ChainSizes holds (with n₂ = 0)

ChainSizes holds (with n₂ = 0).

The following holds (Lean statement, in the scope of module Solutions.KFQ):

theorem NLogN.KFQ.chainSizes : ChainSizes

Context. The lemma belongs to the part KFQ: proof of NLogN.KFM.ChainSizes: For n ≥ n₀ and dP n = D + 1:

• tBD_eq: t_i = 2^{axExp L d i}, hence ∏ t_i = T (prod_tBD) and (∏_{i<D} t_i) · r = T (prod_init);
• win: the source windows (4d−2)t_i < 4d s_i ≤ (4d−1)t_i, s_i odd;
• chainSizes : ChainSizes (with n₂ = 0), from params_ok, L_facts, setup_facts, src_facts, srcS_ok_n and the abstract lemmas of KFQArith.

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: 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.