Writing study: Intrinsic Entropy of Context Length Scaling in LLMs

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← ICLR 2026 oral writing corpus

Jingzhe Shi, Qinwei (Martin) Ma, Hongyi Liu, Hang Zhao, Jenq-Neng Hwang, Lei Li · Official ICLR 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 foundations (primary) · Empirical analysis, diagnosis & auditing.

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

36 physical pages

Before references / appendix

≈ 9.52 page-equivalents

Main figures / tables

7 / 0

Attached appendix figures / tables

11 / 2

Attached appendix span

≈ 20.89 page-equivalents

Before references / appendix: ≈ 26.44%References / intervening material: ≈ 15.53%Attached appendix: ≈ 58.03%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

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	≈ 1.1 pp.	3.06%
2 ASSUMPTIONS, DEDUCTIONS AND OBSERVATIONS FOR LANGUAGE	2	≈ 3.05 pp.	8.47%
3 DEDUCTION: OPTIMAL CONTEXT LENGTH	5	≈ 1.85 pp.	5.14%
4 PROOF OF CONCEPT WITH SYNTHETIC DATA	7	≈ 2.27 pp.	6.31%
5 CONCLUSION AND DISCUSSIONS	9	≈ 0.71 pp.	1.97%
REFERENCES	10	≈ 5.59 pp.	15.53%
A DOWNSTREAM TASK	16	≈ 2.57 pp.	7.14%
B DEFINITION AND PROPERTIES OF CROSS ENTROPY LOSS	18	≈ 1.15 pp.	3.19%
C EXPERIMENTALLY MEASURE NEXT-TOKEN-PREDICTION INFORMATION	19	≈ 1.57 pp.	4.36%
D DEFINITIONS OF INTRINSIC SPACE AND DERIVED PROPERTIES	21	≈ 3.25 pp.	9.03%
E RM (Z1)1HD (Z1) ≤ C1D−1/d + M D−2d, C1 := Cd,L E[q(Z)−1/d]	24	≈ 3.21 pp.	8.92%
F OPTIMAL CONTEXT LENGTH FOR TWO-NEEDLE-IN-HAYSTACK	27	≈ 0.72 pp.	2%
G EXPERIMENTS ON OTHER LANGUAGE MODELS	28	≈ 0.28 pp.	0.78%
H EIGENVALUE-BASED INTRINSIC ENTROPY ESTIMATION	28	≈ 2.59 pp.	7.19%
I EXPERIMENT SETTINGS	31	≈ 0.4 pp.	1.11%
J RELATED WORK	31	≈ 0.98 pp.	2.72%
K INTRINSIC DIMENSION PERSPECTIVE: MEASUREMENTS IN INTRINSIC	32	≈ 3.51 pp.	9.75%
L DISCLOSURE OF LLM USAGE	36	≈ 0.67 pp.	1.86%

All detected figure and table locations

Main text

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

Attached appendix

• Figure 8 · p. 17
• Figure 9 · p. 20
• Table 1 · p. 21
• Table 2 · p. 23
• Figure 10 · p. 27
• Figure 11 · p. 28
• Figure 12 · p. 29
• Figure 13 · p. 30
• Figure 14 · p. 30
• Figure 15 · p. 31
• Figure 16 · p. 34
• Figure 17 · p. 34
• Figure 18 · p. 35

Argument and organization

1. Assumptions, deductions and observations

   Organizes the argument around ideal loss decomposition and measurable proxies.

2. Optimal context length

   Separates training-data effects from fixed-model inference experiments.

3. Proof of concept with synthetic data

   Checks a known Bayes target unavailable in ordinary language data.

Problem, claim and contribution

The paper begins with apparently conflicting long-context observations, then uses Bayes risk plus approximation error to organize them. The question is explanatory, not a claim to introduce a new context-extension architecture.

Introduction and Eq.1, pp.1–3 · Main paper

Argument and section order

Explicit assumption lists and a roadmap make entropy, intrinsic dimension and context length distinct quantities. Appendix D's assumption-to-result table is a useful traceability device for a theory-heavy manuscript.

§§2.1–2.3, pp.3–5; Table 2, p.23 · Main paper

Figures, captions and visual explanation

Figure 1 places a conceptual decomposition beside training and inference measurements. Its training panel subtracts each dataset-size curve's own minimum; the caption and appendix restore the absolute losses needed to interpret that normalization.

Figure 1, p.2; Figure 15, p.31 · Main paper

Tables, comparisons and evidence

Synthetic parity tasks permit comparison against a calculated Bayes loss, unlike language experiments that substitute pretrained-model loss. RecurrentGemma's three excluded short-context points are disclosed; readers should retain that selection when interpreting correlations.

Figure 2, p.5; Table 1, p.21; Appendix G, p.28 · Main paper

Appendix and reproducibility

The supplementary RULER plots include variable tracking, where performance keeps improving with visible context. This is an important boundary to the main narrative of an interior optimum rather than an inconvenient result to omit.

Appendix A.3 and Figure 8, pp.17–18 · Main paper

Clarity and writing conventions

Repeated claims that observations prove the theory exceed the evidence. Monotonic opposing components alone do not guarantee their derivatives cross; nearest-neighbour upper bounds also should not be rewritten as exact learned-model scaling equalities.

§3, pp.5–6; Appendix D.2, pp.23–26 · Main paper

What to reuse—and what not to infer

Transferable lesson: Present an explanatory schematic, raw and normalized evidence, and a transparent assumption map; use synthetic data to test quantities otherwise hidden in real data.

Boundary: Measured representations are model-dependent proxies, not direct Bayes entropy. A finite optimal context is conditional, not universal. The synthetic-task count varies between 100 and 200 in the text, and some theorem statements omit expectations retained in their proofs. Reading these derivations is not an independent proof certification.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-09-19

Reading status

Full text + selected visuals inspected

Main text

pp.1–10, §§1–5; references pp.10–15

Appendix / supplement

pp.16–36, appendices A–L including all formal derivations and experimental settings

Visually inspected PDF pages

2, 17

• official ICLR 2026 proceedings PDF · 36 pages