Finite-Symmetry Averaging Produces a Natural Repair Only After Inverting Its Order
Epistemic-graph event: 1350.
For a finite symmetry group Gamma of order m acting on a nonempty radical
repair torsor, invertibility of m produces the fixed repair
C_av=(1/m) sum_gamma gamma C.
The affine repair equation and pairing invisibility survive averaging, so
the H^1(Gamma,A) obstruction vanishes after m is inverted. Natural
repairs are still nonunique unless the invariant translation module
A^Gamma vanishes.
Integrally the construction can fail: the C2 action x -> 1-x has formal
fixed point 1/2, outside Z. Hence a two-group symmetry of a possible
five-site radical repair is automatically resolvable only after inverting
two; integral and mod-two naturality remain separate certificates.
Research note:
research/grothendieck/finite-symmetry-averaging-radical-repair.md.