For W equal to the sum of Q independent zero-one trials with the same success probability p, Hoeffding gives Pr(|W-Qp|>=a)<=2\*exp(-2a^2/Q), for a>0. This is a standard concentration inequality invoked here, not proved by the example. Because the selected interval contains all integer levels within R/4 of the mean, its outside probability is at most 2\*exp(-R^2/(8Q)). The random choices here count basis directions uniformly. They are not independent values for the repeated original variables in auxiliary copies. The latter still share their original bits.

Applying Hoeffding to the exceptional swap directions

For the swap basis, p=(r-1)/(2r), and the number W of negative choices has mean mu=Q\*p. Set A=floor(mu-(R-1)/2) and keep levels I={A,...,A+R-1}. Its center differs from mu by less than one because of the floor. With r>=2, Q/4<=mu<=Q/2. If R>=32 and 6R<=Q, the interval lies inside {0,...,Q} and contains every integer within R/4 of mu. Therefore a level outside I must deviate from the mean by more than R/4. This converts the fraction of unmeasured-versus-measured basis directions into a tail-probability question.

AI Agents

Fix an integer beta>=1, n>=20, lambda=ceil(log2 n), and nu=ceil(log2 lambda). Assume b,w<=lambda^beta and n<=M<=bn. The recipe is r=floor(sqrt(lambda)/nu^2), Q=floor(lambda/(128\*nu)), R=(beta+3)\*ceil(sqrt(lambda)), tau=1/(r\*lambda), epsilon=1/(b\*r^2\*lambda), and D=r^(2Q). Here r counts groups, Q counts evaluation points, R bounds replacement degree, and epsilon bounds the exceptional basis fraction. This is a candidate recipe, not an automatic valid layout: if r<2 or Q<=R, stop before using its reciprocal parameters. Later checks require R>=32, 6R<=Q, balanced-schedule inequalities, and the tail bound.

The manuscript parameter recipe for a sparse measurement layout

OpenAI. The Exponential Time Hypothesis is False. Manuscript dated October 17, 2026, revised October 25, 2026. This reference pins commit 42907c9e999cc00d6abe7a50bfc1023250edd1b9. The introduction defines the exact-counting task used by this learning path; citing it does not establish that its main theorem is correct.

https://github.com/openai/math/blob/42907c9e999cc00d6abe7a50bfc1023250edd1b9/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf

OpenAI: The Exponential Time Hypothesis is False (October 25, 2026 revision)

Since r<=sqrt(lambda), log2 r<=nu/2. Consequently log2 D=2Q\*log2 r<=lambda/128, so D<=2^(lambda/128). This is a polynomial bound in n because lambda=ceil(log2 n). It does not say that D is small in practice. The layout needs arrays polynomial in D and n; without this bound, calling their construction polynomial in D would not imply polynomial preprocessing in the original input size. The argument applies only after the recipe has passed its basic checks, including r>=2.

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