Residual-Work Compatibility Criteria and Defect-Measure Compactness for Positive-Cone Viscoelastic Reynolds States

Abstract

We prove a residual-work compatibility theory for positive-cone viscoelastic Reynolds states. In Oldroyd--B, the entropy cancellation eliminates incompressible transport and upper-convected stretching against polymeric stress work. After quotienting pressure tensors and spatial means, positive pressure-free residual work can be paid only by the conformation residual through the entropy-dual lever (G=I-A^{-1}). This yields the closed residual-work cone (P+(α/2)\int_{\mathbb T^d}G:S,dx\le 0), exact windowed tests, and the least-cost Hilbert-space repair; constrained closures pay through the projected lever. Strong residual-data limits preserve this cone, while weak--weak limits require the product-defect measure carried by (G:S). The augmented topology is sharp, as shown by localized residual packets. For FENE-P, the lever (G_b(C)=((b-d)/(b-\operatorname{tr}C))I-C^{-1}) makes the finite-extensibility boundary a genuine residual-work boundary.