LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

The phase of a sum lies between upper-half-plane phases

Statement

Let $H = {z \in C : Im z > 0}$ and define $ϕ (z) = \arg (z) / π \in (0, 1)$ . For $z, w \in H$ , the phase of $z + w$ lies between $ϕ (z)$ and $ϕ (w)$ , strictly between them when they are unequal and equal to their common value when they coincide.

Facts & Assumptions

Given: $z, w \in H$ .

[F1]

The stability convention is $H = {z : Im z > 0}$ and $ϕ (z) = \arg (z) / π \in (0, 1)$ (Finite-field stability data for an acyclic quiver).

Proof

Given: The objects and hypotheses in the statement.

1.1F1algebra

Suppose $\arg z > \arg w$ . Then $Im (\overline{w} z) = ∣ w ∣ ∣ z ∣ \sin (\arg z - \arg w) > 0$ . Consequently $Im (\overline{w} (z + w)) > 0$ , which, by [F1], gives $\arg (z + w) > \arg w$ . Also $Im (\overline{(z + w)} z) = Im (\overline{w} z) > 0$ , so [F1] gives $\arg z > \arg (z + w)$ .

2.1F1step 1.1algebra∎

If $\arg w > \arg z$ , interchange $z, w$ in step 1.1. If the arguments are equal, $z, w$ are positive real multiples of the same ray, so their sum has that same argument. These cases prove the stated between-phase rule.

Depends on

Finite-field stability data for an acyclic quiver

Used by

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

Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §4.4 (standard reference, not scraped)
Markus Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Lemma 2.2 (standard reference, not scraped)