Correction: the ε selection law is Thurstone's 1927 comparative judgment, not a discovery

Issued 2026-07-26. This is the canonical correction notice for every ε claim in this repository. Evidence: epsilon-universality/ (three pre-registered experiment files) plus an adversarial prior-art gate. Supersedes the framing in EPSILON-UNIFICATION.md, FINDINGS-and-patentability.md §F1–F2, the two ε papers, and the various roadmap/decision logs.

Policy, same as the ontogenic-mathematics correction: prior art is not a defect — a true claim someone else reached first is still true, and attribution is the point. False or overclaimed statements are a different matter and are corrected.

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What is wrong

1. ε(σ) = Φ(−gap/(σ√2)) is an identity, not a law — and it is 99 years old

For two independent Gaussian scores with equal variance, the difference is N(−gap, 2σ²), so P(wrong wins) = Φ(−gap/(σ√2)) exactly, by algebra.

This is Thurstone's Law of Comparative Judgment, Case V (1927), Psychological Review 34:273–286. It is independently canonical in two other fields: the two-alternative-forced-choice error rate in signal detection theory (Tanner & Swets 1954; Green & Swets 1966), and the BPSK bit-error rate in additive white Gaussian noise in digital communications.

The reported R² = 0.997 was measuring the Monte-Carlo error of a Gaussian random number generator, not the quality of a fit. Verified: exp_u U1, deviation 3×10⁻⁴ across a gap×σ sweep.

2. It is not a universality class, and the three-domain agreement is the CLT

Universality requires robustness to microscopic detail — that is what the word means. Change the noise distribution and the law changes form:

Under Cauchy noise the Gaussian law under-predicts ε by 11× at gap/σ = 4 and 723× at gap/σ = 6 (exp_u U3, exp_w W1). That is a different fixed point, not an irrelevant operator.

The general statement subsuming all of it is ε = F_Δ(−gap) for difference-CDF F_Δ, and the distribution-free treatment already exists as the Mammen–Tsybakov margin condition. The honest name for our "class" is the Gaussian domain of attraction — the most populated and least surprising universality class in probability. The three domains agree because scores everywhere are sums of many small contributions.

3. The class-IL row is a published theorem with a proof

acc ≈ ceiling·(1−ε) is Kim, Xiao, Konishi, Ke & Liu, NeurIPS 2022 (arXiv:2211.02633): CIL = WP × TP, with necessity and sufficiency proved. Our R² = 0.989 was a regression onto an identity our own notes already called "exact." Inter-task confusion as the dominant class-IL failure mode is standard survey terminology (Masana et al. 2020).

4. The analog row is row 1 linearized

"ε ≈ 0.66·noise, R² = 0.87" is the probit curve Taylor-expanded near the origin; 0.66 is a Taylor coefficient. The R² dropping from 0.997 to 0.87 is the signature of forcing a straight line through a probit curve — two fits of one identity is a misspecification, not a generalization.

5. Stronger prior work exists on the Best-of-N side

• Gao, Schulman & Hilton (ICML 2023), arXiv:2210.10760 — quantitatively fitted scaling laws R_bon(d) = d(α−βd), KL_bon = log n − (n−1)/n. These predict where the peak is; ε only predicts that there is one.
• Khalaf et al. 2025, arXiv:2506.19248 — the distribution-free version via a rank/quantile transform, proving the interior optimum is unique. This is the genuinely universal route, and it is the one we did not take.
• The phenomenology is Goodhart's regressional/extremal variants (Manheim & Garrabrant 2018) and the optimizer's curse (Smith & Winkler 2006).

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What is NOT wrong — do not "fix" these

ε* = p/(1−p) is CORRECT. An earlier draft of this notice said otherwise; that was retracted.

A prior-art gate objected that p/(1−p) is an odds ratio while ε is a probability, making the condition vacuous for p > ½. That objection is false, and so was the correction built on it. In verifier-bottleneck/FINDINGS.md F3b, p is the base rate of correct answers among sampled candidates (p = 0.10) — not a Condorcet voter competence — so p/(1−p) = 0.111 is an ordinary probability, and the published empirical crossover (0.11–0.15) brackets it.

Derivation: correct votes ≈ pN, shared-exploit votes ≈ (1−p)qN, so the exploit takes the plurality iff q > p/(1−p). Verified exact for p ∈ {0.05, 0.10, 0.20, 0.30}, max deviation 0.009, reproducing F3b at crossover 0.115 (exp_v, 4/4).

The real defect is notational. ε denotes the conditional exploit rate (among wrong answers) in this formula, threshold p/(1−p); but the unconditional rate elsewhere in the same documents, threshold p. Both are correct for their own reading. Define which ε you mean at each use. Do not change the formula.

verifier-bottleneck/FINDINGS.md already flagged this crossover as "NOT novel as a theorem — List & Goodin 2001" and advised "frame, don't claim." That was right and stands.

Also not wrong

• The measured ε values. ε = 0.38 for a naive class-IL head, the real-RM calibration ε ≈ 0.002–0.05, the AIMC routing numbers — data is not prior art. The values are ours; the functional form is Thurstone's.
• The shipped artifacts. verifier-budget CLI and adaptive_halt.py work, and their value is calibration know-how, which is unaffected by the form being classical.
• The engineering guidance. Select by consensus rather than argmax; watch N; widen the gap or shrink σ to lower ε. All sound within the Gaussian regime.

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What survives, and how to say it

Ship as: "Classical selection theory — Thurstone's Law of Comparative Judgment / signal detection theory — correctly applied to three modern systems, with calibrated constants and a working budget tool." That is defensible, useful, and will survive review.

Do not ship as: a new cross-domain law, a unification, or a universality class. It dies on the first reviewer who has taken a psychophysics or digital communications course.

One genuinely useful operational rule, which does survive

Never extrapolate a Φ fit down to small ε without checking the tail first. The law is uniformly optimistic under heavy tails and catastrophically so exactly where ε looks negligible — which is where safety arguments live. A reported ε = 10⁻⁵ can conceal a true 8×10⁻³. One statistic detects the regime: excess kurtosis, −0.07 (Gaussian) vs +26.8 (Student-t, df=3) at n = 20,000 (exp_w W3). Estimate ε directly; do not fit Φ and extrapolate.

Retracted from our own analysis

The claim that adversarial search manufactures the heavy tails is false (exp_w W2). Argmax selection leaves excess kurtosis at 0.03 — Gaussian — and instead produces a mean shift of 1.25, which is the optimizer's curse. Selection biases; it does not fatten tails.

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The transferable diagnostic

An R² ≈ 0.99+ that holds across unrelated domains should trigger "I have derived an identity," not "I have found a law."

Real empirical laws in messy domains do not fit that well. A fit that good, that portable, almost always means the functional form was implied by a distributional assumption shared by all the simulators. This is the single most useful thing to come out of the ε campaign, and it generalizes far beyond it — see NOVELTY-GATE.md.

The nameable pattern across this whole repository: every "unification" claimed here turned out to be a shared functional form arising from a shared distributional assumption, not a shared mechanism.