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For a complex Gaussian matrix with N=4M-5 rows the phase retrieval map it generates fails to be injective with positive probability.

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2026-06-26 22:18 UTC pith:US5SG2PW

load-bearing objection This note claims to close the missing half of Vinzant's conjecture by showing positive probability of non-injectivity for complex Gaussian matrices at N=4M-5, but the AI-generated proof is not supplied for inspection. the 2 major comments →

arxiv 2606.17922 v1 pith:US5SG2PW submitted 2026-06-16 math.FA cs.ITmath.ITmath.PR

On Injectivity of Phase Retrieval

Zhangsong Li This is my paper
classification math.FA cs.ITmath.ITmath.PR
keywords phase retrievalinjectivityGaussian random matricescomplex vectorsmagnitude measurementsrandom sensing matricessignal recovery
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that an N by M matrix A with i.i.d. complex Gaussian entries where N equals 4M minus 5 produces a phase retrieval map that is non-injective with positive probability. This means there exist distinct vectors that yield identical sets of absolute inner products with the rows of A. The result identifies a concrete dimension at which random measurements no longer guarantee that magnitude data determine the signal uniquely up to phase. A reader would care because the finding marks a threshold below which almost-sure injectivity cannot hold for this ensemble of matrices.

Core claim

If A belongs to the complex N by M matrices with N equal to 4M minus 5 and has i.i.d. standard complex Gaussian entries, then the probability that the phase retrieval map generated by A is not injective is positive. This establishes part (1) of the relevant conjecture on injectivity dimensions.

What carries the argument

The phase retrieval map generated by A, which sends each vector x to the tuple of absolute values of its inner products with the rows of A.

Load-bearing premise

The matrix entries are drawn independently according to the standard complex Gaussian distribution.

What would settle it

An explicit calculation for any fixed small M showing that every matrix with N=4M-5 rows produces an injective phase retrieval map would disprove the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Almost-sure injectivity fails at N=4M-5 for this random ensemble.
  • There exist matrices at this row count for which distinct signals produce identical magnitude measurements.
  • The number of measurements needed to guarantee injectivity for generic Gaussian matrices must exceed 4M-5.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result implies that sampling from the Gaussian ensemble at this dimension will occasionally produce ambiguous recovery instances.
  • For small concrete M one could in principle estimate the measure of the non-injective set by Monte Carlo sampling of matrices.
  • Analogous positive-probability statements may hold when the entries follow other continuous distributions with full support.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to prove that if A ∈ ℂ^{N×M} with N=4M−5 has i.i.d. standard complex Gaussian entries, then the probability that the phase retrieval map generated by A is not injective is positive. This is presented as establishing part (1) of Vinzant's conjecture (restated in BDL+26). The result is stated to have been obtained via generative AI (Rethlas system).

Significance. If the central claim holds, the result would confirm via probabilistic methods that the injectivity threshold for phase retrieval with Gaussian measurements is sharp at N=4M−5, consistent with standard incidence-variety dimension counts in the literature. It would resolve an open conjecture and clarify the transition from almost-sure injectivity to positive-probability failure.

major comments (2)
  1. [Abstract] Abstract and main text: the manuscript asserts a probabilistic proof that the set of non-injective matrices has positive Lebesgue measure (hence positive Gaussian probability), but supplies no lemmas, incidence-variety construction, measure estimate, or error analysis. Without these steps the claim cannot be verified.
  2. The statement that the result was obtained using the Rethlas generative-AI system is given without any accompanying verification, human-checked steps, or reproducibility information, leaving the derivation uninspectable.
minor comments (1)
  1. [Abstract] The citation BDL+26 appears only in the abstract; the bibliography entry (if present) should be checked for completeness and consistency with the journal's style.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review. The manuscript is a concise announcement of a result obtained via the Rethlas generative-AI system. We address the major comments below and commit to revisions that supply the missing technical details.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main text: the manuscript asserts a probabilistic proof that the set of non-injective matrices has positive Lebesgue measure (hence positive Gaussian probability), but supplies no lemmas, incidence-variety construction, measure estimate, or error analysis. Without these steps the claim cannot be verified.

    Authors: We agree that the present short note contains only the statement and does not exhibit the lemmas, incidence-variety construction, measure estimates, or error analysis. In the revised version we will insert a complete proof section that explicitly constructs the relevant incidence variety, computes its dimension, and derives the positive-measure conclusion for the non-injective locus under the Gaussian measure. revision: yes

  2. Referee: The statement that the result was obtained using the Rethlas generative-AI system is given without any accompanying verification, human-checked steps, or reproducibility information, leaving the derivation uninspectable.

    Authors: We concur that the manuscript provides no account of the generative process or subsequent verification. The revision will include a dedicated subsection that records the principal prompts supplied to Rethlas, the intermediate outputs obtained, and the human verification steps (including independent numerical checks on low-dimensional instances) that were performed to confirm the algebraic claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct probabilistic claim

full rationale

The paper asserts that for random matrices A with i.i.d. complex Gaussian entries and N=4M-5, the set of A yielding non-injective phase retrieval has positive probability. This is a direct measure-theoretic statement consistent with incidence-variety dimension counts in the literature. No equations, definitions, or steps in the provided text reduce the claim to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The cited conjecture reference [BDL+26] is external and does not supply the proof. The AI-assisted generation note is irrelevant to circularity. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard model of i.i.d. complex Gaussians and the definition of the phase retrieval map; no additional free parameters or invented entities are introduced.

axioms (1)
  • domain assumption i.i.d. standard complex Gaussian entries for matrix A
    The probability statement is conditioned on this specific random matrix ensemble.

pith-pipeline@v0.9.1-grok · 5609 in / 1087 out tokens · 39724 ms · 2026-06-26T22:18:14.996719+00:00 · methodology

0 comments
read the original abstract

In this short note, we prove that if $A \in \mathbb C^{N \times M}$ with $N=4M-5$ has i.i.d.\ standard complex Gaussian entries, then the probability that the phase retrieval map generated by $A$ is not injective is positive. This proves Part (1) of a conjecture of Cynthia Vinzant, which was later restated by Afonso S. Bandeira in \cite{BDL+26}. The main result of this paper was obtained using generative AI, in particular the Rethlas system.

discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Phase Retrieval in $\mathbb C^4$ Requires Exactly Eleven Measurements

    quant-ph 2026-07 conditional novelty 8.0

    Phase retrieval in C^4 requires exactly 11 intensity measurements: no set of 10 vectors is injective up to phase.

  2. Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory

    cs.AI 2026-07 conditional novelty 7.0

    Danus orchestrates parallel LLM-based proof search around a shared, verifier-checked fact graph, producing six research-level mathematical proofs.

  3. Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory

    cs.AI 2026-07 conditional novelty 6.5

    Danus uses a main planner, parallel workers, and a shared verified fact graph to construct long research-level mathematical proofs across six case studies.

Reference graph

Works this paper leans on

5 extracted references · 2 canonical work pages · cited by 2 Pith papers · 1 internal anchor

  1. [1]

    Bandeira, Jameson Cahill, Dustin G

    Afonso S. Bandeira, Jameson Cahill, Dustin G. Mixon, and Aaron A. Nelson. Saving phase: Injectivity and stability for phase retrieval. Applied and Computational Harmonic Analysis , 37(1):106--125, 2014

    2014

  2. [2]

    arXiv preprint arXiv:2603.29571 , year=

    Afonso S. Bandeira, Daniil Dmitriev, Kevin Lucca, Petar Nizi\'c-Nikolac, and Almut R o dder. Randomstrasse101: Open Problems of 2025. arXiv preprint, arXiv:2603.29571

    work page arXiv 2025

  3. [3]

    An algebraic characterization of injectivity in phase retrieval

    Aldo Conca, Dan Edidin, Milena Hering, and Cynthia Vinzant. An algebraic characterization of injectivity in phase retrieval. Applied and Computational Harmonic Analysis , 38(2):346--356, 2015

    2015

  4. [4]

    Automated Conjecture Resolution with Formal Verification

    Haocheng Ju, Guoxiong Gao, Jiedong Jiang, Bin Wu, Zeming Sun, Leheng Chen, Yutong Wang, Yuefeng Wang, Zichen Wang, Wanyi He, Peihao Wu, Liang Xiao, Ruochuan Liu, Bryan Dai, and Bin Dong. Automated conjecture resolution with formal verification. arXiv preprint, arXiv:2604.03789

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    A small frame and a certificate of its injectivity

    Cynthia Vinzant. A small frame and a certificate of its injectivity. In International Conference on Sampling Theory and Applications (SampTA) , pages 197--200. IEEE, 2015

    2015