Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge 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.

The ordinary Cantor set at its critical exponent

Example

Assume the Axiom of Countable Choice. Let C be the middle-thirds Cantor set and s=log⁡2/log⁡3. Its level-m basic cover has s-cost exactly one. In the small-scale limit,

Hs(C)=1,dim⁡HC=s,H1(C)=0.

Facts & Assumptions

Given: The objects, conventions, and hypotheses in the statement above.

[F1]

Under the standing Countable Choice hypothesis, the middle-thirds Cantor set has critical measure one and dimension s=log⁡2/log⁡3. The Cantor set has dimension log 2 / log 3 and critical measure one

[F2]

Finite measure at exponent s gives zero measure at every larger exponent. Increasing the exponent past finite measure gives zero

Verification

1.1

There are 2m basic intervals of diameter 3−m, and 3s=2. Thus their total cost is 2m3−ms=1. The sharp Cantor theorem provides the matching lower bound, so the infimum cannot fall below one in the limit.

F1
2.1

Since 0<s<1 and the critical measure equals the finite value one, exponent comparison gives H1(C)=0. At level zero the cover is [0,1] and also costs one; shrinking scales require arbitrarily large levels.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources