LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The source maximal-gauge expansion can be used on the same CW slice

Statement

For the supplied end in the preceding lemma, apply the imported stationary maximal-gauge regularity input in the time coordinate tau already fixed there. Choose spatial coordinates y harmonic for the Riemannian quotient metric. This changes only spatial coordinates; the same maximal slice remains $τ = 0$ and $T = \partial_{τ}$ . In these coordinates the induced metric h, lapse N and shift X have full differentiated polyhomogeneous expansions in integer powers of $r^{- 1}$ with finite powers of $\log r$ at each order, and $h - δ$ , $N - 1$ , X are $O_{\infty} (r^{- 1})$ .

Here full differentiated polyhomogeneity means that every finite expansion has a remainder with the corresponding differentiated asymptotic estimates; it is not an analyticity or convergence assertion. This lemma invokes the stated external regularity result, not a locally reconstructed proof of that result and not a replacement of the global maximal hypersurface.

Facts & Assumptions

[F1]

The existing CW end already has a stationary time coordinate tau with that same maximal slice at zero and the required weak end data. Supplied CW logarithmic maximal ends have weak weighted initial data

[F2]

The independent source packet supplies maximal-stationary-gauge polyhomogeneity and the normalized vacuum maximal KID identities. Independent stationary maximal-gauge asymptotic expansion and KID input

[F3]

Work in ZFC, including the inherited analytic choice assumptions. The Axiom of Choice

Proof

**Given:**The actual end hypotheses in the Statement.

Goal: Prove the stated same-slice end consequence.

1.1F1F2givenalgebra

The quotient metric is the positive metric transverse to T on its timelike end. Its coefficients in the original spatial coordinates are $γ_{i j} = g_{i j} - g_{0 i} g_{0 j} / g_{00}$ and are unaffected by a stationary change of time origin. Thus quotient-harmonic spatial coordinates are functions of x alone. Use the source spatial-coordinate choice while retaining tau as the stationary time parameter. Maximality is the geometric zero mean-curvature condition of the supplied $τ = 0$ hypersurface, so it is already the source maximal-time gauge condition; there is no further time equation whose solution would select a different slice.

2.1F1F2step 1.1∎

The logarithmic initial graph and the original regular vacuum end meet the weak initial-data conditions as established in [F1], and initially give power falloff for every exponent strictly below one. Apply the source packet in this already imposed gauge. It supplies full differentiated expansions and the improved order $α = n$ -2=1 in three spatial dimensions. The ADM fields h,N,X are smooth algebraic functions of the stationary metric fields on the uniformly spacelike end, so they inherit these expansions and bounds. This is precisely the one imported regularity step; all subsequent coefficient and logarithm exclusions are authored locally.

Depends on

Used by · 1

Leading stationary modes and a spatial normalization of the same maximal end

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.