ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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 small sensitivity solve error can reverse a two-design ranking

Example

Two scalar candidate designs have exact sensitivities $101 / 100$ and $1$ , so the first is D-optimal. A solver returns $99 / 100$ and $51 / 50$ , each within $1 / 50$ of its exact sensitivity, reversing the computed ranking. Both exact and computed information matrices remain uniformly positive definite, so the conditional design-gap bound applies.

Facts & Assumptions

Given: Candidate set $X = {d_{1}, d_{2}}$ , parameter dimension $p = 1$ , unit precision $W_{d_{1}} = W_{d_{2}} = 1$ , exact sensitivities $J_{d_{1}} = 101 / 100$ , $J_{d_{2}} = 1$ , and computed sensitivities $J_{h, d_{1}} = 99 / 100$ , $J_{h, d_{2}} = 51 / 50$ .

[F1]

Finite nonlinear observation designs, local identifiability, and weighted information defines the scalar information $M (d) = J_{d}^{T} W_{d} J_{d}$ ; Computed Jacobian error controls the weighted information-matrix error defines its computed counterpart $M_{h} (d) = J_{h, d}^{T} W_{d} J_{h, d}$ .

[F2]

Computed sensitivity errors perturb every singular value by at most their operator-norm error: each singular-value error is bounded by the operator norm of the Jacobian error.

[F3]

Computed Jacobian error controls the weighted information-matrix error: for $K = 101 / 100$ , $η = 1 / 50$ , and $w = 1$ , the information error is at most $δ = w (2 K η + η^{2})$ .

[F4]

A uniform positive eigenvalue floor controls log-determinant error: if all matrices on each true/computed segment have eigenvalues at least $α > 0$ , then each log-determinant error is at most $p δ / α$ .

[F5]

Uniform information errors control approximate D-optimal design gaps: a computed maximizer under uniform error $δ$ and eigenvalue floor $α$ has true log-determinant gap at most $2 p δ / α$ when $τ = 0$ .

[F6]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log (1 + x) = \int_{1}^{1 + x} d t / t$ for $x > 0$ and $\log$ is strictly increasing.

Verification

1.1givenF1F6algebra

The exact information values are $M (d_{1}) = (101 / 100)^{2} = 10201 / 10000$ and $M (d_{2}) = 1$ , so the true log-determinant criterion ranks $d_{1}$ above $d_{2}$ . The computed values are $M_{h} (d_{1}) = (99 / 100)^{2} = 9801 / 10000$ and $M_{h} (d_{2}) = (51 / 50)^{2} = 10404 / 10000$ , so the computed criterion ranks $d_{2}$ above $d_{1}$ by strict monotonicity of $\log$ from [F6].

1.2givenF1F2F3algebra

The two Jacobian errors have absolute value $1 / 50$ , and for one-dimensional matrices this is the singular-value error, consistent with [F2]. Here $K = 101 / 100$ , $η = 1 / 50$ , and $w = 1$ , so [F3] gives the uniform bound $δ = 2 (101 / 100) (1 / 50) + (1 / 50)^{2} = 51 / 1250$ . For any approximate design $ξ = (λ, 1 - λ)$ with $0 \leq λ \leq 1$ , $M (ξ)$ is a convex combination of $10201 / 10000$ and $1$ , hence is at least $1$ ; $M_{h} (ξ)$ is a convex combination of $9801 / 10000$ and $10404 / 10000$ , hence is at least $α = 9801 / 10000$ . Their segment also has this lower bound.

2.1step 1.1step 1.2F4F5F6algebra∎

By [F4] each criterion error is at most $δ / α = (51 / 1250) / (9801 / 10000) = 408 / 9801$ . The computed maximizer is $d_{2}$ , so [F5] bounds its true log-determinant gap from the true maximizer $d_{1}$ by $816 / 9801$ . Directly, that gap is $\log (10201 / 10000) \leq 201 / 10000$ by the integral in [F6], and $201 / 10000 < 816 / 9801$ by cross-multiplication. Thus the ranking flip is consistent with the stated uniform error guarantee.

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Sources

Leah R. McLean et al., Estimability Analysis and Optimal Design in Dynamic Multi-scale Models of Cardiac Electrophysiology, §2.3 (standard reference, not scraped)
Stephen Boyd and Lieven Vandenberghe, Convex Optimization, §7.5 (standard reference, not scraped)