Even-Arity Framed Cut Descent Follows by Quadrangulation Induction
Entries 537, 540, and 541 establish framed Cut rigidity at arities eight, ten, and twelve. Their common mechanism extends to every even arity in the cellular fs/Kato sector.
Combinatorial induction
Let (n=2m). Physical Cuts are the non-boundary diagonals joining opposite vertex parities. There are
such diagonals. Each one divides the boundary into two paths of odd edge length, so both resulting polygons have even numbers of vertices. A noncrossing family of (r) physical Cuts therefore decomposes the (n)-gon into (r+1) strictly smaller even polygons.
If one region has more than four vertices, it contains another opposite-parity diagonal internal to that region. Hence the Cut family was not maximal. Conversely a quadrilateral has no further physical Cut. Therefore maximal simplices are exactly quadrangulations, have
Cuts, and decompose the polygon into (m-1) four-point factors.
Framed categorical induction
Assume framed physical lines and their Cut restriction maps are functorial and rigid at all smaller even arities. Every simplex of the (n)-point Cut nerve then carries a product of pointed contractible lower-arity mapping spaces. Its relative deformation and automorphism space is again contractible.
For (r) compatible restrictions, changing their order by (sigmain S_r) contributes
from the Koszul rule. The (r) native marked-normal Thom lines are odd, so their determinant contributes the same character. Thus
for every (r). On a maximal simplex the remaining coefficient is the product of (m-1) primitive four-point units, hence (+1). The top physical obstruction cochain is literally zero, not merely cohomologically trivial.
The unique local framed points therefore form a global coherent section, and the diagram of relative deformation spaces is terminal. Induction from the four-point unit and the established six-point connector gives
Finite audit
The checker exhaustively verifies all strata through (n=14). Its nerve censuses are
At (n=14), all (1428\cdot120=171360) orders of maximal five-Cut restrictions pass.
The scope is essential. The theorem uses the functorial external-product and Cut-restriction structure already established in the cellular fs/Kato model. It does not construct a raw global scheme-level six-functor correspondence, identify a numerical amplitude, or prove that forgetting the logarithmic framing preserves the class.
The remaining foundational frontier is therefore no longer higher-arity cellular coherence. It is comparison of this inductive fs/Kato object with a raw algebraic normalization-sheet/log-DNC six-functor realization.
The executable audit is
research/voevodsky/check_general_even_cut_induction.py.