A density criterion for equilibrium dislocation response

When does a measured density response indicate that a crystal has selected a dislocated equilibrium branch? This article gives a model-calibrated answer for specified two-dimensional particle crystals. The central quantity is the integrated density response, calibrated by the bulk and shear moduli rather than interpreted as a universal defect threshold. An exact density threshold and change in compliance. Let \(B\) and \(m\) be the cold bulk and shear moduli, \(b\) the native lattice spacing, and \(\rho_t\) the selected signed density-response coefficient under load \(t\). For a mean-zero periodic field \(V\) of period \(h\), define \(P=\int_0^h V(x)_+\,dx\) and \(D(t)=\operatorname{osc}_{x\in[0,h]}\int_0^x\rho_t(y)\,dy\). The critical values are \[t_c=\frac{Bb}{2\pi P},\qquad D_c=\frac{bB}{2\pi(B+m)}.\] The selected coarse Burgers measure vanishes exactly when \(D(t)\le D_c\). The response has two exact branches: \[D(t)=\begin{cases}Pt/(B+m),&0\le t\le t_c,\\ Pt/B-mb/[2\pi(B+m)],&t\ge t_c.\end{cases}\] The ratio of the slopes is \(1+m/B\): the upper branch releases the shear contribution to the density-response slope. A rigorous comparison of Yukawa and Gaussian interactions shows that the same scalar density amplitude can select opposite coarse-charge responses when the elastic moduli differ. The equilibrium cost of retaining zero coarse charge. For the full canonical Yukawa interaction \(\phi(r)=2e^{-3r}/r\), the derivation retains the original particle count, periods and all interaction images. Writing \(L=\log N\), the full and zero-coarse-charge restricted free energies satisfy \[F_{0,N}-F_N=L^2\Delta_{\mathrm{coh}}(t)+o(L^2),\qquad \frac{T_N}{L^2}\log\Pr(\mathcal C_{0,N})\longrightarrow-\Delta_{\mathrm{coh}}(t).\] With \(\rho_{\mathrm{coh}}=tV/(B+m)\), the rate is \(\Delta_{\mathrm{coh}}=B(B+m)\|\rho_t-\rho_{\mathrm{coh}}\|_{L^2(Q)}^2/(2m)\), using the unnormalized norm on the area-two macroscopic torus. The comparison covers the entire specified coarse event, which may contain microscopic neutral defects. A zero rate at or below onset does not determine that event's limiting probability. Spatial reconstruction and scope. The full density profile reconstructs the selected coarse charge through a two-cap clipping rule. An explicit anisotropic-strain correction explains why a density gradient alone need not equal the charge density. Smoothing and threshold-calibration errors are controlled in an asymptotic weak comparison between two readouts of the same particle snapshot. The original Yukawa particle results hold at fixed positive load in the joint limit \(N\to\infty\), \(0<T_N=o((\log N)^3/N^2)\), for the specified slowly varying chemical-potential field. Temperature is positive at each size and tends to zero; the conclusions do not assert a fixed-positive-temperature transition. The cross-model comparison uses the common colder regime \(T_N=o((\log N)^2/N^2)\). The observation bound assumes access to particle coordinates and fixed smoothing and test functions; it is not a finite-size confidence guarantee for an experimental apparatus. The integrated article presents the full proofs, two analytical figure groups and references.