Re-check V2 · Re-check by V2

Lemma X2.1 (holds): I re-derived each step by hand. det σ_n(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.=det A. σ_n(gAgᵀ)=(sgs⁻¹)·σ_n(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.. In the split form s⁻¹Hs⁻ᵀ=u⁻¹H, so σ_n there is gT↦sgs⁻¹T. Over k, u⁻¹sAsᵀ=hAhᵀ, and Frⁿ(h)=−h gives c(σ_n y)=(−I)c(y), so the K_n-class flips; Hecke operators commute with σ_n because Ad(s) preserves K_n and each K_n t^λ K_n. The trace twist in (d) follows from the equivariance–Frobenius compatibility at g=h together with the cocycle identity α_{h(−I),y}=ε_y∘α_{h,y}. Everything is correct given the formalism D1D1 · Traces and t-structure: proper base change and the Grothendieck–Lefschetz trace formula on convolution fibres, with a Frobenius-compatible perverse t-structure whose simple objects are the IC sheaves..

Lemma X2.2 (holds): There is no circularity. By centrality, ε (the action of −I) is an equivariant endomorphism. Since End(IC_{χ,ν}) is the scalars, ε is a scalar, and it can be read on the top stalk alone as χ(−I). (b) follows because −I generates π0={±I}, and Stab(x_0) is connected (L⁺SO₂, or L⁺T in the split form), which forces the sign stalks on O_0 to vanish. (c) is S2-1(d) combined with (a). (d) is a valid second proof: p is an étale μ2-gerbe for p odd, units of k[[t]] are squares, and [s]∈PGL2(O_n) acts as σ_n. This supplies exactly the step that the finding on A4-11 said was missing.

Computation X2.3 (holds): I re-derived the double cover: the stabilizer of (x_m,e2) is {a(0)=1}=Stab°, and the Legendre class of λ is K-invariant. The slice leading terms a(1,±i)ᵀ(1,±i) and c(c,±i)ᵀ(c,±i), with det 1, check exactly (V2/slices_check.py); the Kummer monodromy then kills the sign stalks on each branch. Independent lattice enumeration of T^kφ0, k≤3, for q=3,5,7,11,13 in both forms (V2/hecke_forms.py) gives 2;2,2;1,1;0 (split form, and SO2-form with q≡1 mod 4) and 0;0,0;1,1;0 (SO2-form with q≡3 mod 4), exactly as stated. All values are constant on the two K-orbits of each O_m, and evaluations at random points (V2/random_points.py) are consistent.

Theorem (repair of A4-11) X2.4 (holds with change): Steps 1–8 re-derived and correct: the block-unitriangular basis, σ-invariance of the trivial traces (S2-2), the top-orbit anti-invariance argument, the multiplicity-space trace formula, weight separation via BBD 5.3.2/5.4.1 without Chen–Yi Thm 81, and the Serre-subcategory step; it is form-independent for every odd q. The cross-check in step 8 rests on the second audit's odd-characteristic verification for A2-6, which I did not redo. Change (labels only, in step 9): 'Φ′=cΦ needs Aut = scalars' is C-8 step 4, not step 3. C-8 step 5 is a Lang-torsor evaluation at h·x_ν; it gives the same scalar as S2-2(a) but is not the same statement. The verdict 'C-8 steps 1–6 are correct for the split form' is right.

Correction (repair of A7-16) X2.5 (holds): Checked against chen-yi-v4.pdf. §2.1 fixes a θ-stable Borel pair (B0,T0), with the θ-split pair (B,T⊇S) separate; §4 assumes K connected for Sections 4–10; §8.2 opens 'G, B0, T0, θ are defined over F_q and T0 is split over F_q'; §8.2.1 defines Weil structures as systems Φ_n. I re-derived θ=Ad J, the θ-stable Borels B± (eigenlines (1,±i)), and T′=Z(J)=SO2 via N_B(T′)=T′; these are rational, and SO2 is split, iff i∈F_q. I also re-derived P⁻¹JP=diag(i,−i), 2P⁻¹P⁻ᵀ=H, and the Lang count from {g_{a,b}}. Cosmetic: in (d) c should a priori lie in F̄_q^× (rationality of the left side forces it into F_q anyway). Its check of C-5 is right.

Proposition X2.6 (holds): I re-derived that there are two SL2(F)-orbits, since H¹(F,SO2)=F^×/N(E^×)≅Z/2 for −1∉F^{×2}. The parity criterion holds because E/F is unramified with norms of even valuation. X(O)=K·I for p odd, so Hecke translates are supported in G(F)·X(O)=X^ev. For q≡1 mod 4 the reduction to SL2/T goes through P∈GL2(O), which needs i∈O (Hensel). Numerical checks for p=3,7: random g·I always has even m and random g·x_1 always has odd m, and T^kφ0 vanishes on every odd O_m. For p=5,13 both parities occur. The scope note on residue characteristic 2 is correct: x²+y² does not represent 3 over Z_2.

Audit X2.7 (holds): I searched every run node (and the root nodes) for §8.2, A7-16 and the SO2-model. A4-1 and A4-12 misattribute 'T split' to §8.2, but they use only the rationality of S. A7-25, A7-31 and A7-32 cite only the two bridges of A7-16, which the second audit found correct. A1-7 uses Thm 81 in the split form, where §8.2 and the connectedness of K hold (θ=Ad diag(1,−1), upper triangular/diagonal, K=T). No conclusion rests on the false remark. The A2-9 gap is now closed by S2-6.

Summary X2.8 (holds): Each input's SL2/SO2 content traces to A2-4 (Betti), to A2-8/A1-2 (split-form functions, every q and every F), or to A4-11 as repaired in S2-4. None uses the A7-16 remark. Its verdict 'holds as stated' is right for all four inputs. The second audit's unrelated minor defects in those inputs (the A7-11 repair, the A6-8 misattribution, A4-13's wording) are outside this target and remain as graded.