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.

Isometric target areas remain fixed while collar errors vanish

Statement

Assume countable choice ${A C}_{ω}$ . Let $M$ be a smooth manifold and $g$ a smooth symmetric covariant two-tensor that is nondegenerate at every point; its signature is arbitrary. Let $C \subset M$ be a compact smooth embedded two-dimensional manifold, with boundary allowed, whose inclusion $i$ induces a positive-definite metric $h = i^{*} g$ . This positivity is the meaning of spacelike here. For each defined parameter $t$ , let $Φ_{t} : U_{t} \to V_{t}$ be a diffeomorphism between open subsets of $M$ , with $C \subset U_{t}$ , satisfying $Φ_{t}^{*} (g ∣_{V_{t}}) = g ∣_{U_{t}}$ ; this is the ambient isometry condition. For a compact spacelike embedded surface $D$ with inclusion $j_{D}$ , define $∣ D ∣_{g} = \int_{D} μ_{j_{D}^{*} g}$ using its Riemannian density. Then $Φ_{t} (C)$ is again such a surface and $∣ Φ_{t} (C) ∣_{g} = ∣ C ∣_{g}$ for every defined $t$ .

Suppose fixed finite numbers $A, B, L \geq 0$ , with $B = ∣ C ∣_{g}$ , have the following property: for every $ϵ > 0$ and $δ > 0$ there exists an admissible $0 < η < δ$ with a defined translating time $t (η)$ such that $A \leq ∣ Φ_{t (η)} (C) ∣_{g} + ϵ + L η$ . The admissible values of $η$ may depend on $ϵ$ , while $L$ is fixed. Then $A \leq B$ . No convergence of the translating times is needed.

Facts & Assumptions

[A1]

Countable choice ${A C}_{ω}$ is assumed and supplies the stated hypothesis of orientation-free density integration. The Axiom of Countable Choice (AC_ω)

[F1]

Smooth manifolds and smooth tensor fields give the ambient objects. Tensor pullback inserts the differential in each slot; it is smooth and satisfies $(G \circ F)^{*} T = F^{*} (G^{*} T)$ . Smooth manifolds and their smooth charts A smooth tensor field The pullback of a covariant tensor field Pullback of covariant tensors is smooth and functorial

[F2]

An embedded surface, including one with boundary, has a smooth inclusion that is an injective immersion and a homeomorphism onto its image. A diffeomorphism has a smooth inverse. The chain rule gives $d (G \circ F) = d G \circ d F$ ; in boundary charts it follows from the same rule for smooth local extensions. Embedded smooth submanifolds with boundary Immersions and embeddings for manifolds with boundary Diffeomorphisms and local diffeomorphisms of manifolds The chain rule for differentials of smooth maps

[F3]

A smooth symmetric positive-definite covariant two-tensor is a Riemannian metric. Its density in coordinates is $μ_{h} = \sqrt{\det H} ∣ d x^{1} d x^{2} ∣$ , where $H$ is its coordinate matrix; boundary charts are allowed. Riemannian metric and riemannian manifold Riemannian volume density

[F4]

Density pullback by a diffeomorphism uses the absolute Jacobian determinant. Under countable choice, compactly supported smooth density integration is defined, finite, and invariant under diffeomorphisms, including on manifolds with boundary and without an orientation. Pullback of densities by local diffeomorphisms Orientation-free density integration and its properties

Proof

Given: The hypotheses in the Statement, including [A1].

Goal: Prove isometric area preservation and the vanishing-error conclusion.

1.1givenF1F2

Fix a defined $t$ and set $C_{t} = Φ_{t} (C)$ and $f_{t} = Φ_{t} ∣_{C} : C \to C_{t}$ . Transport every chart $x$ of $C$ to $x \circ f_{t}^{- 1}$ on $C_{t}$ . The transition maps are those of $C$ , and the image has the subspace topology because $Φ_{t}$ is an ambient homeomorphism. Its inclusion $j_{t} : C_{t} ↪ M$ is smooth since $j_{t} \circ f_{t} = Φ_{t} \circ i$ . The differential of the right side is injective: $d i$ is injective and $d Φ_{t}$ has inverse $d Φ_{t}^{- 1}$ by the chain rule. Thus $C_{t}$ is smoothly embedded and $f_{t}$ is a diffeomorphism, also at boundary points. An open cover of $C_{t}$ pulls back to an open cover of compact $C$ , and a finite subcover pushes forward, so $C_{t}$ is compact.

2.1givenF1F3F4step 1.1

The induced tensor $h_{t} = j_{t}^{*} g$ is smooth and symmetric. Pullback functoriality and the isometry condition give $f_{t}^{*} h_{t} = i^{*} Φ_{t}^{*} g = i^{*} g = h$ . Since $d f_{t}$ is invertible and $h$ is positive definite, $h_{t}$ is positive definite too. In a chart $x$ of $C$ and the transported chart $y = x \circ f_{t}^{- 1}$ of $C_{t}$ , the metric matrices coincide and $y \circ f_{t} = x$ has identity Jacobian. The density formula and its absolute-Jacobian pullback law therefore give $f_{t}^{*} μ_{h_{t}} = μ_{h}$ . The same coefficient calculation uses smooth extensions in boundary charts.

3.1A1F3F4step 1.1step 2.1

These smooth densities have compact support on the compact surfaces. Under [A1], diffeomorphism invariance of density integration gives $∣ C_{t} ∣_{g} = \int_{C_{t}} μ_{h_{t}} = \int_{C} f_{t}^{*} μ_{h_{t}} = \int_{C} μ_{h} = ∣ C ∣_{g}$ . The absolute-Jacobian law imposes no orientation condition. This holds for each defined $t$ separately.

4.1givenstep 3.1∎

Substitute step 3.1 into each assumed inequality, giving $A \leq B + ϵ + L η$ for the admissible pair. If $d = A - B > 0$ , choose $ϵ = d / 4$ and $δ = d / (4 (L + 1))$ . The hypothesis then gives $0 < η < δ$ , so $ϵ + L η < d / 2 < d$ , contradicting $d \leq ϵ + L η$ . Thus $A \leq B$ . This uses the assumed fixed $L$ and allows the admissible $η$ to depend on $ϵ$ ; there is no fixed- $η$ limit. The translating times disappear by exact area preservation, so their convergence is unnecessary.

Depends on

Used by · 1

Divergence comparison between S and the translated causal cut

Dependency tree · 2 levels

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