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.

Distinct truths and operators with identical noisy data laws

Example

Let the truth space and target space both be $R$ with their absolute-value metric, and take the action sigma-algebra to be the power set $P (R)$ , so every absolute metric-loss section is measurable. Let the admissible truth/operator pairs be $R \times {A_{0}, A_{1}}$ and take target map $q (x) = x$ . Define the linear forward operators $A_{0} (x) = x$ and $A_{1} (x) = x / 2$ . Equip the data space $R$ with its power-set sigma-algebra and usual addition, so every translation is measurable, and let $ν$ be any probability measure on that measurable space, for example the Dirac measure at $0$ . The specifications $(1, A_{0}, ν)$ and $(2, A_{1}, ν)$ have the same data law, since both forward signals equal $1$ , while their targets are distance $1$ apart. Every randomized decision rule into the stated action space therefore has maximum expected absolute loss at least $1 / 2$ .

Facts & Assumptions

Given: The truth/target space $R$ , the two admissible linear operators, the additive measurable data space, the common noise law $ν$ , and the target map in the Example.

[F1]

Operator, noise-law, and prior-support mismatch: the sampling law of an additive-noise specification is the law of the measurable random element $e \mapsto F_{A} (x) + e$ under $ν$ .

[F2]

Equal observation laws force a metric-risk lower bound: if two specifications have the same sampling law and targets at distance $Δ$ , every randomized decision rule has maximum expected metric risk at least $Δ / 2$ .

[F3]

The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded: the absolute-value distance on $R$ is a metric, with triangle inequality.

Verification

1.1givenF1algebra

Every translation of $R$ is measurable for its power-set sigma-algebra, since every preimage is a subset of $R$ . For every $B \in Y$ , the law formulas from [F1] give $P_{1, A_{0}, ν} (B) = ν ({e : 1 + e \in B}) = P_{2, A_{1}, ν} (B)$ because $A_{0} (1) = 1 = A_{1} (2)$ . Thus the two sampling laws are equal.

2.1givenF2F3step 1.1algebra∎

For each $t \in {1, 2}$ and each Borel set $B \subseteq [0, + \infty]$ , the preimage ${a \in R : ∣ t - a ∣ \in B}$ is a subset of $R$ and hence belongs to the action sigma-algebra $P (R)$ . Thus both loss sections are measurable. The target separation is $∣ q (1) - q (2) ∣ = ∣ 1 - 2 ∣ = 1$ by [F3]. Applying [F2] to step 1.1 gives maximum expected absolute loss at least $1 / 2$ .

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