§5 · Agent 5 · Audit: split candidate · 2026-10-17T23:24:13Z

Objection 5.10 · Objection to A3-10: placidity needs no τ (Chen–Yi Remark 37); the gap is elsewhere

refuted correction ★ key step

A3-10's reason (no symmetric embedding \(\tau\)) is wrong. Chen–Yi prove placidity of orbit closures from Drinfeld's theorem and Sakellaridis–Wang degenerations, and state (Remark 37) that it holds for general spherical varieties. The obligations that remain for this \(X\) are three: Sakellaridis–Wang's hypotheses in characteristic \(p\), a dimension theory (Chen–Yi's uses \(\tau\)), and running Appendix A and Prop. 70 verbatim.

Objection 5.10.

Failure of A3-10, step 1. "Chen–Yi's proof (§5) uses the symmetric-space embedding \(\tau\) … no such \(\tau\) is available" is incorrect, and so is step 4's "only the target side lacks a reference". In chen-yi-v4.pdf, Theorem 36 (orbit closures in \(LX\) are placid, with finitely presented embeddings) is deduced from:

Drinfeld's Theorem 35 (\(LY\) is a placid ind-scheme for smooth affine \(Y\));
the affine degeneration \(\mathcal X_\lambda\to\mathbb A^1\) of §5.1;
Lemma 34, which is [SW22, Lemma 2.3.10] plus smoothness of the \(B\)-stabilizer of the open \(B\)-orbit.

Remark 37 then states: "Since [SW22, Lemma 2.3.10] and Theorem 35 are valid for general spherical varieties, Theorem 36 remains true for general spherical varieties."

Corrected obligation (replaces the placidity lemma of A3-10). For \(X=\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) over \(\overline{\mathbb F}_p\), \(p\) odd:

(O1) the Sakellaridis–Wang affine degeneration ([SW22, §2.2]) and [SW22, Lemma 2.3.10] hold in characteristic \(p\). Chen–Yi's footnote 6 records characteristic assumptions in [SW22], which it says [BS25] removes. [BS25] is a paper on symmetric spaces, and whether it covers this non-symmetric \(X\) is unverified. The smoothness of the generic \(B\)-stabilizer is verified here for \(p=3,5,7,11\);
(O2) \(\delta(\nu)=3d_1+3d_2+e\) is a dimension theory on the \(L^+G\)-placid presentation \(LX=\operatorname{colim}\overline{\mathcal O_\nu}\), i.e. it satisfies Chen–Yi (8). Chen–Yi's own Prop. 41 uses \(\tau\), through Lemma 40;
(O3) Chen–Yi's App. A.5–A.7 and the proof of Prop. 70 (via Sun, Thm 1.2) then give H2, PH, D1, D2 for \(X\) verbatim. Pointwise purity, the one place where Chen–Yi say the slices built from \(\theta\) are essential (§1.2.5), is not used by Audit 5.9.

1. What Chen–Yi's §5 uses (read in chen-yi-v4.pdf).

§5.1 constructs the degeneration \(\mathcal X\to\mathcal A\) from the highest-weight filtration of \(k[X]\), following [SW22, §2.2], and \(\mathcal X_\lambda=\mathcal X\times_{\mathcal A,\iota_\lambda}\mathbb A^1\).
Lemma 34: \(LX_\lambda\cong LX\times_{f_\lambda,L\mathcal X_\lambda}L^+\mathcal X^\bullet_\lambda\). Its proof: "the stabilizer of \(B\) on \(X^\circ\) is a smooth subgroup [MS98, §6.3] and the lemma follows from [SW22, Lemma 2.3.10]".
Theorem 36's proof: \(L^iX=LX\cap t^iL^+V\) is placid by Theorem 35, since \(X\) is smooth affine. \(LX_\lambda\) is cut out of \(L^iX\) by an fp-locally closed condition, and Noetherian descent gives placid closures.

Nowhere is \(\tau:X\to G\) used. Remark 37 asserts the general spherical case.

2. Where \(\tau\) or symmetry is used.

Orbit parametrization (Prop. 17) and closure relations (Thm 18). For \(X\), A3-6 replaces these by a direct classification and a finiteness argument (Audit 5.6).
The dimension theory: Lemma 40 shows \(\tau(\mathcal O_u)\subseteq I_0\), hence \(\mathcal O_u\subseteq L^+X\), and Prop. 41 builds \(\delta\) on this.
Transversal slices (§6), which give pointwise purity.

Prop. 70 (semisimplicity of convolutions) uses only placid presentations, Čech nerves and finite-type models.

3. Check of the smoothness hypothesis for \(X\) (computation). At a point \(x=gx_0\), the scheme-theoretic stabilizer \(B_x=B\cap gHg^{-1}\) has Lie algebra \(\mathfrak b\cap\operatorname{Ad}(g)\mathfrak h\), and \(\dim B_x=1\) on the open \(B\)-orbit. For 30 random \(g\in G(\mathbb F_p)\), \(\dim_{\mathbb F_p}(\mathfrak b\cap\operatorname{Ad}(g)\mathfrak h)\) took the minimum value 1 at \(p=3,5,7,11\) (in 7, 14, 18, 23 of the trials). A point with value 1 lies in the open orbit, since \(\dim B\cdot x\ge12-1\). There \(\dim\operatorname{Lie}B_x=\dim B_x\), so the generic \(B\)-stabilizer is smooth for these \(p\).

4. Further remarks.

\(X\) is a closed \(G\)-orbit in the symmetric variety \(\mathrm{Sp}_8/(\mathrm{Sp}_6\times\mathrm{Sp}_2)\) (Audit 5.2), so even "no embedding" is inaccurate. This alone does not give placidity of \(L^+G\)-orbit closures, because \(L^+G\subsetneq L^+\mathrm{Sp}_8\).
A3-10 step 2 (the open embedding \(X\subset\mathrm{IG}\) induces the wrong topology) is correct.
A3-10 step 4 (the source of \(m_\lambda\) is \(L^+H\backslash\mathrm{Gr}_{\le\lambda^*}\), of finite type modulo a pro-unipotent gerbe; \(m_\lambda\) is representable and proper, hence of finite diagonal as Sun's Thm 1.2 requires) is correct.

5. Consequence. The categorical side remains conditional, but on (O1)–(O3) rather than on an unreferenced placidity lemma. (O1) is the only input that is not formal for this \(X\). (O2) is a routine but unproved compatibility, since the relative volumes of Audit 5.6 give the candidate \(\delta\). Until (O1)–(O3) are done, \(\mathcal C_k\) is not even defined on the pages for this non-symmetric \(X\): the brief's definition presupposes Chen–Yi's formalism.

Builds on

Dead end 3.10 · What blocks an unconditional \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) for \(X\)

Built on by

Summary 5.12 · Verdict: (SG) fails, proved; \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) modulo (O1)–(O3)
Theorem 6.3 · Verdict: sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\)

Sources

chen-yi-v4.pdf, §5.1, Lemma 34, Theorem 35, Theorem 36, Remark 37, footnote 6, §1.2.5, Lemma 40, Prop. 41, Prop. 70, App. A
sun-decomposition.pdf, Thm 1.2
scratchpad/solve/A5/bstab.py