Formal note · Reviewed Basis output
Formal Note Example: Unbiasedness of the Sample Mean Under Integrability
A compact Basis manuscript showing that the system can produce a tight proof note with clear assumptions, a clean theorem statement, and an explicit failure case.
This example shows that Basis can stay concise. It is a short formal note, not a padded essay, and it still preserves a real artifact trail and an inspectable manuscript output.
Document facts
- Document
- Formal note
- Length
- 3 pages
- Front matter
- Cover · abstract · proof
- Public edition
- Full viewer with cleaned front matter
- Objective
- Give a rigorous proof that the sample mean is an unbiased estimator under finite expectation, make the key assumption explicit, and include a concrete failure case when the expectation is not defined.
- Outputs
- PDF · TeX manuscript
- Viewer
- Full public edition of the short note with a cleaned title page and no placeholder author text.
Run summary
What this run had to deliver.
- State the minimal assumption that makes the expectation well-defined.
- Prove the unbiasedness identity cleanly with linearity of expectation.
- Show why the claim breaks down for heavy-tailed cases such as the standard Cauchy law.
What persisted after the run
- Public manuscript PDF
- LaTeX source manuscript
- Abstract and main theorem note
- Curated public viewer edition
Inside the output
What the document says.
- Key assumption
- Integrability makes the expectation well-defined and keeps the proof honest.
- Proof move
- Uses linearity of expectation on a finite sum rather than hand-wavy intuition.
- Failure mode
- Explains why the standard Cauchy law makes unbiasedness ill-posed.
What the note covers
The note isolates integrability as the key assumption, proves the unbiasedness identity directly, and makes clear that independence is not required for the expectation calculation itself.
It then closes with a concrete failure case: for the standard Cauchy distribution, the mean is undefined, so the usual unbiasedness statement is not mathematically well-posed.
Actual PDF
Read the output directly.
Full public edition of the short note with a cleaned title page and no placeholder author text.
Human review
What was checked before this became public.
- The public edition keeps the full short manuscript rather than trimming away the title page.
- Placeholder author text was replaced and the document title was rewritten from the raw prompt into a readable note title.
- This example stays narrow and formal, with the scope made explicit on the page.
Where the example comes from
- Prepared from an audited Basis manuscript reviewed on March 8, 2026.
- The public edition preserves the original document shape while removing placeholder front matter.
Use this document as the bar for your own run.
Start with a concrete question, explicit constraints, and the artifact package you expect to review at the end.
Public beta · email verification required · More examples