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arxiv: 2605.23079 · v1 · pith:6IU2TJ3Nnew · submitted 2026-05-21 · 🧮 math.CA · math.CV· math.FA

Discrete Pauli pairs

Torgeir Keun Lysen This is my paper

Pith reviewed 2026-05-25 04:55 UTC · model grok-4.3

classification 🧮 math.CA math.CVmath.FA
keywords discrete Pauli pairsGaussian decaymoduli equalityFourier transformsdensity thresholdsphaseless reconstructionFourier uniqueness
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The pith

Equality of sampled moduli on dense enough discrete sets forces equality of continuous moduli for Gaussian-decaying functions and their Fourier transforms.

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

The paper determines sharp density thresholds above which discrete Pauli pairs with Gaussian decay must coincide with classical Pauli pairs. This means that if two such functions have equal moduli sampled on the discrete set for both the functions themselves and their Fourier transforms, then the moduli agree everywhere in time and frequency. It also identifies the corresponding threshold for weak Pauli pairs, where agreement holds on only one of the two sides. These findings answer a question posed by Ramos and Sousa and serve as phaseless counterparts to known Fourier uniqueness results. The results matter because they clarify the minimal sampling density needed to recover magnitude information uniquely from discrete data.

Core claim

We determine sharp density thresholds for when discrete Pauli pairs with Gaussian decay must be classical Pauli pairs. More precisely, we identify when equality of the sampled moduli of two functions and their Fourier transforms forces equality of the global moduli in time and frequency, thereby answering a question of Ramos and Sousa. We also determine the sharp threshold for when discrete Pauli pairs must be weak Pauli pairs, where either the time-side or the frequency-side moduli agree identically.

What carries the argument

Sharp density thresholds on discrete sets that convert equality of sampled moduli into equality of the continuous moduli, relying on Gaussian decay to control the functions globally.

If this is right

  • Above the threshold density, sampled moduli equality on the discrete set implies full moduli equality for both the function and its Fourier transform.
  • Below the threshold, there exist discrete Pauli pairs that are not classical Pauli pairs.
  • A separate but related density threshold governs when discrete pairs must reduce to weak Pauli pairs with agreement on only one side.
  • The thresholds provide explicit conditions under which phaseless data on discrete sets determine the global magnitude behavior.

Where Pith is reading between the lines

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

  • The same density ideas might apply to other decay classes if suitable entire-function estimates can replace the Gaussian control.
  • These thresholds could inform discrete phase-retrieval algorithms by indicating when magnitude samples alone suffice for uniqueness.
  • Connections may exist to sampling problems in time-frequency analysis where phase information is unavailable or costly to obtain.

Load-bearing premise

The functions possess Gaussian decay.

What would settle it

Exhibiting two distinct functions with Gaussian decay whose sampled moduli agree on a discrete set of density below the identified threshold but whose continuous moduli differ on a positive-measure set.

read the original abstract

We determine sharp density thresholds for when discrete Pauli pairs with Gaussian decay must be classical Pauli pairs. More precisely, we identify when equality of the sampled moduli of two functions and their Fourier transforms forces equality of the global moduli in time and frequency, thereby answering a question of Ramos and Sousa. We also determine the sharp threshold for when discrete Pauli pairs must be weak Pauli pairs, where either the time-side or the frequency-side moduli agree identically. These can both be seen as phaseless versions of the results of Kulikov, Nazarov, and Sodin on Fourier uniqueness pairs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript determines sharp density thresholds under which discrete Pauli pairs (pairs of functions f and their Fourier transforms with equal sampled moduli on discrete sets) with Gaussian decay must be classical Pauli pairs, meaning the global moduli agree in time and frequency. It also identifies the sharp threshold for the weaker conclusion that the pairs are weak Pauli pairs (moduli agree on one side). These results are presented as phaseless analogues of the Kulikov-Nazarov-Sodin theorems on Fourier uniqueness pairs and answer a question posed by Ramos and Sousa.

Significance. If the derivations hold, the work supplies sharp, explicit thresholds in a phaseless setting for controlling global moduli from discrete samples, extending uniqueness-pair theory in a natural direction. The Gaussian decay hypothesis is the key technical device that bridges the discrete sampled moduli to the continuous case, and the clean separation between classical and weak conclusions strengthens the contribution. No free parameters or ad-hoc axioms appear to be introduced.

minor comments (2)
  1. The abstract states the main results but does not name the explicit density thresholds; adding the numerical values (or the precise form of the thresholds) would improve immediate readability.
  2. Notation for the discrete sampling sets and the precise definition of 'Pauli pair' should be introduced with a displayed equation or a short dedicated paragraph early in the introduction for readers unfamiliar with the Ramos-Sousa question.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on sharp density thresholds for discrete Pauli pairs with Gaussian decay, including the recognition that the results provide phaseless analogues of the Kulikov-Nazarov-Sodin theorems and answer the question of Ramos and Sousa. We note the recommendation for minor revision and will incorporate any editorial or presentational improvements in the revised manuscript.

Circularity Check

0 steps flagged

No circularity; derivation extends external uniqueness theorems independently

full rationale

The paper extends cited results of Kulikov-Nazarov-Sodin on Fourier uniqueness pairs to a phaseless setting with Gaussian decay, determining sharp density thresholds for discrete Pauli pairs to be classical or weak. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the thresholds and distinctions (classical vs. weak) are derived from the stated assumptions on moduli equality and decay, without reducing to the paper's own inputs by construction. The work addresses an external question from Ramos and Sousa using standard analytic techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard Fourier analysis results and Gaussian decay assumption; no free parameters, invented entities, or ad-hoc axioms introduced in the abstract.

axioms (2)
  • domain assumption Gaussian decay condition on the functions
    Enables control from discrete samples to global moduli; stated in the main claim.
  • standard math Prior Fourier uniqueness results of Kulikov, Nazarov, and Sodin
    Used as base for phaseless extension.

pith-pipeline@v0.9.0 · 5610 in / 1195 out tokens · 19570 ms · 2026-05-25T04:55:19.018613+00:00 · methodology

discussion (0)

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Reference graph

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