ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

Forward Euler creates an observation-map error

Example

For each parameter $θ \in R$ , solve ${\dot{z}}_{θ} (t) = θ t, z_{θ} (0) = 0 (0 \leq t \leq 1),$ and observe $z_{θ} (1)$ . The exact observation map is $A (θ) = θ / 2$ . Apply one explicit forward-Euler step of size $h = 1$ , $z_{1} = z_{0} + h θ t_{0} with (t_{0}, z_{0}) = (0, 0);$ the resulting discretized observation map is $A_{h} (θ) = 0$ . In the computed inversion error budget with $X = Y = R$ , datum $b = 0$ , truth $θ = 1$ , zero observation error, and $λ = 1$ , the pre-inversion discrepancy is $d_{h} = 1 / 2$ .

Example references

The pre-inversion discrepancy $d_{h} = (A - A_{h}) x^{†}$

Derivative of the square function

Equal derivatives and initial values give equal solutions on an interval

Bounded finite-rank operators are compact

Boundedness of linear operators

Hilbert-space completeness convention

The standard inner product on $R$

Completeness of $R$

Facts & Assumptions

Given: $θ \in R$ , the scalar ODE and initial value in the example, one forward-Euler step with $h = 1$ , datum $b = 0$ , and the computed-inversion error-budget interface.

[F1]

For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term: the function $t \mapsto t^{2}$ is differentiable at every real $t$ with derivative $2 t$ .

[F2]

The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products: $R$ with $⟨ a, b ⟩ = a b$ is a real inner-product space.

[F3]

The reals are complete: every Cauchy sequence in $R$ converges in $R$ .

[F4]

Hilbert space: a complete real inner-product space with its induced norm is a real Hilbert space.

[F5]

A bounded linear operator between normed spaces: a linear map is bounded if some $K \geq 0$ satisfies $∥ T x ∥ \leq K ∥ x ∥$ for every $x$ .

[F6]

Bounded finite rank operators are compact: a bounded linear operator whose range has a finite basis is compact.

[F7]

Error budget for computed Tikhonov inversion: for the stated compact $A$ , bounded finite-dimensional-range $A_{h}$ , supplied adjoints, positive $λ$ , and bijective shifted normal operators, the pre-inversion discrepancy is $d_{h} = (A - A_{h}) x^{†}$ .

[F8]

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant: two functions continuous on an interval and differentiable on its interior with equal derivatives differ by a constant.

Verification

1.1givenF1F8algebra

The function $z_{θ} (t) = θ t^{2} / 2$ has $z_{θ} (0) = 0$ and, by [F1], derivative ${\dot{z}}_{θ} (t) = θ t$ . Any other solution is continuous on $[0, 1]$ and has the same derivative on $(0, 1)$ , so [F8] makes its difference from $z_{θ}$ constant; the common initial value makes this constant zero. Thus the solution is unique and gives the exact endpoint map $A (θ) = z_{θ} (1) = θ / 2$ . At $θ = 1$ the exact observation is $1 / 2$ .

1.2givenF2F3F4F5F6F7algebra

The stated datum $b = 0$ belongs to $Y = R$ . By [F2]–[F4], $X = Y = R$ with the standard inner product is a real Hilbert space. The maps $A (θ) = θ / 2$ and $A_{h} (θ) = 0$ are linear and bounded by [F5], with bounds $1 / 2$ and $0$ ; their ranges have bases $(1)$ and the empty basis, respectively. Hence $A$ is compact by [F6], and $A_{h}$ has finite-dimensional range. The adjoints are multiplication by $1 / 2$ and $0$ , since $⟨ A θ, y ⟩ = θ y / 2 = ⟨ θ, y / 2 ⟩$ and $⟨ A_{h} θ, y ⟩ = 0$ . For $λ = 1$ , the shifted normal operators are multiplication by $5 / 4$ and $1$ , so both are bijective. All hypotheses of [F7] are satisfied.

2.1givenstep 1.1algebra

The stated forward-Euler formula at $(t_{0}, z_{0}) = (0, 0)$ gives $z_{1} = 0 + 1 \cdot θ \cdot 0 = 0$ , so $A_{h} (θ) = 0$ and $A_{h} (1) = 0$ . Thus the two endpoint maps differ even though the observation perturbation is zero.

3.1givenF7step 1.1step 2.1algebra∎

Set $x^{†} = 1$ , $e = 0$ , $δ = 0$ , and $λ = 1$ . Then [F7] gives $d_{h} = (A - A_{h}) x^{†} = 1 / 2 - 0 = 1 / 2$ . This is a nonzero discretization contribution with no observation noise.

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