Writing study: Tessellations of Semi-Discrete Flow Matching
← NeurIPS 2026 oral writing corpus
Emile Pierret, Johannes Hertrich, Samuel Hurault, Julie Delon · Official NeurIPS 2026 oral record
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
- 28 physical pages
- Before references / appendix
- ≈ 10.55 page-equivalents
- Main figures / tables
- 6 / 0
- Attached appendix figures / tables
- 1 / 0
- Attached appendix span
- ≈ 15.66 page-equivalents
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 | ≈ 1.69 pp. | 6.04% |
| 2 Background on Flow Matching and Semi-discrete Optimal Transport | 3 | ≈ 1.26 pp. | 4.5% |
| 3 Some Simplification Results | 4 | ≈ 1.93 pp. | 6.89% |
| 4 Topology of the Flow Matching Cells | 6 | ≈ 1.63 pp. | 5.82% |
| 5 Geometry of the Flow Matching Cells | 8 | ≈ 2.47 pp. | 8.82% |
| 6 Conclusion and Discussion | 10 | ≈ 0.87 pp. | 3.11% |
| Acknowledgments | 11 | ≈ 0.13 pp. | 0.46% |
| References | 11 | ≈ 1.79 pp. | 6.39% |
| A Existence and Uniqueness of the flow map in the Semi-discrete case | 13 | ≈ 0.3 pp. | 1.07% |
| B Proofs of the Simplifications | 13 | ≈ 4 pp. | 14.29% |
| C Proofs for Topology of the FM Cells | 17 | ≈ 8.07 pp. | 28.82% |
| D Proofs for Geometry of the FM Cells | 25 | ≈ 2.38 pp. | 8.5% |
| E Higher Dimensional Training Example (MNIST) | 28 | ≈ 0.91 pp. | 3.25% |
All detected figure and table locations
Main text
Attached appendix
Argument and organization
- Problem and contribution
Isolates the exact semi-discrete flow field to ask which geometry belongs to the objective itself, explicitly excluding neural optimization effects from the main theorem setting.
- Argument sequence
After simplification results, topology and geometry have separate sections; a constructive counterexample follows positive topological results rather than mixing unlike claims.
- Evidence and verification
Theorem 8 enumerates four precise geometric differences immediately after the visual counterexample. The later trained MNIST illustration is explicitly supplementary, not a proof of the neural case.
Problem, claim and contribution
Isolates the exact semi-discrete flow field to ask which geometry belongs to the objective itself, explicitly excluding neural optimization effects from the main theorem setting.
Introduction and Section 2, pp. 1–3 · Main paper
Argument and section order
After simplification results, topology and geometry have separate sections; a constructive counterexample follows positive topological results rather than mixing unlike claims.
Sections 3–5, pp. 4–9 · Main paper
Figures, captions and visual explanation
Figure 5 aligns optimal-transport cells, flow-matching cells and the lines used in the proof for the same four-point configuration, keeping point colors and panel coordinates consistent. Figure 1 holds the ten target points and assignment-color identities fixed across optimal transport, exact-velocity flow matching and learned-velocity flow matching. This aligned geometric comparison makes curved versus affine cell boundaries visible before the formal contribution statements; it is an illustrative construction, not an empirical performance plot.
Figure 5, p. 8 (render inspected); Figure 1, p. 2 (additional rendered inspection) · Main paper
Tables, comparisons and evidence
Theorem 8 enumerates four precise geometric differences immediately after the visual counterexample. The later trained MNIST illustration is explicitly supplementary, not a proof of the neural case.
Theorem 8 and Figure 5, p. 8; Appendix E and Figure 7, p. 28 · Main paper
Appendix and reproducibility
Proof appendices mirror the main simplification, topology and geometry sections, while the additional learned example is kept separate from the exact-field proof chain.
Appendices A–E, pp. 13–28 · Main paper
Clarity and writing conventions
The section introduction tells readers that geometric findings are primarily negative, and states additional assumptions beside the strongest topological conclusion.
Sections 4–5, pp. 6–8; conclusion, p. 10 · Main paper
What to reuse—and what not to infer
Transferable lesson: For a theory paper, draw the construction that makes the theorem necessary and separate positive properties, counterexamples and empirical illustrations.
Boundary: A colored region plot supports intuition but does not replace the analytic argument. Exact semi-discrete results do not automatically establish learned-model behavior.
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 1–6, pp. 1–10; references pp. 11–12 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–E, pp. 13–28: flow existence, simplification/topology/geometry proofs, and a higher-dimensional trained example. Source code, data, model outputs and external video assets were not executed or independently validated.
- Visually inspected PDF pages
- 2, 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.