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.

A PReLU slope changes both initialization scales

Example

For the PReLU source scale calculation, take negative-side slope $a = 1 / 2$ , fan-in $n = 5$ , and fan-out $\hat{n} = 10$ . Compare the exact forward recurrence coefficient $(1 + a^{2}) n σ^{2} / 2$ with the atomless conditional backward coefficient $(1 + a^{2}) \hat{n} σ^{2} / 2$ . Compute the two unit-coefficient weight variances and determine whether they agree. The backward coefficient has the corollary's independence assumptions; it is not asserted to describe an arbitrary actual-network gradient, and a zero atom with a specified kink gate changes its factor.

Facts & Assumptions

Given: The dimensions, slope, and forward and atomless conditional backward coefficients stated in the Example.

[F1]

The PReLU forward and backward scale factors are $(1 + a^{2}) / 2$ times fan-in or fan-out and the corresponding weight variances are $2 / ((1 + a^{2}) n)$ or $2 / ((1 + a^{2}) \hat{n})$ . (PReLU fan-in and fan-out variance scales)

Verification

1.1givenF1algebra

Since $1 + a^{2} = 1 + 1 / 4 = 5 / 4$ , the forward-preserving variance is $2 / ((5 / 4) 5) = 8 / 25$ . The forward recurrence factor is $(5 / 8) 5 (8 / 25) = 1$ .

1.2givenF1algebra

The backward-preserving variance is $2 / ((5 / 4) 10) = 4 / 25$ . The backward recurrence factor is $(5 / 8) 10 (4 / 25) = 1$ .

2.1step 1.1step 1.2algebra∎

The two variances $8 / 25$ and $4 / 25$ differ because the fan-in and fan-out differ, so no shared weight variance makes both recurrence coefficients equal to one.

Depends on

PReLU fan-in and fan-out variance scales

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, Delving Deep into Rectifiers, §2.2 (standard reference, not scraped)
Simon J. D. Prince, Understanding Deep Learning, §7.5.1–7.5.3 (standard reference, not scraped)
Johns Hopkins University, Handout #8: Neural Network Training and Tokenization, §3 (standard reference, not scraped)