CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

The sharp real phase-retrieval count for full-spark measurements

Statement

Let $d \geq 1$ , $m \geq 0$ , and let $Φ = (a_{i})_{i \in [m]}$ be a finite family in $R^{d}$ , with $[m] = {1, \dots, m}$ and $[0] = \emptyset$ . Write $⟨ u, v ⟩ = \sum_{k < d} u_{k} v_{k}$ . If $m \leq 2 d - 2$ , then there are $x, z \in R^{d}$ with $z \neq x$ and $z \neq - x$ but $⟨ a_{i}, x ⟩^{2} = ⟨ a_{i}, z ⟩^{2}$ for every $i \in [m]$ . If $m \geq 2 d - 1$ and every list of $d$ distinct measurement labels indexes linearly independent vectors, then equality of all these squared measurements implies $z = x$ or $z = - x$ . For every $d \geq 1$ , this count is attained at $m = 2 d - 1$ : for any distinct real numbers $t_{1}, \dots, t_{m}$ , the vectors $a_{i} = (1, t_{i}, t_{i}^{2}, \dots, t_{i}^{d - 1})^{T}$ satisfy the stated independence condition and hence the squared measurements determine the signal up to global sign.

Statement references

Real global-sign phase retrieval

Complement property

Full-spark condition

Complement-property characterization

Vandermonde determinant and moment-curve independence

Radu Balan, Pete Casazza, and Dan Edidin, Corollaries 2.6–2.7

Radu Balan, Corollary 3.3

MIT OpenCourseWare 18.013A, Chapter B §1

Facts & Assumptions

Given: A finite labeled family $Φ = (a_{i})_{i \in [m]}$ in $R^{d}$ , with $d \geq 1$ and $m \geq 0$ .

[F1]

Real phase retrieval is equivalent to the complement property: real intensity measurements are injective modulo global sign exactly when every partition has one side spanning $R^{d}$ .

[F2]

Complement property for a real finite measurement family: the complement property is the stated condition over every subset of labels and its labeled complement.

[F3]

Full-spark real measurement families: full spark means every list of $d$ distinct measurement labels gives a linearly independent list of vectors.

[F4]

Linear map between vector spaces over the same field, Kernel and image of a linear map, Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , Linear subspace of a vector space, $span (S)$ is exactly the set of linear combinations of finite lists of elements of $S$ , and $span (\emptyset) = {0_{V}}$ , Rank-nullity: $\dim_{F} V = nullity T + rank T$ : the coefficient map and its image are defined, subspaces are closed under linear combinations, the span is the smallest subspace containing the columns and equals their finite linear combinations, and rank-nullity computes image dimension from a finite-dimensional domain.

[F5]

The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ , Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis: $R^{k}$ has dimension $k$ , including $k = 0$ , and $R^{d}$ has dimension $d$ .

[F6]

Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent: an independent list of columns has no nonzero coefficient vector in the kernel of its coefficient-to-column map.

[F7]

If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ : a linear subspace of $R^{d}$ has dimension at most $d$ and equals $R^{d}$ exactly when its dimension is $d$ .

[F8]

The Vandermonde determinant for real moment-curve vectors: for distinct real nodes $t_{1}, \dots, t_{d}$ , the $d$ moment-curve columns $(1, t_{j}, \dots, t_{j}^{d - 1})^{T}$ are linearly independent.

[F9]

The reals form a field: the real coordinate and coefficient operations satisfy the field axioms used in the finite linear maps and the Vandermonde construction.

Proof

1.1givenF4F5F9algebra

Let $I \subseteq [m]$ have $q$ labels and enumerate them in increasing order as $i_{0}, \dots, i_{q - 1}$ . Define $T_{I} : R^{q} \to R^{d}$ by $T_{I} (c) = \sum_{j < q} c_{j} a_{i_{j}}$ . This map is linear by distributivity of finite sums. Its image is a subspace: it contains zero, and if $u = T_{I} (c)$ and $v = T_{I} (c^{'})$ , then $α u + β v = T_{I} (α c + β c^{'})$ . For each $j < q$ , the $j$ th standard unit vector $e_{j}$ satisfies $T_{I} (e_{j}) = a_{i_{j}}$ , so the smallest-subspace property gives $span (Φ_{I}) \subseteq im T_{I}$ . Conversely every $T_{I} (c)$ is a finite linear combination of the listed measurement vectors, so [F4] gives $im T_{I} \subseteq span (Φ_{I})$ . Thus the image equals the span, also when $q = 0$ because both are ${0}$ .

2.1step 1.1F4F5

If $q < d$ , rank-nullity and [F5] give $\dim (im T_{I}) \leq q < d$ , so by step 1.1 this span cannot equal $R^{d}$ .

3.1step 2.1F1F2constructalgebra

Suppose $m \leq 2 d - 2$ . Choose $S \subseteq [m]$ with $∣ S ∣ = \min (m, d - 1)$ . Then $∣ S ∣ \leq d - 1$ ; if $m > d - 1$ , the bound on $m$ gives $∣ [m] ∖ S ∣ = m - d + 1 \leq d - 1$ , and if $m \leq d - 1$ the complement is empty. Applying step 2.1 to both sides shows that neither side spans, so the complement property fails by [F2]. Hence phase retrieval fails by [F1].

4.1step 1.1step 3.1givenF1F2F3F4F5F6F7F9choosealgebra

The lower count is established in step 3.1. For the sufficient count, suppose $m \geq 2 d - 1$ and $Φ$ is full spark. For any $S \subseteq [m]$ , the two sides cannot both have fewer than $d$ labels, since that would imply $m \leq 2 d - 2$ . Thus one side contains $d$ distinct labels. By [F3], their vectors are independent. The coefficient map $T : R^{d} \to R^{d}$ for these $d$ columns has trivial kernel by [F6]; by step 1.1 its image is their span, and rank-nullity [F4] and [F5] give that image dimension $d$ . Hence the image, and therefore the span of the chosen vectors, is $R^{d}$ by [F7]. The containing side also spans $R^{d}$ . Since $S$ was arbitrary, the complement property holds, and [F1] gives phase retrieval.

5.1step 3.1step 4.1F3F8constructalgebra∎

Set $m = 2 d - 1$ and choose distinct real nodes $t_{1}, \dots, t_{m}$ . Every $d$ -label subfamily is independent by [F8], so the family is full spark by [F3]. Since $m \geq 2 d - 1$ , step 4.1 proves it performs phase retrieval. For $d = 1$ , the same argument uses one node and the one-entry vector $(1)$ ; for all $d$ , step 3.1 rules out any smaller count.

Depends on

Used by

The complement property is not sufficient for complex phase retrieval Counterexample
A two-dimensional three-vector full-spark real frame Example

Dependency tree · two levels

77 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), Corollaries 2.6–2.7 (standard reference, not scraped)
Radu Balan, Frames and Phaseless Reconstruction, full lecture notes (2015), Corollary 3.3 (standard reference, not scraped)
MIT OpenCourseWare 18.013A, Chapter B §1, Vandermonde determinant (standard reference, not scraped)