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 nuisance-score projection

Example

Let $U, V$ be the coordinate functions on $Ω = {- 1, 1}^{2}$ , with $P$ uniform and counting measure as a common dominator. For $2 ∣ θ ∣ + ∣ η ∣ < 1$ , define $P_{θ, η} {(u, v)} = \frac{1}{4} (1 + θ (u + v) + η u) .$ Take $θ$ as the scalar parameter of interest and $η$ as nuisance. At $(0, 0)$ designate the straight one-sided paths $(θ_{t}, η_{t}) = (a t, b t)$ as allowed paths, with those having $a = 0$ designated as nuisance paths. Then the interest score is $S_{θ} = U + V$ , the nuisance tangent space is $T_{η} = span {U}$ , and the full tangent space is $T = span {U, V}$ . The efficient score is $V$ , the efficient information is $1$ , and the canonical gradient for $θ$ is $V$ .

Facts & Assumptions

Given: The four-point sample space, uniform law $P = P_{0, 0}$ , and the displayed parameterized family.

[F1]

Efficient score and efficient information: $S_{e f f} = S_{θ} - Π_{T_{η}} S_{θ}$ and $I_{e f f} = E_{P} [S_{e f f}^{2}]$ in the scalar case.

[F2]

Pathwise differentiability and parameter influence curves: one-sided DQM paths define tangent scores, their closed linear span is the tangent space, and the canonical gradient is the unique representer in that space.

[F3]

Score and Fisher information: in a dominated differentiable parametric submodel, score coordinates are derivatives of the log density.

[F4]

Efficient score gives the canonical gradient: under Axiom of Choice and Countable Choice, if the full score space is the interest-score plus nuisance-tangent sum, the target derivative along score $a^{T} S_{θ} + t$ is $a$ , and $I_{e f f}$ is nonsingular, then $I_{e f f}^{- 1} S_{e f f}$ is the canonical gradient.

[F5]

Probability measures and probability spaces: a probability measure has total mass one.

[F6]

Expectation of a nonnegative or integrable random variable: expectations of integrable functions are their integrals against the law.

[A1]

The Axiom of Choice: the general score-density convention assumes the Axiom of Choice.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed by the Hilbert projection and canonical-gradient results used here.

Verification

Given: $U (u, v) = u$ , $V (u, v) = v$ , $P (u, v) = 1 / 4$ , and the displayed model.

1.1givenF5algebra

For each parameter in the stated domain, $∣ θ (u + v) + η u ∣ \leq 2 ∣ θ ∣ + ∣ η ∣ < 1$ , so each mass is positive. Their sum is one because $\sum_{u, v} (u + v) = \sum_{u, v} u = 0$ and the four constant terms sum to four. Thus the family consists of probability laws by [F5].

1.2givenF2F3F6algebra

Finite averaging, using [F6], gives $E_{P} U = E_{P} V = 0$ , $E_{P} U^{2} = E_{P} V^{2} = 1$ , and $E_{P} U V = 0$ . Thus $w = a (U + V) + b U$ is bounded and centered. Along $(θ_{t}, η_{t}) = (a t, b t)$ the likelihood ratio is $1 + t w$ , which is nonnegative and integrates to one for sufficiently small $t \geq 0$ . The square-root identity $\sqrt{1 + z} - 1 - z / 2 = - z^{2} / [2 (1 + \sqrt{1 + z})^{2}]$ gives an $L^{2} (P)$ remainder bounded by $t^{2} ∥ w^{2} ∥_{2} / 2 = o (t)$ , so the path is DQM with score $w$ by [F2]. Direct differentiation of the log density gives interest score $U + V$ and nuisance score $U$ by [F3]. Thus the designated nuisance scores span ${U}$ , which is closed since $∥ c U ∥_{2} = ∣ c ∣$ , and all designated scores $a (U + V) + b U$ span ${U, V}$ , which is closed because $∥ a U + b V ∥_{2}^{2} = a^{2} + b^{2}$ .

2.1givenF1step 1.2algebra

Since $U \in T_{η}$ and $E_{P} [(U + V - U) U] = E_{P} [V U] = 0$ , $U$ is the orthogonal projection of $S_{θ} = U + V$ onto $T_{η}$ . Hence $S_{e f f} = V$ and $I_{e f f} = E_{P} V^{2} = 1$ by [F1].

3.1givenF2F4step 1.2step 2.1A1A2algebra∎

Along each direction $(a t, b t)$ , the target parameter is $θ_{t} = a t$ , so its right derivative is $a$ . Step 1.2 gives the full score-space decomposition $T = \overline{{a S_{θ} + b U : a, b \in R}}$ ; step 2.1 gives nonsingular $I_{e f f} = 1$ . The finite densities are explicit, and the theorem's stated axiom inputs are Axiom of Choice [A1] and Countable Choice [A2]. Therefore the hypotheses of [F4] apply, and the canonical gradient is $I_{e f f}^{- 1} S_{e f f} = V$ .

Depends on

Used by

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