Alphabeta MathHow 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 signed half-system for modulo gives
Example
For and , the signed half-system is
There are two negative signs, so Gauss's lemma gives .
Facts & Assumptions
Given: The odd prime , multiplier , and the half-system .
Multiplication by a unit modulo an odd prime reduces the half-system to unique signed representatives whose absolute values permute the half-system (Multiplication by with permutes an odd prime's signed half-system up to sign).
If counts the least positive residues of that exceed for , then (Gauss's quadratic-residue lemma).
Verification
The products reduce modulo to the signed representatives .
Their absolute values are , a permutation of as [L1] requires, and exactly two signs are negative.
Fact [L2] gives .
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 57 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
- H. Hackman, Elementary Number Theory, Chapter D, Section D.IV (standard reference, not scraped)