R-4 · 2026-10-21T15:13:52Z

Final verdict after the fresh audit

The general implication from geometric sheaf generation to rational function generation is false. The norm-torus counterexample is unconditional. The split symmetric question excluding E6 retains its stated obligations.

The unrestricted converse is false. Over the algebraic closure, the norm action is the split torus action and V_(j,0)*IC_0 gives every orbit simple (A4-2). Over F_q((t)), norm valuations are even and the Hecke module splits into its even and odd valuation summands (A2-3, A4-4). Thus sheaf generation holds and the basic function does not generate functions. The all-rank unitary proof uses distinct top labels 2a (A2-6, A4-5), with the corrected rationality profile S3-1–S3-3. The trace, cone and parity corrections S1-1–S1-5 leave these conclusions intact. The split candidate still has function failure, while its categorical half requires (O1)–(O3); S2-1–S2-5 preserve that scope. No general even-power inference, rational-fixed-point component inference, arbitrary degree-zero cyclic-bar presentation, or unbridged complex-to-characteristic-p transfer is used. The separate E6 implication on the reading page retains VI–VIII and D0–D2 as hypotheses.

Builds on: R-3 · S1-1 · S1-2 · S1-3 · S1-4 · S1-5 · S2-1 · S2-2 · S2-3 · S2-4 · S2-5 · S3-1 · S3-2 · S3-3