HvdH machine lemma FIN.GK_le (Final assembly, part 3: the cost of the step for `q = mkParams K N`, u)

The following holds (Lean statement, in the scope of module NLogN.HFin3):

theorem NLogN.FIN.GK_le {K N β W : ℕ} (h : HB K N β W) (Φf Φm : ℕ) :
    GK K N Φf Φm ≤ 3 * 2 ^ (mkParams K N).logOuter * Φf +
      Φm * (100 * (K + 1) * (W + 2) * 2 ^ (mkParams K N).logT) +
      (10 ^ 22 * ((K + 1) ^ 2 * ((2 ^ (mkParams K N).logT + 1) * (β + 2) ^ 2)) +
        10 ^ 64 * ((K + 1) ^ 2 * ((2 ^ (mkParams K N).e + 1) ^ 3 * (β + 2) ^ 12)) +
        10 ^ 7 * ((N + 1) * (β + 2)))

Context. The lemma belongs to the part Final assembly, part 3: the cost of the step for q = mkParams K N, under explicit bounds.

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 $|z|$.