C-G2 · 2026-10-21T15:07:39Z

Review G2 · Agents 3, 5

A3-1 (minor): Cyclicity, the injection of periods into eigenfunctions and the invariant-measure pairing hold in either characteristic. The inverse-variable substitution in Step 4 should be performed in Haar-integral form; a coset summand is not right K-invariant before its inner integration. A5-1 clarifies this notation. The conclusion does not use Sakellaridis's characteristic-zero-only module theorem.

A3-3 (error): Part (d) is false with its universal condition a != b^{+/-1}. The quotient SL2^3/{+/-1} also identifies simultaneous central signs. Take a=2,b=-2: the triples have identical quotient images despite satisfying its stated exclusion. The triple/isotropic-graph model, stabilizer, transitivity, dimension and cited dual embedding hold. A5-2 gives the correct generic nonconjugacy condition. The error does not enter the independent two-period proof.

A3-4 (minor): The four flag orbits, smooth closed stabilizers, Mackey filtration and generic lifting re-derive over any nonarchimedean local field. The Ext identity applies because the target is admissible/reflexive, induction from a closed subgroup is exact and right adjoint, and unipotent coinvariants are exact. The displayed reducibility list is incomplete as an exact description, but genericity only requires a proper closed exceptional set; A5-3 enlarges it. Casselman 1980 Prop. 3.5 applies on the regular generic locus with no characteristic-zero restriction.

A3-8 (error): The semismall defect identity, relevance condition, branching and exhaustive relevance hold, but (b) wrongly calls c the number of top-dimensional components of EVERY nonempty fibre. It counts components of expected dimension f only. For lambda=((1,0),(1,0)), nu=(0,0,2), the fibre has dimension 2 and two top components, with count 2q^2+3q-1, whereas f=3 and c=0. A5-8 explicitly uses f-dimensional components. The SO5/SO4 integral-weight branching hypotheses hold, also confirmed by 21 independent exact Weyl-character checks.

A3-9 (error): Step 2 identifies the multiplicity with ordinary H^{2f} of the open Gr^lambda fibre, whereas the !-stalk is H_c^{2f}. For lambda=((1,1),0), nu=(0,0,2), this is a real difference: the nonproper dimension-2 SL2 Grassmannian orbit has H^4=0 and one-dimensional H_c^4. A5-8/A5-9 correct it. The conditional perversity, expected-dimensional branching multiplicities, generation and trace obstruction survive. Its universal nonzero fusion evaluation also needs A5-2/A5-9's central-sign exclusion.

A3-10 (error): The missing-placidity diagnosis contradicts Chen-Yi Thm. 36 and Rem. 37: the latter extends the theorem to general spherical varieties. Its proof uses affine degeneration, Lemma 34 and Drinfeld Thm. 35, not the asserted symmetric embedding. Bare placidity still does not verify a compatible dimension theory or finite-type purity/decomposition descent. Sun Thm. 1.2 requires finite-type finite-field stacks with affine stabilizers and a proper finite-diagonal map. A5-10 correctly leaves (O1)-(O3). The isometry bundle has fibre SL2, but no constant-SL2 torsor action is supplied.

A3-11 (error): The main relevant-multiplicity and all-n obstruction assertions hold. The closing remark is false: one trace-zero multiplicity space need not make an entire row vanish, since other columns/degrees may survive; a zero row at n=1 need not recover at every even n. Eigenvalues 1,zeta_3,zeta_3^2 still have trace zero at n=2. Row recovery also does not imply column spanning. A5-12 calls for dropping the remark. For irrelevant fibres, c=0 plus O(q^{n(f-1)}) is a valid asymptotic bound, not their actual leading term.

A3-13 (error): The principal verdict remains conditional: unconditional function failure, conditional sheaf generation, no complete split symmetric counterexample. However, its claim that the one missing input is the placidity lemma repeats A3-10's false diagnosis. Equal-characteristic affine degeneration, dimension-theory compatibility and transfer of the sheaf/decomposition formalism remain (O1)-(O3), as A5-10/A5-12 state. The generic fusion and conditional branching conclusions survive the existing sign, H_c and expected-dimension corrections.

A5-2 (error): The corrected condition a != +/-b^{+/-1} is right. However, the opening claim that quotient images COINCIDE for both a=-b and a=-b^{-1} is false in the inverse case: they are conjugate, generally unequal. At a=-1/2,b=2, their SO5 torus coordinates -1/4 and -4 are interchanged, so the images differ; simultaneous inversion in the subgroup Weyl group relates them. The a=-b counterexample and all generic-fusion uses hold. The odd-characteristic embedding in the Sp8 symmetric variety is valid.

A5-5 (minor): The quotient maps, invariant self-dual pairings, Goursat surjectivity and independent pullbacks prove noncyclicity over every local field and every unramified extension. No characteristic-zero module theorem is used. The final heuristic sentence about a generic (chi1,chi2) having empty fibre must mean the full four-dimensional pair of G parameters; it is false on the displayed diagonal family (chi,chi), which has the two generic lifts. This imprecision is unused in the proof.

A5-11 (error): The principal n-independent branching obstruction and explicit retract example hold conditionally. Step 4 still falsely asserts that at even n the row is nonzero merely from trace zero at n=1. A 3-cycle with eigenvalues 1,zeta_3,zeta_3^2 is trace zero at n=2, and a single trace-zero space need not force a whole zero row. The actual split example has scalar Frobenius on relevant cohomology, so its all-n failure is unaffected. The generic parity assertion must not be retained as a theorem.

A5-12 (minor): The conclusion correctly leaves the split sheaf example conditional on (O1)-(O3) while proving the function half unconditionally. Its listed repairs cover signs, H_c, genericity and the geometric obligation. The blanket assertion that A3-8 holds needs the expected-dimensional convention; saying only two assertions were false overlooks the H versus H_c error and the parity remark. These bookkeeping defects do not change the final mathematical verdict.

Builds on: A3-3 · A3-8 · A3-9 · A3-10 · A3-11 · A3-13 · A5-2 · A5-11