feat(nonlinear-static-newton-raphson): step 11 - equilibrium-projection-and-reference-gate

docs/nonlinear-static-newton-raphson/formulation.md

@@ -1599,7 +1599,15 @@ on any increment/recovery/output failure:
 ### 14.1 Four convergence channels
 
 Let free translation and rotation-coordinate subvectors carry superscripts \(u\)
-and \(\phi\). Residual channels use
+and \(\phi\). Define the matching external residual-channel scales
+
+\[
+S_{ext}^u=\|\mathbf f_{ext,f}^u\|_\infty,
+\qquad
+S_{ext}^\phi=\|\mathbf f_{ext,f}^\phi\|_\infty.
+\]
+
+When the matching external scale is nonzero, residual channels retain
 
 \[
 \eta_R^u=
@@ -1615,10 +1623,31 @@ and \(\phi\). Residual channels use
 \|\mathbf f_{ext,f}^\phi\|_\infty)}
 \]
 
-when the matching denominator is nonzero. They pass iff
-\(\eta_R^u\le10^{-2}\) and \(\eta_R^\phi\le10^{-2}\).
-An exact-zero denominator instead passes iff the numerator is at most
-\(10^{-3}\) model-consistent force or \(10^{-3}\) model-consistent force-length.
+for \(S_{ext}^u>0\) and for \(S_{ext}^\phi>0\), respectively. Each nonzero-external-scale
+channel passes iff its ratio is \(\le10^{-2}\). An exact-zero matching external
+scale instead selects the existing typed absolute residual gate:
+
+\[
+\boxed{
+S_{ext}^u=0
+\ \Longrightarrow
+\|\mathbf r_f^u\|_\infty\le10^{-3}
+\quad\text{model-consistent force},}
+\]
+
+\[
+\boxed{
+S_{ext}^\phi=0
+\ \Longrightarrow
+\|\mathbf r_f^\phi\|_\infty\le10^{-3}
+\quad\text{model-consistent force-length}.}
+\]
+
+The exact-zero branch selector is the matching external scale itself, not
+\(\max(\|\mathbf f_{int,f}\|_\infty,
+\|\mathbf f_{ext,f}\|_\infty)\). Therefore a nonzero internal residual does not
+select the relative branch when its matching external scale is exactly zero. When
+the external scale is nonzero, only the retained relative gate applies.
 
 Let