Writing study: Free Decompression with Algebraic Spectral Curves

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← NeurIPS 2026 oral writing corpus

Siavash Ameli, Chris van der Heide, Liam Hodgkinson, Michael Mahoney · 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: Theory & formal analysis (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

62 physical pages

Before references / appendix

≈ 10.09 page-equivalents

Main figures / tables

6 / 1

Attached appendix figures / tables

12 / 4

Attached appendix span

≈ 48.86 page-equivalents

Before references / appendix: ≈ 16.27%References / intervening material: ≈ 4.92%Attached appendix: ≈ 78.81%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. Appendix span includes its explicit cover/contents/list-of-figures frontmatter; actual lettered subsection boundaries use their substantive starts, not TOC links.

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	≈ 1.66 pp.	2.68%
2 Background on Free Decompression	3	≈ 0.79 pp.	1.27%
3 Implementing the Polynomial Method	4	≈ 4.27 pp.	6.89%
4 Empirical Results	8	≈ 1.46 pp.	2.35%
5 Conclusions	9	≈ 1.27 pp.	2.05%
References	11	≈ 3.04 pp.	4.9%
Contents	14	≈ 0.68 pp.	1.1%
Appendix A	14	≈ 1.64 pp.	2.65%
Appendix B	16	≈ 1.22 pp.	1.97%
Appendix C	17	≈ 2.96 pp.	4.77%
Appendix D	20	≈ 3.74 pp.	6.03%
Appendix E	24	≈ 5.09 pp.	8.21%
Appendix F	29	≈ 9.21 pp.	14.85%
Appendix G	38	≈ 8.93 pp.	14.4%
Appendix H	47	≈ 3.99 pp.	6.44%
Appendix I	51	≈ 9.12 pp.	14.71%
Appendix J	60	≈ 2.29 pp.	3.69%

All detected figure and table locations

Main text

• Figure 1 · p. 2
• Figure 2 · p. 6
• Figure 3 · p. 7
• Figure 4 · p. 8
• Figure 5 · p. 9
• Figure 6 · p. 9
• Table 1 · p. 10

Attached appendix

• Table A.1 · p. 15
• Figure A.1 · p. 16
• Figure E.1 · p. 26
• Figure E.2 · p. 28
• Figure F.1 · p. 37
• Figure H.1 · p. 51
• Figure I.1 · p. 53
• Figure I.2 · p. 54
• Figure I.3 · p. 55
• Figure I.4 · p. 56
• Figure I.5 · p. 57
• Figure I.6 · p. 58
• Figure I.7 · p. 59
• Table I.1 · p. 59
• Table I.2 · p. 60
• Table I.3 · p. 61

Argument and organization

1. Problem and contribution

   Motivates extrapolating large-matrix spectra from smaller matrices, then identifies restrictive assumptions in prior free decompression.

2. Argument sequence

   Defines the algebraic representation and physical root selection before evolution, spectral quantities and synthetic/model-derived examples.

3. Evidence and verification

   The runtime table separates initial eigendecomposition from evolution time and places distribution, moment and edge errors alongside speed.

Problem, claim and contribution

Motivates extrapolating large-matrix spectra from smaller matrices, then identifies restrictive assumptions in prior free decompression.

Introduction · Main paper

Argument and section order

Defines the algebraic representation and physical root selection before evolution, spectral quantities and synthetic/model-derived examples.

Sections 2–4 · Main paper

Figures, captions and visual explanation

Complex-plane panels and a two-sheet surface visualize why following the physical branch is a computational issue rather than a symbolic afterthought.

Figure 2, p. 6 · Main paper

Tables, comparisons and evidence

The runtime table separates initial eigendecomposition from evolution time and places distribution, moment and edge errors alongside speed.

Table 1, p. 10 · Main paper

Appendix and reproducibility

The long appendix separates reconstruction constraints, root selection, continuation, spectral edges and concrete matrix constructions.

Appendices D–I · Main paper

Clarity and writing conventions

A dedicated limitations paragraph names heavy tails, finite-size effects and near-singular edges rather than ending with an unrestricted scalability claim.

Limitations, p. 10 · Main paper

What to reuse—and what not to infer

Transferable lesson: For mathematical computation, explain branch-selection geometry and report approximation error alongside runtime savings.

Boundary: An algebraic modeling assumption and numerical edge cases remain. The 62-page author version is not a conference-required appendix size.

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 distributed manuscript scan of Sections 1–5 and limitations, pp. 1–10: abstract/framing, section sequence, substantive text windows, captions, evidence organization and conclusion; not exhaustive technical verification.

Appendix / supplement

Writing-oriented scan of Mathematical and numerical appendices through p. 62, including root selection, curve evolution and experiment construction; code and experiments were not executed.

Visually inspected PDF pages

6, 10

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 2605.03634v1; not verified identical to the accepted or camera-ready OpenReview revision · 62 pages