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.

Invariance theorem for plain Kolmogorov complexity

Statement

An optimal description machine exists. Consequently, after fixing one $U$ , $C (x) : = C_{U} (x)$ is invariant under replacement of $U$ by another optimal machine up to an additive constant.

Proof

Given: an effective enumeration $(M_{e})$ of description machines.

1.1

Choose a computable prefix-free self-delimiting coding $e \mapsto ⟨ e ⟩$ and define $U (⟨ e ⟩ p) = M_{e} (p)$ ; malformed inputs diverge. Prefix-freeness makes the split into $⟨ e ⟩$ and $p$ unique, so this is partial computable.

given

2.1

For a fixed $M = M_{e}$ , every $M$ -description $p$ becomes a $U$ -description of length $∣ p ∣ + ∣ ⟨ e ⟩ ∣$ , so $U$ is optimal by Optimal description machines. Applying this twice to two optimal machines gives the additive-constant comparison.

step 1.1given∎

Depends on

Optimal description machines

Used by

False: Kolmogorov complexity is an absolute integer False statement
Basic conditional-complexity inequalities Lemma
Symmetry of information up to logarithmic terms Theorem

Dependency tree · two levels

2 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 1 (standard reference, not scraped)