LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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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.

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Finite masking-chain transitions define a probability law

Statement

For a finite absorbing-mask chain with finite clean vocabulary, mask symbol, sequence length, strict grid, and survival schedule as specified above, each sequence-state transition row is a probability vector, and the displayed joint path mass $μ (x) 1_{{z_{0} = x}} \prod_{j = 1}^{N} \prod_{i = 1}^{L} a_{j} (z_{j - 1, i}, z_{j, i})$ sums to one over $x \in V^{L}$ and $z_{0}, \dots, z_{N} \in S$ . Thus it defines a probability law on the finite clean-sequence and path space.

Statement references

Finite masking diffusion chain: fixes $V, m, L, N, α, μ$ , the one-coordinate rows, product sequence kernels, and the path-mass formula.

Facts & Assumptions

Given: A finite nonempty $V$ , $m \notin V$ , $L, N \geq 1$ , the strict survival schedule, and a probability vector $μ$ on $V^{L}$ as in the defining item. Prove each row and the joint path mass are normalized.

[F1]

Finite masking diffusion chain: $r_{j} = α_{j} / α_{j - 1}$ and $a_{j} (v, v) = r_{j}$ , $a_{j} (v, m) = 1 - r_{j}$ , $a_{j} (m, m) = 1$ , with all other entries zero; $Q_{j} (z, z^{'}) = \prod_{i} a_{j} (z_{i}, z_{i}^{'})$ .

[F2]

Measure kernel and probability kernel: a probability kernel has probability-measure rows and measurable evaluation functions.

Proof

1.1givenF1F2algebra

The strict schedule gives $0 \leq r_{j} \leq 1$ . For a clean token $v$ , its one-coordinate row has total mass $r_{j} + (1 - r_{j}) = 1$ ; for $m$ , the row mass is $1$ . Therefore every product row satisfies $\sum_{z^{'} \in S} Q_{j} (z, z^{'}) = \prod_{i = 1}^{L} \sum_{w \in V \cup {m}} a_{j} (z_{i}, w) = 1$ . Its entries are nonnegative. Since $S$ is finite with its power-set sigma-algebra, [F2] makes $Q_{j}$ a probability kernel.

2.1F1step 1.1algebra

For fixed $x$ , sum the conditional path mass backward: summing first over $z_{N}$ uses the row sum of $Q_{N}$ from step 1.1 and removes that factor; repeating for $z_{N - 1}, \dots, z_{1}$ leaves $1_{{z_{0} = x}}$ . Summing over $z_{0}$ then gives $1$ , so each conditional path mass sums to one.

3.1givenF1step 2.1algebra∎

Multiplying these conditional path laws by $μ (x)$ and summing over the finite set $V^{L}$ gives total mass $\sum_{x} μ (x) = 1$ . All masses are nonnegative and the path space is finite, so the joint mass is a probability vector.

Depends on

Used by

Cited to discharge well-definedness by Finite masking diffusion chain.

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)
David A. Levin and Yuval Peres, Markov Chains and Mixing Times, second edition (standard reference, not scraped)