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Theorem · explicit cocycle primitive

QA-BK-T02

The receipt explicitly kills the capping cocycle

Exact statement

For any normalized set-section σ of π, its factor-set cocycle satisfies c_σ=δ(D∘σ); correcting σ by that 1-cochain produces the unique zero-D homomorphic section.

StatusProved by direct substitution; internally audited
External reviewNo documented external or specialist review of this FPRD deduction is recorded.

Context

A section can fail to respect multiplication. Its failure is a 2-cocycle valued in the forgotten boundary-twist lattice.

Definitions

  • c_σ(h₁,h₂) is defined by σ(h₁)σ(h₂)σ(h₁h₂)⁻¹=k(c_σ(h₁,h₂)).

Hypotheses and scope

  • The direct-product capping theorem QA-BK-T01.

Proof or evidence

Applying D to the factor-set identity gives c_σ(h₁,h₂)=r(h₁)+r(h₂)−r(h₁h₂), with r=D∘σ. The corrected section k(−r(h))σ(h) has zero factor set.

Cocycle calculation →

Verification notes

Signs and normalization were checked symbolically over finite groups and free-abelian receipt dimensions one through four.

Limitations

  • Vanishing of the group-extension class does not imply that a positive factorization can be corrected without negative twists.

Open work

Extend the calculation to the semidirect coefficient system arising from permuted boundary labels.