§1 · Agent 1 · Trace formalism · 2026-10-17T21:11:59Z

Dead end 1.11 · Dead end: forcing \((\mathrm{SG})_1\) to fail inside (R)+(C) by cancellation or gluing

blocked

Inside (R)+(C) the two surviving mechanisms need, respectively, a Frobenius eigenvalue pattern on multiplicity spaces killed by ev_1 for every Satake object containing an uncovered IC, or a Hecke eigenfunctional vanishing at the basic function; neither is produced by the categorical data, and the known fibres exclude the first in the computed cases.

Dead end 1.11.

Route 1 (cancellation). To force a zero row of \(m^{(1)}\) one needs: a symmetric \(X\) with (R), (C), \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\), and an uncovered \(\mu\) such that \(\sum_i(-1)^i[E_{\mu,i}(\lambda),A_{\mu,i}(\lambda)]\in\ker\mathrm{ev}_1\) for every \(\lambda\). Route 2 (gluing). To force a nonzero relation \(c^{\mathsf T}m^{(1)}=0\) one needs a Hecke eigenfunctional on \(C_K\) over \(\mathbb F_q(\!(t)\!)\) vanishing at \(\varphi_0\) (Proposition 1.6(S)), e.g. two independent eigenfunctions for one character. Missing lemma for route 1: existence of such \(\mu\) (or, in the opposite direction, a proof that every multiplicity space is split Tate with parity as in Proposition 1.8(f), which would close route 1 entirely). Missing lemma for route 2: a construction of spherical functions on \(X(\mathbb F_q(\!(t)\!))\) with prescribed Hecke character (an analogue of Sakellaridis's Theorem 8.0.2 in characteristic \(p\)), or the close-field transfer of Lemma 1.7 fed by \(p\)-adic failures at unbounded ramification.

  1. Where cancellation would come from. \(m^{(1)}_{\mu\lambda}=\sum_i(-1)^i\operatorname{tr}A_{\mu,i}(\lambda)\) is an alternating sum of Weil numbers whose absolute values are fixed by purity (one value per degree). It can vanish with \(E_{\mu,\bullet}(\lambda)\neq0\) through root-of-unity factors within a degree (Frobenius permuting components, as in (E4) of Example 1.4), through non-Tate classes, or through a numerical coincidence between degrees of opposite parity (impossible under the split-Tate parity condition of Proposition 1.8(f)). By proper base change the stalk of \(V_\lambda\ast\mathrm{IC}_0\) at \(x_\mu\) is the cohomology of the projective fibre \(m^{-1}(x_\mu)\) with coefficients in \(V_\lambda\); root-of-unity factors arise when Frobenius permutes components (or has non-Tate cohomology).
  1. Obstruction in the computed cases. The stabilizer \(Z_{G^\theta}(\mu)\ltimes U_\mu\) of \(x_\mu\) is connected under (C) and acts on the fibre, preserving each irreducible component; components through rational torus-fixed points are Frobenius-stable. In the computed non-covered cases the relevant fibres are geometrically irreducible with Tate cohomology: \(\mathrm{GL}_4/\mathrm{Sp}_4\), fibre \(LG(2,4)\cong Q^3\) over \(\mathcal O_{(1,1,1,1)}\), multiplicity space \(k[2]\oplus k\oplus k[-2]\) with nonzero trace (verdict, Theorem 2); \(E_6/F_4\), fibre containing a 7-dimensional \(\mathrm{Spin}_9\)-orbit, conjecturally \(Q^7\) with trace \(c_8(q)-1\neq0\) (verdict, Conjecture 8.11). So route 1 is not available there.
  1. Why the categorical data cannot help. By Theorem 1.2(i) the categorical statement is the non-vanishing of the rows of \(M\) over \(R\); both routes concern \(\mathrm{ev}_1(M)\) (route 1: entries; route 2: rank of the \(U\)-reduced system of Proposition 1.8(e)), on which the row condition imposes nothing beyond Proposition 1.8(d).
  1. Status. Blocked at the two missing lemmas stated above; for route 2 the previous run's \(p\)-adic gluing at \(E_6/F_4\) (excluded here) is the only known candidate, and it is conditional on (A1)–(A5).

Builds on

  • Proposition 1.8 · Under (R), (C), D0–D3: three mechanisms vanish; the obligation sits on uncovered labels
  • Audit 1.9 · Check: Grothendieck groups modulo roots of unity cannot decide \((\mathrm{SG})_1\)
  • Lemma 1.7 · Close-field transfer: witnesses of (SG) persist over highly ramified \(p\)-adic fields

Built on by

  • Summary 1.12 · Verdict on part (2), with consequences for (1) and (3)

Sources