preB_fsp (KFE: fine binary embedding / extraction and fine high-digit swaps)

The following holds (Lean statement, in the scope of module Solutions.KFESwap):

theorem NLogN.KFE.preB_fsp (s : IState HS) (F : Fin HS → List ℤ) (v : Fin HS → ℕ) {M B : ℕ}
    (hM : zsz (M : ℤ) ≤ B) (hv : ∀ i ∈ regD, v i ≤ M) {Pre G Suf : Shape} {T Q : ℕ}
    {LP LG LS : List ℤ}
    (FP : F kP = (vol Pre : ℤ) :: LP) (FG : F kG = (vol G : ℤ) :: LG)
    (FS : F NLogN.KIL.kS = (vol Suf : ℤ) :: LS)
    (vP : v kP = vol Pre) (vG : v kG = vol G) (vS : v NLogN.KIL.kS = vol Suf) (vT : v aT2 = T)
    (vQ : v aQE = Q) (hP : 1 ≤ vol Pre) (hg : 1 ≤ vol G) (hS : 1 ≤ vol Suf) (hT : 1 ≤ T)
    (hTQ : T ≤ Q) (hVM : vol (binShape Pre G Suf Q) ≤ M) (xs : List (ZMod 2))
    (F0 : F 0 = enc2 xs) (F15 : F 15 = []) (F20 : F 20 = []) (F21 : F 21 = [])
    (hxs : xs.length = vol (binShape Pre G Suf T)) (hB1 : 1 ≤ B) :
    ∃ v' : Fin HS → ℕ, (∀ i, i ∉ [pX, pB, pZ, pT] → v' i = v i) ∧ (∀ i ∈ regD, v' i ≤ M) ∧
      FSp preB (encK s (regState F v regD))
        (encK s (regState (Function.update F 0
          (enc2 (AModelL12.embed (binShape Pre G Suf T) (binShape Pre G Suf Q) 0 xs))) v' regD))
        (2 * (10 ^ 9 * (B + 2) ^ 2 + cPadF (vol (binShape Pre G Suf Q)) B))

Context. The lemma belongs to the part KFE: fine binary embedding / extraction and fine high-digit swaps.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs18.