The Gluing Method

The Gluing Method is proposed as a simple procedure for constructing a global relation from local relations that are valid at different points, branches, regions, or asymptotic regimes. The present work treats finite collections of local relations and refers to this case as finite gluing. Given local equations $F_j(z)=0$ and positive localized weights $\rho_j(z)$, the glued relation is $F_{glue}(z)=\frac{\sum_j \rho_j(z)F_j(z)}{\sum_j\rho_j(z)}=0.$ The inputs need not be Taylor series or explicitly solved functions: implicit and multibranch equations, differential equations, asymptotic laws, finite-size models, numerical local fits, and scaling relations can be glued directly without first resolving them into single-valued branches. Gaussian, Cauchy, logistic/Fermi, Bose-type, and parastatistical kernels can be used as positive weight generators, and the weights may depend on both parameters and state variables. The method is illustrated at several levels. Two asymptotic constant relations at $x=\pm\infty$ generate a $\tanh$ crossover exactly under symmetric Gaussian gluing. Local differential equations can be glued directly: two constant-coefficient asymptotic ODEs generate a variable-coefficient global ODE, and adding one finite-point local ODE substantially improves both the equation residual and its solution in a nontrivial $\erf$ test. In a zero-dimensional $\phi^4$ partition function, low-order weak- and strong-coupling differential relations reconstruct the intermediate-coupling regime with percent-level accuracy without increasing perturbative order. In the one-dimensional Ising chain and one-dimensional Anderson model, the method connects finite-size, thermodynamic, weak/strong-disorder, and mesoscopic regimes. For Anderson transport, a small number of local anchors added to weak-disorder scaling, a finite-size boundary relation, and a strong-disorder relation reduce the withheld maximum relative error of the mean transmission to about $3.2\%$ for the tested range $N/\xi\le3.5$. The purpose of this preprint is methodological rather than axiomatic. No general convergence theorem is claimed. The numerical and analytic examples are intended to show that finite gluing can reuse low-complexity local theories across regimes, while also making clear that the user must respect the validity region, normalization, tail suppression, and physical composition structure of the local models.