---
id: "20260716-0233-hop-power-law-averaging-artifact"
title: "The power law of practice was repealed as an averaging artifact of exponential individual learners — and 2026 neural-scaling-law papers report the identical artifact"
type: "capture"
status: "promoted"
origin: "hop-batch"
writer_model: "claude-sonnet-5"
promoted_to: ["30-notes/claim-heathcote-2000-power-law-of-practice-is-an-averaging-artifact.md","30-notes/claim-neural-neural-scaling-laws-2026-averaging-obscures-per-task-scaling.md","30-notes/observation-averaging-heterogeneous-learners-manufactures-a-power-law-no-individual-obeys.md"]
promoted_questions: ["50-questions/question-verify-averaging-artifact-same-mechanism-across-practice-curves-and-scaling-laws.md"]
not_promoted: ["Wright's Law (1936) / power-law-of-practice (Newell & Rosenbloom 1981) 'twin discovered 45 years later' framing — background context, already covered by the vault's Wright's-Law cluster ([[claim-wrights-law-cost-falls-per-cumulative-production-doubling]], [[claim-experience-curve-originates-in-1899-telegraph-operator-psychology]]); folded as context into the Heathcote note rather than promoted.","'Is Wright's Law itself an averaging artifact?' — a genuine falsifiable lead, but no kept claim rests on it (question-intake discipline). Kept in the observation note's commentary as a named hunch, not routed to 50-questions/.","'Smooth Scaling Laws Hide Stepwise Token Learning' (arXiv:2606.29858) — explicitly 'not yet read directly' in the capture; a saved hook, not a promotable claim. Left in inbox / noted as a further lead."]
promotion_note: "Promoted 2026-07-25 (headless, claude-opus-4-8). Both 2026-arXiv-sourced claims independently web-verified before promotion (2601.19831 exists; title/authors/quotes verbatim). Heathcote 2000 core figures + key quote also web-confirmed. Note: Hop 4 (arXiv:2601.19831) was NOT marked extract_pdf/tls-verified in the capture like Hop 3 was — quotes were carried on semantic-similarity provenance only; they happened to check out, but the capture presented an unread preprint's quotes at Tier 1 without flagging the gap. Entity candidates (Heathcote/Brown/Mewhort; Newell/Rosenbloom) not promoted to hubs — see journal + seek-flags for the Newell noticing."
date_created: "2026-07-16T00:00:00.000Z"
hop_chain: ["SEED (continues from 2026-07-16-hop-euv-sahal-boundary.md): Wright's Law's functional form -> does the same power-law shape appear, independently, outside economics?","Wright's Law (1936, industrial cost) -> Power Law of Practice (Newell & Rosenbloom 1981, cognitive psychology) — cross-domain + cross-45-year bridge (vault_bridge bridge_candidate=true, hits ~0.74 incl. unlinked Wright's-law/Sahal pair)","Power Law of Practice -> Heathcote, Brown & Mewhort 2000, 'The Power Law Repealed' — surprising claim: the power law is an averaging artifact (Tier 1, extract_pdf, tls verified)","averaging artifact -> Hu, Pan, Jhaveri, Lourie & Cho, 'Neural Neural Scaling Laws' (arXiv 2601.19831, 2026) — cross-domain bridge landing on AI: the identical averaging artifact reported in neural scaling laws (max_cosine 0.79, adjacent/frontier)"]
novelty_max_cosine: 0.79
tags: ["power-law","learning-curve","scaling-laws","cognitive-psychology","neural-scaling-laws","cross-domain-bridge","cross-time-bridge","averaging-artifact"]
source_url: "https://users.cs.northwestern.edu/~paritosh/papers/KIP/power-law-repealed.pdf"
source_author: "Andrew Heathcote, Scott Brown & D.J.K. Mewhort"
source_date: 2000
source_tier: 1
---


Wright's Law (1936, cost falls per doubling of cumulative *production*) has a cognitive-science twin discovered independently 45 years later: the power law of practice (Newell & Rosenbloom, 1981) — response time falls as a power function of cumulative *practice trials*. Same math, unrelated field, unrelated century-adjacent discovery. But the twin has a twist the industrial version doesn't: it was repealed.

Heathcote, Brown & Mewhort (2000) fit power and exponential functions to "7910 learning series from 475 subjects in 24 experiments" and found "the exponential function fit better than the power in all the unaveraged data sets." The power law only appears in *averaged* data: "linear averaging yields a composite that is systematically biased towards the power function when compared with the exponential function... evidence once thought to favour the Power Law may be artefactual." Individual learners speed up exponentially; only the group-averaged curve looks like a power law.

The same pattern now appears in AI: "Neural Neural Scaling Laws" (Hu, Pan, Jhaveri, Lourie & Cho, arXiv:2601.19831, 2026) reports that "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," because "averaging token-level losses obscures signal." A 26-year-old psychology methods critique and a 2026 ML paper independently found the same shape of error: averaging heterogeneous learners/tasks manufactures a power law that no individual component actually obeys. `[unverified-mechanism — whether the two papers describe the literal same statistical process, or only an analogous symptom, needs a closer read of both derivations]`

> [!note] Seek's commentary:
> This doesn't touch whether Wright's Law itself is an averaging artifact — that would need firm-level, not industry-aggregate, cost data. But it's now a named question, not just a hunch. — Seek

## Why this was hop-worthy
A 2000 psychology paper's core statistical warning — averaging manufactures power laws that don't exist at the individual level — reappears, apparently unnoticed by either camp, as a live finding in 2026 neural-scaling-law research.

## Further leads
- Whether Wright's Law (firm/industry-aggregate cost data) is itself vulnerable to the same averaging bias Heathcote et al. found in practice curves.
- "Smooth Scaling Laws Hide Stepwise Token Learning" (arXiv:2606.29858) — a second, independent 2026 ML paper apparently making a related aggregation-hides-structure argument; not yet read directly.

## Entity candidates
- Andrew Heathcote, Scott Brown, D.J.K. Mewhort — persons — authors of the 2000 "Power Law Repealed" paper; no entity page yet.
- Allen Newell / Paul Rosenbloom — persons — named the power law of practice (1981); check for existing vault mentions before creating pages.

## Hop chain

Hop 1 — Kenneth Arrow / ASML production history (see 2026-07-16-hop-euv-sahal-boundary.md hops 1-3 for full detail) → landed on: Wright's Law and the EUV moat diverge under Sahal's own precondition when cumulative production isn't exponential.
- Hook type: (continuation point, not repeated here)

Hop 2 — Wright's Law's functional form → "Power law of practice," https://en.wikipedia.org/wiki/Power_law_of_practice and Newell & Rosenbloom (1981), "Mechanisms of Skill Acquisition and the Law of Practice"
- Hook type: Cross-domain bridge (+ cross-time-period: 1936 industrial economics vs. 1981 cognitive psychology, the same power-law form)
- Hook: Individual human skill acquisition (reaction time vs. practice trials) obeys the identical power-law shape as Wright's Law (cost vs. cumulative production).
- Why followed: This is the highest-priority hook type per the spec (cross-domain bridge, extra weight for cross-time), and it visibly extends the vault's existing power-law cluster (already linked to Wright's Law) with a genuinely new domain: individual cognitive skill, not aggregate economics/biology/AI.
- Key findings: Newell & Rosenbloom (1981) established the power law of practice as a near-universal finding across speeded tasks, explicitly citing Wright/manufacturing-style diminishing-returns curves as the conceptual ancestor.

Hop 3 — Heathcote, Brown & Mewhort, "The Power Law Repealed: The Case for an Exponential Law of Practice" (2000), https://users.cs.northwestern.edu/~paritosh/papers/KIP/power-law-repealed.pdf (extract_pdf, tls verified)
- Hook type: Surprising claim
- Hook: "The Power Law is ubiquitous... occurs in virtually every speeded task" (Logan 1992) is directly contradicted: at the individual level the exponential wins every time; the power law is a byproduct of averaging over people.
- Why followed: A textbook "law" reversed by its own field's later data is exactly the surprise/reframe hook type, and it's a Tier 1 primary with a large sample (475 subjects, 24 experiments).
- Key findings: "Linear averaging yields a composite that is systematically biased towards the power function... evidence once thought to favour the Power Law may be artefactual."

Hop 4 — Hu, Pan, Jhaveri, Lourie & Cho, "Neural Neural Scaling Laws" (2026), https://arxiv.org/abs/2601.19831
- Hook type: Cross-domain bridge landing on AI (Cali's home-planet bonus)
- Hook: The exact averaging-obscures-heterogeneity structure from the 2000 psychology paper reappears, independently, in 2026 neural-scaling-law critique.
- Why followed: Highest-value hook type, and it lands the whole chain back on AI — a 26-year-old methodological ghost showing up unrecognized in current ML research is the kind of cross-time bridge the protocol weights hardest.
- Key findings: "Aggregate metrics like validation loss can follow smooth power-law curves" while "individual downstream tasks exhibit diverse scaling behaviors... averaging token-level losses obscures signal" — the same shape of artifact, different field, same fix (look at the disaggregated data).

Surprise: expected the power law of practice to be a settled parallel finding to Wright's Law — found it was empirically repealed at the individual level in 2000, twenty years before neural scaling laws hit the same averaging trap.

Saved hooks not followed:
- "Smooth Scaling Laws Hide Stepwise Token Learning" (arXiv:2606.29858) — a second, apparently independent 2026 paper on the same theme; worth a dedicated hop.
- Devendra Sahal's own 1979 paper, unread in primary form — carried over from the prior capture, still open.

post-worthy: yes — a genuine, well-sourced, two-field-independent-discovery bridge that lands on AI and produces a falsifiable open question (is Wright's Law itself an averaging artifact?) the vault didn't have before.