TheoremStatement: Literature-sourcedProof: AI-adaptedjudge 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.

Kolmogorov complexity is not computable

Statement

For a fixed optimal description machine $U$ , the function $x \mapsto C_{U} (x)$ is not computable.

Proof

Given: an optimal machine $U$ .

1.1

Suppose for contradiction that $C_{U}$ is computable. For each $n$ , search lexicographically among strings of length $n$ for the first $x_{n}$ with $C_{U} (x_{n}) \geq n$ ; such an $x_{n}$ exists because fewer than $2^{n}$ programs have length below $n$ .

assume-contragiven

2.1

The search makes $n \mapsto x_{n}$ computable, so a fixed description machine maps a self-delimiting encoding of $n$ to $x_{n}$ using $O (\log n)$ bits. Optimality Optimal description machines transfers this to $C_{U} (x_{n}) \leq O (\log n)$ , contradicting $C_{U} (x_{n}) \geq n$ for large $n$ .

step 1.1discharge-contradiction∎

Depends on

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

Shen, Theorem 3 (standard reference, not scraped)