Static Defects Carry No Nieh–Yan Charge: A Legs Theorem for Torsion Defect Networks in Einstein–Cartan Gravity

In Einstein–Cartan gravity, a torsion defect offers a natural loophole in arguments making the Nieh–Yan charge vanish in smooth regions. The Nieh–Yan form is exact, NY = d(e ∧ T), so its flux through a smooth vacuum horizon is zero — but exactness is a smoothness statement, and around a singular line, as with Aharonov–Bohm flux, the erased charge might survive concentrated on the line. This paper tests that loophole for the entire static sector and closes it by a theorem, not an estimate. A static dislocation's torsion 2-form is supported, as a differential form, on the transverse 2-plane of its line. In three spatial dimensions any two such transverse planes share a leg. The Nieh–Yan density, N = T^a ∧ T_a - R_ab ∧ e^a ∧ e^b, is built from wedge products of those 2-forms, and every such wedge — diagonal, cross-term, at a crossing, for disclinations as well as dislocations — vanishes identically by antisymmetry, before any question about products of distributions arises. A static defect network therefore carries a Nieh–Yan density that is identically zero, not small. The boundary charge of an eternal piercing is also zero: the flux of e^a ∧ T_a through a constant-time cap is extensive, bL for a segment of length L, and the two caps of a closed spacetime slab contribute +bL and -bL and cancel. Exact antisymmetrization verifies the vanishing, and a control — time-legged torsion returns a wedge norm of 0.33 ≠ 0 — proves the arithmetic can see a wedge when the legs allow one: the vanishing is a consequence of staticity alone. Equivalently, a static network's axial current is exactly conserved, and the Nieh–Yan density, its divergence, is zero. The defect's genuine quantum is not the Nieh–Yan charge. Each screw carries the Burgers holonomy b, Φ_B = ∫∫ T^z = b — quantized, frame-free, a holonomy-species analogue of Aharonov–Bohm flux, distinct from a divergence integral. EC fixes no value for b and assigns it no entropy: its only torsion length, ℓ_T, is a bulk density scale, and b = ℓ_T is a postulate, not a Cartan-equation consequence. Adopting it anyway, with q_post = b^2/(4l_P^2) = 5.276535 x 10^{36} nats, the density required for 1/(4l_P^2) is n = 1.813725 x 10^{32} m^{-2} = T_A^2 exactly, n ℓ_T^2 = 1.000000 — one piercing per twist cell: defect counting is the cell tiling it was meant to replace. For the figure eight (b, -b), signed counting gives s_pair = 0, unsigned gives 2q with the postulate inherited, and the crossing wedge is exactly zero. The sector's complete output in s/a = C/(4l_P^2) is C ∈ {0, 1, 2}, each previously seen or degenerate, none equal to the grading target 3Omega_Lambda = 2.0550. The theorem is a positive result whose content is the vanishing. The static sector joins the smooth bulk and the smooth boundary on the closed side of the map, and the proof isolates the only channel for any Nieh–Yan content: time-legged, dynamical torsion — not treated here, since it counts events, not states, and requires a statistical bridge the frame does not contain.