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.

Full-spark real measurement families

Definition

Let $d \geq 1$ , $m \geq d$ , and let $Φ = (a_{i})_{i \in [m]}$ be a finite family in $R^{d}$ . The family is full spark if for every list of distinct labels $i_{1}, \dots, i_{d} \in [m]$ , the vectors $a_{i_{1}}, \dots, a_{i_{d}}$ are linearly independent over $R$ .

Definition references

Finite real measurement families and labels

Linear independence over a specified field

Radu Balan, full-spark definition, §3.2

Radu Balan, Pete Casazza, and Dan Edidin, full-spark sufficiency, Corollary 2.6

Depends on

Used by

Dependency tree · two levels

16 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, Frames and Phaseless Reconstruction, full lecture notes (2015), §3.2 (standard reference, not scraped)
Radu Balan, Pete Casazza, and Dan Edidin, On signal reconstruction without phase, Applied and Computational Harmonic Analysis 20(3) (2006), Corollary 2.6 (standard reference, not scraped)