ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

An explicit saddle of the oscillator action at $T = 2 π$ : the two variations $\sin (t / 2)$ and $\sin (3 t / 2)$

Example

Fix $ω = 1$ and $T = 2 π$ in the chart $L (q, v) = \frac{1}{2} (v^{2} - q^{2})$ on $O = R \times R \times R$ over $[0, 2 π] \times R$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations), and consider the variations

$η_{1} (t) = \sin \frac{t}{2}, η_{3} (t) = \sin \frac{3 t}{2}$

of the zero curve $q_{0} \equiv 0$ . The proposition Second variation of the oscillator action on a finite interval: both signs for $T > π / ω$ and the value at $T = π / ω$ gives

$δ^{2} S (q_{0}) [η_{1}, η_{1}] = - \frac{3 π}{4} < 0, δ^{2} S (q_{0}) [η_{3}, η_{3}] = \frac{5 π}{4} > 0,$

and with the exact identity $S (ε η) = \frac{ε^{2}}{2} δ^{2} S (q_{0}) [η, η]$ of that proposition,

$S (ε η_{1}) = - \frac{3 π ε^{2}}{8} < 0 < \frac{5 π ε^{2}}{8} = S (ε η_{3}) (ε \neq 0) .$

Thus at the duration $T = 2 π > π = π / ω$ the stationary zero curve has variations that decrease and variations that increase the action arbitrarily close to it: it is a saddle, and neither a local minimiser nor a local maximiser. Every number below is exact; no measurement or numerical integration is involved.

Facts & Assumptions

Given: The chart $L (q, v) = \frac{1}{2} (v^{2} - q^{2})$ over $[0, 2 π] \times R$ , the zero curve $q_{0}$ , the variations $η_{1}, η_{3}$ , and the action $S$ and second variation $δ^{2} S$ of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations and The fixed-endpoint second variation of a time-dependent action at a curve.

[L1]

Second variation values and exact action (Second variation of the oscillator action on a finite interval: both signs for $T > π / ω$ and the value at $T = π / ω$ , The second variation of a time-dependent action: twice differentiability of the fixed-endpoint family and the Hessian integral formula): for this chart with parameters $ω = 1$ , $T = 2 π$ one has $δ^{2} S (q_{0}) [η_{n}, η_{n}] = \frac{T}{2} ((n π / T)^{2} - ω^{2})$ for the variations $η_{n} (t) = \sin (n π t / T)$ , and $S (ε η_{n}) = \frac{ε^{2}}{2} δ^{2} S (q_{0}) [η_{n}, η_{n}]$ for every real $ε$ .

[L2]

Trigonometric integral data (The derivatives of sine and cosine are cosine and minus sine, Sine and cosine defined by their real power series, The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ , The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): the double-angle formulas give $\cos^{2} u = \frac{1}{2} (1 + \cos 2 u)$ and $\sin^{2} u = \frac{1}{2} (1 - \cos 2 u)$ ; $\sin$ and $\cos$ are continuous, hence integrable on $[0, 2 π]$ ; the fundamental theorem evaluates the integrals below; and $\sin 0 = \sin π = \sin 2 π = \sin 3 π = 0$ .

Verification

Proof technique: direct; substitute $T = 2 π$ , $ω = 1$ into the proposition and evaluate the defining integrals independently.

1.1givenL2algebra

The variations: $η_{1}$ and $η_{3}$ are $C^{\infty}$ on $R$ with $η_{1} (0) = η_{1} (2 π) = 0$ and $η_{3} (0) = η_{3} (2 π) = 0$ by [L2], so both are variations of $q_{0}$ ; their derivatives are $η_{1}^{'} (t) = \frac{1}{2} \cos \frac{t}{2}$ and $η_{3}^{'} (t) = \frac{3}{2} \cos \frac{3 t}{2}$ .

2.1step 1.1givenL1algebra

Second variations: substituting $T = 2 π$ and $ω = 1$ into [L1] gives $δ^{2} S (q_{0}) [η_{n}, η_{n}] = π (n^{2} / 4 - 1)$ , hence $δ^{2} S (q_{0}) [η_{1}, η_{1}] = π (1 / 4 - 1) = - \frac{3 π}{4}$ and $δ^{2} S (q_{0}) [η_{3}, η_{3}] = π (9 / 4 - 1) = \frac{5 π}{4}$ .

3.1step 2.1L2algebra

Independent integral check: by [L2], $\int_{0}^{2 π} η_{1}^{'} (t)^{2} d t = \frac{1}{4} \int_{0}^{2 π} \frac{1 + \cos t}{2} d t = \frac{1}{4} \cdot π, \int_{0}^{2 π} η_{1} (t)^{2} d t = \int_{0}^{2 π} \frac{1 - \cos t}{2} d t = π,$ so $δ^{2} S (q_{0}) [η_{1}, η_{1}] = π / 4 - π = - \frac{3 π}{4}$ ; and with $η_{3}^{'} (t)^{2} = \frac{9}{4} \cos^{2} \frac{3 t}{2}$ the same computation gives $\int_{0}^{2 π} η_{3}^{' 2} = \frac{9}{4} \cdot π$ and $\int_{0}^{2 π} η_{3}^{2} = π$ , so $δ^{2} S (q_{0}) [η_{3}, η_{3}] = \frac{9 π}{4} - π = \frac{5 π}{4}$ , in agreement with step 2.1.

4.1step 2.1step 3.1L1algebra∎

Action values and saddle: by the exact identity in [L1], $S (ε η_{1}) = \frac{ε^{2}}{2} \cdot (- \frac{3 π}{4}) = - \frac{3 π ε^{2}}{8} < 0$ and $S (ε η_{3}) = \frac{5 π ε^{2}}{8} > 0$ for every $ε \neq 0$ , while $S (q_{0}) = 0$ ; since both families tend to $q_{0}$ as $ε \to 0$ , the stationary zero curve is neither a local minimiser nor a local maximiser, as claimed.

An explicit saddle of the oscillator action at $T = 2 π$ : the two variations $\sin (t / 2)$ and $\sin (3 t / 2)$

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

An explicit saddle of the oscillator action at T=2πT=2π: the two variations sin⁡(t/2)sin(t/2) and sin⁡(3t/2)sin(3t/2)

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

An explicit saddle of the oscillator action at $T = 2 π$ : the two variations $\sin (t / 2)$ and $\sin (3 t / 2)$