Alphabeta Math
RemarkRemark: AI-generatedProof: Not applicableaudited 2026-08-01
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 exponential roadmap and its circularity hazards

Remarks

The equivalence theorem The power-series, product-limit, IVP, functional-equation, and Picard definitions agree is an identification after independent proofs, not a list of mutually supporting definitions. The power series first defines exp⁡; the addition law is proved by an absolutely convergent Cauchy product; differentiation is then termwise. Only after those steps are the differential-equation and functional-equation characterizations invoked.

The bijection onto the positive reals (The exponential is a continuous bijection from R onto (0,∞)), polynomial domination (The exponential dominates every fixed nonnegative integer power at +∞), and irrationality of e (The number e is irrational) use no logarithm. Each result is therefore independent of an inverse function and of real exponentiation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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