Equivariant Radical Repair Has a First-Cohomology Obstruction
Epistemic-graph event: 1349.
Let a symmetry group Gamma act on the nonempty torsor of radical repair
classes, whose translation module is
A=H^0(Hom(C(G),R)). For any repair C_0, the displacement
b(gamma)=gamma C_0-C_0
defines a canonical class [b] in H^1(Gamma,A). An automorphism-natural
repair exists exactly when this class vanishes; natural repairs then form a
torsor under A^Gamma.
The hostile integral control is C2 acting on Z by m -> 1-m. Its class
is the nonzero element of H^1(C2,Z_sign)=Z/2, so no integral fixed repair
exists. After inverting two, the fixed repair 1/2 appears.
Therefore existence and naturality of a radical repair are independent certificates, with the same averaging and integrality hazards already seen for finite-deck transfer.
Research note:
research/grothendieck/equivariant-radical-repair-h1-obstruction.md.