Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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.

A Banach mean revisited as a charge

Example

Assume AC. A Banach mean determines a positive charge ν with

ν(N)=1,ν({n})=0(n∈N),

so ν is not countably additive.

Facts & Assumptions

[A1]

AC holds (The Axiom of Choice).

[L1]

Under AC there is a positive normalized shift-invariant mean L on real ℓ∞ (Existence of a shift-invariant mean on bounded sequences).

[L2]

A bounded functional corresponds to the charge ν(A)=L(1A) (The dual of ell-infinity is ba).

Verification

technique · direct

Given: The objects and hypotheses in the Statement.

1.1

Use [A1] exactly through [L1] and define ν by [L2]. Positivity of L [given, A1, L1, L2] makes ν positive, and normalization gives ν(N)=L(1)=1.

A1L1L2
2.1

Shift invariance makes all singleton masses equal, say to c≥0. [given, L1, L2, step 1.1] Finite additivity gives Nc≤ν(N)=1 for every positive integer N, hence c=0. If ν were countably additive, the disjoint singleton decomposition of N would give ν(N)=∑n0=0, a contradiction.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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