Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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.

Two to the square root of two from rational suprema and from exp(sqrt(2) log 2)

Example

Let s=2s=\sqrt2. The two constructions give the same number: 2[s]=sup⁡{2q:q∈Q, q<s}=exp⁡(slog⁡2)=lim⁡qn→s, qn∈Q2qn.2^{[s]}=\sup\{2^q:q\in\mathbb Q,\ q<s\}=\exp(s\log2)=\lim_{q_n\to s,\ q_n\in\mathbb Q}2^{q_n}.

Facts & Assumptions

Given: The nonnegative square root s=2s=\sqrt2.

[L1]

The rational-supremum and exponential constructions agree (The rational-supremum construction of real powers agrees with the exponential construction).

[L2]

Rational approximations to an exponent converge to its real power (Every rational approximation to a real exponent gives the same limiting real power).

[L3]

2\sqrt2 exists and is positive (Square roots exist: a unique a≥0\sqrt{a} \ge 0 with (a)2=a(\sqrt{a})^2 = a; the positives are {x2:x≠0}\{x^2 : x \neq 0\}).

Verification

technique · direct
1.1

By [L3], ss is a real exponent, so [L1] gives 2[s]=exp⁡(slog⁡2)2^{[s]}=\exp(s\log2).

L1L3
1.2

For every rational sequence qn→sq_n\to s, [L2] gives 2qn→2s=exp⁡(slog⁡2)2^{q_n}\to2^s=\exp(s\log2).

L2
2.1

Thus the supremum, exponential, and rational-limit descriptions agree.

step 1.1step 1.2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 12 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