HvdH machine lemma CR.stC_cost (Parameter computation, part 5: costs of stages B–D and of `parP` (name)

The following holds (Lean statement, in the scope of module NLogN.HCrPar5):

theorem NLogN.CR.stC_cost (K e : ℕ) (hK : 1 ≤ K) (he : 1 ≤ e) (t : IState HS) (LR LH : List Item)
    (hR : t.G hRr = .int ((2 ^ e : ℕ) : ℤ) :: LR) (hH : t.G hRh = .int ((2 ^ (e - 1) : ℕ) : ℤ) :: LH)
    (B : ℕ) (hV : 2 * (facN K).factorial * facN K + 8 * K * copQ K (2 ^ e) + 4 * K * 2 ^ e + 8 * K + 8
      < 2 ^ B) :
    (stC K).cost t ≤ (facN K + 20) * 10 ^ 9 * (B + 2) ^ 2

Context. The lemma belongs to the part Parameter computation, part 5: costs of stages B–D and of parP (namespace NLogN.CR).

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 $|z|$.