ExampleConstruction: Literature-sourcedVerification: 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.

A pretraining–adaptation interface with no shift guarantee

Example

Let $X = {a, b}$ and $Y = {0, 1}$ , each with its full power-set sigma-algebra. Give the source and target examples the joint laws $P_{S} = δ_{(a, 0)}$ and $P_{T} = δ_{(b, 1)}$ , so their input marginals are $μ_{S} = δ_{a}$ and $μ_{T} = δ_{b}$ . Here $δ_{z}$ denotes unit point mass at $z$ . Supply the conditional label kernels $K_{S} (x, \cdot) = δ_{0}$ and $K_{T} (x, \cdot) = δ_{1}$ for each $x \in X$ . This is a finite train/evaluation distribution shift with disjoint input support; it is not covariate shift.

For a downstream training sample $D = {(x_{i}, y_{i})}_{i = 1}^{n}$ drawn from $P_{S}$ , with $n \geq 1$ , a simple parameter empirical-risk-minimization formulation chooses $γ$ from a nonempty parameter set $Γ$ to minimize ${\hat{R}}_{D} (γ) = n^{- 1} \sum_{i = 1}^{n} ℓ_{t a s k} (f_{γ} (x_{i}), y_{i})$ when that infimum is attained; $f_{γ} : X \to Y$ is a task predictor and $ℓ_{t a s k} : Y \times Y \to [0, \infty)$ is a task loss. Fine-tuning is a parameter-based adaptation that updates some or all pretrained parameters using task information. In-context learning instead adds task demonstrations and a test input to the prompt while keeping the pretrained parameters fixed. It is an adaptation route by input conditioning, not empirical-risk minimization over changed model parameters.

The report’s 2021 theory discussion asks when a small population pretraining loss implies a small minimal adaptation loss and identifies that interface as a theory gap involving the two data distributions, objectives, training methods, and architecture. A pretraining cross-entropy scaling fit does not by itself give a downstream task-loss or reliability guarantee for this example. This is a conceptual interface example and makes no empirical comparison of prompting and fine-tuning.

The diagram shows the support points of the two laws with their unit masses. It is only a support illustration; the verification derives the shift condition directly from the singleton set ${b}$ .

Example references

Foundation models as broad-pretraining and adaptation systems: distinguishes prompt conditioning from parameter-based adaptation and records that the pretraining-loss-to-adaptation-loss relationship is not a general guarantee.

Source and target joint laws, risks, and covariate shift: defines source and target joint laws, input marginals, supplied conditional label kernels, and the additional domination and kernel-equality hypotheses for covariate shift.

Bommasani et al., On the Opportunities and Risks of Foundation Models, §§4.3 and 4.10.1–4.10.3: adaptation modes, the optimization-based interface, in-context learning without parameter optimization, and the report’s theory question.

Stanford CS324, Large Language Models, “Adaptation” lecture notes: probing, fine-tuning, prompting, and downstream task adaptation distinctions.

Facts & Assumptions

Given: The finite spaces, joint laws, and conditional kernels are as stated in the example. The ERM notation is defined only when the sample is nonempty; no attainment or optimizer-convergence claim is assumed. The pretrained parameter vector is held fixed by the in-context route.

[F1]

Foundation models as broad-pretraining and adaptation systems: the report’s 2021 interface question is whether small population pretraining loss implies small minimal adaptation loss; the definition states no such general guarantee. It defines in-context learning as prompt conditioning with pretrained parameters fixed.

[F2]

Source and target joint laws, risks, and covariate shift: covariate shift requires supplied conditional label kernels that agree outside a $μ_{T}$ -null set and $μ_{T} ≪ μ_{S}$ ; the definition does not assert existence of conditional kernels.

[F3]

Foundation models as broad-pretraining and adaptation systems: in-context learning conditions on a prompt while keeping pretrained parameters fixed; parameter-based adaptation optimizes some or all parameters using task information.

Verification

1.1givenF2constructalgebra

For every nonnegative $g : X \times Y \to [0, + \infty]$ , $\int g d P_{S} = g (a, 0) = \int_{X} \int_{Y} g (x, y) K_{S} (x, d y) μ_{S} (d x)$ , and likewise $\int g d P_{T} = g (b, 1) = \int_{X} \int_{Y} g (x, y) K_{T} (x, d y) μ_{T} (d x)$ . Thus the supplied kernels and stated marginals give the two joint laws under the source/target convention.

2.1givenF2step 1.1algebra

For $B = {b}$ , $μ_{S} (B) = 0$ while $μ_{T} (B) = 1$ , so $μ_{T} ≪̸ μ_{S}$ . This violates a required covariate-shift hypothesis in [F2], regardless of whether the two kernels agree.

3.1givenF1F3step 1.1step 2.1∎

By [F3], the in-context route keeps the pretrained parameters fixed and changes the prompt input, whereas a parameter-ERM formulation varies $γ \in Γ$ to minimize the stated sample loss. By [F1], the 2021 report does not supply a general implication from low pretraining loss to low adaptation loss. The finite shift example therefore carries no task-performance guarantee.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Rishi Bommasani et al., On the Opportunities and Risks of Foundation Models, §§4.3 and 4.10.1–4.10.3 (standard reference, not scraped)
Stanford CS324, Large Language Models, Adaptation lecture notes (standard reference, not scraped)