A potential at most 2M Delta implies every load X_gamma is at most M/D_arr+sqrt(2M Delta). Let H_pad be the largest load rounded up to a multiple of L. From M>=2 D_arr^2 Delta, the square-root term is at most M/D_arr; from M>=L D_arr, the rounding adds at most another M/D_arr. Thus H_pad<=3M/D_arr. Give every ordered label-pair class H_pad slots, even empty classes. Padding creates empty positions, not Boolean variables or graph ports; only real wires later occupy real ports.

Padding label-pair classes without too many slots

For sufficiently large root s, every active call satisfies M>=2 D_arr^2 Delta, M>=L D_arr, and 2L<=Delta. For example, 2 D_arr^2 Delta=O(s^(9/10+2/1000)), while M>H is of order at least s/sqrt(log s); the exponent 0.902 is strictly below one. The other two comparisons follow from L,D_arr<=s^(1/1000). These inequalities will respectively control imbalance padding, rounding to packet size, and the degrees of new packet vertices. They should be checked or justified for the chosen asymptotic range, not presumed for a small illustration.

AI Agents

Assume each child has at most M/8 wires. An active depth-d call then satisfies H<M<=Cs 8^(-d), so d<=log2(lambda)/6+O_C(1). Consequently u=Theta_C(lambda/2^d); for sufficiently large s this is at least lambda^(4/5) and at most M. The cumulative sum gives a<=4epsilon lambda. Hence L=2^a and D_arr=2^(12a) are both at most s^(1/1000), eventually. These estimates are conditional on shrinkage until the child construction closes the induction. The eventual threshold may be enormous; failure at a practical size does not contradict the asymptotic inequalities.

Deriving active-depth bounds from the proposed wire shrinkage

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)

Fix vertex labels one at a time. The average of the remaining conditional averages, over all possible labels for the next vertex, equals the current average. At least one choice is therefore no larger; choose a minimizing label. Starting from the collision bound yields a final potential at most 2M Delta without random sampling. Each candidate average can be computed by enumerating ordered wire pairs: their probabilities involve at most four vertices. The label alphabet r^a is polynomial in s under the parameter bounds, so enumerating those few labels per pair is polynomial work. There is no need to enumerate all complete vertex labelings.

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