HvdH machine lemma FIN.theorem_1_1_nlogn (Final assembly, part 5: the `O(n log n)` theorem)

The following holds (Lean statement, in the scope of module NLogN.HFin5):

theorem NLogN.FIN.theorem_1_1_nlogn : IntMul.MulTimeBound fun n => n * Real.log n

Context. The lemma belongs to the part Final assembly, part 5: the O(n log n) theorem: For K ≥ 10^4 and N huge, HB K N (200 b) (1000 K γ³) holds (part 4), so GK_le bounds the non-multiplication cost lin by O(N log N) and the two recursive-call families by a quarter of TT' N each. Top.hv_mulTimeBound with D = 2 then gives MulTimeBound (n log n).

Formalization Note. This is a lemma of the Lean development NLogN, a multitape-Turing-machine implementation of the Harvey–van der Hoeven $O(n\log n)$ integer multiplication algorithm, used in the proof of NLogN.FIN.theorem_1_1_nlogn : IntMul.MulTimeBound fun n => n * Real.log n. All names refer to the definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8 (namespace NLogN), built on the platform model Def_IntMul_MultitapeModel and the published reference Def_IntMul_HvdH_MachineRef. The machine layer: IState HS is a state of 64 item stacks; RM programs have a semantics p.run s, a definedness predicate p.ok M s and a cost p.cost M s relative to a multiplication spec M : MulSpec; zsz z is the bit length of $|z|$.