ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 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 shared-link Gaussian tree with nonidentifiable link means

Example

Assume AC for the library's standard-normal-law convention. Let a rooted tree have root $o$ , one shared link $e_{0} = (o, v)$ , and two branch links $e_{1} = (v, r_{1})$ and $e_{2} = (v, r_{2})$ to leaf receivers. On each probe, the three link delays are independent with laws $X_{i} \sim N (μ_{i}, 1)$ , $i = 0, 1, 2$ , and the observed path delays are $Y_{1} = X_{0} + X_{1}$ and $Y_{2} = X_{0} + X_{2}$ . The distinct link-mean vectors $μ = (0, 0, 0)$ and $μ^{'} = (1, - 1, - 1)$ induce the same joint law of $(Y_{1}, Y_{2})$ ; hence the three link means are not identifiable. The same equality holds for any number of independent repeated probes with these fixed link laws.

The diagram labels the shared and branch links by their delay variables; the two rightmost vertices are the receivers.

Example references

Standard normal and normal laws

Axiom of Choice

Identifiability of an estimand

Lawrence, Michailidis, and Nair, Example 3

Facts & Assumptions

Given: The stated rooted tree, independent link delays with fixed means $μ_{i}$ and variance $1$ , and the two observed path sums. Assume AC only for the cited library definition of the normal laws.

[A1]

The Axiom of Choice: AC says every family of nonempty sets has a choice function; it is assumed here only because the cited normal-law definition declares it.

[F1]

Standard normal and normal laws: $N (μ, 1)$ is the pushforward of the standard normal law under $z \mapsto μ + z$ ; the definition assumes AC.

[F2]

Identifiability of an estimand: The identity target is identifiable exactly when equal sampling laws force equal parameter vectors.

Verification

1.1givenA1F1construct

On one probability space carrying independent $Z_{0}, Z_{1}, Z_{2} \sim N (0, 1)$ , set $X_{i}^{μ} = μ_{i} + Z_{i}$ and $X_{i}^{μ^{'}} = μ_{i}^{'} + Z_{i}$ . By [F1], each vector has the required independent link-delay laws.

2.1givenF1step 1.1algebra

For $μ = (0, 0, 0)$ the observation vector is $(Z_{0} + Z_{1}, Z_{0} + Z_{2})$ . For $μ^{'} = (1, - 1, - 1)$ it is $((1 + Z_{0}) + (- 1 + Z_{1}), (1 + Z_{0}) + (- 1 + Z_{2})) = (Z_{0} + Z_{1}, Z_{0} + Z_{2})$ . Thus the two observation vectors are equal on this coupling and have the same joint law; independent repetitions also have the same law.

3.1givenF2step 2.1algebra∎

The mean vectors are distinct, so [F2] implies that the identity link-mean target is not identifiable from the observed path-delay law.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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.

Sources

E. Lawrence, G. Michailidis, and V. N. Nair, Statistical Inverse Problems in Active Network Tomography, IMS Lecture Notes–Monograph Series 54 (2007), 24–44 (standard reference, not scraped)