ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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 concrete geometric majorant

Example

For $f (x, y) = x^{2} + 1 / (1 - x - y)$ at the origin, $r = 1 / 2$ and $M = 2$ give $f ≪ 2 / (1 - 2 x - 2 y)$ .

Facts & Assumptions

Given: The explicit two-variable germ and proposed geometric majorant in the Example; their coefficient comparison is to be calculated.

[F1]

Compare ordinary coefficients at each multi-index. (Coefficientwise majorisation).

Verification

1.1

For $∣ x ∣ + ∣ y ∣ < 1$ , the geometric expansion and binomial formula give $[x^{a} y^{b}] (1 - x - y)^{- 1} = (\binom{a + b}{a})$ . Hence $[x^{a} y^{b}] f = (\binom{a + b}{a}) + 1_{(a, b) = (2, 0)}$ . The proposed majorant has coefficient $2^{a + b + 1} (\binom{a + b}{a})$ .

givenalgebra

2.1

Outside $(a, b) = (2, 0)$ the comparison follows from $2^{a + b + 1} \geq 2 > 1$ . At $(2, 0)$ the coefficients are 2 and 8, respectively. In particular the constant coefficients are 1 and 2. Every coefficient is nonnegative, so these inequalities give the claimed majorisation by F1.

step 1.1F1algebra∎

Source notes

Gantumur, §3 Exercise 14, printed p. 8, supplies the geometric-majorant construction; this concrete polynomial perturbation and calculation are adapted locally.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Gantumur, Math 580 Lecture Notes 2: The Cauchy-Kovalevskaya Theorem (standard reference, not scraped)