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.

Level-set area transfer retains generator multiplicity

Statement

Use the preceding flow, level identity and endpoint regularity, with the selected domains $A_{z}$ and endpoint maps $F_{z}$ of the truncated-height construction on its compact smooth source surface $C$ . Equip $C$ with its induced metric, $Σ_{z}$ with its induced metric, and the collar with $h_{0}$ . Fix $- 2 δ < z < 0$ . Suppose the nonnegative Gram two-Jacobians satisfy $0 \leq J_{2} F_{z} \leq J_{2} F_{0} \leq 1$ on $A_{z} \subset A_{0}$ . If the inverse-flow two-dilation is at most $1 + ϵ (δ)$ uniformly for $∣ z ∣ \leq 2 δ$ , then $H_{h_{0}}^{2} ({u = z}) \leq (1 + ϵ (δ)) \int_{A_{z}} J_{2} F_{z} d μ_{C} \leq (1 + ϵ (δ)) \int_{A_{0}} J_{2} F_{0} d μ_{C} .$ The level set is countably two-rectifiable with finite area. This uses surjectivity and counting multiplicity, with no injectivity or smooth-image assumption.

Facts & Assumptions

[F1]

The inverse-flow two-dilation is supplied for $∣ z ∣ \leq 2 δ$ , and the exact level-set identity ${u = z} = Ψ_{- z} (D_{z})$ is supplied for $∣ z ∣ < 2 δ$ . A compact normal-flow collar has uniformly small area distortion The truncated past height is Lipschitz and has exactly the pulled-back cuts as levels

[F2]

The domains are compact/Borel and the onto endpoint maps have countably many locally smooth Lipschitz extensions. Every negative-time cut has a closed selected domain and smooth endpoint extensions

[F3]

For such endpoint maps the area formula counts multiplicity and gives image area at most the Jacobian integral. Lipschitz height functions, cut measure and Jacobians Area formula and endpoint bound with multiplicity

[F4]

The truncated height $u$ , cuts $D_{z}$ , selected geodesic hit domains $A_{z} \subset C$ and unique endpoint maps $F_{z} : A_{z} \to D_{z}$ are the supplied construction. Truncated causal-boundary height for smooth approximation

Proof

Given: The hypotheses in the Statement, with $- 2 δ < z < 0$ .

Goal: Establish the stated conclusions.

1.1givenF1F2F4algebra

With the selected maps of [F4], form $G_{z} = Ψ_{- z} \circ F_{z}$ on $A_{z}$ . Since $- 2 δ < z < 0$ , [F1] applies to this interior slab time. By [F2] the endpoint map is onto $D_{z}$ ; by [F1] its inverse-flow image is exactly ${u = z}$ . The composition has local smooth extensions after restricting the endpoint extensions to the open inverse-flow domain around their hits. Its differential lands first in the smooth section $Σ_{z}$ before inverse-flow transfer. For any two input tangent vectors the inverse-flow area-dilation bound therefore gives $J_{2} G_{z} \leq (1 + ϵ (δ)) J_{2} F_{z}$ , including rank-deficient derivatives. This is the Gram-determinant definition of two-dimensional dilation.

2.1givenF2F3step 1.1

Apply [F3] to the Borel domain $A_{z}$ and $G_{z}$ . Counting multiplicity is at least one at every point of its image, so $H_{h_{0}}^{2} ({u = z}) \leq \int_{A_{z}} J_{2} G_{z} d μ_{C}$ . The countable Lipschitz extensions also make this image countably two-rectifiable. Now use step 1.1, the pointwise assumed optical comparison, and $A_{z} \subset A_{0}$ . All integrands are nonnegative and measurable by local smooth extension. These are the asserted inequalities with their multiplicity and metric conventions.

3.1givenF2F3step 2.1∎

Apply [F2] at time zero: $A_{0}$ is compact and $F_{0}$ has local smooth extensions into $Σ_{0}$ . Around each point of $A_{0}$ , shrink an extension neighborhood to have compact closure within its extension domain. The smooth extension and smooth source and target metrics give a continuous, bounded Gram Jacobian on that closure. A finite subcover of $A_{0}$ therefore gives a finite common bound $M$ for $J_{2} F_{0}$ almost everywhere on $A_{0}$ , using the Jacobian convention in [F3]. The compact smooth source surface $C$ has finite area: cover it by finitely many relatively compact coordinate disks, on which its smooth area density is bounded and the coordinate disks have finite Euclidean area. Consequently $\int_{A_{0}} J_{2} F_{0} d μ_{C} \leq M μ_{C} (C) < \infty$ . If $A_{0}$ is empty the integral is zero. Step 2.1 now gives finite level area. This argument does not extend the assumed bound $J_{2} F_{0} \leq 1$ from $A_{z}$ to $A_{0}$ .

Depends on

Used by · 1

Lipschitz causal height and level-set Jacobian bound

Dependency tree · 2 levels

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