LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

A transverse predictor hyperplane selects one nearby point of a regular branch

Statement

Let $c : [- δ, δ] \to U \subseteq R^{n + 1}$ be a unit-speed reparameterization of the unique locally normalized solution curve supplied by A regular solution curve has a unique local arclength chart and a quadratic predictor defect, with $c (0) = z_{0}$ . Let $τ \in R^{n + 1}$ be a unit vector, let $0 < r_{-}, r_{+} \leq δ$ , and suppose that for some $κ > 0$ , $τ^{T} c^{'} (s) \geq κ (- r_{-} \leq s \leq r_{+}) .$ Let $z_{p} \in R^{n + 1}$ satisfy $d : = τ^{T} (z_{p} - z_{0}), 0 \leq d \leq κ r_{+} .$ Then there is exactly one $s_{*} \in [- r_{-}, r_{+}]$ such that $τ^{T} (c (s_{*}) - z_{p}) = 0.$ Thus the predictor hyperplane has exactly one intersection with this local branch segment containing its basepoint. The statement is local to the supplied chart.

Facts & Assumptions

Given: The chart $c$ , $z_{0}$ , unit normal $τ$ , $r_{-}$ , $r_{+}$ , $κ$ , and $z_{p}$ with the conditions in the Statement.

[F1]

A regular solution curve has a unique local arclength chart and a quadratic predictor defect: $c$ is a $C^{1}$ unit-speed reparameterization of the unique local solution curve on an interval containing $[- r_{-}, r_{+}]$ .

[F2]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : the derivative of a linear functional composed with a differentiable curve is the functional applied to the curve derivative.

[F3]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion and Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative: a $C^{1}$ scalar function's increment on a compact interval is the integral of its derivative.

[F4]

If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ : an integrable function bounded below by $κ$ on an interval of length $ℓ$ has integral at least $κ ℓ$ .

[F5]

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ : a continuous real function takes every value between its endpoint values.

Proof

Given: The chart and inequalities in the Statement.

1.1F1F2F3F4F5givenalgebra

Define $g (s) : = τ^{T} (c (s) - z_{p})$ on $[- r_{-}, r_{+}]$ . By [F2], $g^{'} (s) = τ^{T} c^{'} (s) \geq κ$ . The fundamental theorem in [F3] therefore gives $g (r_{+}) - g (0) = \int_{0}^{r_{+}} g^{'} (u) d u \geq κ r_{+}$ by [F4]. Since $g (0) = - d$ and $d \leq κ r_{+}$ , we have $g (0) \leq 0$ and $g (r_{+}) \geq 0$ . If $d = 0$ , then $s_{*} = 0$ is a zero; if $d > 0$ , the intermediate value theorem [F5] gives $s_{*} \in (0, r_{+}]$ with $g (s_{*}) = 0$ .

2.1F2F3F4step 1.1algebra∎

If $- r_{-} \leq s_{1} < s_{2} \leq r_{+}$ , [F2]–[F4] give $g (s_{2}) - g (s_{1}) = \int_{s_{1}}^{s_{2}} g^{'} (u) d u \geq κ (s_{2} - s_{1}) > 0.$ Thus $g$ is strictly increasing, so it has at most one zero on $[- r_{-}, r_{+}]$ . The zero found in step 1.1 is therefore unique.

Depends on

Used by

Dependency tree · two levels

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Sources

Eugene L. Allgower and Kurt Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics 13 (1990) (standard reference, not scraped)
K. A. Cliffe, A. Spence, and S. J. Tavener, The numerical analysis of bifurcation problems with application to fluid mechanics, Acta Numerica 9 (2000), 39–131 (standard reference, not scraped)