ExampleConstruction: AI-adaptedVerification: 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.

Prompt and distribution-sensitive transfer

Example

Use $R^{2}$ with $e_{1} = (1, 0)$ , $e_{2} = (0, 1)$ , $⟨ (a, b), (c, d) ⟩ = a c + b d$ , and scale $s > 0$ . For one image $x$ with normalized image embedding $u = e_{1}$ , consider classes $A, B$ and two prompt templates. Under the first wording, their normalized text embeddings are $v_{A 1} = e_{1}$ , $v_{A 2} = e_{2}$ , and $v_{B 1} = v_{B 2} = q_{B} = (4 / 5, 3 / 5)$ . Embedding-space prompt averaging gives $q_{A} = (1 / \sqrt{2}, 1 / \sqrt{2})$ and $q_{B} = (4 / 5, 3 / 5)$ . The class-prototype classifier scores $s / \sqrt{2}$ for $A$ and $4 s / 5$ for $B$ . Define individual-text retrieval here to rank the four prompt embeddings by their dot product with $u$ ; it selects the first $A$ prompt, while the class-prototype classifier selects $B$ . If a second prompt wording gives both $A$ prompts embedding $e_{1}$ and leaves the $B$ prompts unchanged, then the classifier selects $A$ . These are possible encoder outputs, demonstrating that prompt wording and the candidate construction can change the result.

For a second fixed zero-shot classifier, let the normalized class prototypes be $q_{A} = e_{1}$ and $q_{B} = e_{2}$ , and let two distinct images $x_{0} \neq x_{1}$ both have normalized image embedding $e_{1}$ . Give them the fixed true labels $A$ and $B$ , respectively, so the classifier predicts $A$ for both. Let $d_{A}, d_{B}$ be class-specific descriptions: $d_{A}$ is true of $A$ and false of $B$ , while $d_{B}$ is true of $B$ and false of $A$ ; their normalized text embeddings are $q_{A}, q_{B}$ . The $B$ image $x_{1}$ paired with the incorrect $A$ description $d_{A}$ is semantically mismatched but has cosine similarity $1$ ; paired with its correct $B$ description $d_{B}$ , it has cosine similarity $0$ . Thus this construction shows that the embeddings need not preserve semantic compatibility. Define the finite source evaluation law by assigning mass $3 / 4$ to $(x_{0}, A)$ and $1 / 4$ to $(x_{1}, B)$ , and the target evaluation law by assigning mass $1 / 4$ to $(x_{0}, A)$ and $3 / 4$ to $(x_{1}, B)$ . The label assigned to each image is the same under both laws, while the input frequencies differ. Define 0–1 error under either law as the total mass of its misclassified labelled inputs. The errors are $1 / 4$ on the source law and $3 / 4$ on the target law, illustrating input-distribution sensitivity for this fixed classifier.

The vector geometry in the first construction is shown schematically below: the horizontal component is the similarity with the image embedding $e_{1}$ . The exact coordinates above determine the rankings; no proof relies on the drawing.

As separate empirical context, Radford et al. trained CLIP on 400 million WebImageText image/text pairs. Unless otherwise specified, their reported “CLIP” results use ViT-L/14@336px, trained for 32 epochs with one additional high-resolution epoch. On their 27-dataset suite, zero-shot CLIP outperformed a regularized logistic-regression classifier on ResNet-50 features on 16 datasets. Their default “A photo of a {label}” prompt improved ImageNet accuracy by 1.3%, and ensembling 80 prompts added a further 3.5 percentage points over that default. These are measured outcomes under the paper's data, models, training, evaluation suite, and prompt choices.

In §3.3, the authors report that zero-shot CLIP improved effective robustness relative to ImageNet models across seven natural distribution shifts, reducing the robustness gap by up to 75%. Their dataset-specific class prompts increased average effective robustness by 5%, with gains concentrated on a few datasets. They caution that these results do not isolate the cause of the robustness gap and that pretraining and evaluation distributions may share spurious correlations.

In §5, the authors report median detected overlap of 2.2% and mean overlap of 3.2% across 35 evaluation datasets; only seven datasets had an overall accuracy change above 0.1%, and the largest detected increase was 0.6% on Birdsnap. They also state that detector recall could not be checked across all 400 million training pairs and that the Overlap and Clean subsets could differ in distribution or difficulty. Section 6 notes that the benchmark suite was partly co-adapted through repeated validation-set queries and reports weak results on fine-grained, systematic, novel, and truly out-of-distribution tasks. These results and limitations supply no general guarantee of transfer, out-of-distribution accuracy, reasoning, fairness, or semantic alignment.

Example references

Radford et al., §§2.5 and 3.1.2–3.1.5: training data/model scope, zero-shot results, and prompt measurements

Radford et al., §3.3 “Robustness to Natural Distribution Shift,” §5 “Data Overlap Analysis,” and §6 “Limitations”

Machine Learning II lecture notes, zero-shot transfer, prompt engineering/ensembling, measured evaluation, overlap, and limitations

Facts & Assumptions

Given: The two finite embedding setups, the class descriptions with their stated meanings, the source/target evaluation laws, a positive scale $s$ , and the specified individual-text retrieval and finite 0–1 error rules. Verify the prompt-dependent classifier outputs, the retrieval/classification difference, the semantic-pair similarity contrast, and both finite error values.

[F1]

Zero-shot prompt classification: For each class, normalize each prompt embedding, average and renormalize those vectors to form a prototype, score it by the scaled inner product with the normalized image embedding, and return the least-index class maximizing the score.

[A1]

In the first setup, $u = e_{1}$ , $v_{A 1} = e_{1}$ , $v_{A 2} = e_{2}$ , and $v_{B 1} = v_{B 2} = (4 / 5, 3 / 5)$ . Under the alternate wording, $v_{A 1} = v_{A 2} = e_{1}$ and the $B$ vectors remain unchanged. Retrieval ranks the four individual prompt embeddings by inner product with $u$ .

[A2]

In the second setup, $x_{0} \neq x_{1}$ , $q_{A} = e_{1}$ , $q_{B} = e_{2}$ , $u (x_{0}) = u (x_{1}) = e_{1}$ , and the fixed labels are $y (x_{0}) = A$ , $y (x_{1}) = B$ . The source law places masses $3 / 4, 1 / 4$ on $(x_{0}, A), (x_{1}, B)$ , and the target law places masses $1 / 4, 3 / 4$ on those same labeled inputs. Error is the total mass assigned to misclassified inputs.

[A3]

The class-specific descriptions $d_{A}, d_{B}$ are true only of classes $A, B$ , respectively, among these two classes, and their normalized text embeddings are $q_{A} = e_{1}$ and $q_{B} = e_{2}$ .

Verification

1.1givenF1A1algebra

Under the first wording, the normalized mean of $e_{1}$ and $e_{2}$ is $q_{A} = (1 / \sqrt{2}, 1 / \sqrt{2})$ , while the two $B$ prompts average to $q_{B} = (4 / 5, 3 / 5)$ .

1.2givenF1A1algebra

Under the alternate wording, the $A$ prototype is $e_{1}$ and the $B$ prototype remains $(4 / 5, 3 / 5)$ , giving scores $s$ and $4 s / 5$ . Thus the same image is classified as $A$ after the prompt change.

1.3givenF1A2algebra

In the second setup both image embeddings equal $e_{1}$ , so their scores against $(q_{A}, q_{B}) = (e_{1}, e_{2})$ are $(s, 0)$ and the classifier predicts $A$ on both images. It is correct on $x_{0}$ and incorrect on $x_{1}$ .

2.1givenF1A1step 1.1algebra

Their scores are $z_{A} = s / \sqrt{2}$ and $z_{B} = 4 s / 5$ . Since both sides are positive and $(4 / 5)^{2} = 16 / 25 > 1 / 2 = (1 / \sqrt{2})^{2}$ , $z_{B} > z_{A}$ , so the class-prototype classifier returns $B$ .

2.2givenA2step 1.3algebra

Since $x_{0} \neq x_{1}$ , the source law gives input mass $3 / 4$ to $x_{0}$ and the target law gives it $1 / 4$ , with the reverse masses at $x_{1}$ ; the label assigned to each image is fixed. The only error atom has source mass $1 / 4$ and target mass $3 / 4$ . Therefore the source and target errors are $1 / 4$ and $3 / 4$ , respectively, with difference $1 / 2$ .

3.1givenA1step 2.1algebra

The individual retrieval dot products are $1, 0, 4 / 5, 4 / 5$ . Since $1 > 4 / 5$ , retrieval returns the first $A$ prompt, unlike the class-prototype classifier.

4.1givenA2A3algebra∎

By [A2] and [A3], $x_{1}$ is a $B$ image and $d_{A}$ is false of $B$ , so $(x_{1}, d_{A})$ is semantically mismatched. The stated inner-product formula $⟨ (a, b), (c, d) ⟩ = a c + b d$ gives similarity $1$ for $(x_{1}, d_{A})$ and $0$ for the matched pair $(x_{1}, d_{B})$ .

Depends on

Zero-shot prompt classification

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Alec Radford et al., Learning Transferable Visual Models From Natural Language Supervision (standard reference, not scraped)
Machine Learning II, Lecture notes — 2021 Clip (standard reference, not scraped)