Character classes and the contradiction
Finite partitions of the character set \(\Lambda\) of \(p\)-power roots of unity give orthogonal projectors on the test object away from the point at infinity. Their germs in the quotient by a contractible neighborhood define integral character classes, natural under maps and homotopies of tests. Over finite quotient sheets, a locally constant multiplier cyclically permutes the character parts, equating their rational images; the sum formula then forces each image to \((4/p^k)\bar u_d\), contradicting the fixed lattice.
All sheaves and pairings are over \(\R\); \(\C\) denotes the real coefficient plane with its complex scalar operations.
Projectors and classes
There are canonical isomorphisms \(Rq_*\underline\C_{W^+} \simeq \mathcal H \simeq \bigoplus_{\lambda \in \Lambda} \mathcal H_\lambda\), where \(\mathcal H = q_*\underline\C_{W^+}\) and \(\mathcal H_\lambda = \ker(T - M_\lambda)\) for the shift operator \(T\) and scalar multiplication \(M_\lambda\). For every continuous \(t : Y \to X\) to a compact Hausdorff space and every \(D \subset \Lambda\), the natural map \(\bigoplus_{\lambda \in D} Rt_*\mathcal H_\lambda \to Rt_*\mathcal H_D\) is an isomorphism in the bounded-below derived category; the left sum is a coproduct in the ambient derived category.
Proof sketch. The orbit pullback gives the first isomorphism by additivity. Stalks are locally constant functions on \(G/H\), factoring through finite cyclic quotients, which diagonalize since \(z^{p^j}-1\) has distinct roots. For the derived sum, sections over a compact set involve finitely many indices, so global sections commute with sums and sums of soft sheaves are soft; proper base change identifies stalk cohomology, and termwise sums compute derived coproducts.
If \(D, E \subset \Lambda\) are disjoint, the cross pairing between \(B_D\) and \(B_E\) induced by the restricted test form is zero. In particular, the projectors of any finite partition of \(\Lambda\) are self-adjoint.
Proof sketch. The orientation-preserving \(G\)-action acts by isometries, with \(T^\dagger = T^{-1}\) and \(M_\lambda^\dagger = M_{\lambda^{-1}}\). For \(\lambda \notin E\), \(T - M_\lambda\) vanishes on \(B_{\{\lambda\}}\) while \(T^{-1} - M_{\lambda^{-1}}\) is invertible on \(B_E\) (componentwise inverses \((\mu^{-1}-\lambda^{-1})^{-1}\)); adjointness kills the cross pairing. Coproduct universality extends this from singletons to arbitrary disjoint \(D, E\), including conjugate pairs.
Dependencies
Fix a finite partition \(\Lambda = D_0 \sqcup \cdots \sqcup D_{m-1}\) and a compact test \(t : Y \to X\) whose value \(b_t\) has a contractible neighborhood basis. The open projectors define quotient endomorphisms \(e_i\) of \(A_t\) in \(\mathcal Q_U(X) = \T(X)/\T(U)\) satisfying \(e_ie_j = \delta_{ij}e_i\), \(\sum_i e_i = \mathrm{id}\), and \(e_i^\dagger = e_i\). With the integral localization isomorphism \(\ell_U : E_d(X) \to W_d(\mathcal Q_U(X))\), the class \(z_i(t) \in E_d(X)\) is defined by \(\ell_U(z_i(t)) = [A_t,\alpha_t] + [A_t,\alpha_t \circ (2e_i - \mathrm{id})]\). The definition is independent of \(U\).
Well-definedness. The twisted form is nonsingular and symmetric since \(2e_i - \mathrm{id}\) is a self-adjoint involution; no image object for \(e_i\) is needed. Independence of \(U\) follows by comparing through a smaller contractible neighborhood and using injectivity of the localization maps.
Dependencies
For a compact test \(t\) and any finite partition of \(\Lambda\), the character classes satisfy the integral identity \(\sum_i z_i(t) = 2[A_t,\alpha_t]\) in \(E_d(X)\).
Proof sketch. In the quotient, the explicit self-adjoint involution \(S = \sum_j E_jP_j\) on \(A^{\oplus m}\) is an isometry from the sum of the twisted forms to \((A,\alpha) \perp (A,-\alpha)^{\perp(m-1)}\), whose class is \((2-m)[A,\alpha]\); adding the \(m\) untwisted terms gives \(2[A,\alpha]\). Injectivity of \(\ell_U\) lifts the identity integrally. No quotient idempotents are split.
Dependencies
Naturality and the endgame
Let \(t : Y \to X\) be a compact test and \(g : X \to X'\) a continuous map of compact tests, with contractible neighborhood bases at \(b_t\) and \(g(b_t)\). For every fixed finite partition of \(\Lambda\), \(g_*z_i(t) = z_i(gt)\) in \(E_d(X')\).
Proof sketch. Pushforward gives a duality-preserving quotient functor commuting with the localization isomorphisms, carrying the total form to \((A_{gt},\alpha_{gt})\). Representing a projector by a roof and pushing through the neighborhood \(N' = X' \setminus g(X \setminus N)\) computes it entirely where it agrees with its character description; open base change identifies this with the projector for \(gt\), and the germ theorem gives equality in the quotient.
Dependencies
For a fixed partition of \(\Lambda\), homotopic maps \(t_0, t_1 : Y \to S^d\) satisfy \(z_i(t_0) = z_i(t_1)\) in \(E_d(S^d)\).
Proof sketch. Uniform continuity subdivides the homotopy into close stages landing in the compact semialgebraic test \(C_d = \{(x_0,x_1) : \langle x_0,x_1\rangle \geq 0\}\), a smooth \(2d\)-manifold with boundary having contractible neighborhood bases everywhere. Its two projections are homotopic via normalized interpolation, so covariance plus homotopy invariance equates the classes at consecutive stages; iteration gives the result.
Dependencies
Fix \(k \geq 1\), \(m = p^k\), the cyclic partition \(D_i = \zeta^iD_0\) of \(\Lambda\), and the sheet region \(V\) with multiplier \(\beta\) from finite quotient sheets. If \(h : Y \to S^d\) is constant on \(Y \setminus V\), then \(z_0(h) = \cdots = z_{m-1}(h)\) in \(E_d(S^d)\).
Proof sketch. Multiplication by the locally constant unit \(\beta\) is a real isometry of the restricted coefficient sheaf; with the paper's \(T\) convention it shifts \(\lambda\)-eigensheaves to \(\zeta^{-1}\lambda\)-eigensheaves, cyclically permuting the partition. After the proper restricted pushforward it conjugates \(e_i\) to \(e_{i-1}\) modulo \(m\); the germ theorem promotes the isometry to the quotient, where the twisted (and untwisted) forms become isometric, and integral localization transfers the equality.
Dependencies
For every \(n \geq 1\) and every prime \(p\), an orientation-preserving continuous \(\Zp\)-action on a connected open subset of \(\R^n\) has nonzero kernel.
Proof sketch. Suppose the action faithful. Fix the lattice \(L_d\) first, then choose \(k\) with \(m = p^k > 4L_d\) and the cyclic partition. Moving the test gives \(h \simeq D_sa\) constant off the sheet region; equal parts, homotopy, covariance, and the degree action give \(D_{s*}z_i(a) = D_{s*}z_j(a)\), and cancelling the nonzero rational scalar \(s\) equates all \(\bar z_i(a)\) while the classes stay integral in the fixed lattice. The coefficient form is two positive real lines, so \([A_a,\alpha_a]\) has rational image \(2\bar u_d\) and the sum formula gives each \(\bar z_i(a) = (4/m)\bar u_d\), forcing \(4L_d/m \in \Z\) with \(0 < 4L_d/m < 1\): contradiction.