Lemma 3.4: uniform graph regularity at the bifurcation sphere

PDF p. 17: Lemma 3.4 and proof; PDF p. 18: Lemma 3.4 and proof; PDF p. 19: Lemma 3.4 and proof

Work in ZFC, including the Axiom of Choice. Adopt the regular mass-one stationary vacuum construction with $T{\mathrm{\mid }}_S\equiv ̸0$. Assume the actual original exterior section $C={\mathrm{\Sigma }}_0\cap \mathcal{E}$ is maximal for $g$, and use its closure ${\mathrm{\Sigma }}_1=C\cup S$ as the reference section. The spherical identification and the physical stationary end chart remain independent. Use the same selected smooth single-primitive Harrison family $g_q$ and stationary product as in the compact-height contract.

Fix $\delta \in (0,1)$ and one supplied family of connected smooth spacelike maximal sections ${\mathrm{\Sigma }}_q$, $q\in [\delta ,1]$, smooth up to compact boundary $S$. Their interiors are stationary graphs $t=u_q(x)$ over ${\mathrm{\Sigma }}_1^{\circ }$, and the family satisfies the complete candidate-section, normalized-end and uniform-estimate hypotheses of the compact-height result, including those of its radial height-decay and transformed-lapse inputs and positive stationary lapse. Fix the common end-estimate radii and constants for this family in these hypotheses; in particular, for any chosen decay exponent used by compact height, its normalized height bound holds beyond a common radius with a common constant. The normalized end height tends to zero at infinity for each fixed $q$.

Assume two-sided access to sufficiently far selected original ends: for some $R_a$ and every $R\ge R_a$, let $U_R=\mathbb{R}\times \{r>R\}$ in the selected physical stationary end chart and require $\mathcal{E}\subset I_{(\mathcal{E},g)}^{-}(U_R)\cap I_{(\mathcal{E},g)}^{+}(U_R)$. Both witnessing timelike paths lie in the unchanged original metric and exterior. This is the explicit access hypothesis of the compact-height supplier.

Let $V$ be the common reference collar and $K=T+{\mathrm{\Omega }}_{H}Z$ its complete Hawking field on the collar saturation. There is a smaller common collar of $S$ in ${\mathrm{\Sigma }}_1$ on which every ${\mathrm{\Sigma }}_q$ is represented by the unique $K$-flow height $u_q^K$: for each interior base point $x$, ${\varphi }_{u_q^K(x)}^K(x)\in {\mathrm{\Sigma }}_q$. These heights extend smoothly to $S$ for each $q$ and obey a common bound $\mathrm{\mid }{u}_q^K\mathrm{\mid }\le C_{\delta }$.

After a further common shrinking, each candidate near $S$ is a one-to-one graph along future normal geodesics of ${\mathrm{\Sigma }}_1$ for $g_q$, with graph height zero on $S$. The graphs are uniformly spacelike and their heights have uniform $C^{k,\alpha }$ bounds up to $S$ for every finite integer $k\ge 0$ and $0<\alpha <1$. The collar and bounds may depend on $\delta$, the fixed background and the fixed supplied family through its common uniform-estimate data, including the end-estimate radii and constants; the Hölder bound may also depend on $k,\alpha$. They are independent of $q\in [\delta ,1]$. No independence across families with unspecified uniform-estimate data is asserted.

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Mass-one convention and Harrison denominator

Stationary maximal graphs and normalized graph heights

Spacelike sections, maximality and lapse–shift data

Normal graphs and the nonlinear mean-curvature map

Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

Lemma 3.3: common spacelike reference collar

Lemma 3.2: compact stationary-height bounds

Existence and boost-coordinate description of Hawking-flow heights

K-flow barriers with boundary tangency comparison and uniform height bound

Boundary angle bound and normal-graph cone

Cutoff maximum principle and uniform collar regularity

Remark 1.3: lapse bounds and stationary product

Causal exterior, chronological boundaries and Cauchy data

The Axiom of Choice

The article citation attributes Lemma 3.4 and its horizon graph conclusions; it supplies no prerequisite proof. This generated conditional formulation retains those conclusions for a fixed supplied family, makes actual physical exterior-section maximality and two-sided far-end access explicit, and records dependence on the supplied uniform-estimate data. The previous generated assertion of independence across all candidate families was not established by the compact-height/barrier contracts. This is a generated-draft repair, not a finding against the original author. The immutable original extraction and its coverage are unchanged, including original Theorem 1.1 and its unresolved source-assessment status. The argument is a local composition of the stated suppliers; it asserts no candidate existence.