Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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 elementary numerical bound 2<e<32<e<3

Statement

2<e<3.2<e<3.

Facts & Assumptions

Given: e=∑k≥01/ι(k!)e=\sum_{k\ge0}1/\iota(k!) (The real exponential function and the number ee by a power series).

[L1]

The factorial tail bound is A geometric bound for tails of the exponential series.

Proof

technique · direct
1.1

The first three terms give e>1+1+1/2>2e>1+1+1/2>2.

given
1.2

The term at index 22 is 1/21/2. Apply [L1] with x=1,N=2x=1,N=2: the tail from index 33 onward is at most 2/3!=1/32/3!=1/3. Thus the whole tail after index 11 is at most 1/2+1/3=5/6<11/2+1/3=5/6<1, and hence e<2+1=3e<2+1=3.

L1algebra
2.1

Combining the strict bounds gives the claim.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 65 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources