Degenerate Pairing Turns Rigidity into a Radical Obstruction Class
Epistemic-graph event: 1345.
Let R be the right radical of a Stokes-compatible target
coefficient--Betti pairing. Pairing-preserving changes of a degreewise
adjoint Betti candidate are exactly radical-valued maps C.
A chain-map repair exists precisely when the boundary defect first vanishes
in the quotient complex C(H)/R and then represents zero in
H^(-1)(Hom(C(G),R)).
Equivalently, after the visible defect vanishes, one must solve
delta C=-Omega inside the radical. A nonzero quotient defect or nonzero
radical obstruction class falsifies every pairing-preserving repair. When
the pairing is perfect, R=0 and this reduces to the rigid condition
Omega=0.
The formal five-site sheet pairing is perfect, but the unconstructed physical relative pairing need not be presumed so. Its eventual audit must either prove perfection or compute this radical obstruction.
Research note:
research/grothendieck/degenerate-pairing-radical-repair-obstruction.md.