Failure of Antisymmetry and of the Cyclic Jacobi Identity for a Specified Finite-Fourier Tethered Operation (Companion paper II)

Companion paper II, version 1.1 (finite-Fourier model, machine-checked). For the vector-valued tethered operation {a,b} = [a,b] + T(a,b) with tether axis u_Y = (sin y, 0, 0) and the divergence-free field a_Z = (0, 0, sin x), the Jacobiator J(a_Z, a_Z, a_Z) is nonzero when κw ≠ 0: its z-component at mode eⁱ⁽ˣ⁺ʸ⁾ equals (3i/8)κw. A separate finite-dimensional witness (u = e_1, cross-product bracket on ℝ^3) gives the nonzero Jacobiator (κw)e_2. Because T is symmetric the operation is not antisymmetric, and both witnesses repeat arguments. This is a valid algebraic failure of the cyclic Jacobi identity for this specified operation. It is not a failure of the Lie part (which satisfies Jacobi), not a Poisson-bracket failure, and not a geometric refutation. What Lean checks. shearY_transverse_coeff, shearY_jacobi_fails_transverse, brT_uY_aZ_aZ, brT_not_antisymm, JacobiWitness.mixed_sum_eq, tethered_jacobiator_ne_zero and paperII_nonvacuous. The hypothesis-to-declaration table (README.md) found no mismatch. 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.