False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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 distance function is smooth on all of m times m

Statement

The Riemannian distance function is smooth everywhere on $M \times M$ .

Facts & Assumptions

Given: $M = R$ with $g = d x^{2}$ .

[F1]

Riemannian distance on a connected manifold: On a connected Riemannian manifold define $d_{g} (p, q) = \inf {L_{g} (γ) : γ is piecewise C^{1} from p to q}$ . Lengths are those of def-riemannian-speed-and-length. For each pair $p, q$ , lem-any-two-points-in-a-connected-smooth-manifold-can-be-joined-by-a-piecewise-c-one-curve supplies a curve, so the set of lengths is nonempty, contains a finite real number and is bounded below by zero. Applying the least-upper-bound property cor-cauchy-reals-lub-complete to the negatives gives a finite nonnegative infimum. On the empty connected manifold this defines the empty distance function; there are no pairs to evaluate. No minimizing curve is part of this definition.

[F3]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative: Let $a < b$ . Suppose $G : [a, b] \to R$ is continuous on $[a, b]$ and differentiable on $(a, b)$ . If $f : [a, b] \to R$ is Riemann integrable and $f (x) = G^{'} (x) (a < x < b),$ then $\int_{a}^{b} f = G (b) - G (a) .$ No derivative of $G$ at either endpoint is assumed, and the two endpoint values assigned to the integrable extension $f$ do not enter the conclusion.

Refutation

technique · direct

1.1

For any piecewise $C^{1}$ curve $γ$ from $x$ to $y$ , speed is $∣ γ^{'} ∣$ , and integration and Newton–Leibniz on the pieces give $L (γ) = \int ∣ γ^{'} ∣ \geq ∣ \int γ^{'} ∣ = ∣ y - x ∣$ . The affine path $γ (t) = x + t (y - x)$ , $0 \leq t \leq 1$ , has length $∣ y - x ∣$ . Therefore the infimum defining distance equals $d (x, y) = ∣ y - x ∣$ .

F1F3given

2.1

If $d$ were smooth on $R^{2}$ , its restriction along the smooth map $x \mapsto (x, 0)$ would be differentiable at zero. That restriction is $∣ x ∣$ ; its difference quotient at zero is $1$ for $x > 0$ and $- 1$ for $x < 0$ . The unequal one-sided limits contradict differentiability.

step 1.1∎

Source locator

Lee, p. 338, Euclidean Riemannian distance; the nonsmoothness is the displayed absolute-value difference quotient.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)