LemmaStatement: 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.

Most finite strings are incompressible

Statement

For every description machine $M$ and integers $0 \leq c \leq n$ , at most $2^{n - c} - 1$ strings of length $n$ have $C_{M} (x) < n - c$ . Under the uniform distribution on ${0, 1}^{n}$ , their probability is less than $2^{- c}$ .

Proof

Given: $0 \leq c \leq n$ and a description machine $M$ .

1.1

There are $\sum_{i < n - c} 2^{i} = 2^{n - c} - 1$ programs shorter than $n - c$ , and each has at most one output under Description machines and plain Kolmogorov complexity.

given

2.1

Therefore no more strings can have such a description; division by $2^{n}$ in the uniform finite probability space Finite probability spaces, outcome weights, events, and event probabilities gives the bound.

step 1.1given∎

Depends on

Used by

The incompressibility method Definition
Counting incompressible strings of a fixed length Example

Dependency tree · two levels

8 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, §5 (standard reference, not scraped)