shiftU_runs (KIL2: the Lemma 10 controlled shift with unit coefficient `u = ±c` (`2 ≤ c ≤ 20`)

The following holds (Lean statement, in the scope of module Solutions.KIL2Unit):

theorem NLogN.KIL.shiftU_runs [NeZero Q] (hpos : ∀ a ∈ sh, 1 ≤ a) (t : IState HS) (R : List Item)
    (Bb : ℕ) (hV : zsz ((AModel.vol sh : ℕ) : ℤ) ≤ Bb) (hQ2 : zsz ((2 * Q : ℕ) : ℤ) ≤ Bb)
    (i : Fin n) (u : ZMod Q) (hu : SmallUnit Q u) (c : Fin n → ZMod Q)
    (hc : ∀ j, i ≤ j → c j = 0) (ys : List ℤ) (hy : DOK sh Bb ys) :
    IRuns I iprog (scodeU Λ (.shift i u c)) (t.upd 0 (ints ys ++ R))
      (t.upd 0 (ints ((Step.shift i u c).arr Λ 0 ys) ++ R)) (cShiftU sh.length (AModel.vol sh) Bb)

Context. The lemma belongs to the part KIL2: the Lemma 10 controlled shift with unit coefficient u = ±c (2 ≤ c ≤ 20): shiftArr Λ i u c with u = c (resp. u = -c), c a small positive integer prime to Q, is done as the paper's Lemma 4: first the fixed multiplicative permutation y ↦ c·y (mod Q) of the target field (mulPc: split into c increasing streams, merge by residue), then the rotation shP (resp. the reflection shN) by the offsets Σ_j c_j·d_j.

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.