The fine uniform shift body

The fine uniform shift body.

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

theorem NLogN.KFD.shFracU_fspec [NeZero Q] (i : Fin n) (c : Fin n → ZMod Q)
    (hc : ∀ j, i ≤ j → c j = 0) (A C : Shape) (hsh : sh = A ++ Q :: C)
    (hA : (Λ.pos i : ℕ) = A.length) (k d : ℕ) (neg : Bool) (hk : 1 ≤ k) (hkQ : k ≤ Q)
    (hd : 1 ≤ d) (hdQ : d ≤ Q) (hQk : Nat.Coprime Q k) (hQd : Nat.Coprime Q d)
    (t : IState HS) (xs : List ℤ) (w Bb : ℕ) (R : List Item)
    (hlen : xs.length = vol sh) (hb : Bnd w xs) (hB1 : 1 ≤ vol C) (hQ1 : 1 ≤ Q)
    (hP1 : 1 ≤ vol A) (hV : zsz ((vol A * (Q * vol C) : ℕ) : ℤ) ≤ Bb)
    (hQ2 : zsz ((2 * Q : ℕ) : ℤ) ≤ Bb) (h0 : t.G 0 = ints xs ++ R)
    (ac : List (ℕ × ℕ)) (hac : ∀ p ∈ ac, 1 ≤ p.1 ∧ p.2 < Q)
    (hvol : vol (ac.map Prod.fst) = vol A) (htbl : NLogN.KIL.tbl Q ac = NLogN.KIL.tbl Q (A.zip (csOf Λ c A)))
    {LA LC LM LAM LN : List Item}
    (hS : StgOK t Q ac LA LC LM LAM) (hNQ : t.G sNQ = .int (-(Q : ℤ)) :: LN)
    (hp : StP t (vol A) Q (vol C)) :
    FSp (shFracU k d ac.length neg) t
      (postSt t (shiftArr Λ i (fracU Q k d neg) c 0 xs) R LA LC LM LAM)
      (cShFF k d ac.length (vol A * (Q * vol C)) w Bb)

Context. The lemma belongs to the part KFD: closed bounds for the fine gather / scatter, fine shifts.

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.