LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Space-bounded universal simulation

Statement

A fixed universal simulator, given a machine description and input of total length $n$ , can simulate a computation using $s (n)$ all-tapes space in $O (n + s (n))$ all-tapes space. In particular, this is $O (s (n))$ when $n = O (s (n))$ .

Facts & Assumptions

Given: a description/input pair of total length $n$ and a simulated run using at most $s (n)$ visited cells.

Proof

technique · direct

1.1

Retain the description and input, and store the active state, tape contents of visited cells, and head positions; update this finite configuration one transition at a time.

givenconstruct

2.1

The description/input portion costs $O (n)$ cells and the current simulated configuration costs $O (s (n))$ cells. Updating one configuration at a time requires no history of the run and no halting clock. Hence the simulator uses $O (n + s (n))$ space, which is $O (s (n))$ under $n = O (s (n))$ .

step 1.1algebra∎

Depends on

Used by

The space-diagonal machine halts Lemma
The deterministic space hierarchy theorem Theorem

Dependency tree · two levels

5 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, §3.2 (standard reference, not scraped)