Writing study: Detecting the Semantic Fixed Point: A Geometric Framework for Efficient Inference

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← ICML 2026 oral writing corpus

Jiawei Gu, Ziyue Qiao, Xiao Luo · Official ICML 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

34 physical pages

Before references / appendix

≈ 8.4 page-equivalents

Main figures / tables

3 / 10

Attached appendix figures / tables

7 / 10

Attached appendix span

≈ 22.91 page-equivalents

Before references / appendix: ≈ 24.71%References / intervening material: ≈ 7.91%Attached appendix: ≈ 67.38%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Source-bound correction: Source-checked physical-page text restores wrapped top-level heading labels on pp. 2. Heading starts and measured positions are unchanged; this is label completeness, not a new section. Newly detected unnumbered impact/singular-acknowledgment statements are retained from the size-aware source extraction; authored heading corrections are unchanged.

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	≈ 1.54 pp.	4.53%
2. Phenomenon: Two-Phase Structure of Hidden State Trajectories	2	≈ 1.17 pp.	3.44%
3. Method	4	≈ 1.87 pp.	5.5%
4. Experiments	5	≈ 3.14 pp.	9.24%
5. Conclusion	9	≈ 0.17 pp.	0.5%
Acknowledgement	9	≈ 0.05 pp.	0.15%
Impact Statement	9	≈ 0.12 pp.	0.35%
References	9	≈ 2.68 pp.	7.88%
A. Related Work	12	≈ 1 pp.	2.94%
B. Theoretical Analysis	13	≈ 10.34 pp.	30.41%
C. Experimental Details	23	≈ 2.41 pp.	7.09%
D. Deep Dissection of Representation Geometry	25	≈ 3.72 pp.	10.94%
E. Systematic Validation Across Conditions	29	≈ 5.44 pp.	16%

All detected figure and table locations

Main text

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

Attached appendix

• Table 11 · p. 23
• Table 12 · p. 26
• Table 13 · p. 26
• Table 14 · p. 27
• Table 15 · p. 27
• Figure 4 · p. 28
• Table 16 · p. 28
• Figure 5 · p. 29
• Figure 6 · p. 30
• Table 17 · p. 31
• Table 18 · p. 31
• Figure 7 · p. 31
• Figure 8 · p. 32
• Table 19 · p. 32
• Table 20 · p. 32
• Figure 9 · p. 33
• Figure 10 · p. 34

Argument and organization

1. Problem and contribution

   Contrasts output confidence with internal representation convergence, emphasizing the vocabulary-sized projection cost of repeated confidence checks.

2. Argument sequence

   An empirical phase pattern motivates the predicate; the inference pipeline states request-level depth selection before reporting retained performance and actual overhead.

3. Evidence and verification

   Matched-average-depth comparison tests adaptivity, while latency breakdown tests whether avoided layers survive decision overhead; appendix certificates require a specified contraction regime.

Problem, claim and contribution

Contrasts output confidence with internal representation convergence, emphasizing the vocabulary-sized projection cost of repeated confidence checks.

Introduction, p. 2 · Main paper

Argument and section order

An empirical phase pattern motivates the predicate; the inference pipeline states request-level depth selection before reporting retained performance and actual overhead.

Sections 2-4, pp. 2-8 · Main paper

Figures, captions and visual explanation

Performance-retention rows remain distinct from raw task scores. A diagnostic multipanel separates exit distributions, failure modes, static-depth controls, fitted decay and measured/theoretical speed.

Tables 2-5, p. 5; Figure 3, p. 8; both PNGs inspected · Main paper

Tables, comparisons and evidence

Matched-average-depth comparison tests adaptivity, while latency breakdown tests whether avoided layers survive decision overhead; appendix certificates require a specified contraction regime.

Section 4.5, p. 8; Appendix B.8-B.11, pp. 20-23 · Main paper

Appendix and reproducibility

The supplement records last-token state storage, numerical small-step handling, per-option scoring and tokenizer-boundary checks, alongside broader geometry diagnostics.

Appendix C.1-C.4, pp. 23-24; E, pp. 29-34 · Main paper

Clarity and writing conventions

The title's fixed-point idea is made concrete through step size and directional stability, but semantic convergence is stronger language than stable internal geometry alone supports.

Sections 2-3, pp. 2-5 · Main paper

What to reuse—and what not to infer

Transferable lesson: Pair a cheap stopping signal with matched-compute controls and end-to-end latency, not only a reduction in layers.

Boundary: Prompt-level truncation is not token-adaptive exit; empirical small updates do not certify correctness or establish all the assumptions of the appendix bounds.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-10-07

Reading status

Writing-focused manuscript scan + selected visuals inspected

Main text

Writing-focused distributed manuscript scan, not exhaustive technical/proof validation. Sections 1-5, pp. 1-9: geometric phenomenon, stopping predicate, request-level inference, accuracy/compute comparisons, ablations and diagnostics.

Appendix / supplement

Writing-focused appendix scan; code, data and external assets not executed. Appendices A-E, pp. 12-34: related work, conditional error/certificate derivations, implementation/evaluation, representation dissection and cross-condition/architecture plots.

Visually inspected PDF pages

5, 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.

• Published ICML 2026 proceedings PDF, PMLR volume 306; retrieved 2026-10-07. Public OpenReview decision notes were not independently reverified. · 34 pages