ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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.

Poisson modification flattens a radial quadratic on the chosen inner disc

Example

Fix $0 < r < 2$ and consider the subharmonic function $u (z) = ∣ z ∣^{2}$ on the disc $∣ z ∣ < 2$ . Its Poisson modification on the inner disc $D (0, r)$ is $P_{D (0, r)} u (z) = {\begin{cases} r^{2}, & ∣ z ∣ < r, \\ ∣ z ∣^{2}, & r \leq ∣ z ∣ < 2. \end{cases}$

Facts & Assumptions

Given: The function $u (z) = ∣ z ∣^{2}$ on $∣ z ∣ < 2$ and an inner radius $0 < r < 2$ .

[L1]

The Poisson modification is harmonic on the chosen inner disc, equals the original function outside it, and majorizes the original function (Poisson modification is subharmonic and majorizes the original function, Poisson modification on a compactly contained disc).

[L2]

A bounded-domain harmonic extension of fixed continuous boundary data is unique (The bounded plane Dirichlet problem has at most one continuous harmonic solution).

Verification

technique · direct

1.1

On the circle $∣ z ∣ = r$ , the boundary values of $u$ are the constant $r^{2}$ . Hence the constant function $h (z) = r^{2}$ is harmonic on $D (0, r)$ and has exactly the boundary values required by the Poisson modification.

givenalgebra

2.1

By [L1], the modified function agrees with $u$ outside $D (0, r)$ and is harmonic inside. Since step 1.1 gives a harmonic candidate with the correct boundary data on the inner disc, [L2] forces the inside harmonic piece to be exactly $h (z) = r^{2}$ . This gives the displayed formula.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, 2nd ed. (standard reference, not scraped)