prod_tB (KFY 11: size facts of the padded box)
ProvedNLogN.KFY.prod_tBcomplexity-theoryinteger-multiplicationkappa-improvement
The following holds (Lean statement, in the scope of module Solutions.KFY11Facts):
theorem NLogN.KFY.prod_tB {n : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) : (List.ofFn (NLogN.KFM.tB n)).prod = 2 ^ IntMul.OpenAI.Glue.LP n
Context. The lemma belongs to the part KFY 11: size facts of the padded box.
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 set_option linter.unusedSimpArgs false set_option linter.unusedVariables false open NLogN open NLogN.KFY open IntMul IntMul.OpenAI.Glue IntMul.OpenAI.RP NLogN.KFM
Formal statement
theorem NLogN.KFY.prod_tB {n : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) : (List.ofFn (NLogN.KFM.tB n)).prod = 2 ^ IntMul.OpenAI.Glue.LP n := by sorrySource
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.KFY.prod_tB` of the Lean development Solutions/KFY11Facts.lean (lines 48–65), integer-multiplication-kappa campaign.