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 scalar state-space layer has matching recurrent and convolutional outputs

Example

Take the scalar state-space model $A = 0$ , $B = C = 1$ , direct term $D = 0$ , step size $Δ = 1$ , and initial state $x_{- 1} = 0$ . For any integer $L \geq 1$ and any length- $L$ scalar input sequence $(u_{0}, \dots, u_{L - 1})$ , the bilinear matrices are $\overset{ˉ}{A} = 1$ and $\overset{ˉ}{B} = 1$ , so the recurrence gives $x_{t} = x_{t - 1} + u_{t}$ and $y_{t} = \sum_{s = 0}^{t} u_{s}$ for $0 \leq t < L$ . The length- $L$ impulse kernel is $(1, \dots, 1)$ , and its causal convolution $\sum_{j = 0}^{t} K_{j} u_{t - j}$ gives the same outputs for $0 \leq t < L$ .

Example references

DPLR structured state-space parameterization

S4 state-space recurrence and convolution procedure

Gu, Goel, and Ré, “Efficiently Modeling Long Sequences with Structured State Spaces,” §§2.3–2.4

Facts & Assumptions

Given: The scalar parameters and zero initial state in the Example, with a length- $L$ input sequence.

[F1]

DPLR structured state-space parameterization: when $I - Δ A / 2$ is invertible, $\overset{ˉ}{A} = (I - Δ A / 2)^{- 1} (I + Δ A / 2)$ , $\overset{ˉ}{B} = (I - Δ A / 2)^{- 1} Δ B$ , and the zero-state kernel has entries $C^{*} {\overset{ˉ}{A}}^{j} \overset{ˉ}{B}$ .

[F2]

S4 state-space recurrence and convolution procedure: the recurrent mode solves $(I - Δ A / 2) x_{t} = (I + Δ A / 2) x_{t - 1} + Δ B u_{t}$ and returns $C^{*} x_{t}$ .

Proof

1.1givenF1algebra

Here $I - Δ A / 2 = 1$ and $I + Δ A / 2 = 1$ , so [F1] gives $\overset{ˉ}{A} = 1$ and $\overset{ˉ}{B} = 1$ .

2.1givenF2step 1.1induction

By [F2] and step 1.1, $x_{t} = x_{t - 1} + u_{t}$ and $y_{t} = x_{t}$ . The base value is $x_{- 1} = 0$ ; if $x_{t - 1} = \sum_{s = 0}^{t - 1} u_{s}$ , then $x_{t} = x_{t - 1} + u_{t} = \sum_{s = 0}^{t} u_{s}$ . Induction proves $y_{t} = \sum_{s = 0}^{t} u_{s}$ .

3.1F1step 1.1step 2.1algebra∎

By [F1] and step 1.1, every kernel coefficient is $C^{*} {\overset{ˉ}{A}}^{j} \overset{ˉ}{B} = 1$ . Therefore its causal convolution at time $t$ is $\sum_{j = 0}^{t} u_{t - j} = \sum_{s = 0}^{t} u_{s}$ , which equals the recurrent output in step 2.1 after substituting $s = t - j$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Gu, Goel, and Ré, Efficiently Modeling Long Sequences with Structured State Spaces, §§2.3–2.4 (standard reference, not scraped)