You haven't created information. The information was always in the data. You have just made use of it. By your logic, everything generated with a model is information.

I don't know why you don't just try to use one of the available resources, but here is ChatGPT's answer when I gave it this comment thread:

Thank you for the examples and the discussion. I appreciate the points you’re making about the use of synthetic data. Let's break this down further:

1. Synthetic Data Generation:
2. Example of Relationships: You mentioned augmenting data with relationships, such as identifying "Tony's mother is Fay" and then inferring "Fay's son is Tony." This is essentially generating new data points based on logical extensions of existing information. The relationships themselves are indeed derived from the "organic" data, but the value comes from the structured and comprehensive augmentation.
3. Math Datasets: In the case of creating "reasoning" steps from known pairs of "math questions" and "answers," this process involves a form of synthesis where the model is trained to fill in the intermediary steps. The steps themselves are new data, but they are inferred based on the patterns and rules learned from the original dataset.
4. Creation of Information:
5. The term "creation of information" can be contentious. In your examples, synthetic data can be seen as new information in the sense that new data points or reasoning steps are generated. However, these new pieces of information are fundamentally rooted in the patterns and structures of the original data. Thus, while the specific instances (like reasoning steps) may not have existed before, the underlying principles and relationships were present in the organic data.
6. Limiting Factors:
7. The crux of my argument is that the capability to generate useful synthetic data still relies heavily on the richness and comprehensiveness of the original dataset. If the organic data lacks certain patterns or relationships, the synthetic data generated will also reflect these limitations. Therefore, the ultimate performance ceiling is influenced by the quality and extent of the original data.
8. Practical Implications:
9. Utilizing synthetic data, especially for augmenting datasets and improving model training, is undeniably valuable. The advancements in fine-tuning models with such data show significant improvements. However, it’s important to acknowledge that these improvements are constrained by the foundational organic data’s quality and diversity.

In summary, while synthetic data can indeed add new layers of information and enhance model performance, it does so based on the original organic data’s inherent information. The creation of new data points or reasoning steps exemplifies advanced generalization and structuring rather than the creation of fundamentally new information from nothing. The limitations of the initial data still play a crucial role in defining the extent of these improvements.