ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

An unvalidated calibration offset leaves the true quantity unidentified

Example

Suppose one instrument reading $R$ is modeled by $R = q + b,$ where $q \in R$ is the true quantity and $b \in R$ is an unvalidated calibration offset. Take sample space $(R, P (R))$ , parameter set $Θ = R^{2}$ , and sampling law $P_{(q, b)} = δ_{q + b}$ . The target is $ψ (q, b) = q$ . The recorded reading determines the sum $q + b$ , but the experiment does not separately determine $q$ and $b$ . Thus the unvalidated assumption $b = 0$ is not established by this reading model; an external calibration standard or additional model structure would be needed to support it.

Facts & Assumptions

Given: The experiment and target just specified, with the offset $b$ unrestricted in $Θ$ .

[F1]

Statistical experiments: an experiment consists of a measurable space, a parameter set, and a family of probability measures on that space.

[F2]

Parametric statistical models: a parametric model is the map from its parameter set to the probability measures of its experiment; the map need not be injective.

[F3]

The Dirac set function at a point: $δ_{x} (A) = 1$ exactly when $x \in A$ and is zero otherwise.

[F4]

A Dirac set function is a probability measure: a Dirac set function is a probability measure on the chosen sigma-algebra.

[F5]

Estimands on a statistical model: an estimand is a specified target-valued function on the parameter set.

[F6]

Identifiability of an estimand: an estimand is identifiable when equal sampling laws imply equal target values.

[F7]

Scientific statistical model contracts: a formal contract does not prove that a model or calibration assumption is scientifically or physically applicable.

Verification

1.1givenF1F2F4construct

For each $(q, b) \in Θ$ , [F4] makes $P_{(q, b)}$ a probability measure. Hence the family is an experiment by [F1], and its parameter map is a parametric model by [F2].

1.2givenF5F6

The map $ψ : Θ \to R$ , $(q, b) \mapsto q$ , is the specified estimand by [F5]. The criterion in [F6] therefore applies to this target.

2.1givenF3F6F7algebrastep 1.2∎

For any $(q, b) \in Θ$ , let $(q^{'}, b^{'}) = (q + 1, b - 1)$ . Both parameter pairs lie in $Θ$ and $q^{'} + b^{'} = q + b$ , so for every $A \subseteq R$ , $P_{(q, b)} (A) = 1_{{q + b \in A}} = 1_{{q^{'} + b^{'} \in A}} = P_{(q^{'}, b^{'})} (A)$ by [F3]. But $ψ (q^{'}, b^{'}) = q + 1 \neq q = ψ (q, b)$ . Thus $ψ$ is not identifiable by [F6]; by [F7], this model conclusion does not establish any physical calibration claim.

Remarks

The calibration offset is substantively untestable from this one-reading law alone. If calibration data identify $b$ at a known value $b_{0}$ , or the model fixes $b = b_{0}$ , then the observation $R = q + b_{0}$ determines $q = R - b_{0}$ in the resulting model. Additional data or nontrivial restrictions on $b$ may still leave $q$ unidentified. The proof uses no axiom of choice.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.