Alphabeta Math
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.

Kraft inequality and effective prefix-code allocation

Statement

If D⊆{0,1}∗ is prefix-free, then ∑p∈D2−∣p∣≤1. Conversely, a computably enumerable request list (xi,ni) with ∑i2−ni≤1 has an effective prefix-free allocation of codewords of lengths ni.

Proof

Given: a prefix-free set or a Kraft-bounded request list.

1.1

At depth N, a word p excludes exactly 2N−∣p∣ leaves; prefix-freeness makes these excluded sets disjoint. Divide their total by 2N and let N grow to obtain the inequality.

given
2.1

For requests, assign the leftmost currently available node of the requested depth. The Kraft bound says fewer than all leaves are occupied, so induction never gets stuck and keeps the allocated set prefix-free, as required by Prefix-free machines and prefix complexity.

step 1.1construct∎

Depends on

Used by

Dependency tree · two levels

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Sources