ExampleConstruction: AI-generatedVerification: 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.

Two-step scalar Gaussian forward chain and posterior

Example

Assume AC and take a one-dimensional chain with $T = 2$ , $β_{1} = \frac{1}{4}$ , $β_{2} = \frac{1}{2}$ , and fixed data value $x_{0} \in R$ . Then $α_{1} = \frac{3}{4}$ , $α_{2} = \frac{1}{2}$ , ${\overset{ˉ}{α}}_{1} = \frac{3}{4}$ , and ${\overset{ˉ}{α}}_{2} = \frac{3}{8}$ . The forward marginal at step two is $q (x_{2} ∣ x_{0}) = N (\sqrt{3 / 8} x_{0}, 5 / 8)$ , and the posterior given $x_{2}, x_{0}$ is $q (x_{1} ∣ x_{2}, x_{0}) = N ⁣ (\frac{2 \sqrt{3}}{5} x_{0} + \frac{\sqrt{2}}{5} x_{2}, \frac{1}{5}) .$ At step one, $q (x_{0} ∣ x_{1}, x_{0})$ assigns probability one to every Borel set containing $x_{0}$ and zero to every Borel set not containing $x_{0}$ , so it is not a Lebesgue density.

Example references

Scalar specialization of the forward marginal and posterior formulas: Ho, Jain, and Abbeel, Eqs. (4), (6), and (7).

Independent Gaussian-kernel derivation: Prince, Chapter 18, “The diffusion kernel” and “The tractable conditional”.

Facts & Assumptions

Given: AC, $T = 2$ , $β_{1} = \frac{1}{4}$ , $β_{2} = \frac{1}{2}$ , and fixed $x_{0} \in R$ .

[A1]

AC is inherited from the Gaussian law and posterior interfaces. The Axiom of Choice.

[F1]

The schedule definitions are $α_{t} = 1 - β_{t}$ and ${\overset{ˉ}{α}}_{t} = \prod_{s = 1}^{t} α_{s}$ . Finite Gaussian forward chain and Gaussian reverse model.

[F2]

The forward marginal and conditioned posterior have the exact formulas in the theorem. Closed Gaussian forward marginal and conditioned posterior.

[F3]

A scalar $N (m, v)$ is a normal law with mean $m$ and variance $v$ . Multivariate normal law, including singular covariance.

Verification

technique · substitute the schedule values into the exact formulas

Given: The hypotheses in the Example.

1.1A1F1givenalgebra

The schedule gives $α_{1} = 1 - \frac{1}{4} = \frac{3}{4}$ , $α_{2} = 1 - \frac{1}{2} = \frac{1}{2}$ , ${\overset{ˉ}{α}}_{1} = \frac{3}{4}$ , and ${\overset{ˉ}{α}}_{2} = \frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8}$ .

2.1F2F3step 1.1algebra

By [F2] and the scalar normal convention [F3], $q (x_{2} ∣ x_{0})$ has mean $\sqrt{3 / 8} x_{0}$ and variance $1 - \frac{3}{8} = \frac{5}{8}$ .

2.2A1F1F2F3step 1.1algebra

The posterior variance is ${\tilde{β}}_{2} = \frac{(1 / 2) (1 - 3 / 4)}{1 - 3 / 8} = \frac{1 / 8}{5 / 8} = \frac{1}{5}$ . The $x_{0}$ coefficient is $\frac{\sqrt{3 / 4} (1 / 2)}{5 / 8} = \frac{2 \sqrt{3}}{5}$ , and the $x_{2}$ coefficient is $\frac{\sqrt{1 / 2} (1 / 4)}{5 / 8} = \frac{\sqrt{2}}{5}$ . Substitution into [F2] gives the displayed posterior.

3.1givenstep 2.2algebra∎

Since the final conditioning includes the fixed value $x_{0}$ , the conditional law of $x_{0}$ itself is concentrated at that value. It is therefore a point mass and has no Lebesgue density for the reconstruction term.

Source notes

Gaussian marginal and posterior theorem: Closed Gaussian forward marginal and conditioned posterior.

Depends on

Used by

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Sources

Jonathan Ho, Ajay Jain, and Pieter Abbeel, Denoising Diffusion Probabilistic Models (2020) (standard reference, not scraped)
Simon J. D. Prince, Understanding Deep Learning, Chapter 18: Diffusion models (2026), pp. 363–387 (standard reference, not scraped)