S2-5 · 2026-10-21T15:10:57Z
Cluster-local rederivation of the affected verdict inputs
The three replacements S2-1, S2-2, S2-3 repair A1-11, A2-11, A5-2, respectively. None of their false assertions is required by the affected verdict inputs. The exact remaining obligation under full (R),(C) is still spanning on the uncovered-label quotient, or a concrete example violating that spanning condition while satisfying categorical generation. The non-symmetric split \(\mathrm{Sp}_4^2/\mathrm{SL}_2^3\) candidate still has only a conditional categorical half, subject to (O1)--(O3).
- A1-12. Its named remaining obligations use the matrix/triangular criteria, not the false rational-fixed-point claim. Replace its reference to A1-11 by S2-1. There is still no established split (R),(C) counterexample from this route. The matrix and categorical-trace assertions require the separate trace-cluster repairs; this return makes no new claim about those assertions.
- A2-12. The trace-column versus geometric-retract distinction and unitary forcing mechanism do not use the full-torus categorical claim deleted in S2-2. The analysis of injectivity modulo ineffective cocharacters is now proved by S2-4. Read the phrase about twisting as label twisting only: (R) removes that twist, not multiplicity-space Frobenius. For the displayed cyclicity identity, explicitly use the convention \(\beta:M\to\mathrm{Fr}^*M\), with inclusion \(i:M\to N\), retraction \(p:N\to M\); then \(\mathrm{Fr}^*(i)\beta p:N\to\mathrm{Fr}^*N\) is well typed. Under the opposite convention the corresponding morphism is \(i\beta\mathrm{Fr}^*(p)\).
- A2-13. The unitary counterexamples are proved in A2-3--A2-10, independently of A2-11. Its nonsplit-torus exclusion is supplied by S2-2. The sentence about a Weil structure not coming from a rational Satake trace must refer to the designated non-even target in A2-8, not to all Satake images; rational images and the unit plainly have induced Weil structures. The full-(R) question remains unresolved by this approach.
- A4-12. Delete its unproved categorical claim for \(\mathrm{GL}_2/T_E\); the dead-end conclusion is proved without it in S2-2. Its independently reviewed unitary counterexamples retain their characteristic-\(p\) proofs. No complex-to-finite-field transfer is introduced by this repair.
- A4-19, A4-20. Replace the ineffective-factor quotient sketch by S2-4. If literal injectivity on the full dominant cone is discussed, retain the positive-rank condition; the rank-zero case is trivial. The conclusion about uncovered-label spanning remains. Limit any assertion that twisting vanishes to twisting of labels; do not exclude multiplicity cancellation or arbitrary nongeometric module autoequivalences. The categorical-trace part of A4-20 remains for the other salvage cluster and root to validate.
- A5-12. Use the exact equality/conjugacy distinction of S2-3; generic fusion is unchanged. The characteristic-\(p\) function-level failure is independent of the erroneous exceptional-point equality. The categorical half stays conditional on (O1) characteristic-\(p\) spherical degeneration/placidity inputs, (O2) the dimension theory, and (O3) the compatible sheaf formalism/decomposition theorem. When using A3-8 or A5-9, retain the expected-dimensional convention and compactly supported top cohomology; this return does not upgrade any of those conditional statements. It also makes no claim that the run contained only two defects.
- Consequently the cluster changes no mathematical scope: nonsplit unitary examples settle the unrestricted converse negatively; the full-(R),(C) question needs the precise uncovered-label argument; and the split nonsymmetric candidate requires its explicit geometric formalism. These are dependency conclusions, not a new audit of the independent branches already reviewed by G1/G2.