ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Hardwiring a length-indexed advice string

Statement

Let the advised machine $M$ receive one advice bit $a_{n}$ and output $x_{1} \oplus a_{n}$ on every nonempty input $x$ . At length $2$ , take $a_{2} = 1$ . Hardwiring this advice gives the circuit $C_{2} (x_{1}, x_{2}) = \neg x_{1}$ , and encoding that circuit as advice recovers the same computation.

Facts & Assumptions

Given: the displayed machine, $a_{2} = 1$ , and the input $x = 00$ .

[L1]

The P/poly/advice theorem supplies both circuit encoding and hardwiring directions. by P/poly equals polynomial time with polynomial advice.

Verification

technique · direct

1.1

Replacing the advice wire by the constant $1$ turns the XOR gate into a NOT gate on $x_{1}$ ; the unused input $x_{2}$ remains an input wire. On $00$ , both $M (00, 1)$ and $C_{2} (00)$ output $1$ .

givenconstruct

2.1

Conversely encode $C_{2}$ by the topological list ``inputs $x_{1}, x_{2}$ ; NOT $x_{1}$ ; designate the NOT gate as output.'' A fixed evaluator given this list as advice returns $1$ on $00$ and, for every two-bit $x$ , returns $\neg x_{1}$ ; the declared input $x_{2}$ is simply unused.

L1step 1.1construct

3.1

The same finite instance therefore exhibits both hardwiring and circuit- encoding directions of [L1].

L1step 1.1step 2.1∎

Depends on

P/poly equals polynomial time with polynomial advice

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Arora and Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
Lance Fortnow, Counting Complexity (standard reference, not scraped)