ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

A ReLU kink has no classical backpropagated derivative

Example

Let $r (u) = \max {0, u}$ , with $r (0) = 0$ , and consider the one-step scalar objective $J (θ) = r (θ)$ represented by initial state $h_{0} (θ) = θ$ , transition $h_{1} = r (h_{0})$ , and loss $ℓ (θ, h_{1}) = h_{1}$ . The ReLU transition is $C^{1}$ on each of the open regions $u < 0$ and $u > 0$ , with its value specified at $u = 0$ . At $θ = 0$ , $J$ has no classical derivative, and the stated forward-, reverse-, and recurrence-backpropagation procedures return NONDIFFERENTIABLE with no tangent or gradient rather than selecting a subgradient.

Facts & Assumptions

Given: The one-step graph at $θ = 0$ with ReLU transition $r (u) = \max {0, u}$ .

[F1]

Finite straight-line computational programs and real-arithmetic tapes: for a piecewise- $C^{1}$ primitive, a branch-boundary input returns NONDIFFERENTIABLE; the definition does not silently choose a subgradient or generalized derivative.

[F2]

Backpropagation through a finite shared-parameter recurrence: on a piecewise- $C^{1}$ map the recurrence procedure uses a derivative only on a unique strict active branch and returns NONDIFFERENTIABLE at a boundary.

[F3]

Forward-mode automatic differentiation on a finite computational graph: a forward-mode input on a branch boundary returns NONDIFFERENTIABLE and no tangent.

[F4]

Reverse-mode automatic differentiation for a scalar graph output: a reverse-mode input on a branch boundary returns NONDIFFERENTIABLE and no gradient.

Verification

1.1givenchoosealgebra

Since $J (0) = 0$ , for every $h < 0$ the difference quotient $(J (h) - J (0)) / h$ equals $0$ , while for every $h > 0$ it equals $1$ . The two one-sided limits differ, so $J$ is not differentiable at $0$ .

1.2F1given

The executed ReLU primitive receives $h_{0} = 0$ , the boundary between its $u < 0$ and $u > 0$ branches. By [F1] this input has no selected strict branch.

2.1F2F3step 1.2

At that boundary, the recurrence procedure in [F2] and the forward procedure in [F3] both return NONDIFFERENTIABLE and no derivative value.

2.2F4step 1.2

The reverse procedure in [F4] also returns NONDIFFERENTIABLE and no gradient at this boundary.

3.1step 1.1step 2.1step 2.2∎

This boundary status agrees with step 1.1: there is no classical derivative for these procedures to propagate, and their specified output does not invent one.

Depends on

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