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.
has no finite Cauchy principal value at zero
Example
Symmetry does not rescue the nonnegative singularity : does not exist as a finite real number.
Facts & Assumptions
Given: The function away from zero.
Principal value uses the sum of the two symmetric truncations (Cauchy principal values at a finite singularity and on the real line).
The rational-power formula evaluates each proper truncation (Truncated integrals of rational powers).
Verification
For , symmetry and [L2] give [L2]
This tends to as , not to a finite real. Therefore the principal value in [L1] diverges.
Depends on
- Cauchy principal values at a finite singularity and on the real line
- Truncated integrals of rational powers
- The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$
- Inverses of positives are positive, and reciprocation reverses order
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: 90 results over 22 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
- William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)