NRR.EMP.VariableBody — variable-body equal-area power partitions

This public aggregator exposes the stable variable-body API for the Akopyan–Avvakumov–Karasev power-partition development. All results are stated over a compact metric parameter space with a continuous site family:

[MetricSpace X] [CompactSpace X]    sites : C(X, Config n)

The parent body varies in the Hausdorff subbody space BodySpace K A of a fixed planar parent K. Over this compact family the development provides:

• continuity of each power-cell area (continuous_cellArea);
• the closed equal-area relation and the resulting continuity of the canonical normalized equal-area weight (isClosed_isNormalizedEqualAreaWeight, weightBound, continuous_normalizedWeight_compactFamily);
• continuity of the canonical cells and their packaging as subbodies (continuous_canonicalCell);
• the canonical children child, each lying in BodySpace K (A / (n : ℝ)), with continuous bodies, areas, and perimeters (continuous_child, continuous_child_perimeter);
• the pointwise equal-area partition exposed only as a dependent witness (witness), with no topology placed on ConvexPartition.

The continuity results hold on this compact family only; Config n itself is not claimed to be compact. The proofs go through the equal-area existence and uniqueness cores reached via EMP.normalizedWeight, and do not invoke EMP.continuous_normalizedWeight_core.