1 A variational obstruction in Geometric Unity Abstract. Symmetry of the Hessian constrains exact action gradients that are linear in curvature, with connection-independent coefficients and affine lower-order terms. For unrestricted u(p,q) connections, pointwise Spin equivariance forces the curvature term to vanish in dimension at least four. Applied to Geometric Unity’s 2021 operator construction, this makes the claimed nonzero curvature response incompatible with the asserted first-order bosonic variation. 1. Variational rigidity With the metric and auxiliary field ε fixed, Geometric Unity (GU) asserts [1, (9.7)] (1) δI A ( α ) = ⟨ α , Υ⟩, Υ : = S ε F A + κ * ( A − B ). The trace duality pairing is ⟨α, η⟩ = ∫ Y Re Tr(α ∧ η), for α a one-form and η a thirteen-form, both ad-valued. Here S ε is the Shiab operator, κ is constant, A = ∇ 0 + π, and B = ∇ 0 + ε −1 d 0 ε. The field π ranges over Ω¹(Y, ad), so A varies freely while B and S ε remain fixed [1, (5.5), (6.22), (9.3)]. Work on an n-dimensional coordinate patch. Let g = u(p,q), N = p + q ≥ 2, with b(X,Y) = Re Tr(XY). On an open set of unrestricted g -connections, suppose a C² functional satisfies (2) δI A ( α ) = ∫ b ( α i , G i ( A )) dμ , G i ( A ) = 1 2 L ijk F A , jk + M ik A k + c i Variations have compact support; the positive density dμ is fixed. Repeated indices are summed. The real-linear maps L, M and the source c are smooth and independent of A, with L ijk = −L ikj . Write SF for the curvature term. Fixed Hodge factors are absorbed into the coefficients, and the density identifies the output (n − 1)-forms with vectors. Lemma. Let P s , P z be the b-orthogonal projections onto s = su(p,q) and the one-dimensional center z . Then (3) L ijk = a ijk P s + d ijk P z , where a and d are real and completely alternating in i,j,k. Proof. Let † denote the b-adjoint. Hessian symmetry makes the linearization (4) ( K A β ) i = L ijk (∂ j β k + [ A j , β k ]) + M ik β k formally self-adjoint. Its principal coefficients give L ijk = −(L kji ) † . Vary only A j by tZ near a point and subtract the self-adjointness identities. Fixed terms cancel; using (ad Z ) † = −ad Z gives (5) L ijk ad Z = ( L kji ad Z ) † = ad Z L ijk Thus each L commutes with every adjoint action. It preserves the center, and its central component on s = [ s , s ] vanishes. The complexification sl N (ℂ) has irreducible adjoint representation; Schur’s lemma yields a real scalar on s . The center contributes one further real scalar, proving (3). Since both projections are self-adjoint, the principal relation gives a ijk = −a kji , and likewise for d. Together with antisymmetry in j,k, this gives complete alternation. 2 2. Spin symmetry forces the curvature term to vanish Theorem. Let V be the n-dimensional tangent space with any nondegenerate metric, n ≥ 4. Let Spin(V) act internally by conjugation through a representation in U(p,q). Under (2), if the pointwise map S is equivariant for the simultaneous Spin action on its form and internal indices, then S = 0. Explicitly, if D k (h) denotes the induced action on k-forms tensored with internal conjugation, the required identity is (6) SD 2 ( h ) = D n − 1 ( h ) S ( h ∈ Spin( V )). Proof. The projections in (3) commute with conjugation. The fixed density is preserved by the Spin action. Equivariance therefore makes a and d invariant elements of Λ³V. For n ≥ 4, (7) (Λ 3 V ) Spin( V ) = 0. Indeed, in an orthonormal basis take distinct i,j,k and choose ℓ outside this triple. For the infinitesimal orthogonal transformation in the (i,ℓ)-plane, the (ℓ,j,k)-component of its action on a is a nonzero signed multiple of a ijk . Invariance forces a ijk = 0. This works for rotations and boosts, hence in every signature. The same holds for d, so every L ijk vanishes. In dimension three, the invariant volume tensor survives, as in the Chern-Simons curvature gradient. 3. Application to the stated GU operator In GU, n = 14 and g = u(64,64). Definition (8.7) in [1] makes the Φ i Spin(7,7)-invariant. The wedge, matrix and Hodge operations in (9.3) therefore make S 1 satisfy (6). This is pointwise equivariance of the operator and requires no background isometries. For C ε (X) = εXε −1 , acting only on internal indices, (9.3) gives (8) S ε = C ε −1 S 1 C ε At ε = 1, the exact variation (1) meets the theorem’s hypotheses, so S 1 = 0. Equation (8) then gives S ε = 0 for every ε. Hence the nonzero Ricci/scalar response asserted after [1, (9.3)] is incompatible with the exact first variation [1, (9.7)]. Scope. The result requires correcting (9.7) or changing a hypothesis of the construction. It does not exclude the second-order action I 2 = ‖Υ‖² in [1, (9.11)]: every zero of Υ is stationary for that action, although additional stationary points may exist. GU’s field content and Υ = 0 are not ruled out. Reference [1] Eric Weinstein. Geometric Unity: Author’s Working Draft. Version 1.0, 1 April 2021. Official draft.