One-vertex composition of phase-difference landscapes

Every edge of $E_1\cup E_2$ has endpoints in $V=V_1\cup V_2$, so the glued network has the same unordered-edge convention, with any common loop at $v$ occurring once in the edge union. Its energy is the displayed sum by definition. Write $W_i=V_i\setminus \{v\}$. The map sending $\theta$ to $(c,z_1,z_2)$, where $c={\theta }_v$ and $z_i=({\theta }_x-{\theta }_v{)}_{x\in W_i}$, has inverse ${\theta }_v=c$ and ${\theta }_x=c+(z_i{)}_x$ for $x\in W_i$. The sets $W_1,W_2$ are disjoint and cover $V\setminus \{v\}$, so these formulas are mutually inverse. In angular charts both maps are represented locally by linear maps, hence are smooth with smooth inverses. If $W_i$ is empty, its factor is the one-point empty product. Thus the shared phase and the two relative-phase factors give a product diffeomorphism.

Let $u_i(z_i)$ be $U_i$ evaluated with phase $0$ at $v$ and phases $z_i$ on $W_i$. Rotation invariance gives $U(c,z_1,z_2)=u_1(z_1)+u_2(z_2)$, independently of $c$. To transfer derivatives, take any local affine map $B(y)=My+b$ between the angular-coordinate open sets and any $C^2$ scalar function $f$. Applying [F5] gives ${\mathrm{\partial }}_j(f\circ B)(y)={\sum }_r{M}_{rj}({\mathrm{\partial }}_{r}f)(B(y))$. Applying [F5] again to each ${\mathrm{\partial }}_{r}f$ gives ${\mathrm{\partial }}_k{\mathrm{\partial }}_j(f\circ B)(y)={\sum }_{r,s}{M}_{rj}{M}_{sk}({\mathrm{\partial }}_s{\mathrm{\partial }}_{r}f)(B(y))$. Thus the composition is $C^2$, and $\mathrm{\nabla }(f\circ B)=M^T(\mathrm{\nabla }f)\circ B$ and ${\mathrm{\nabla }}^2(f\circ B)=M^T(({\mathrm{\nabla }}^{2}f)\circ B)M$. These formulas apply to the component sections defining $u_i$ and to the inverse coordinate maps in step 1.1; all domains can be restricted to angular-chart open neighborhoods. When $M$ is invertible, the gradient vanishes in one coordinate system exactly when it vanishes in the other, and the Hessian quadratic form on $k$ in the new coordinates equals the old form on $Mk$. Empty relative-phase factors have no derivatives, and the same formulas use empty sums. For the displayed sum, the relative-factor derivatives are those of $u_i$ and the $c$ derivative is zero. Consequently a point is critical for $U$ exactly when both $z_i$ are critical for $u_i$; applying the invertible component coordinate map also makes this equivalent to criticality of each corresponding $U_i$.

A point is a local minimum of $U$ exactly when both $z_i$ are local minima of their $u_i$. Indeed, by [F6] a neighborhood witnessing a minimum contains a product $N_c\times N_1\times N_2$ of open neighborhoods of its coordinates. Fixing $c$ and the other relative factor in the energy inequality shows $u_i(z_i)\le u_i(w_i)$ for every $w_i\in N_i$, proving necessity. Conversely, choose open neighborhoods witnessing the two factor minima and any open neighborhood of $c$. Their product is a neighborhood by [F6], and adding the factor inequalities proves the minimum property for $U$. Applying the same argument to the component coordinates $({\theta }_v,z_i)$ shows that a minimum of $u_i$ is exactly a minimum of $U_i$ at the corresponding configuration. The diffeomorphisms in step 1.1 carry these neighborhoods to neighborhoods in the original phase tori. This also covers a one-point relative factor. Consequently every minimum of $U$ restricts to synchronized configurations on both components. Their common phase at $v$ is the same, so the whole glued configuration is synchronized.

At a critical point of $U$ that is not a local minimum, step 3.1 gives a factor that is not a local minimum. By step 2.1 its corresponding component configuration is critical, so [A1] gives a tangent direction $h$ with strictly negative component Hessian quadratic form. Let $L_i$ be the invertible derivative of the inverse component coordinate map $(c,z_i)\mapsto \theta {\mathrm{\mid }}_{V_i}$, and write $L_i^{-1}h=(a,b)$. The Hessian formula in step 2.1 shows that the component form in these coordinates on $(a,b)$ is the same negative number. Since $U_i$ is independent of $c$, its quadratic form on $(a,b)$ equals that on $(0,b)$. Extend $(0,b)$ to the glued product by taking zero velocity in the other relative-phase factor and in $c$. The sum in step 2.1 has block-diagonal second derivatives, so its quadratic form on this extended direction is the same negative number. Applying the inverse glued coordinate derivative to that direction and using the same Hessian formula gives a negative direction in the original angular coordinates of $U$. The coordinate maps are locally affine, so their second derivatives vanish and no additional term occurs. This proves the remaining assertion.