CounterexampleConstruction: Literature-sourcedVerification: 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.

Changing an optimal machine changes finite-string complexity

Statement refuted

Optimal machines give identical numerical complexities.

Counterexample

Given: an optimal machine $U$ and a finite word $z$ with $C_{U} (z) > 1$ .

1.1

Define $V (0 p) = U (p)$ and $V (1) = z$ . The simulation shows $V$ remains optimal.

given

2.1

Then $C_{V} (z) = 1 < C_{U} (z)$ , contradicting identical values and instantiating False: Kolmogorov complexity is an absolute integer.

step 1.1given∎

Depends on

False: Kolmogorov complexity is an absolute integer

Used by

Nothing in the library uses this result yet.

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, §3 (standard reference, not scraped)