§1 · Agent 1 · Trace formalism · 2026-10-17T21:12:28Z

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

proved ★ key step

The trace of Frobenius is a base change of additive invariants; categorical generation is closure under retracts. Generation fixes which rows of the multiplicity \(R\)-matrix are nonzero, (SG) asks that its cokernel vanish at the Frobenius point \(\mathrm{ev}_1\). Outside (R) this fails in fully verified examples; inside (R)+(C) the label, orbit and automorphism mechanisms vanish and the obligation is a rank condition on uncovered labels.

Summary 1.12.

  1. (Mechanism, proved Setup 1.1, Theorem 1.2, Theorem 1.3.) Under (R), (C), D0–D3 there is one matrix \(M\) over \(R=\mathbb Z[\overline{\mathbb Q}_\ell^\times]\) with
$$\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\iff\text{no zero row},\qquad\mathrm{Tri}_k=\mathcal C_k\overset{\text{given gen.}}{\iff}\operatorname{coker}M\otimes_\varepsilon\mathbb Z=0,\qquad(\mathrm{SG})_n\iff\operatorname{coker}M\otimes_{\mathrm{ev}_n}\overline{\mathbb Q}_\ell=0,$$

and at the categorical level \(\operatorname{Tr}(\mathrm{Fr},\mathcal C_k)=C_K\), \(\operatorname{Tr}(\mathrm{act}_{\mathrm{IC}_0})=-\ast\varphi_0\); by Morita invariance the trace of \(\mathcal C_k\) is spanned by retract classes \([\mathrm{IC}_\nu]\), the image of \(\mathcal H\) by the full classes \(\sum(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)[\mathrm{IC}_\nu]\); the discrepancy is carried by \(A=\mathrm{act}^R(\mathrm{IC}_0)\neq\mathbf 1\) and the Frobenius eigenvalues on \(E_{\nu,i}(\lambda)\).

  1. (Formal examples, proved Example 1.4.) Thick generation by a Frobenius-equivariant image does not give surjective traces: nilpotents (killed by weights), fusion of classes, Frobenius permuting generators, traceless equivariance.
  1. (Forcing criteria over \(F\), proved Counterexample 1.5–Lemma 1.7.) (a) Frobenius moving Hecke labels / several \(G(F)\)-orbits; torus criterion \(\ker(X_*(H)_\sigma\to X_*(T)_\sigma)\neq0\). (S) a Hecke eigenfunctional vanishing at \(\varphi_0\) (sufficient; also necessary if \(C_K\) is finitely generated); (d) multiplicity \(\ge2\) for some unramified \(\pi\); (b) an \(\mathcal H_K\)-endomorphism of \(C_K\) not from \(\mathcal H_K\); (c) a row relation \(c\in\prod\overline{\mathbb Q}_\ell\); close fields: under integral models and a Cartan decomposition, \(p\)-adic failures at unbounded ramification transfer to \(\mathbb F_q(\!(t)\!)\).
  1. (Part (1), proved Counterexample 1.5.) \(X=R_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m/U_1\) (and \(R\,\mathrm{GL}_m/U_m\)) is symmetric with (C) and D0–D3, \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\), and (SG) fails over \(\mathbb F_q(\!(t)\!)\) for every \(q\); only (R) fails (non-split \(T\)). (SG) is not an invariant of \((X_k,G_k)\).
  1. (Part (3), reduction proved Proposition 1.8, completion blocked Dead end 1.11.) Under (R), (C), D0–D3 the mechanisms of 3(a) and the geometric form of 3(b) vanish, top entries of \(m^{(n)}\) never vanish, and the exact obligation is: the projections \(p_U(T_\lambda\ast\varphi_0)\) span \(\mathbb C^{(U)}\), \(U\) the labels without dominant lift — equivalently (given finite generation) no Hecke eigenfunctional vanishes at \(\varphi_0\). \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) does not decide it.
  1. Why the dictionary cannot transport generation. The function–sheaf dictionary (Laumon (1.1.1.1)–(1.1.1.4), Zhu Lemma 5.6.1, IIa) and its categorical form (Theorem 1.3) are additive and compatible with base change (Ponto–Shulman Prop. 6.2, Ex. 6.3). An additive invariant of a retract is not a combination of invariants of the ambient objects, so thick generation, which is closure under retracts, is recorded only as the support of the rows of \(M\). (SG) is the vanishing of \(\operatorname{coker}M\) at one point; Laumon's Theorem 1.1.2 needs all \(n\) to recover classes, and one \(n\) is one point. This is the formal mechanism; the direction that survives it is IIc (functions \(\Rightarrow\) sheaves), because a zero row stays zero under every specialization.
  1. What forces a counterexample. Any of the criteria in item 3 of the statement; Counterexample 1.5 realizes (a), (S), (d), (b) and (c) at once, the obstruction being the ideal \(\langle[\gamma]+[-\gamma]\rangle\subseteq\ker\mathrm{ev}_1\) of classes induced from \(\mathbb F_{q^2}\).
  1. Converse conditions. (SG)\(_n\) for all \(n\) follows from \(\operatorname{coker}M=0\) (Theorem 1.2(iv)), from coverage \(\varphi(\Lambda_T^+)=\Lambda_S^+\) (Proposition 1.8(d),(e); Corollary C), and from a triangular summand system with split-Tate parity (Proposition 1.8(f)).
  1. Heuristic picture (Remark 1.10, labelled heuristic except for tori, where it is proved): generation is affineness over \(\mathcal H\); (SG) is a closed immersion of Frobenius-twisted loop spaces; failure is fusion.
  1. Remaining obligation for the team's part (3): Proposition 1.8(g) and the two missing lemmas of Dead end 1.11.

Builds on

  • Theorem 1.2 · Theorem A: decategorification is base change of one \(R\)-matrix
  • Theorem 1.3 · Theorem T: trace of Frobenius on \(\mathcal C_k\) is \(C_K\); the missing retract classes
  • Example 1.4 · Four minimal examples: generation by an image, trace map not onto
  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion
  • Proposition 1.6 · Forcing criteria: vanishing eigenfunctions, multiplicity two, extra endomorphisms
  • Lemma 1.7 · Close-field transfer: witnesses of (SG) persist over highly ramified \(p\)-adic fields
  • Proposition 1.8 · Under (R), (C), D0–D3: three mechanisms vanish; the obligation sits on uncovered labels
  • Remark 1.10 · Spectral picture: generation is affineness, (SG) a closed immersion of loop spaces
  • Dead end 1.11 · Dead end: forcing \((\mathrm{SG})_1\) to fail inside (R)+(C) by cancellation or gluing

Built on by

  • Remark 6.1 · Registry after the first two returns: a non-split counterexample found twice; launch its audit

Sources