Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 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.

A two-branch computation and its parsimonious tableau formula

Statement

On empty input, let N make one nondeterministic choice b∈{0,1}, record b in its state, and accept. Its two accepting paths give two legal padded tableaux, and the exact Cook--Levin encoding has exactly two satisfying assignments.

Facts & Assumptions

Given: the displayed one-choice machine N.

[L1]

The Cook--Levin construction can be made parsimonious. by The Cook--Levin construction can be made parsimonious.

Verification

technique · direct
1.1

The paths are start $\xrightarrow{0}$ accept-with-tag-$0$'' and start →1 accept-with-tag-1''. Padding repeats the final configuration, so these give two and only two legal accepting tableaux.

givenalgebra
2.1

In the exact encoding of [L1], the tableau variables are uniquely fixed by one of these tableaux and every auxiliary variable is constrained by a biconditional with the subformula it names. Equivalently, after eliminating those uniquely determined variables, the formula is the tautology b∨¬b with declared variable b. Its two assignments correspond bijectively to the two paths.

L1step 1.1
3.1

Therefore the source accepting-path count and the formula's satisfying- assignment count are both exactly 2.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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