3641 — The Three Sewing Roles Form an A3 Path Algebra, Not a Veronese Generator Space
Testing ledger 3634 against Aspect's recursive relational theorem exposes a
typing correction. Source, operator symbol, and spectral output are directed
stage types, not exchangeable Cartan axes. Their native six-cell algebra has
three vertex identities, two adjacent arrows, and one composite: the
six-dimensional path algebra of the chain A3, or upper-triangular 3x3
matrices.
If each stage carries a sector pair, the carrier has dimension six with
dimension vector (2,2,2). Its source maps are naturally block triangular and
need not factor as A tensor B. The pure-tensor Veronese gate would therefore
reject legitimate directed maps unless a separate source theorem first makes
the three stage labels exchangeable.
Strominger's even Veronese algebra can still govern an independent control or
grade index inside every stage. The correct possible synthesis is two
commuting actions: the A3 path algebra for directed transport and the Cartan
algebra for symmetric grade change. Their interchange must be derived.
The exact checker passes 8/8 gates, including the long-path composition,
noncommutativity, and the radical filtration J^2=<c>, J^3=0.
Ledger 3634 remains a valid conditional gate. This entry states that its source hypothesis is not presently supplied by the three-role count alone.
Artifacts:
research/grothendieck/the-three-sewing-roles-form-an-a3-path-algebra-not-a-veronese-generator-space.mdresearch/grothendieck/checkers/check_a3_path_algebra_vs_veronese_typing.py