LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge 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.

Finite masked-diffusion joint model is normalized

Statement

Let $V$ be finite and nonempty, $m \notin V$ , $L, N \geq 1$ , and $S = (V \cup {m})^{L}$ . Let $P_{j}$ be probability kernels on $S$ , put prior mass one at $z_{N} = m^{L}$ , and let $D (x ∣ z_{0}) = 1_{{x = z_{0}}}$ when $z_{0} \in V^{L}$ and $D (x ∣ z_{0}) = ∣ V ∣^{- L}$ otherwise. Then the finite joint mass $p (x, z_{0}, \dots, z_{N}) = 1_{{z_{N} = m^{L}}} (\prod_{j = 1}^{N} P_{j} (z_{j}, z_{j - 1})) D (x ∣ z_{0})$ is nonnegative and sums to one over $x \in V^{L}$ and $(z_{0}, \dots, z_{N}) \in S^{N + 1}$ .

Statement references

Finite masked-diffusion generative model: defines the terminal point mass, reverse-kernel product, normalized decoder, and joint mass.

Facts & Assumptions

Given: The finite generative joint mass and the sampler's probability kernels. Prove the total joint mass is one.

[F1]

Finite masked-diffusion generative model: $p (x, z_{0}, \dots, z_{N}) = 1_{{z_{N} = m^{L}}} (\prod_{j = 1}^{N} P_{j} (z_{j}, z_{j - 1})) D (x ∣ z_{0})$ .

[F2]

Finite-grid masked sampler defines a normalized law: each reverse-kernel row $P_{j} (z, \cdot)$ is a probability vector.

Proof

1.1F1givenalgebra

For fixed $z_{0} \in S$ , if $z_{0} \in V^{L}$ then $D (\cdot ∣ z_{0})$ is a point mass; otherwise it assigns $∣ V ∣^{- L}$ to each of the $∣ V ∣^{L}$ clean outputs. In both cases $\sum_{x \in V^{L}} D (x ∣ z_{0}) = 1$ .

2.1F1F2step 1.1algebra

Sum [F1] first over $x$ , using step 1.1. Then sum over $z_{0}$ in $P_{1} (z_{1}, z_{0})$ , over $z_{1}$ in $P_{2} (z_{2}, z_{1})$ , and successively over $z_{N - 1}$ in $P_{N} (z_{N}, z_{N - 1})$ . Each inner sum is one by [F2], leaving $\sum_{z_{N}} 1_{{z_{N} = m^{L}}} = 1$ . All sums are finite, so this iterated summation equals the total joint mass.

3.1F1step 2.1algebra∎

Every factor in [F1] is nonnegative, and step 2.1 shows the finite total mass is one. Therefore the displayed function is a probability law.

Depends on

Used by

Cited to discharge well-definedness by Finite masked-diffusion generative model.

Dependency tree · two levels

5 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)