HvdH machine lemma Rsp.dTabBP_all (Building the `D̃'` table `dTab p s t α` (one row `dTilde p s t α l`, `)

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

theorem NLogN.Rsp.dTabBP_all (s0 : IState HS) (p s t α Bq : ℕ) (LP LS LT LA LK LG : List Item) (hs : 0 < s)
    (hP : s0.G TP = .int (p : ℤ) :: LP) (hS : s0.G TS = .int (s : ℤ) :: LS)
    (hT : s0.G NLogN.Rsp.TT = .int (t : ℤ) :: LT) (hA : s0.G TBe = .int ((α ^ 2 : ℕ) : ℤ) :: LA)
    (hK : s0.G TK2 = .int ((s ^ 2 : ℕ) : ℤ) :: LK)
    (hG : s0.G TSg = .int ((2 * α ^ 2 - 2 : ℕ) : ℤ) :: LG)
    (hBq : 4 * (α ^ 2 + s + t + 2) ^ 6 < 2 ^ Bq) (hpB : zsz (p : ℤ) ≤ Bq)
    (hKB : zsz ((s ^ 2 : ℕ) : ℤ) ≤ Bq) (hgB : zsz ((2 * α ^ 2 - 2 : ℕ) : ℤ) ≤ Bq)
    (hEB : ∀ j, zsz (HvdH.expPos p (s ^ 2) j (2 * α ^ 2 - 2)) ≤ Bq) :
    dTabBP.ok s0 ∧ dTabBP.run s0 = s0.upd W3 (ints (dTab p s t α).flatten ++ s0.G W3) ∧
      dTabBP.cost s0 ≤ tabCost 1 s (dTWCost p (2 * α ^ 2 - 2) Bq) + 10 ^ 9

Context. The lemma belongs to the part Building the D̃' table dTab p s t α (one row dTilde p s t α l, l < s) on W3: Scalars: TP = p, TS = s, TT = t, TK2 = s², TBe = α², TSg = 2α² - 2.

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 $|z|$.