Writing study: Escaping the Cognitive Well: Efficient Competition Math with Off-the-Shelf Models

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← NeurIPS 2026 oral writing corpus

Xingyu Dang, Rohit Agarwal, Rodrigo Porto, Anirudh Goyal, Liam Fowl, Sanjeev Arora · Official NeurIPS 2026 oral record

Observed writing choices, not acceptance rules. This study examines the inspected manuscript version. It does not establish why the paper was accepted or that its presentation choices should be copied unchanged.

Figure, table and section ledger

Editorial contribution type: Efficiency, inference & systems (primary).

Layout extraction, not a required paper template. Counts are distinct caption labels; a multi-panel figure counts once. Section spans are estimated from heading positions, including floats and whitespace. Method and limitations.

Canonical PDF

35 physical pages

Before references / appendix

≈ 9.2 page-equivalents

Main figures / tables

3 / 2

Attached appendix figures / tables

4 / 8

Attached appendix span

≈ 23.91 page-equivalents

Before references / appendix: ≈ 26.29%References / intervening material: ≈ 5.4%Attached appendix: ≈ 68.31%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Source-bound correction: Top-level heading and reference/appendix boundaries checked against source PDF text. Bibliography titles/author initials, numbered contribution/property bullets, diagram steps, equations and TOC entries are not substantive headings. Section spans are page-coordinate estimates, not ink area or prose-only length. This is an author manuscript mirror, not a verified accepted-version template. Source layout uses physical single-column y/page-height, including equation-heavy pages.

Section-by-section lengths and source pages

The title/abstract/front matter precede the first numbered heading. A span ends at the next detected top-level heading, not at the last sentence; end-page whitespace is included. References and statements are listed separately. These are not measurements of text-only or artwork area.

Top-level heading	Starts on PDF page	Estimated span	Share of entire PDF
1 Introduction	1	≈ 2.7 pp.	7.71%
2 Method Overview	4	≈ 1.72 pp.	4.91%
3 Results	6	≈ 3.6 pp.	10.29%
4 Conclusion	9	≈ 0.4 pp.	1.14%
Acknowledgments and Disclosure of Funding	10	≈ 0.11 pp.	0.31%
References	10	≈ 1.89 pp.	5.4%
A Algorithm Details	12	≈ 0.51 pp.	1.46%
B Ablations	12	≈ 4.89 pp.	13.97%
C Compute details	17	≈ 2.09 pp.	5.97%
D Escaping the Cognitive Well: Qualitative analysis	19	≈ 1.25 pp.	3.57%
E Limitations of Our Pipeline	20	≈ 0.47 pp.	1.34%
F System Prompts	21	≈ 11.23 pp.	32.09%
G Miscellaneous	32	≈ 3.47 pp.	9.91%

All detected figure and table locations

Main text

• Figure 1 · p. 2
• Figure 2 · p. 3
• Table 1 · p. 8
• Table 2 · p. 8
• Figure 3 · p. 9

Attached appendix

• Figure 4 · p. 12
• Figure 5 · p. 15
• Table 3 · p. 16
• Table 4 · p. 16
• Table 5 · p. 17
• Table 6 · p. 17
• Figure 6 · p. 18
• Table 7 · p. 18
• Table 8 · p. 19
• Table 9 · p. 19
• Figure 7 · p. 19
• Table 10 · p. 20

Argument and organization

1. Problem and contribution

   Separates a stalled proof search from a more deceptive failure where an internal grader accepts flawed reasoning, making grader reliability an explicit design target.

2. Argument sequence

   A concrete failure analysis motivates context detachment and conjecture isolation; methodology distinguishes generation, critique and final aggregation before multi-backbone and grader evaluations.

3. Evidence and verification

   Accuracy and expert assessment are not treated as interchangeable; dedicated grading tables report false positives and false negatives, while an equal-token example illustrates the proposed failure mechanism.

Problem, claim and contribution

Separates a stalled proof search from a more deceptive failure where an internal grader accepts flawed reasoning, making grader reliability an explicit design target.

Sections 1.1–1.2, pp. 2–3 · Main paper

Argument and section order

A concrete failure analysis motivates context detachment and conjecture isolation; methodology distinguishes generation, critique and final aggregation before multi-backbone and grader evaluations.

Sections 1–3, pp. 1–9 · Main paper

Figures, captions and visual explanation

Figure 2 places accuracy against logarithmic USD cost and uses model-specific markers with text labels. Its caption states that the displayed performance is autograder-measured and that pricing is from January 2026. The two grader tables state whether a reference solution is available and put metric directions in the headers. The intervening definitions distinguish false-positive from false-negative costs, while the prose explains why a better standalone grading score need not make a better controller for the solution pipeline.

Figure 2, p. 3 (render inspected); Tables 1–2, p. 8 (additional rendered inspection) · Main paper

Tables, comparisons and evidence

Accuracy and expert assessment are not treated as interchangeable; dedicated grading tables report false positives and false negatives, while an equal-token example illustrates the proposed failure mechanism.

Figure 1, p. 2; Tables 1–2, p. 8; Figure 3, p. 9 · Main paper

Appendix and reproducibility

An algorithm/architecture overview, sensitivity studies and literal prompts make the multi-stage method inspectable; a detailed cost table supports the main efficiency plot.

Appendices A–C, pp. 12–20; Appendix F, pp. 21–31 · Main paper

Clarity and writing conventions

The prose distinguishes an explanatory hypothesis about grader feedback from the measured benchmark outcome; the limitations keep small benchmark size and scoring uncertainty visible.

Section 1.2, p. 3; Appendix E, p. 20 · Main paper

What to reuse—and what not to infer

Transferable lesson: For test-time reasoning systems, show accuracy versus monetary cost, disclose pricing date, and validate the grader rather than treating its score as ground truth.

Boundary: The Pareto plot uses an autograder; do not substitute those values for expert-scored accuracy or imply that dated API costs are current prices.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-10-02

Reading status

Writing-focused manuscript scan + selected visuals inspected

Main text

Writing-oriented scan across manuscript framing, section sequence, figure/table captions, selected argument/evaluation prose and conclusion; not exhaustive technical or proof verification. Sections 1–4, pp. 1–9, followed by acknowledgments and references pp. 10–11. Source is the exact public author version listed below, not a verified accepted/camera-ready revision.

Appendix / supplement

Writing-oriented appendix/documentation scan: Appendices A–G, pp. 12–35: algorithm overview, controlled ablations, pricing/costs, examples, limitations and literal prompts. Source code, data, model outputs and external video assets were not executed or independently validated.

Visually inspected PDF pages

3, 8

This is a writing-focused scan of the manuscript and the stated supplementary scope: framing, organization, visual explanation, evidence presentation and wording. It is not a line-by-line proof audit, reproduction of results or exhaustive scientific review.

• Public author arXiv manuscript 2602.16793v2; not verified identical to the accepted or camera-ready OpenReview revision · 35 pages