Numerical budget (44), natural-number form
ProvedNLogN.KFM.budget_natNumerical budget (44), natural-number form.
The following holds (Lean statement, in the scope of module Solutions.KFMErr2):
theorem NLogN.KFM.budget_nat (b γ S E : ℕ) (hb : 100 ≤ b) (hγ : 4 * γ ≤ b) (hS : S ≤ 2 ^ b)
(hE : E ≤ b ^ 2 * 2 ^ b) :
2 * 2 ^ (2 * b + 4) * S ^ 2 * (3 * 2 ^ γ * (3 * E + 22) + 2) < 2 ^ (6 * b)
Context. The lemma belongs to the part KFM 14: the whole error chain and exact recovery (§9.7), abstract form.
Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine -per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs20.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction. import Definitions.Def_IntMul_Kappa_K_Defs20 set_option autoImplicit false set_option relaxedAutoImplicit false set_option maxHeartbeats 1000000 open NLogN open NLogN.KFM open Complex
theorem NLogN.KFM.budget_nat (b γ S E : ℕ) (hb : 100 ≤ b) (hγ : 4 * γ ≤ b) (hS : S ≤ 2 ^ b)
(hE : E ≤ b ^ 2 * 2 ^ b) :
2 * 2 ^ (2 * b + 4) * S ^ 2 * (3 * 2 ^ γ * (3 * E + 22) + 2) < 2 ^ (6 * b) := by sorry