ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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 finite grid can miss a continuous-time crossing

Example

On $[0, 1]$ , let the deterministic continuous path be $x (t) = 1 - 2 ∣ t - 1 / 2 ∣$ , and define the target crossing event as ${\sup_{0 \leq t \leq 1} x (t) \geq 3 / 4}$ . Sampling only at grid times ${0, 1}$ misses the crossing.

Facts & Assumptions

Given: The path $x (t) = 1 - 2 ∣ t - 1 / 2 ∣$ , the threshold $3 / 4$ , and the endpoint grid ${0, 1}$ .

[F1]

Finite-budget stopped-event kernel with a cemetery outcome: A finite-budget status reports a target hit only when that target is reached at an inspected state within the budget.

[F2]

Finite-budget stopped-kernel measurability and event normalizer: The finite-budget normalizer is the probability of a target hit by the budget.

[F3]

Truncation and total-variation errors for stopped crossings: A grid-detected event contained in the continuous target has probability bias equal to the probability of the missed-crossing event; no rate follows without an additional bound.

Verification

technique · direct

1.1givenalgebra

The path satisfies $x (0) = x (1) = 0 < 3 / 4$ , while $x (1 / 2) = 1 \geq 3 / 4$ . Thus the continuous crossing event occurs and endpoint-grid detection fails.

2.1F1F2F3step 1.1given∎

The model is deterministic, so its continuous crossing probability is $1$ and the grid-detected probability is $0$ ; their absolute bias is $1$ . This is exactly the missed-crossing event in [F3], and no error rate in grid spacing can be inferred from this single example.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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