# Context experiment — n=5 run (2026-07-19)

**Target paper:** *On Language Generation in the Limit with Bounded Memory* (arXiv 2605.30324)
**Reference paper:** Kleinberg & Mullainathan, *Language Generation in the Limit* (arXiv 2404.06757)
**Model:** `gpt-5.4` all roles · **Scale:** {200, 300, 400 words} × {without PDF, with PDF} × **5 trials** (30 reconstructions)
**Frozen summaries measured:** 200→`c200-f`, 300→`c300-c`, 400→`c400-d` (same seeds as the earlier n=3 run).

## How we scored

Per-trial pass/fail is brittle: judging each reconstruction alone against a fixed bar,
an LLM judge applies that bar inconsistently, so near-identical work flips pass↔fail on
noise. So the authoritative result uses a **single judge that ranked all 30
reconstructions together**, blinded to which had the PDF. We report the rank as a
**quality score from 1 to 30, higher is better** (best of the 30 scores 30).

## Result — single-judge ranking (higher = better)

| Summary length | Without PDF | With PDF | Effect of the PDF |
|---:|:--:|:--:|:--|
| 200 words | 9.2 | **21.2** | **helps a lot** |
| 300 words | 8.8 | 10.2 | helps slightly |
| 400 words | **23.6** | 20.0 | hurts slightly |

Per-trial scores (of 30):

| Cell | trial scores |
|---|---|
| 400 without | 30, 27, 26, 25, 10 |
| 400 with | 29, 28, 17, 15, 11 |
| 200 with | 24, 23, 21, 20, 18 |
| 300 with | 19, 16, 13, 2, 1 |
| 200 without | 22, 9, 8, 4, 3 |
| 300 without | 14, 12, 7, 6, 5 |

## What this run says

1. **400 words is the strongest budget, with or without the PDF.** The four best
   reconstructions of all 30 are all 400-word trials (both conditions). More words
   reliably produce better reconstructions — this is the run's most robust signal.
2. **The PDF helps at short budgets and is redundant at long ones.** With PDF beats
   without at 200 (21.2 vs 9.2, a large gap) and 300 (10.2 vs 8.8), but slightly trails
   at 400 (20.0 vs 23.6). Coherent reading: the reference paper supplies scaffolding a
   thin summary lacks, but adds nothing once the summary is already rich.
3. **Pass/fail would have told the opposite story.** The per-trial pass counts (in the
   appendix) put "with PDF" at or below "without PDF" in every cell — the reverse of the
   ranking. The two scorers disagree on the *sign* of the context effect; the joint
   ranking is the one to trust, and it favors the PDF at short budgets.

## The catch — this contradicts the earlier n=3 run

In the earlier n=3 study, **200-with-PDF was the worst cell**; here at n=5 it is one of
the best (score 21.2). Same prompt, same condition, opposite result. The between-run
swing is large enough that the short-budget context effect is **not yet safe to bank**:
one run says the PDF hurts at 200, the other says it helps a lot there. What *does*
reproduce across both runs is that **400 words is reliably the best budget**.

## Why the scores move so much (mechanism)

Every reconstruction recovers the main positive theorem and the arbitrary-text
characterization. The judge's separators are all in the delicate boundary results —
chiefly the **element-generator impossibility proof** (needs an exhaustive case split),
plus using invalid constant texts for the arbitrary-text necessity direction, and
stating the index-generator result as exact identification rather than eventual subset
containment. Whether an attempt nails these is close to a coin-flip, which is why cells
span a wide range and why run-to-run results swing.

## Bottom line

At n=5, the authoritative ranking says: **more words help (400 is best), and the
reference paper helps when the summary is short but not when it is long.** But this
short-budget benefit reverses versus the earlier run, so it is a suggestive signal on
one paper, not a settled effect. The only claim stable across both runs is that 400
words is the reliable budget.