docs(changelog): local interpolation with a linear-reproduction guarantee

docs/developer/CHANGELOG.md

@@ -6,6 +6,49 @@ This log tracks significant development work at a conceptual level, suitable for
 
 ## 2026 Q3 (July – September)
 
+### Local Interpolation That Reproduces Linear Fields (July 2026)
+
+**The local scattered-point interpolator now has a linear-reproduction
+guarantee**, and the swarm proxy variables use it by default (#430).
+
+Underworld's local interpolator was inverse-distance (Shepard) weighting.
+Its weights are positive and sum to one, so it reproduced a *constant* exactly
+but not a linear field: any field with a gradient was smeared, and the error
+did not fall as the points crowded together. Measured on an exactly linear
+field — which lies inside both the P1 and P2 proxy space, so the finite element
+discretisation contributes nothing and all of the error is particle-to-node
+transfer — the swarm proxy carried 1e-3 to 2e-2 relative error, falling only
+first order with refinement and **not at all** with stencil size.
+
+- New `order=1` scheme on `uw.kdtree.KDTree.rbf_interpolator_local`: a
+  polyharmonic kernel with an affine tail, solved per target point on its own
+  nearest-neighbour stencil. Constants and linear fields are exact by
+  construction, and the result stays sparse at `nnn` non-zeros per row. The
+  existing inverse-distance path is unchanged and remains the KDTree default.
+- Proxy error on a linear field falls to round-off; on a quadratic field it
+  improves roughly ninety-fold in 2D and thirty-fold in 3D. Swarm proxy
+  variables now default to `order=1`.
+- `KDTree.interpolation_matrix()` returns the transfer as a sparse operator.
+  The weights depend only on geometry, so one build serves every field and
+  component — the form a multigrid prolongation or a remesh transfer wants.
+- Stencils that cannot support an affine fit (collinear in 2D, coplanar in 3D)
+  are detected, retried on a wider neighbourhood, and only then fall back to
+  inverse distance with a warning. They never return `NaN`, and never silently.
+- An opt-in limiter bounds the non-affine part of the interpolant while leaving
+  the linear reconstruction untouched, so limiting does not cost the guarantee.
+
+Two deliberate exclusions, both measured rather than assumed:
+`MeshVariable.rbf_interpolate` keeps inverse distance because it is the
+fallback rung of the point-location ladder, whose documented contract is that
+it is bounded; and `IndexSwarmVariable` material level sets keep it because a
+material indicator is piecewise constant — there is nothing for linear
+exactness to gain at a discontinuity, and signed weights push level sets
+outside `[0, 1]`.
+
+Related: swarm proxy refresh no longer fails under an active units model
+(#426, #434); the units the proxy advertises are tracked separately (#439).
+New subsystem documentation: `subsystems/interpolation.md`.
+
 ### Purposeful Adapt / Redistribution Naming (July 2026)
 
 **User-facing mesh-modification names now state the capability** (maintainer