Exact mixed-rod boundary class for the balance theorem

An admissible HKWX mixed-rod datum is a reconstructed vacuum-map datum with its supplied ordered rods satisfying all the conditions below. Write $\mathrm{\Phi }=(u,v)$, $e(u,v)=\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+e^{4u}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$ and $e^{\mathrm{\prime }}(U,v)=\mathrm{\mid }\mathrm{\nabla }U{\mathrm{\mid }}^2+e^{4u}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$. Locally integrable means integrable on every compact subset of the specified neighborhood, with Euclidean volume. In the trace conditions below, use a continuous extension of $v$ to each compact subinterval of the indicated open axis rod, equal there to $c_j$; the source provides such representatives in Proposition 2.1(i).

(V) Require local minimality of renormalized energy under competitors preserving the specified singularity data and rod traces, in the variational class of HKWX §2.1. Adopt this source hypothesis independently of the harmonic-map equations; no implication from those equations alone is asserted here.

(A1a) Near each compact subinterval $I$ of an open axis rod, $U,v$ are bounded, $e^{\mathrm{\prime }}(U,v)$ is integrable on a neighborhood of $I$, and $v=c_j$ in trace on $I$.

(A1b) Near every compact subinterval inside a nondegenerate horizon rod, $u,v$ are bounded and $e(u,v)$ is integrable.

(A1c) At the lower and upper pole of a nondegenerate rod, respectively put $\zeta =z-z_i^{\pm }$ and $r=\sqrt{{\rho }^2+{\zeta }^2}$. The renormalized functions are ${\widehat{U}}_i^{-}=U-\frac{1}{2}\log (r-\zeta )$ and ${\widehat{U}}_i^{+}=U-\frac{1}{2}\log (r+\zeta )$. Require ${\widehat{U}}_i^{\pm },v$ bounded and $\mathrm{\mid }\mathrm{\nabla }{\widehat{U}}_i^{\pm }{\mathrm{\mid }}^2+e^{4u}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$ integrable near the pole.

(A2) At each degenerate puncture use centered polar coordinates $(r_i,{\theta }_i,\varphi )$. For some ${ϵ}_i,{\mathrm{\Lambda }}_i>0$, require $\mathrm{\mid }u+\log \sin {\theta }_i\mathrm{\mid }\le {\mathrm{\Lambda }}_i$ off the axis in $B_{{ϵ}_i}(p_i)$ and $e^{\mathrm{\prime }}(U,v)$ locally integrable on $B_{{ϵ}_i}(p_i)\setminus \{p_i\}$. The adjacent axis traces are $c_i,c_{i+1}$ and satisfy $c_{i+1}-c_i=4{\mathcal{J}}_i\mathrm{\ne }0$.

(A3) Outside a sufficiently large ball, $U,v$ are bounded, $e^{\mathrm{\prime }}(U,v)$ is locally integrable including across the ordinary axis, $v$ has traces $c_1,c_{N+1}$ on the two semi-infinite rods, and $U\to 0$ at infinity. These are explicit input conditions; calling a spacetime regular does not, by this definition alone, verify them.

(H) For every compact subinterval $I\subset \mathrm{Int}{\mathcal{H}}_i$ of a nondegenerate rod, $u,v$ extend smoothly and axisymmetrically to a neighborhood of $I$ in ${\mathbb{R}}^3$. For a real constant ${\kappa }_i$ common to that rod, require $\alpha =\log \rho +2u+{\kappa }_i+O({\rho }^2)$, uniformly near each such $I$. Here $O({\rho }^2)$ means that the remainder has absolute value at most $C_I{\rho }^2$ on a sufficiently small neighborhood off the axis. This specifies the horizon-interior behavior using Proposition 2.1(iii), equation (2.23).

Ordered horizon rods and twist-jump labels

Harmonic-map and Weyl reconstruction comparison data

Given: The reconstructed functions, neighborhoods and finite rod labels in [F1], with the independently adopted source predicates in [F2].

Goal: Check the elementary renormalizations and interval cases, while retaining the imported variational and boundary requirements as inputs.

1.1 All logarithms are evaluated where their arguments are positive off the axis. Indeed $r>\mathrm{\mid }\zeta \mathrm{\mid }$ there, and $\sin {\theta }_i>0$. Each displayed renormalization therefore defines a smooth function on the punctured off-axis domain; its boundedness and the specified energy integrals are definite scalar conditions. [given, F1, algebra]

2.1 The interval classification determines which conditions apply: poles and interior-rod conditions apply only to strict intervals, and puncture conditions only to singletons. The trace constants and nonzero differences are finite numerical requirements. The horizon condition specifies smooth extensions and an explicit uniform remainder bound on each compact interior interval. Conjoin these scalar and extension requirements with the separately adopted variational predicate [F2]; none is proved merely by naming a horizon regular. This checks the elementary formulation and does not prove existence, local minimality, or geometric spacetime membership. [given, F1, F2, step 1.1] ∎

HKWX §2.1 supplies the variational predicate (V) and the boundary/energy conditions (A1)–(A3). Proposition 2.1(i),(iii) supplies the axis representative and the explicit horizon-interior regularity used above, under the full §2.1 hypotheses. These premises are imported; the Justification checks only the local formulation.