ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

Uniform density gives exact midpoint compander distortion

Example

Take the uniform source density $f (x) = 1$ and point density $g (x) = 1$ on $[0, 1]$ . For every integer $N \geq 1$ , the compander uses $N$ equal cells and their midpoints as reproduction values.

Facts & Assumptions

Given: The interval $[0, 1]$ , densities $f = g = 1$ , and the compander definition of Positive C1 compact one-dimensional compander.

[F1]

Positive C1 compact one-dimensional compander: the cells are bounded by $ϕ (j / N)$ and the reproduction values are $ϕ ((j - \frac{1}{2}) / N)$ , where $ϕ$ is the inverse cumulative coordinate.

[F2]

Exact high-rate limit for a positive compact one-dimensional compander: the limit coefficient is $\frac{1}{12} \int_{0}^{1} f (x) / g (x)^{2} d x$ under the positive normalized $C^{1}$ hypotheses.

Verification

1.1givenF1algebra

Here $G_{g} (x) = x$ and $ϕ (u) = u$ , so cell $j$ is $[(j - 1) / N, j / N)$ (with the final endpoint included) and its reproduction value is $(j - \frac{1}{2}) / N$ by [F1].

2.1givenstep 1.1algebra

Put $h = 1 / N$ . On each cell, translation to its midpoint gives $\int_{(j - 1) / N}^{j / N} (x - (j - \frac{1}{2}) / N)^{2} d x = \int_{- h / 2}^{h / 2} t^{2} d t = h^{3} / 12$ . Summing the $N$ equal cell integrals yields $D_{N} (1, 1) = 1 / (12 N^{2})$ , including $N = 1$ .

3.1F2step 2.1algebra∎

Therefore $N^{2} D_{N} (1, 1) = 1 / 12$ for every $N$ , while [F2] gives the same limit coefficient $(1 / 12) \int_{0}^{1} 1 d x = 1 / 12$ .

Depends on

Used by

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Dependency tree · two levels

12 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

Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)
Robert Gallager, 6.450 Principles of Digital Communications I, Chapter 3: Quantization (standard reference, not scraped)
Robert M. Gray and David L. Neuhoff, Quantization, IEEE Transactions on Information Theory 44(6) (1998) (standard reference, not scraped)