Introductory Harrison asymptotic claims

PDF p. 6: §1.4, first two paragraphs

In the article's mass-one stationary vacuum setting, with the chosen Harrison family, §1.4 (PDF p.6) prints the spatial one-form estimate $(W_q{)}_i=(q+r^{-1})O(r^{-3}),$ and the uniform denominator expression $D_q=4(q+r^{-1}{)}^2(1+O_5(r^{-1}))$ for $q\in [0,1]$. It states that every $g_q$ for $q>0$ is asymptotically flat in normalized coordinates $t_q\asymp t\mathrm{/}q$, $x_q=qx$, and that these ends recede to infinity in the original spacetime coordinates. It also states that $g_0$ is no longer asymptotically flat and that its spatial sections are asymptotically cylindrical with limiting area $4\pi$.

These are preserved as attributed introductory claims. The one-form order differs from the detailed §2.3 estimate on PDF p.10, $(W_q{)}_i=(1-q{)}^2(W_0{)}_i-4q(1-q)g_{0i}=(q+r^{-1})O_5(r^{-2})\mathrm{.}$ The latter follows from the chosen primitive bound $(W_0{)}_i=O_5(r^{-3})$ and $g_{0i}=O_6(r^{-2})$, through the local weighted-decay lemma. At $q=0$ the introductory expression asks for $O(r^{-4})$, whereas this detailed argument supplies $O_5(r^{-3})$. The extra order in the introductory formula remains unestablished here: neither a proof of it nor a geometric counterexample under the article's complete hypotheses is supplied. The two formulas are retained distinctly; this remark is not a supplier for the stronger estimate and does not assert an article defect.

For the denominator, coordinate and spatial-limit mechanisms, the five local lemmas give complete conditional calculations from the explicitly supplied end, normalized potential and primitive bounds. Under those hypotheses, the positive-parameter normalization lemma establishes an asymptotically flat designated spacetime end for each fixed $q>0$. At $q=0$, the cylinder lemma establishes the asymptotically cylindrical geometry of the specified spatial coordinate sections and their limiting coordinate-sphere area; this spatial calculation alone does not establish the broader assertion excluding every possible asymptotically flat spacetime end or slicing. The exact positive-parameter time normalization is $t_q=d_{q}t\mathrm{/}(4q)$, not just a choice up to scale; the same pole-evaluation constant $d_q=D_q(p)$ used in the one-form contraction is required. The $O_5$ constants for normalized coordinates are controlled on $q\in [\delta ,1]$ with $\delta >0$ fixed. A fixed $r_q=R$ represents original radius $R\mathrm{/}q$, while $q=0$ is handled directly in the original spatial coordinates, not by substituting zero in $t_q,x_q$.

Given: The attributed introductory formulas and the declared end data and suppliers.

Goal: Explain the conditional calculations and the limits of the stronger printed estimate.

1.1 The introductory formulas and their detailed counterpart are source-attribution content, not imported premises proving themselves. The one-form comparison in [F3] identifies exactly the additional decay requested at zero parameter, and leaves it unresolved without promoting the scalar non-implication example to a spacetime counterexample. [given, F1, F2, F3]

2.1 The denominator and the end conclusions in [F3]–[F4] are conditional consequences of explicit end data and the supplied primitive; [F2] records the scope in which the independently authored construction provides that primitive. They do not infer a new, stronger one-form bound or a weaker construction hypothesis. In particular, the normalized component bounds in [F4] establish an asymptotically flat designated end for every fixed $q>0$; at $q=0$, [F4] establishes the cylinder only for the specified spatial sections. The article's assertion that $g_0$ is no longer asymptotically flat is retained as attribution; the spatial calculation alone does not exclude every other possible spacetime end or slicing. Radu's independent §3.3 (3.28) prints the same mass-one normalized potential expansion, and (3.19) with the subsequent normalized time gives $\tau =d_{q}t\mathrm{/}4$ and $s=\tau \mathrm{/}q$, matching the exact time scale used here. Those passages are contextual comparison, not an extra unrecorded logical input. Accordingly all introductory claims, including the stronger printed one-form order, remain visible with their distinct evidence and limits. [F2, F3, F4, step 1.1] ∎