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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.
AutocompleteWith cards better111 / 280 · beu-f53fd785d3756c8d

Long-time continuum limits for twisted Schr\"{o}dinger lattices

Mathematics · 2609.34081v1

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

Long-time continuum limits for twisted Schr\"{o}dinger lattices

Introduction / related work

ndent modulation instability [ 31]. Twist-induced geometric phases and suppression of optical tunneling have also been observed experimentally [35]. Although (1) is conjugate to the cubic NLS by (6), this time-dependent translation has no direct analogue for (2). Consequently, our analysis must address this nonautonomous lattice dynamics. Our proof combines Birkhoff coordinates and modified energies to control the propagation and size of the consistency error in the nonlinear comparison. We exploit the complete integrability of cubic NLS through its Birkhoff representation as coordinatewise rotations [ 19, 24], using their action-dependent frequencies and the regularity of the coordinate maps to derive linear bounds of the tangent flow. A related Birkhoff argument for Strang splitting appears in [18]. Since the lattice approximation does not preserve this integrable structure, we use modified energies to control its Sobolev growth. In [ 12], modified energies based on higher time derivatives yield polynomial discrete Sobolev bounds for an exponential-in-time continuum-limit estimate, motivating our optimized error bounds. To accommodate fractional Sobolev regularities, we adapt the

↓ 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: With 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 mid-sentence with 'To accommodate fractional Sobolev regularities, we adapt the' and clearly needs continuation. Output A directly completes this sentence by specifying what is adapted: 'same modified-energy framework to fractional Sobolev norms,' with additional coherent elaboration about 'low- and high-frequency regimes' that naturally extends the prior discussion of modified energies and the nonautonomous twist structure. Output B introduces 'Birkhoff normal form techniques' as an alternative framework, which creates a problematic pivot: the draft already established that Birkhoff coordinates are used for the cubic NLS part, while modified energies handle the lattice approximation. Shifting to 'Birkhoff normal form techniques' for fractional Sobolev regularities contradicts the draft's stated structure and leaves the sentence fragment ungrammatically abandoned. Output A maintains logical continuity with the manuscript's actual argument.

Evidence

Same model and position · retrieved passages supplied · first output

Preferred by blind review

same modified-energy framework to fractional Sobolev norms, where the discrete dispersion relation and the nonautonomous twist term require separate control of low- and high-frequency regimes.

Passages supplied to the Evidence version

Card 1 · Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schr{\"o}dinger equations on $h\mathbb{Z}$

We consider the discrete nonlinear Schr{\"o}dinger equations on a one dimensional lattice of mesh h, with a cubic focusing or defocusing nonlinearity. We prove a polynomial bound on the growth of the discrete Sobolev norms, uniformly with respect to the stepsize of the grid. This bound is based on a…

Read full passage excerpt

We consider the discrete nonlinear Schr{\"o}dinger equations on a one dimensional lattice of mesh h, with a cubic focusing or defocusing nonlinearity. We prove a polynomial bound on the growth of the discrete Sobolev norms, uniformly with respect to the stepsize of the grid. This bound is based on a construction of higher modified energies.

Card 2 · Mobile Localization in nonlinear Schrodinger lattices

Using continuation methods from the integrable Ablowitz-Ladik lattice, we have studied the structure of numerically exact mobile discrete breathers in the standard Discrete Nonlinear Schrodinger equation. We show that, away from that integrable limit, the mobile pulse is dressed by a background of r…

Read full passage excerpt

Using continuation methods from the integrable Ablowitz-Ladik lattice, we have studied the structure of numerically exact mobile discrete breathers in the standard Discrete Nonlinear Schrodinger equation. We show that, away from that integrable limit, the mobile pulse is dressed by a background of resonant plane waves with wavevectors given by a certain selection rule. This background is seen to be essential for supporting mobile localization in the absence of integrability. We show how the variations of the localized pulse energy during its motion are balanced by the interaction with this background, allowing the localization mobility along the lattice.

Card 3 · Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro Potential

We derive a class of discrete nonlinear Schr{ö}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of discretizations contains subclasses conserving classical norm or…

Read full passage excerpt

We derive a class of discrete nonlinear Schr{ö}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of discretizations contains subclasses conserving classical norm or a modified norm and classical momentum. These equations are interesting from the physical standpoint since they support stationary discrete solitons free of the Peierls-Nabarro potential. As a consequence, even in highly-discrete regimes, solitons are not trapped by the lattice and they can be accelerated by even weak external fields. Focusing on the cubic nonlinearity we then consider a small perturbation around stationary soliton solutions and, solving corresponding eigenvalue problem, we (i) demonstrate that solitons are stable; (ii) show that they have two additional zero-frequency modes responsible for their effective translational invariance; (iii) derive semi-analytical solutions for …

Card 4 · Statistical mechanics of general discrete nonlinear Schr{ö}dinger models: Localization transition and its relevance for Klein-Gordon lattices

We extend earlier work [Phys.Rev.Lett. 84, 3740 (2000)] on the statistical mechanics of the cubic one-dimensional discrete nonlinear Schrodinger (DNLS) equation to a more general class of models, including higher dimensionalities and nonlinearities of arbitrary degree. These extensions are physicall…

Read full passage excerpt

We extend earlier work [Phys.Rev.Lett. 84, 3740 (2000)] on the statistical mechanics of the cubic one-dimensional discrete nonlinear Schrodinger (DNLS) equation to a more general class of models, including higher dimensionalities and nonlinearities of arbitrary degree. These extensions are physically motivated by the desire to describe situations with an excitation threshold for creation of localized excitations, as well as by recent work suggesting non-cubic DNLS models to describe Bose-Einstein condensates in deep optical lattices, taking into account the effective condensate dimensionality. Considering ensembles of initial conditions with given values of the two conserved quantities, norm and Hamiltonian, we calculate analytically the boundary of the 'normal' Gibbsian regime corresponding to infinite temperature, and perform numerical simulations to illuminate the nature of the localization dynamics outside this regime for various cases. Furthermore, we show quantitatively how this …

Card 5 · Discrete Breather and Soliton-Mode Collective Excitations in Bose-Einstein Condensates in a Deep Optical Lattice with Tunable Three-body Interactions

We have studied the dynamic evolution of the collective excitations in Bose-Einstein condensates in a deep optical lattice with tunable three-body interactions. Their dynamics is governed by a high order discrete nonlinear Schrodinger equation (DNLSE). The dynamical phase diagram of the system is ob…

Read full passage excerpt

We have studied the dynamic evolution of the collective excitations in Bose-Einstein condensates in a deep optical lattice with tunable three-body interactions. Their dynamics is governed by a high order discrete nonlinear Schrodinger equation (DNLSE). The dynamical phase diagram of the system is obtained using the variational method. The dynamical evolution shows very interesting features. The discrete breather phase totally disappears in the regime where the three-body interaction completely dominates over the two-body interaction. The soliton phase in this particular regime exists only when the soliton line approaches the critical line in the phase diagram. When weak two-body interactions are reintroduced into this regime, the discrete breather solutions reappear, but occupies a very small domain in the phase space. Likewise, in this regime, the soliton as well as the discrete breather phases completely disappear if the signs of the two-and three-body interactions are opposite. We h …

Model-reported card use

The model did not report that any card materially affected this completion.

No Evidence

Same model and position · no retrieved passages · first output

Not preferred in this comparison

Birkhoff normal form techniques developed for semilinear Hamiltonian lattice equations, which provide the analytic framework needed to control the nonautonomous dynamics.

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.

This case does not have both with-card continuation text and a model-reported use record. It remains among all 280 tests but does not enter the reported-use comparison.

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.