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.

The analyzed residual network skips at every layer after the first

Example

For depth $H = 3$ , the analyzed residual recurrence is $x^{(1)} = \sqrt{c_{σ} / m} σ (W^{(1)} x)$ , $x^{(2)} = x^{(1)} + \frac{c_{r e s}}{3 \sqrt{m}} σ (W^{(2)} x^{(1)})$ , and $x^{(3)} = x^{(2)} + \frac{c_{r e s}}{3 \sqrt{m}} σ (W^{(3)} x^{(2)})$ . Each layer after the first adds its nonlinear branch to the immediately preceding representation.

Example references

Du et al., residual architecture, equation (3)

Residual recurrence convention

Facts & Assumptions

Given: The residual recurrence, depth $H = 3$ , and $0 < c_{r e s} < 1$ .

[F1]

At every residual layer $h \geq 2$ , the new representation is the sum of the previous representation and a scaled activation branch. Fully connected and residual networks with the Du Gram kernel

Verification

Given: $H = 3$ in the stated recurrence.

1.1F1givenalgebra

Substituting $h = 2$ gives an identity contribution $x^{(1)}$ and nonlinear contribution $\frac{c_{r e s}}{3 \sqrt{m}} σ (W^{(2)} x^{(1)})$ in $x^{(2)}$ .

2.1F1givenalgebra∎

Substituting $h = 3$ gives an identity contribution $x^{(2)}$ and nonlinear contribution $\frac{c_{r e s}}{3 \sqrt{m}} σ (W^{(3)} x^{(2)})$ in $x^{(3)}$ . Thus the recurrence contains the skip at both post-initial layers.

Depends on

Fully connected and residual networks with the Du Gram kernel

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Du et al., residual architecture, equation (3) (standard reference, not scraped)