Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07
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Effective dimension is the liminf prefix-complexity rate

Statement

For every X∈2ω, dim⁡(X)=lim inf⁡n→∞K(X↾n)/n.

Proof

Given: X∈2ω.

1.1

If an effective s-gale succeeds, its threshold prefix sets yield effective covers of X with s-weight bounded; Kraft allocation Kraft inequality and effective prefix-code allocation assigns infinitely many prefixes descriptions of length at most sn+O(log⁡n).

given
1.2

Conversely, infinitely many prefixes with K(X↾n)≤sn provide a c.e. Kraft-bounded request family; its allocated codes define an effective s+ε-gale succeeding on X.

given
2.1

Taking infima over s in the definition Effective Hausdorff dimension and letting ε↓0 proves the equality; Invariance theorem for prefix complexity changes only O(1)/n.

step 1.1step 1.2∎

Depends on

Used by

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Sources