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This case was judged by an anonymized AI reviewer and has not been independently reviewed by people; inspect the draft, cards, and both outputs to assess it.
AutocompleteWithout cards better118 / 280 · beu-bb142f881070fd15

Quantitative Constraints for Stable Sampling on the Sphere

Mathematics · 2601.04119v1

FLOWING EVIDENCE BENCHMARK

How do we compare writing with and without cards?

At the same writing position, the same model produces one first result with cards and one without. All 280 pairs enter the overall anonymized review, which assesses whether a decision not to continue fits the draft.

Overall: 138 with-card wins, 26 ties, 107 no-card wins, 5 pairs where neither was usable, and 4 inconclusive judgments. Both sides wrote text in 218 pairs; 48 had a deliberate stop and 14 included an invalid output.

SAME DRAFT · WITH OR WITHOUT CARDS

01 · FIXED WRITING POSITION

MANUSCRIPT

Same writing position ▌

Matched autocomplete at the same draft position

02 · TWO MATCHED VERSIONS

SHARED BY BOTH

Draft, writing task, and generation model

A · With cards

Relevant literature cards supplied

B · Without cards

No literature cards supplied

Generation model

Same model and version

A → first output

B → first output

Anonymized AI judge

First outputs are anonymized as X and Y

Continuation review: accuracy, fit to the writing task and usability
Returns: X preferred / tie / Y preferred / both unusable

Review again with the answer order swapped

HOW DOES BLIND JUDGING WORK?

① Anonymize both outputs
The AI judge sees the same draft and both first outputs without knowing which received cards.

② Compare and swap order
The judge applies task-specific criteria in X/Y and then Y/X order.

③ Review disagreements
A third pass resolves disagreements. Incomplete reviews remain in the denominator.

How are positions and cards selected?

We selected 40 writing positions in each of seven fields, for 280 comparisons. The position and relevant literature cards were set before each comparison, so both versions began from the same draft.

The main result retains every frozen position, including places where cards might not add a useful next move. This avoids selecting cases based on their outputs, but 49% describes the whole cohort, not a win rate where evidence clearly had an opportunity to help. This cohort did not record that opportunity label for each position before generation, so those two groups cannot yet be separated reliably.

CARD USE · 140/280

Reported card use in continuations with text

The model reports whether a card affected its continuation. In 140 of 280 tests it produced continuation text and reported card use. The previous count of 150 included 10 cases without continuation text and has been corrected.

COMPARISON · 70 / 7 / 30

Which written version was better?

Of the 107 reported-use cases where both versions produced text, the blinded AI judge preferred the with-card version 70 times, found 7 about the same, and preferred the without-card version 30 times.

How are the two table percentages calculated?

ADOPTION COVERAGE

140reported use with text
÷
280all positions
=
50%

WITH-CARD VERSION PREFERRED

70with cards better
÷
107comparable tests
=
65%

The percentages use different groups: 50% counts continuations with text reporting card use among all 280 tests; 65% covers the 107 such cases with text on both sides. The generating model reports card use, while an anonymized AI judge compares writing quality.

Card use is self-reported by the generator. The anonymized AI judge sees the draft and source snippets, so source-aware preference remains possible; there was no independent human review. The overall result describes these 280 frozen positions, not a win probability for any writing task.

Manuscript writing position

Text excerpt · not a PDF page

Research manuscript · excerpt

Quantitative Constraints for Stable Sampling on the Sphere

Introduction / related work

d the results are genuinely quantitative. Moreover, these bounds yield quantitative constraints for thes-dimensional Haus- dorff volume of Marcinkiewicz-Zygmund sampling sets and, in particular, optimal lower bounds for the length of Marcinkiewicz-Zygmund curves. 1.Introduction Marcinkiewicz-Zygmund inequalities are often referred to as “sampling inequal- ities” and form the foundation of stable sampling, see [12, 28] and the references therein. They appear, sometimes under different names, in several areas of mathe- matics. In approximation theory, for instance, they are also known as approximate quadrature rules or approximatet-designs, and are special types of frame inequali- ties. In complex analysis these conditions correspond to Logvinenko-Sereda sets and Carleson measures [26]. Necessary conditions for Marcinkiewicz-Zygmund inequalities have been known for some time on the torus and the sphere [21, 25, 30], and more generally on Rie- mannian manifolds [26, 27] and on metric measure spaces [9, 24]. However, these results are typically qualitative, and the relevant constants are either left unspeci- fied [3, 9, 26] or do not appear in density formulations [11, 13, 21, 27]. For

↓ Next paragraph was generated here

The highlighted text is near the writing position. This is a layout of the evaluation input, not a PDF screenshot.

A matched comparison at the same writing positionOverall AI review: Without cards better

Both arms use the same model, task and manuscript context. Only the retrieved passages differ. The first recorded output from each arm is shown below.

Read overall AI review rationale

A = with cards · B = without cards

The draft ends with 'For' and clearly needs completion to finish the contrast begun with 'However, these results are typically qualitative...'. Output B directly continues this contrast with 'In contrast, the bounds derived here are explicit and quantitative' — this is a natural, argument-preserving completion that mirrors the opening claim that 'the results are genuinely quantitative.' Output A, while topically related, introduces an unsupported specific claim: it states that quantitative MZ inequalities on the sphere have been established 'only in specific settings, such as sufficient conditions in terms of mesh norm and separation radius for spherical harmonic spaces.' The source mentioning mesh norm and separation radius ([3] in sources) discusses *sufficient conditions for sampling and interpolation*, not established quantitative MZ inequalities. This misrepresents the source material and introduces an unsupported bibliographic claim about what has been 'established.' The phrase 'only in specific settings' is also an unsupported narrowing. Output B makes no concrete claim requiring source verification; it simply restates the paper's own contribution in the explicit/quantitative framing already established in the draft's opening sentence.

Evidence

Same model and position · retrieved passages supplied · first output

Not preferred in this comparison

the sphere, quantitative Marcinkiewicz-Zygmund inequalities have been established only in specific settings, such as sufficient conditions in terms of mesh norm and separation radius for spherical harmonic spaces.

Passages supplied to the Evidence version

Card 1 · Sampling, Marcinkiewicz-Zygmund Inequalities, Approximation, and Quadrature Rules

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L^2$, we derive approximation theorems and quadrature rules. The derivation is completely elementary and requires only the definition of Marcinkiewicz-Zygmund inequality, Sobolev spaces, and the solution of least square problems.

Card 2 · On the reconstruction of functions from values at subsampled quadrature points

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect bo…

Read full passage excerpt

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with structured quadrature points (QMC), which provide stable $L_2$ reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a RKHS of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the $d$-dimensional …

Card 3 · Sufficient conditions for sampling and interpolation on the sphere

We obtain sufficient conditions for arrays of points, $\mathcal{Z}=\{\mathcal{Z}(L) \}_{L\ge 1},$ on the unit sphere $\mathcal{Z}(L)\subset \mathbb{S}^d,$ to be Marcinkiewicz-Zygmund and interpolating arrays for spaces of spherical harmonics. The conditions are in terms of the mesh norm and the sepa…

Read full passage excerpt

We obtain sufficient conditions for arrays of points, $\mathcal{Z}=\{\mathcal{Z}(L) \}_{L\ge 1},$ on the unit sphere $\mathcal{Z}(L)\subset \mathbb{S}^d,$ to be Marcinkiewicz-Zygmund and interpolating arrays for spaces of spherical harmonics. The conditions are in terms of the mesh norm and the separation radius of $\mathcal{Z}(L)$.

Card 4 · Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of $N$ samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of ove…

Read full passage excerpt

Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of $N$ samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to $J$, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.

Card 5 · Near Optimal Reconstruction of Spherical Harmonic Expansions

We propose an algorithm for robust recovery of the spherical harmonic expansion of functions defined on the d-dimensional unit sphere $\mathbb{S}^{d-1}$ using a near-optimal number of function evaluations. We show that for any $f \in L^2(\mathbb{S}^{d-1})$, the number of evaluations of $f$ needed to…

Read full passage excerpt

We propose an algorithm for robust recovery of the spherical harmonic expansion of functions defined on the d-dimensional unit sphere $\mathbb{S}^{d-1}$ using a near-optimal number of function evaluations. We show that for any $f \in L^2(\mathbb{S}^{d-1})$, the number of evaluations of $f$ needed to recover its degree-$q$ spherical harmonic expansion equals the dimension of the space of spherical harmonics of degree at most $q$ up to a logarithmic factor. Moreover, we develop a simple yet efficient algorithm to recover degree-$q$ expansion of $f$ by only evaluating the function on uniformly sampled points on $\mathbb{S}^{d-1}$. Our algorithm is based on the connections between spherical harmonics and Gegenbauer polynomials and leverage score sampling methods. Unlike the prior results on fast spherical harmonic transform, our proposed algorithm works efficiently using a nearly optimal number of samples in any dimension d. We further illustrate the empirical performance of our algorithm …

Model-reported card use

Card 3 · fact

Affected output: quantitative Marcinkiewicz-Zygmund inequalities have been established only in specific settings, such as sufficient conditions in terms of mesh norm and separation radius for spherical harmonic spaces

Model explanation: The row supplied the concrete sphere-specific quantitative result (mesh norm and separation radius conditions for spherical harmonic spaces) that the sentence names as the existing exception to the qualitative-only situation.

No Evidence

Same model and position · no retrieved passages · first output

Preferred by blind review

In contrast, the bounds derived here are explicit and quantitative.

Earlier card-use comparison

The earlier diagnostic compared only cases with reported card use and text on both sides; its scope differs from the overall AI review above.

The blinded model review preferred the no-Evidence version. Its original rationale appears below.

Read the original model judgment

Output A provides a coherent, useful continuation that directly contrasts with the preceding text's critique of prior results as 'typically qualitative' with 'unspecified' constants. The phrase 'explicit and quantitative' mirrors the draft's own language ('genuinely quantitative,' 'quantitative constraints') and logically advances the argument without adding unsupported specifics. Output B, however, introduces multiple unsupported claims: (1) that quantitative MZ inequalities on the sphere have been established 'only in specific settings' — the sources do not establish this exclusivity claim; (2) it conflates sufficient conditions from one source [Sufficient conditions for sampling and interpolation on the sphere] with a general claim about where quantitative bounds exist, when the draft actually states that necessary conditions on the sphere have been known but were qualitative; (3) it incorrectly implies that the mesh norm/separation radius conditions are quantitative, when the cited source's snippet does not characterize them as such and the draft explicitly says sphere results were qualitative. The B output also creates a non-sequitur by jumping to 'the sphere' without the needed transition, and its claim about 'only in specific settings' is an unsupported exclusivity assertion not found in any source.

The passages are input material; their presence does not mean an output used them or that they support every claim. Draft and source passages are excerpted; no simulated scores or PDF appear here.