ExampleConstruction: AI-adaptedVerification: AI-adapted

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 pulsed horseshoe on declared data: $a = 100$ , the radicand box $[74, 126]$ , the contraction factor $1 / \sqrt{74}$ , the limiting quadratic fixed points $1 \pm \sqrt{101}$ , and the physical bounds of the coded states at $ℓ_{0} = 1 m$ , $m = 1 k g$ , $ω_{0} = 2 π s^{- 1}$

Physical scope

A declared-data check of the constants and bounds in the page's pulsed-horseshoe theorem: at amplitude a=100 the coding box is R=13, the radicands lie in [74,126], the contraction factor is 1/sqrt(74), the limiting quadratic map has fixed points 1 plus or minus sqrt(101), the limiting kick satisfies K composed with the rotation = F, and with length scale 1 m, mass 1 kg and frequency scale 2 pi per second the coded states on the invariant axis are bounded by 13 m in axial position and 26 pi kg m per second in axial momentum, with period (pi/2+delta)/(2 pi) seconds. All quantities are declared model data; the pulse-width threshold is not numerically evaluated and no measurement is involved.

Example

Declared model data. Take the full physical pulsed oscillator model of A smooth periodically forced one-label Cartesian model with a transverse homoclinic horseshoe: the physical stroboscopic map has a compact invariant set conjugate to the shift, chaotic in the stated sense clause (i), and restrict its coded states to the invariant axis, where transverse coordinates and momenta vanish. Use the declared data $a : = 100, ℓ_{0} : = 1 m, m : = 1 k g, ω_{0} : = 2 π s^{- 1},$ and a pulse width $0 < δ < δ_{0}$ with $δ_{0}$ the supplier threshold (not evaluated numerically here). Let $R : = \sqrt{a} + 3$ , $L : = \sqrt{a - 2 R}$ and let $r_{\pm}$ be the fixed points of the limiting quadratic map.

Assertions to check. (i) $R = 13$ and the coding radicands $a + x_{n - 1} + x_{n + 1}$ lie in $[74, 126] = [a - 2 R, a + 2 R]$ , with $\sqrt{126} < 13 = R$ ; (ii) the contraction factor of the coding map is $1 / L = 1 / \sqrt{74}$ and $r_{\pm} = 1 \pm \sqrt{101}$ ; (iii) for the limiting pulse, $K \circ R_{0} = F$ with $R_{0} (q, p) = (p, - q)$ , $K (q, p) = (q, p + q^{2} - a)$ and $F (x, y) = (y, y^{2} - a - x)$ ; (iv) every coded state satisfies $∣ Q ∣ \leq 13 m$ and $∣ Π ∣ \leq 26 π k g m s^{- 1}$ , and the forcing period is $T_{p} = (π / 2 + δ) / (2 π) s$ .

Facts & Assumptions

Given: the constants $a = 100$ , $R = \sqrt{a} + 3$ , $L = \sqrt{a - 2 R}$ and the scales $ℓ_{0}, m, ω_{0}$ above; the dimensionless model, its rotation $R_{0}$ , pulse endpoint maps $K_{δ}$ and stroboscopic map $G_{δ} = K_{δ} \circ R_{0}$ of The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map; the limiting kick $K$ .

[L1]

The horseshoe theorem and its bounds (A smooth periodically forced one-label Cartesian model with a transverse homoclinic horseshoe: the physical stroboscopic map has a compact invariant set conjugate to the shift, chaotic in the stated sense clauses (ii)-(iv)): for $0 < δ < δ_{0}$ the full physical stroboscopic map $Ψ_{δ}$ has a compact invariant coded set on the invariant axis, with transverse coordinates and momenta zero; each state satisfies $∣ Q ∣ \leq R ℓ_{0}$ and $∣ Π ∣ \leq m ℓ_{0} ω_{0} R$ , and its coded trajectories exist for all times.

[L2]

The coding contraction (The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution) and the limiting pulse (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 coding map $T_{σ}$ on the ball of bounded sequences with $∣ x ∣_{\infty} \leq R$ is a contraction with factor $L^{- 1}$ when the radicands stay positive and the derivative bound holds, and $K_{δ} \to K$ in $C^{1}$ as $δ \to 0$ with $K (q, p) = (q, p + q^{2} - a)$ .

[L3]

Elementary arithmetic: $R^{2} - a - 2 R = 4 \sqrt{a} + 3 > 0$ for $a = 100$ , so $R = \sqrt{a} + 3$ satisfies $\sqrt{a + 2 R} < R$ ; the fixed points of $x \mapsto$ the constant-code recurrence $x^{2} - a = 2 x$ solve $x^{2} - 2 x - a = 0$ .

Verification

technique · direct

1.1givenL2L3algebra

Arithmetic of the coding box: for $a = 100$ , $R = \sqrt{100} + 3 = 13$ ; hence $a - 2 R = 100 - 26 = 74$ and $a + 2 R = 100 + 26 = 126$ , so the radicands of [L2] lie in $[74, 126]$ whenever the coded sequence satisfies $∣ x_{n} ∣ \leq R$ . Moreover $\sqrt{126} < 13$ because $126 < 169$ , and $\sqrt{74} > 0$ , so all square roots in the coding recurrence are defined with the derivative bound required by [L2]. Also $R^{2} - a - 2 R = 169 - 100 - 26 = 43 = 4 \sqrt{100} + 3 > 0$ , which is the supplier's inequality ensuring $T_{σ}$ maps the ball into itself.

1.2givenL2algebra

Limiting pulse composition: with $R_{0} (q, p) = (p, - q)$ and $K (q, p) = (q, p + q^{2} - a)$ one computes $K (R_{0} (q, p)) = K (p, - q) = (p, - q + p^{2} - a)$ and $F (q, p) = (p, p^{2} - a - q)$ , which agree; this is the algebraic core of the $C^{1}$ -closeness statement $G_{δ} \to F$ of [L2].

2.1step 1.1givenL2L3algebra

Contraction and fixed points: the radicand bound of step 1.1 gives the derivative bound $\frac{1}{2 \sqrt{74}}$ for the square root, hence the contraction factor $(\frac{1}{2 \sqrt{74}}) \cdot 2 = \frac{1}{\sqrt{74}} = L^{- 1}$ for the coding map, as in [L2] with $L = \sqrt{74}$ . Constant codes satisfy $x_{n + 1} + x_{n - 1} = x_{n}^{2} - a$ with $x_{n} \equiv r$ , that is $2 r = r^{2} - a$ or $r^{2} - 2 r - a = 0$ , whose solutions are $r = 1 \pm \sqrt{1 + a} = 1 \pm \sqrt{101}$ . The branch conditions hold with exact inequalities: $\sqrt{101} > 8$ because $101 > 64$ , so $∣ r_{-} ∣ = \sqrt{101} - 1 > 7$ ; $\sqrt{101} > 1 + \sqrt{74}$ because $101 > 75 + 2 \sqrt{74}$ reduces to $13 > \sqrt{74}$ , that is $169 > 74$ ; so $∣ r_{\pm} ∣ > L$ ; and $r_{+} \leq 13$ , $∣ r_{-} ∣ \leq 13$ because $\sqrt{101} \leq 12$ and $\sqrt{101} - 1 \leq 13$ , both from $101 \leq 144$ .

2.2step 1.1givenL1algebra

Physical bounds: by [L1] clause (iv) every coded physical state satisfies $∣ Q ∣ \leq R ℓ_{0} = 13 m$ and $∣ Π ∣ \leq m ℓ_{0} ω_{0} R = 13 \cdot 2 π k g m s^{- 1} = 26 π k g m s^{- 1}$ ; the forcing period is $T_{p} = P / ω_{0} = (π / 2 + δ) / (2 π) s$ and the pulse duration is $δ / (2 π) s$ .

3.1step 2.1step 1.2step 2.2givenalgebra∎

Reading: every number above is declared model data or an exact surd; the threshold $δ_{0}$ is not numerically evaluated, no decimal approximation is substituted into a proof step, and no measurement or apparatus is involved (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

Remarks

Status of the numerical values. The values $13$ , $74$ , $126$ , $1 / \sqrt{74}$ , $1 \pm \sqrt{101}$ and $26 π$ are exact consequences of the declared data; they are quoted as a check of the supplier's constants, not as an empirical estimate of any oscillation. In particular the threshold $δ$ below which the horseshoe exists is left symbolic because the supplier does not compute it.
Unit consistency. $[ℓ_{0}] = ℓ$ , $[Π] = m ℓ_{0} ω_{0} = μ ℓ τ^{- 1}$ , $[T_{p}] = τ$ , and the coded box is bounded in both position and canonical momentum (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

The pulsed horseshoe on declared data: $a = 100$ , the radicand box $[74, 126]$ , the contraction factor $1 / \sqrt{74}$ , the limiting quadratic fixed points $1 \pm \sqrt{101}$ , and the physical bounds of the coded states at $ℓ_{0} = 1 m$ , $m = 1 k g$ , $ω_{0} = 2 π s^{- 1}$

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The pulsed horseshoe on declared data: a=100a=100, the radicand box [74,126][74,126], the contraction factor 1/741/74​, the limiting quadratic fixed points 1±1011±101​, and the physical bounds of the coded states at ℓ0=1 mℓ0​=1m, m=1 kgm=1kg, ω0=2π s−1ω0​=2πs−1

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The pulsed horseshoe on declared data: $a = 100$ , the radicand box $[74, 126]$ , the contraction factor $1 / \sqrt{74}$ , the limiting quadratic fixed points $1 \pm \sqrt{101}$ , and the physical bounds of the coded states at $ℓ_{0} = 1 m$ , $m = 1 k g$ , $ω_{0} = 2 π s^{- 1}$