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.

Absolute-loss RKHS regularization has an explicit stability envelope

Example

Let $X = {x_{0}}$ and $Y = {0, 1}$ with their discrete sigma-algebras, and let $P (X = x_{0}, Y = 1) = q$ and $P (X = x_{0}, Y = 0) = 1 - q$ for $q \in [0, 1]$ . Take $H = R$ as the RKHS of the one-point kernel $K (x_{0}, x_{0}) = 1$ , so $f_{t} (x_{0}) = t$ and $∥ f_{t} ∥_{H} = ∣ t ∣$ . Use absolute loss $c (t, y) = ∣ t - y ∣$ and regularization parameter $λ > 0$ . For a size- $m$ sample and retained index set $J \subseteq [m]$ , use the fixed-denominator objective $m^{- 1} \sum_{j \in J} ∣ t - y_{j} ∣ + λ t^{2}$ . If $p_{J}$ and $r_{J}$ count the labels $1$ and $0$ in $J$ , respectively, its unique minimizer is $t_{J} = \min {1, \max {0, (p_{J} - r_{J}) / (2 λ m)}}$ . Thus all full and deleted outputs lie in $[0, 1]$ , every loss is bounded by $M = 1$ , and the deletion-stability coefficient satisfies $β_{m} \leq 1 / (2 λ m)$ . For $0 \leq t \leq 1$ , the population and empirical risks are $R_{P} (f_{t}) = q (1 - t) + (1 - q) t$ and ${\hat{R}}_{S} (f_{t}) = m^{- 1} [p_{[m]} (1 - t) + r_{[m]} t]$ .

Example references

The full-size deletion convention and RKHS stability conclusion are given by Strongly convex RKHS regularized ERM is stable.

Uniform deletion stability and its coefficient are defined by Uniform algorithmic stability under deletion.

The supervised prediction and loss setting is specified by Supervised prediction problems.

Population risk is an integral and empirical risk is the finite sample average by Population and empirical risk.

Facts & Assumptions

Given: The one-point supervised model, $q \in [0, 1]$ , $m \geq 1$ , $λ > 0$ , and a sample with labels in ${0, 1}$ .

[F1]

Population risk is the integral under $P$ and empirical risk is the sample average. Population and empirical risk

[F2]

Uniform deletion stability is the supremum loss change from a full output to its indexed deletion output. Uniform algorithmic stability under deletion

[F3]

For a real RKHS with $K (x, x) \leq κ^{2}$ and convex $σ$ -Lipschitz loss, attained fixed-denominator regularized ERM minimizers have $β_{m} \leq σ^{2} κ^{2} / (2 λ m)$ . Strongly convex RKHS regularized ERM is stable

[A1]

Here $K (x_{0}, x_{0}) = 1$ and $c (t, y) = ∣ t - y ∣$ ; each retained sample uses denominator $m$ and $λ > 0$ . Strongly convex RKHS regularized ERM is stable

Verification

1.1givenA1algebra

For $t \in [0, 1]$ , the objective is $p_{J} / m + (r_{J} - p_{J}) t / m + λ t^{2}$ , whose derivative vanishes at $(p_{J} - r_{J}) / (2 λ m)$ ; strict convexity gives the clipped formula as the unique minimizer on this interval. For $t < 0$ the objective is $p_{J} / m - ∣ J ∣ t / m + λ t^{2}$ and is decreasing toward $0$ ; for $t > 1$ it is $∣ J ∣ t / m - p_{J} / m + λ t^{2}$ and is increasing away from $1$ . Hence no global minimizer lies outside $[0, 1]$ , including when $J = \emptyset$ .

2.1F1A1step 1.1

Since $t_{J}, y \in [0, 1]$ , $∣ t_{J} - y ∣ \leq 1$ , so the common loss bound is $M = 1$ . Formula [F1] gives $R_{P} (f_{t}) = q (1 - t) + (1 - q) t$ and ${\hat{R}}_{S} (f_{t}) = m^{- 1} [p_{[m]} (1 - t) + r_{[m]} t]$ for $t \in [0, 1]$ . The map $t \mapsto ∣ t - y ∣$ is convex and satisfies $∣ ∣ t - y ∣ - ∣ s - y ∣ ∣ \leq ∣ t - s ∣$ , so $σ = 1$ ; the one-point RKHS has $κ = 1$ .

3.1F2F3step 2.1algebra∎

By [F2] and [F3], the learner has uniform deletion-stability coefficient $β_{m} \leq 1 / (2 λ m)$ ; step 2.1 independently verifies the bounded-loss premise required by the high-probability theorem.

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