zsz_one (Linear bounds on the costs of item operations)
ProvedNLogN.zsz_onecomplexity-theoryinteger-multiplicationkappa-improvement
The following holds (Lean statement, in the scope of module NLogN.Costs):
theorem NLogN.zsz_one : zsz 1 = 1
Context. The lemma belongs to the part Linear bounds on the costs of item operations.
Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below : the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.
Preamble
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction. import Definitions.Def_IntMul_Kappa_K_Defs33 set_option autoImplicit false set_option relaxedAutoImplicit false set_option maxHeartbeats 1000000 open NLogN
Formal statement
theorem NLogN.zsz_one : zsz 1 = 1 := by sorry
Source
OpenAI, Integer multiplication below n log n (preprint, 23 Sep 2026), https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026. Lemma `NLogN.zsz_one` of the Lean development NLogN/Costs.lean (lines 59–60), integer-multiplication-kappa campaign.