Papers · William Alexander Aldred

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

by William Alexander Aldred — Independent Researcher

Abstract

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 ∧ Tₐ - Rₐb ∧ 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 ∧ Tₐ 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 qrm pₒst = b²/(4ℓ_P²) = 5.276535 × 10³⁶ nats, the density required for 1/(4ℓ_P²) is n = 1.813725 × 10³² rm m⁻² = 𝒯_A² exactly, n ℓ_T² = 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 srm 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_P²) is C ∈ 0, 1, 2, each previously seen or degenerate, none equal to the grading target 3Ω_Λ = 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.

Keywords: Einstein–Cartan gravity; torsion; Nieh–Yan charge; dislocations; Burgers vector; defect networks; horizon entropy

The program

• The Aldred Limit
• Two Bounds and an Integration Constant
• Supplementary Memos for “Two Bounds and an Integration Constant”
• The Constant Changes Address
• The Crossing Charges Nothing
• No Derivation in the Frame
• Static Defects Carry No Nieh–Yan Charge (this page)
• Creation Events Are Nieh–Yan Charged
• There Is No Twisted Vacuum
• Propagation Is Not Enough
• Axiom H
• One Flaw, Two Residues
• The ε_eff Turnover
• The Generation Map Has a Fixed Point
• Three Fences Any Λ Mechanism Must Pass