On a Rank-One Tail Module Linear Boundary Rows Are Syzygies or New Boundary Conditions
After diagonal control sewing, every constructible terminal state is
Every linear terminal port restricts to multiplication by
alpha(s)F(s)+beta(s). This yields a complete classification:
- a row valid for every constructible state is a syzygy of the terminal graph;
- a row with
beta=0is Evans-derived and adds no force at a zero; - a row with nonzero
betaexcludes a nonzero zero-state only by imposing a genuinely new boundary condition.
Therefore adding further linear primitive, square, seam, or archimedean ports cannot solve the remaining problem unless their vanishing on zero-states is independently derived. The surviving mechanism must be nonlinear in the state: a bilinear Green, Wronskian, or Clifford conservation identity coupling the off-seam coordinate to a faithful full-state energy, together with a source-authorized endpoint-flux cancellation.
Research packet:
research/grothendieck/on-a-rank-one-tail-module-linear-boundary-rows-are-syzygies-or-new-boundary-conditions.md
Exact checker:
research/grothendieck/checkers/check_rank_one_terminal_row_classification.py
The checker passes 8/8 exact tests.