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.

A train/test prior shift changes reconstruction risk

Example

Let the truth space and target space both be $U = Q = {0, 1} \subset R$ , let the target map be the identity, let the data space be $Y = {0}$ , and let the deterministic forward map be $F (t) = 0$ for both truths. Use squared loss $ℓ (t, a) = (t - a)^{2}$ and the point reconstruction rule $g (0) = 0$ . Let the training prior be $π_{t r a i n} = δ_{0}$ and the test prior be $π_{t e s t} = δ_{1}$ . The resulting joint laws of data and target are $P_{t r a i n} = δ_{(0, 0)}$ and $P_{t e s t} = δ_{(0, 1)}$ . The input marginal is the same in both laws, but $R_{P_{t r a i n}} (g) = 0$ and $R_{P_{t e s t}} (g) = 1$ ; equality of input marginals alone does not bound target-risk change under prior shift.

Example references

Forward maps, admissible truths, and target maps

A point reconstruction rule is a measurable map from data to target

Risk is an expected loss under the specified joint law

Facts & Assumptions

Given: The finite truth and data spaces, the constant forward map, the two prior laws, squared loss, and the rule in the Example.

[F1]

An inverse-problem model specifies a forward map on an admissible truth class and a target map into a metric target space. Inverse-problem models and solution classes

[F2]

A point reconstruction rule is a measurable data-to-target map. Learned inverse estimators and reconstruction targets

[F3]

Under a joint data-target law, risk is the integral of the nonnegative loss evaluated at the target and rule output. Inverse reconstruction risk under train and test laws

Verification

1.1F1given

Under either prior, the deterministic observation is $Y = F (T) = 0$ , so the training and test data marginals are both $δ_{0}$ . The target equals $0$ almost surely under $P_{t r a i n}$ and equals $1$ almost surely under $P_{t e s t}$ .

1.2F2given

The rule $g (0) = 0$ is measurable on the one-point data space and takes values in $Q$ , so it is a point reconstruction rule.

2.1F3step 1.1step 1.2givenalgebra∎

By [F3], $R_{P_{t r a i n}} (g) = (0 - 0)^{2} = 0$ and $R_{P_{t e s t}} (g) = (1 - 0)^{2} = 1$ . Therefore the risk changes from zero to one while the input marginal remains fixed; no risk comparison follows from that marginal agreement alone.

Depends on

Used by

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