§1 · Agent 1 · Literature and Table 1 audit · 2026-10-17T03:06:06Z

Objection 1.8 · Comparability: the same pair \((G,X)\) in every row, but three different fields

proved audit: minor defect

Minor defect (second audit). Substance correct. Imprecisions: Chen–Yi Thm. 18 is listed among the ℓ-adic inputs, but over ℂ it is Nadler's theorem (Chen–Yi Remark 19) and Chen–Yi's proof is algebraic in every characteristic ≠ 2; 'for every homogeneous row X(F) is one G(F)-orbit' fails for the rows GLₙ/Oₙ and PGL₂/PO₂ (several orbits — the very reason (SG) fails there) and holds for the rows with G^θ connected, the standing setting; for E₆/F₄ in positive characteristic, H¹(F, F₄) = 1 is Bruhat–Tits rather than Kneser; 'Proposition 1.7 proves FN fails ⟹ SH fails' should read 'Proposition 1.7 proves that SH fails, using the function-level symmetry'.

Every row compares the same group acting on the same variety, but the sheaf statement is proved over \(\C\) with ℓ-adic inputs, the function statement under Sakellaridis's hypotheses over \(p\)-adic fields, and the only setting where the two levels are linked by a functor is \(\F_q(\!(t)\!)\) against ℓ-adic sheaves. A divergence read off the pages' table would therefore compare statements with no map between them.

Objection 1.8.

  1. In every row of Table 1 the same pair \((G,X)\) acts at both levels. The rows \(\SL_2/T\), \(\GL_2/T\), \(\PGL_2/T\) are one variety under different groups (Proposition 1.2); \(\Spin_4/\Spin_3\) and \((\SL_2\times\SL_2)/\SL_2\), \(\Spin_6/\Spin_5\) and \(\SL_4/\Sp_4\) are the same pair up to the exceptional isomorphisms. The conventions (left actions, \(\operatorname{vol}K=1\), Sakellaridis's \(H\backslash G\) exchanged by inversion) do not affect (SG).
  2. Three settings are in play, and the pages compare only two of them:
  3. A cell "sheaves hold, functions fail" or the reverse, read off Table 1, compares (S1) with (S2). Such a divergence would not be a contradiction, only a discrepancy between two shadows. The meaningful test is (S3).
  1. Setting (S1). The sheaf statement is formulated for Betti sheaves over \(\C\). Every non-formal input used to decide it — orbit closures, placidity, pointwise purity, the decomposition of convolutions, the semisimplicity criterion — is proved by Chen–Yi for ℓ-adic sheaves over \(\overline{\F}_p\), \(p\ne2\). Carrying these to \(\C\) is open item 1 of Proposition III; Chen–Yi's §1.2.6 asserts that the arguments carry over to mixed Hodge modules but does not write this out.
  2. Setting (S2). Theorem E assumes (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit., and (A1)(A1) Field and group: F is a local field of characteristic 0 (p-adic), and G is split over the integers, with connected fibres and simply connected derived group. Its characteristic-p counterpart is (A1ₚ). forces \(\operatorname{char}F=0\). Lemma A(1) also uses Borel–Serre finiteness, which needs characteristic \(0\); but for every homogeneous row on the pages \(X(F)\) is one \(G(F)\)-orbit by direct arguments (Hilbert 90, Kneser, Witt).
  3. Setting (S3). For \(F=\F_q(\!(t)\!)\) and ℓ-adic sheaves over \(\overline{\F}_q\), the function attached to a Weil sheaf by the trace of Frobenius relates the two levels. Here Theorem E is unavailable, so every "under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies." function cell is silent; the unconditional function results — Propositions D, G, J, the spheres (Proposition 1.3) and \(\GL_4/\Sp_4\) (Proposition 1.4) — do hold there.
  4. Consequence. The meaningful divergence test is (S3), where Theorem 1.6 proves "functions hold \(\Rightarrow\) sheaves hold" under connected stabilizers, and Proposition 1.7 proves "functions fail \(\Rightarrow\) sheaves fail" for \(\SL_2/\SO_2\).
  5. Models. \(\SL_2/\SO_2\) over \(\F_q\) requires choosing the split model \(\theta=\operatorname{Ad}\operatorname{diag}(1,-1)\), \(G^\theta=T\) split, since \(\SO_2\) of \(x^2+y^2\) is split only when \(-1\) is a square. The sheaf statement is insensitive to this choice; the function statement (Proposition J) is formulated for the split model.

Builds on

  • Setup 1.1 · Table 1 of the Summary, re-derived row by row

Built on by

  • Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide
  • Audit 7.16 · Audit: the comparability objection is correct — three fields, two bridges, and what each bridge needs
  • Correction 10.5 · The \(\SO_2\)-model of \(\SL_2/\SO_2\) over \(\F_q\) and Chen–Yi's splitness hypothesis
  • Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\); Lang’s count holds in both
  • Proposition X2.6 · (SG) fails for \(\mathrm{SL}_2/\mathrm{SO}(x^2+y^2)\) over every \(F\), \(q\) odd
Original record, as the agent wrote it at 03:06 UTC

Comparability of the two statements: same X and G per row, but three different fields

Checked for every row: the same pair \((G,X)\) acts at both levels (the rows \(\mathrm{SL}_2/T\), \(\mathrm{GL}_2/T\), \(\mathrm{PGL}_2/T\) are the same variety under different groups, A1-2; \(\mathrm{Spin}_4/\mathrm{Spin}_3\) and \((\mathrm{SL}_2\times\mathrm{SL}_2)/\mathrm{SL}_2\), \(\mathrm{Spin}_6/\mathrm{Spin}_5\) and \(\mathrm{SL}_4/\mathrm{Sp}_4\) are the same \((G,X)\) up to the exceptional isomorphisms, so the identifications in Table 1 are legitimate). Conventions (left actions, \(\operatorname{vol}K=1\), Sakellaridis's \(H\backslash G\) exchanged by inversion) are stated on classical.html and do not affect (SG).

Fields: three settings are in play and the pages compare only two of them. (1) Sheaf statement as defined: Betti \(\mathcal C\) over \(k=\mathbb C\), \(F=\mathbb C(\!(t)\!)\). All non-formal inputs (Chen–Yi Thm 18, 36, 81, Prop 70, Thm 87) are ℓ-adic over \(\overline{\mathbb F}_p\), \(p\ne2\); the transfer is the pages' open item 1 (Chen–Yi §1.2.6 only asserts it). (2) Function statement under (A): \(F\) \(p\)-adic, \(\operatorname{char}F=0\) ((A1)); also Lemma A(1) uses Borel–Serre finiteness (char 0), though for every homogeneous row on the pages \(X(F)\) is one \(G(F)\)-orbit by direct arguments (Hilbert 90, Kneser, Witt). (3) The only setting where the two levels are literally related by a functor (trace of Frobenius): \(F=\mathbb F_q(\!(t)\!)\) versus ℓ-adic sheaves over \(\overline{\mathbb F}_q(\!(t)\!)\). Here Theorem E is unavailable, so every "under (A)" function cell is silent in this setting; the unconditional function results (Props D, G, J; A1-3; A1-4) hold there. Consequences for the question: a cell "SH holds, FN fails" or "SH fails, FN holds" read off Table 1 would compare statements over \(\mathbb C(\!(t)\!)\) and \(\mathbb Q_p\) with no map between them; such a divergence would not be a contradiction, only a discrepancy between two shadows. The only meaningful divergence test is setting (3), where A1-6 proves FN ⟹ SH under connected stabilizers and A1-7 proves FN fails ⟹ SH fails for \(\mathrm{SL}_2/\mathrm{SO}_2\). Also: \(\mathrm{SL}_2/\mathrm{SO}_2\) over \(\mathbb F_q\) requires choosing a split model (\(\theta=\operatorname{Ad}\operatorname{diag}(1,-1)\), \(G^\theta=T\) split), since \(\mathrm{SO}_2\) for \(x^2+y^2\) is split only when \(-1\) is a square; the sheaf statement is insensitive to this, the function statement is formulated for the split model (Prop J).

Sources