---
title: "The line no one walks"
status: "draft"
started: "2026-07-18T00:00:00.000Z"
writer_model: "claude-opus-4-8"
tags: ["learning-curve","scaling-laws","wrights-law","averaging-artifact","history-of-psychology","neural-scaling-laws","cross-domain-bridge"]
---


Right now, in the middle of 2026, a lot of money is riding on a curve. You have seen it even if you have not seen the paper: model loss falling as a smooth power law of compute, a clean line on a log-log plot, and the whole industry leaning forward to read where it goes next. Spend more, get less loss, the line says so. The line is the argument for the next data center.

The line is a hundred and twenty-seven years old, and the first time anyone drew it, it had bumps.

In 1899 two psychologists, William Bryan and Noble Harter, published *Studies on the Telegraphic Language* — a study of how Morse operators get good at Morse. They tracked receiving speed over months of practice and drew the result. That drawing is the ancestor of the curve now used to forecast the cost of intelligence. William Nordhaus, tracing the modern experience curve back to its root, lands on exactly them: telegraph operators, 1899, before anyone thought a factory could learn.

And their curve had *plateaus*. Long flat stretches where the operator stopped getting faster, then a jump, then another flat stretch. Bryan and Harter read the flats as reorganization — the operator mastering letters, then going still, then suddenly hearing whole words, then still again, then whole phrases. Learning as a staircase, not a slope.

< the plateaus are contested — Keller's 1958 paper is literally titled "The phantom plateau" and argues they were a measurement artifact. so the honest claim is that Bryan and Harter *reported* flats, not that flats are settled fact. hold that thought; it comes back >

Then, in 1936, an aircraft engineer named T.P. Wright flattened them.

Wright's law — cost falls a constant fraction every time you double cumulative production — is the same learning curve with the staircase sanded off. No plateaus. A clean log-linear descent, the same slope forever. It is a beautiful object and it forecasts well and it became the spine of a century of cost projection. Devendra Sahal later proved something that should make you trust it more and does the opposite: when production grows exponentially in calendar time, Wright's law and Moore's law become mathematically the same curve, read off two different axes. Cost-falls-with-scale and cost-falls-with-time stop being rival explanations. They are one line wearing two labels. That is why the AI inference number everyone quotes — roughly 280× cheaper per token from late 2022 to late 2024 — sits so comfortably next to Moore's law. Same shape. Same confidence.

The confidence is the problem. Because the smoothing was a choice, made once, and we have spent ninety years mistaking the choice for the terrain.

Here is where a psychology paper from 2000 turns the light on.

Andrew Heathcote, Scott Brown, and D.J.K. Mewhort took 7,910 individual learning series — 475 people across 24 experiments — and did the thing almost nobody does. They fit the curve to each *person* before averaging. The paper is called "The Power Law Repealed," and its finding is flat and devastating: "the exponential function fit better than the power in all the unaveraged data sets." Individual learners do not follow a power law. Each one speeds up exponentially. The power law only appears when you average them together — and worse than that, the averaging *manufactures* it: "linear averaging yields a composite that is systematically biased towards the power function."

Read that twice. The smooth power-law learning curve is not a fact about learning. It is a fact about crowds. Take a room full of exponential learners, each on their own jagged path, average them, and out falls a graceful power law that describes no one in the room. The smoothness is the artifact. It is the shadow the crowd casts, not the shape of any body in it.

< this is the same move the vault keeps finding under a different name. a suspiciously clean result is usually a sign that some structure got washed out on the way to the page. perfection as the fingerprint of a missing witness. here the missing witness is the individual >

And once you know to look, the smooth curve fails in more than one direction.

Nordhaus, in the same body of work, shows the curve's central coefficient is statistically unidentified: because cumulative production and calendar time trend upward together, a regression cannot tell learning-by-doing apart from plain exogenous progress. His demonstration is the sharp end — build a case with *zero* actual learning and the fitted coefficient still comes out at 0.2. Across 34 industries, two reasonable ways of fitting the curve correlate at 0.009. Effectively no agreement about which technologies even learn. The curve, he argues, is not reading supply-side learning at all. It is reading whether output kept growing — which is a fact about demand. A demand thermometer in a supply-side lab coat.

And even a real curve can run backward. Benkard's study of the Lockheed L-1011 found that when production slowed, unit cost *rose* — the organization forgot what it had learned to do cheaply. The ratchet slips. A cost gain treated as permanent turns out to be stored not in the technology but in a workforce doing the thing often enough not to forget it, and forgetting is a rate, not an event.

< Nordhaus has critics I have not read yet, and the Benkard depreciation number rests on a secondary summary I could not trace to his primary. so take the last two paragraphs as loaded, not fired. the load-bearing one is Heathcote, and that one is a Tier-1 primary with the sample in the title >

So: three separate ways the clean line lies. It hides the plateaus. It hides the individuals. It hides whether there is any learning under it at all.

Now walk back to 2026.

A paper posted this year — "Neural Neural Scaling Laws," Hu, Pan, Jhaveri, Lourie, and Cho — reports that in large models "aggregate metrics like validation loss can follow smooth power-law curves" while "individual downstream tasks exhibit diverse scaling behaviors: some improve monotonically, others plateau, and some even degrade with scale." The reason they give is the oldest one in the file: "averaging token-level losses obscures signal."

Some plateau. Some degrade.

That is Bryan and Harter's staircase, rediscovered by hand, inside a transformer. It is Heathcote's warning, printed again twenty-six years later by people who — as far as I can tell — had never read him. The smooth scaling law that the industry is extrapolating is an average over a bag of tasks that individually do everything *except* trace the smooth line. Some are climbing. Some have gone flat. Some are getting worse as you spend more.

< whether the 2026 mechanism is literally Heathcote's averaging bias or only a cousin of it, I can't yet say — the paper's derivation needs a closer read than I've given it, and I'm flagging that rather than asserting the identity. what I can say is that both papers point at the same fix: stop averaging, look at the components >

I am not saying the scaling laws are wrong. Averaged curves are real; they forecast aggregates, and sometimes an aggregate is exactly what you're buying. I am saying the smooth line is the one shape in the whole picture guaranteed to belong to no individual member of it — and the thing you'd most want to know before betting the next data center is which tasks are on a plateau and which are quietly degrading, and that is precisely the information averaging is built to erase.

The learning curve was born jagged, in a study of people learning Morse. It got its bumps sanded off once, in 1936, to make a law. It has been re-roughening ever since — in 1958, in 2000, and now in 2026 — every time someone bothers to look under the average and finds the staircase still there.

The line is smooth because we drew it over a crowd. No one walks it. That was true of the telegraph operators, and it is true of the tasks inside the model, and the only new thing in 2026 is how much we are willing to spend on the belief that someone does.

## Sources

- [[claim-experience-curve-originates-in-1899-telegraph-operator-psychology]] — Bryan & Harter 1899, the learning curve's origin in psychology, its plateaus, and Wright's 1936 smoothing; the Keller "phantom plateau" caveat.
- [[claim-wrights-law-cost-falls-per-cumulative-production-doubling]] — Wright's law as the smooth log-linear experience curve (`[unverified-mechanism]` on the 1936 primary; concept uncontested).
- [[claim-sahals-identity-equates-wrights-law-and-moores-law]] — Sahal's identity, Wright's law and Moore's law collapsing to one curve under exponential production (Tier 1, verified-verbatim).
- [[claim-inference-cost-collapsed-280x]] — the ~280× per-token inference-cost fall, 2022–2024, as the AI instance of a Moore/Wright-shaped decline.
- [[claim-nordhaus-learning-coefficient-statistically-unidentified]] — the coefficient is unidentified and biased up; zero learning still fits 0.2; the curve reads demand (Tier 1; Nordhaus's critics not yet read).
- [[claim-organizational-forgetting-can-reverse-the-learning-curve]] — Benkard's L-1011: the ratchet slips, forgetting runs the curve backward (`[unverified-quant]` on the depreciation rate).
- [[observation-population-scaled-improvement-hits-a-sublinear-brake-across-domains]] — the discipline of correcting a resemblance rather than confirming it; the AI scaling-law leg and its softness.
- [[2026-07-16-hop-power-law-averaging-artifact]] (inbox capture, not yet promoted) — Heathcote, Brown & Mewhort 2000 "The Power Law Repealed" (Tier 1, tls-verified quotes); Hu et al. 2026 "Neural Neural Scaling Laws," arXiv:2601.19831 (`[unverified-mechanism]` on whether the two describe the literal same statistical process).

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## References

*The 7 sources this piece rests on — tiers as recorded, not all primary — generated from the frontmatter of the claim-notes it cites. Every field copied, none composed.*

- New Things Under the Sun (Matt Clancy); cross-referenced against Our World in Data's learning-curve explainer (ourworldindata.org/learning-curve). 2026. [document title not recorded in the note — see the claim-note].  
  https://www.newthingsunderthesun.com/pub/6b1c38y9/release/5  ·  *Tier 3*
- Andrew Heathcote, Scott Brown & D.J.K. Mewhort. 2000. [document title not recorded in the note — see the claim-note].  
  https://users.cs.northwestern.edu/~paritosh/papers/KIP/power-law-repealed.pdf  ·  *Tier 1*
- C. Lanier Benkard, 'Learning and Forgetting: The Dynamics of Aircraft Production' (AER 90(4):1034-1054). 2000. [document title not recorded in the note — see the claim-note].  
  https://web.stanford.edu/~lanierb/research/Learning_and_Forgetting_AER.pdf  ·  *Tier 1*
- Lafond, Bailey, Bakker, Rebois, Zadourian, McSharry & Farmer (2017), citing Sahal (1979). 2017. "How well do experience curves predict technological progress? A method for making distributional forecasts."  
  https://arxiv.org/abs/1703.05979  ·  *Tier 1*
- Seek (synthesis across Kremer 1993, Bloom/Jones/Van Reenen/Webb 2020, Desai/Fisher/Murray 2007, and Kaplan 2020); representative source_quote and source_url are the Desai/Fisher/Murray brake leg. 2026. "The speed of evolution and maintenance of variation in asexual populations."  
  https://pubmed.ncbi.nlm.nih.gov/17331728/  ·  *Tier 1*
- Stanford HAI AI Index (280x figure, direct); Epoch AI — Cottier, Snodin, Owen, Adamczewski (9x-900x range, direct). 2026. "The 2025 AI Index Report."  
  https://hai.stanford.edu/ai-index/2025-ai-index-report  ·  *Tier 3*
- William D. Nordhaus (attribution); Bryan & Harter (1899), 'Studies on the Telegraphic Language', archived via gwern.net. 2009. [document title not recorded in the note — see the claim-note].  
  https://www.nber.org/system/files/working_papers/w14638/w14638.pdf  ·  *Tier 1*

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