenever all three arguments are divergence-free and point along e_x. On this class the Lie bracket vanishes; T need not vanish, but it takes values in ℂu, which lies in the class, so every nested term vanishes. Lean proves that the hypotheses are inhabited with a nonzero bracket inside the class (non-vacuity) and that ⟨u,u⟩ ≠ 0 cannot be dropped: for the isotropic axis eⁱʸe_x the Jacobiator on (2cos y, 0, 0) is 6(κw)^2 eⁱʸe_x. This is an elementary, machine-checked boundary case, not a new Jacobi theorem. Companion paper II shows the e_x-direction hypothesis cannot be dropped. What Lean checks. shear_jacobi_holds_of_pr_ne_zero, shearY_jacobi_holds, paperI_nonvacuous, shear_jacobi_fails_isotropic and the supporting lemmas in FTVerified/May/FieldJacobi.lean and FTVerified/Companion/JacobiCompanion.lean. The paper's hypothesis-to-declaration table (README.md) matches every Lean hypothesis to a stated assumption; no mismatch was found. The isotropic guard relies on Lean's convention x/0 = 0, which the paper states. Open gap (not addressed here). No identification of this finite-Fourier operation with the state-dependent geometric functional bracket has been proved, in Lean or on paper. The geometric bracket in BenFrohman/frohmanian-tether-geometric has a state-dependent axis u, a state-dependent projection Π_u and an enstrophy prefactor with κ = 3/2 and no free parameter w; the operation here has a frozen axis, free complex κ and w, a complex-bilinear pairing and Lean's division convention. Neither result is claimed to transport to the geometric setting, and nothing here bears on Navier–Stokes or any Clay problem. Reproducibility. Full Lean source: BenFrohman/frohmanian-tether-verified at a761aed99fba58ed4ac9bd5970577b2fc089824d (tip of main on 11 November 2026; a git archive zip of that commit is attached). The paper cites code commit cf04afd; the Lean sources, scripts, lakefile, manifest and toolchain are byte-identical between that commit and a761aed. Lean leanprover/lean4:v4.30.0-rc1, Mathlib 0c154d67103f74be3a0f2c509f72ccbf5be9f2a7. One-command rebuild from the repository root: lake exe cache get && lake build && python3 scripts/check_jacobi_axioms.py. GitHub Actions on main at a761aed passed (run 42932429909). On 11 November 2026 the pinned commit was rebuilt locally from scratch for the project modules (attached build.log); the attached axiom output (axiom-output.txt: PASS, 49 declarations) shows only propext, Classical.choice and Quot.sound (or a subset), and the attached scan found no sorry, admit, native_decide or axiom declarations. SHA256SUMS lists every attached file. Limits. A kernel check shows that the stated Lean propositions follow from the stated Lean definitions. It does not show that those definitions model any geometric or physical object.