DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

Complement property for a real finite measurement family

Definition

Let $d \geq 1$ , $m \geq 0$ , and let $Φ = (a_{i})_{i \in [m]}$ be a finite family in $R^{d}$ . For $S \subseteq [m]$ , write ${span}_{R} (Φ_{S}) : = {span}_{R} {a_{i} : i \in S},$ with ${span}_{R} (Φ_{\emptyset}) = {0}$ . The family $Φ$ has the complement property if for every $S \subseteq [m]$ , ${span}_{R} (Φ_{S}) = R^{d} or {span}_{R} (Φ_{[m] ∖ S}) = R^{d} .$ The definition applies to labeled families, including repeated or zero vectors.

Definition references

Finite real measurement families and index set

Linear span and the empty span

Finite-combination description and empty-span value

Radu Balan, Pete Casazza, and Dan Edidin, complement property, §2

Philipp Grohs, Sarah Koppensteiner, and Martin Rathmair, complement property, §2.1

Depends on

Used by

The sharp real phase-retrieval count for full-spark measurements Corollary
A two-vector real family with a failed partition and an ambiguous pair Counterexample
Real phase retrieval is equivalent to the complement property Theorem

Dependency tree · two levels

21 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

Radu Balan, Pete Casazza, and Dan Edidin, On signal reconstruction without phase, Applied and Computational Harmonic Analysis 20(3) (2006), §2 (standard reference, not scraped)
Philipp Grohs, Sarah Koppensteiner, and Martin Rathmair, Phase Retrieval: Uniqueness and Stability, arXiv:1901.07911v2, §2.1 (standard reference, not scraped)