ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

One-token posterior on a finite mask schedule

Example

Take $V = {a, b}$ , $L = 1$ , $N = 2$ , $α_{0} = 1$ , $α_{1} = 1 / 2$ , $α_{2} = 0$ , and clean token $X = a$ . At the reverse pair $(s, t) = (1, 2)$ , observing $Z_{2} = m$ has probability one and the true posterior assigns probability $1 / 2$ to $Z_{1} = a$ and $1 / 2$ to $Z_{1} = m$ . At the reverse pair $(s, t) = (0, 1)$ , conditional on the positive-probability event $Z_{1} = m$ , the true posterior removes the mask with probability one.

Example references

Finite masking diffusion chain: gives the one-token survival schedule and absorbing mask.

Posterior of a finite masking chain: gives the exact conditional probabilities for each reverse pair.

Facts & Assumptions

Given: The one-token finite masking chain in the Example. Verify both posterior rows.

[F1]

Finite masking diffusion chain: $r_{j} = α_{j} / α_{j - 1}$ and the mask is absorbing.

[F2]

Posterior of a finite masking chain: at a masked later state, the earlier clean-token probability is $(α_{s} - α_{t}) / (1 - α_{t})$ ; an unmasked token is copied.

Verification

1.1givenF1F2algebra

For $(s, t) = (1, 2)$ , $c_{1, 2} = (1 / 2 - 0) / (1 - 0) = 1 / 2$ , and $1 - c_{1, 2} = (1 - 1 / 2) / (1 - 0) = 1 / 2$ . Since $α_{2} = 0$ , [F1] gives $Z_{2} = m$ with probability one, so [F2] yields the displayed posterior row.

2.1givenF1F2step 1.1algebra∎

For $(s, t) = (0, 1)$ , $c_{0, 1} = (1 - 1 / 2) / (1 - 1 / 2) = 1$ . The conditioning event $Z_{1} = m$ has probability $1 - α_{1} = 1 / 2 > 0$ , so [F2] gives probability one to $Z_{0} = a$ and zero to the mask, with no zero-denominator endpoint used.

Depends on

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

Subham Sekhar Sahoo et al., Simple and Effective Masked Diffusion Language Models, arXiv v1 (standard reference, not scraped)