Nonsplit stages carry curvature

Exact statement

The real three-dimensional Heisenberg extension has nonzero central splitting curvature: for any linear lifts of the two quotient basis vectors, their bracket is the central basis vector. Therefore a naive zero-curvature split-stage model cannot represent this extension.

StatusExact, checked, and domain-corrected

External reviewNo documented external or specialist review of this Quotient Arsenal result is recorded.

Context

The general calculus treats a failed lift as a cocycle, its cohomology class as the obstruction, and a primitive—when one exists—as the receipt that repairs the presentation.

Proof or evidence

For $(x,y,z)(x',y',z')=(x+x',y+y',z+z'+xy')$ , the central Maurer–Cartan form satisfies $d(dz-x\,dy)=-dx\wedge dy$ ; every tested central shift leaves the bracket $e_Z$ .

Local-to-global and reduction-in-stages proof and evidence →

Verification notes

The archived totalization and stage proofs were reconstructed, their domains and choice indeterminacies were checked, and an independent finite implementation tested strict presheaf descent, secondary-coset invariance, split exterior dimensions, and Heisenberg curvature.

Limitations

This is a real Lie-group/Lie-algebra witness, not an integral-lattice theorem.
No novelty or specialist-review claim is made; the statement is limited to the displayed presentation and hypotheses.

Open work

Carry the extension curvature as a stage receipt.