Resolve reduced derivative trend with a recorded four-step diagnostic grid

experiments/GRAVITY_REDUCTION_IMPLEMENTATION_2026-09-17.md

@@ -91,3 +91,27 @@ The focused test source `tests/solver/gravity_reduction.cpp` covers:
 These algebraic tests establish the reduction contract. They do not establish
 FEM convergence, a faster implementation, or an additional virial-qualified
 stellar equilibrium; those require the integrated runtime experiments.
+
+## Resolved FEM difference sampling
+
+The first frozen n=1.5 probe completed all gravity solves and preparations. Its
+central-difference errors at steps 0.01, 0.001, and 0.0001 were respectively
+`1.7761e-9`, `1.9661e-11`, and `7.1208e-11`. Only the first exceeded twenty times
+the recorded arithmetic scale. The observed first-interval order was about
+1.956, but the conservative checker correctly left the trend unresolved because
+the finer endpoint did not meet its resolution criterion.
+
+The diagnostic now uses the explicit, validated step array `{0.1, 0.03, 0.01,
+0.001}` and records it in the summary. The larger steps test whether truncation
+dominates sufficiently to reveal a central-difference trend; retaining 0.001
+also preserves an independent small-step accuracy check. This adds one central
+pair, hence at most four extra gravity correction solves under the existing
+two-correction completion budget. The inner tolerance, prepared completion gate,
+and checker thresholds are unchanged. There is no assumption that larger steps
+will prepare successfully or automatically pass the trend check.
+
+These remain finite-range empirical derivative checks. The recorded arithmetic
+scale is heuristic, and a gravity residual bound by itself is not a bound on its
+amplified effect in the retained equations. The larger grid is intended to
+resolve a measurable trend without claiming such an unmeasured amplification
+bound or interpreting floor-limited differences as derivative failures.