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.

The $δ = 0$ endpoint computation: the free rotation $(q, p) \mapsto (p, - q)$ , the kick $(q, p) \mapsto (q, p + q^{2} - a)$ and $K \circ R_{0} = F$

Example

In the model of The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map:

the free oscillator $q^{'} = p$ , $p^{'} = - q$ on the interval $[0, π / 2]$ carries the initial data $(q_{0}, p_{0})$ to

$(q, p) (\frac{π}{2}) = (q_{0} \cos \frac{π}{2} + p_{0} \sin \frac{π}{2}, - q_{0} \sin \frac{π}{2} + p_{0} \cos \frac{π}{2}) = (p_{0}, - q_{0}) = R_{0} (q_{0}, p_{0});$

the $δ = 0$ rescaled equation $q_{s} = 0$ , $p_{s} = w (s) (q^{2} - a)$ has the exact solution $q (s) = q_{0}$ , $p (s) = p_{0} + (q_{0}^{2} - a) \int_{0}^{s} w$ , so its endpoint map is

$K (q_{0}, p_{0}) = (q_{0}, p_{0} + q_{0}^{2} - a)$

because $\int_{0}^{1} w = 1$ ; and

$(K \circ R_{0}) (q_{0}, p_{0}) = K (p_{0}, - q_{0}) = (p_{0}, - q_{0} + p_{0}^{2} - a) = F (q_{0}, p_{0}) .$

Moreover $β_{δ} \geq 0$ with $\int_{0}^{P} β_{δ} = 1$ and the restriction of the pulse to $[0, P]$ has support contained in $(π / 2, P)$ . On $R$ its support is the periodic union $⋃_{j \in Z} (π / 2 + δ supp w + j P)$ .

Facts & Assumptions

Given: The model data $w$ , $δ > 0$ , $β_{δ}$ , $R_{0}$ , the rescaled equation and the maps $K, F$ of The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map.

[L1]

Trigonometric solution (The derivatives of sine and cosine are cosine and minus sine, Sine and cosine defined by their real power series, Parity and the Pythagorean identity for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Pi is the first positive zero of sine): the functions $q (t) = q_{0} \cos t + p_{0} \sin t$ , $p (t) = - q_{0} \sin t + p_{0} \cos t$ satisfy $q^{'} = p$ , $p^{'} = - q$ and the initial conditions; $\cos (π / 2) = 0$ , $\sin (π / 2) > 0$ , and the Pythagorean identity gives $\sin (π / 2) = 1$ .

[L2]

Pulse data (The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map): $w : R \to R$ is smooth, nonnegative and compactly supported inside $(0, 1)$ with $\int_{0}^{1} w = 1$ ; $β_{δ}$ is $P$ -periodic with $\int_{0}^{P} β_{δ} = 1$ ; $β_{δ} = 0$ on $[0, π / 2]$ and $β_{δ} (t) = δ^{- 1} w ((t - π / 2) / δ)$ on $[π / 2, P]$ .

[L3]

Since $w$ is continuous on $[0, 1]$ , it is integrable (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion); its integral function $W (s) : = \int_{0}^{s} w (u) d u$ satisfies $W^{'} (s) = w (s)$ on $[0, 1]$ by the first part of the fundamental theorem of calculus (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive), and $W (1) = 1$ by [L2].

[L4]

Uniqueness of solutions (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval): the free oscillator field $(q, p) \mapsto (p, - q)$ is continuous and locally Lipschitz, so its initial value problems have a unique maximal solution; every solution with the same initial data is a restriction of that maximal solution.

Verification

direct; differentiate the displayed trigonometric solutions, integrate the constant right-hand side, and substitute the two endpoint maps.

1.1givenL1L4algebra

Free rotation: by [L1] the displayed $(q, p)$ solves the free equations with the stated initial data, so by the uniqueness clause of [L4] it is the solution and the time- $π / 2$ map is $(q_{0}, p_{0}) \mapsto (q_{0} \cos \frac{π}{2} + p_{0} \sin \frac{π}{2}, - q_{0} \sin \frac{π}{2} + p_{0} \cos \frac{π}{2}) = (p_{0}, - q_{0}) = R_{0} (q_{0}, p_{0})$ .

1.2givenL2L3algebra

Kick: for $δ = 0$ the rescaled equation is $q_{s} = 0$ , $p_{s} = w (s) (q_{0}^{2} - a)$ . By [L3], $q (s) = q_{0}$ and $p (s) = p_{0} + (q_{0}^{2} - a) W (s)$ solve it with the stated initial data; evaluating at $s = 1$ and using $W (1) = 1$ gives $K (q_{0}, p_{0}) = (q_{0}, p_{0} + q_{0}^{2} - a)$ .

2.1step 1.1step 1.2algebra

Composition: substituting $(q_{0}, p_{0}) \mapsto (p_{0}, - q_{0})$ into $K$ and expanding gives $(K \circ R_{0}) (q_{0}, p_{0}) = (p_{0}, - q_{0} + p_{0}^{2} - a) = F (q_{0}, p_{0})$ , which is the displayed identity.

3.1givenL2∎

Pulse support and normalization: by [L2], on the single period $[0, P]$ the coefficient vanishes outside the compact set $π / 2 + δ supp w \subset (π / 2, P)$ , is nonnegative, and has integral $1$ . Periodicity translates that set by each $j P$ , $j \in Z$ , giving the stated global support union. The integral on one period is $1$ ; the global periodic coefficient does not have bounded support.

Remarks

The $δ = 0$ computation is a formal comparison object: the endpoint map $K$ is the endpoint map of the limiting pulse of unit total strength concentrated in zero time. The genuine model has $δ > 0$ , and the comparison is quantified in The rescaled pulse endpoint map converges to the kick $K (q, p) = (q, p + q^{2} - a)$ in $C^{1}$ , and the period map is smooth, symplectic and $C^{1}$ -close to $F$ .

The $δ = 0$ endpoint computation: the free rotation $(q, p) \mapsto (p, - q)$ , the kick $(q, p) \mapsto (q, p + q^{2} - a)$ and $K \circ R_{0} = F$

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The δ=0δ=0 endpoint computation: the free rotation (q,p)↦(p,−q)(q,p)↦(p,−q), the kick (q,p)↦(q,p+q2−a)(q,p)↦(q,p+q2−a) and K∘R0=FK∘R0​=F

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The $δ = 0$ endpoint computation: the free rotation $(q, p) \mapsto (p, - q)$ , the kick $(q, p) \mapsto (q, p + q^{2} - a)$ and $K \circ R_{0} = F$