The Eight-Point Framed Physical Line Is Rigid
Entry 536 separated the ambient derived (mathbb Z^5) from the uniqueness question for the distinguished physical section. This entry computes the correct relative deformation diagram with all physical boundary values held fixed.
On each of the eight Cut charts, the local object is the unique six-point framed connector of Entries 388 and 421 tensored with the primitive four-point unit. Its fixed data are the generic (Q)-roof, Cartier residue, endpoint swap, Tor orientation, and primitive coefficient (+1). The local relative deformation and automorphism groups are both zero.
On each of the twelve compatible pair overlaps, the object is the tensor of three primitive four-point units from Entry 442. Both ordered restrictions and their Koszul sign are fixed. A relative endomorphism must kill the tensor unit and is therefore zero; the overlap has no residual gauge automorphism.
Consequently the Čech diagram of relative endomorphism complexes has ranks
Its degree-zero homology and its higher automorphism groups vanish. Entry 533 supplies an existing global primitive section, so the framed section space is nonempty and has exactly one component:
The ambient free (mathbb Z^5) of Entry 536 does not act on this section. Its representatives are derived edge cochains of the unfixed loaded object; they do not preserve the predeclared primitive boundary values and hence are absent from the relative endomorphism diagram.
This is a rigidity theorem in the same cellular fs/Kato sector as Entries 435--446 and 533. It does not promote the construction to a raw global scheme-level correspondence, nor does it establish higher-arity factorization beyond the first octagon.
The next nonvacuous direction is no longer eight-point Cut coherence. It is either:
- construct the raw algebraic/log six-functor comparison beyond the Kato sector; or
- formulate the (n=10) physical Cut nerve and test whether the native Thom local system again cancels its Koszul holonomy before attempting a full cellular lift.
The executable audit is
research/voevodsky/check_n8_framed_physical_line_rigidity.py.