Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

Distinct unit summands need not attain the sum bound

Statement refuted

The assertion “for all n>0 and all unit complex numbers z1,…,zn, ∣∑izi∣=n” is false.

Facts & Assumptions

Given: The universal equality assertion for unit complex summands.

[F1]

The upper bound is attained exactly when all unit summands agree (Equality in the unit-complex finite-sum bound).

Counterexample

technique · direct
1.1

Take n=2, z1=1 and z2=−1 in C. Their moduli are 12+02=1 and (−1)2+02=1, so both satisfy the unit-modulus hypothesis and n>0.

givenalgebra
2.1

Their sum is 1+(−1)=0, whose modulus is zero. Consequently ∣z1+z2∣=0<2=n, so this instance fails the asserted equality and refutes its universal quantifier. The summands are distinct, in agreement with F1.

step 1.1F1algebra∎

Sources

Etingof et al., Lemma 5.4.5 proof, p. 101, motivates the strictness interface. The pair 1,−1 is the locally specified counterexample.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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