LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-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 full global partial-model hypotheses force a constant penalty coefficient

Statement

Let $W : Θ \to [0, \infty)$ be bounded and $C^{1}$ , and suppose $V (w, θ) = ∣ w ∣ W (θ)$ is globally $λ$ -semiconvex on $R \times Θ$ for one finite coefficient $λ$ . Suppose either $Θ = R^{q}$ , or $Θ$ is the closure of a nonempty bounded connected open convex set with $C^{1}$ boundary and $d W (n) = 0$ on that boundary. Then $W \equiv c$ for some $c \geq 0$ . In the full partial-model hypotheses, the Neumann assumption for $G_{f} = ⟨ f, φ ⟩ + W$ at $f = 0$ supplies the latter condition. Consequently, writing $H_{+} = ⟨ f, φ ⟩ + c$ and $H_{-} = - ⟨ f, φ ⟩ + c$ , one has $H_{-} = 2 c - H_{+}$ and $\nabla H_{-} = - \nabla H_{+}$ .

Statement references

Feature-measure energy and its regularity assumptions

Facts & Assumptions

Given: AC and the hypotheses in the Statement.

[F1]

Global semiconvexity means that adding a fixed quadratic makes the potential convex on its convex domain. Feature-measure energy and its regularity assumptions

Proof

Given: The data and assumptions in the Statement.

1.1F1givenalgebra

Fix $s > 0$ and two points $θ_{0}, θ_{1} \in Θ$ . Restrict the semiconvex inequality [F1] to the slice $w = s$ and divide it by $s$ . Its quadratic error is bounded by $∣ λ ∣ t (1 - t) ∣ θ_{1} - θ_{0} ∣^{2} / (2 s)$ . Letting $s \to \infty$ gives the convexity inequality for $W$ , for every $0 \leq t \leq 1$ .

2.1step 1.1givenalgebra

If $Θ = R^{q}$ , restrict $W$ to any line. A nonzero secant slope of a convex function forces its values to grow without bound in one of the two directions: for $a < b < c$ , convexity makes the secant slope on $[b, c]$ at least that on $[a, b]$ , and the reversed assertion treats a negative slope. Boundedness therefore makes every secant slope zero. Any two points lie on a line, so $W$ is constant.

3.1step 1.1givenalgebrastep 2.1∎

In the bounded case $W$ attains its maximum at a point $x$ of the compact closure. At an interior maximum $d W_{x} = 0$ . At a boundary maximum its tangential differential is zero, because local $C^{1}$ boundary charts give two-sided differentiable curves in every tangent direction, while the Neumann hypothesis gives zero normal differential. These tangent and normal directions span the ambient space, so again $d W_{x} = 0$ . For any $y \in Θ$ , the one-variable convex function $t \mapsto W (x + t (y - x))$ has right derivative $d W_{x} (y - x) = 0$ ; its secant slope is at least this derivative. Hence $W (y) \geq W (x)$ , and maximality gives equality. The final reflected formulas follow by direct differentiation.

Source note

This is a consequence of the full inherited global-semiconvexity assumptions, not an added hypothesis restricting the approved claim. Chizat–Bach Assumptions 3.4 alone do not imply the global semiconvexity in their Assumptions 2.1: on $Θ = [- 1, 1]$ , take $φ = 0$ , $R = 0$ , and $W (θ) = 2 + \cos (π θ)$ . These satisfy bounded $C^{1, 1}$ , positivity, Neumann and dense-regular-value conditions, but $\partial_{θ θ}^{2} V (s, 0) = - s π^{2}$ for $s > 0$ , ruling out any global lower Hessian bound. This diagnoses that source implication; the locally proved result retains the actual full library hypotheses.

Depends on

Feature-measure energy and its regularity assumptions

Used by

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Chizat and Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport, locally reconstructed auxiliary proof (standard reference, not scraped)