HvdH machine lemma Rsp.XE_bnd (Building the `S` weight table and the step table)

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

theorem NLogN.Rsp.XE_bnd (σ : IState HS) (s t m i k c Bq : ℕ) (ht : 0 < t) (hi : i ≤ 2 * m) (hk : k < t)
    (hc : c ≤ s) (hcv : ((s : ℤ) * (((k + 1 : ℕ) : ℤ) + -1) + ((t : ℤ) + -1)) / (t : ℤ) = c)
    (hBq : 4 * (s + t + 2 * m + 2) ^ 4 < 2 ^ Bq)
    (hI : σ.top TCI = ((i + 1 : ℕ) : ℤ)) (hK : σ.top TCK = ((k + 1 : ℕ) : ℤ))
    (hS : σ.top TS = s) (hT : σ.top NLogN.Rsp.TT = t) (hM : σ.top TM = m) : NLogN.Rsp.XE.Bnd σ Bq

Context. The lemma belongs to the part Building the S weight table and the step table: sTabP pushes ints (sTab p s t α).flatten on BW; sDelP pushes ints (sDel s t) on BD. Scalar parameters (registers): TP = p, TS = s, TT = t, TM = m = sWidth p α, TBe = betaT p α, TK2 = α² t², TRI = 2m+1; counters TCI, TCK.

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 $|z|$.