Lemma 12 swap wrapper

Lemma 12 swap wrapper. Stack 0 is enc2 xs ++ R; the two [2^u] fields are interchanged; cost ≤ K (V + 1) u^τ.

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

theorem NLogN.KFL.swI_spec :
    ∃ T q₀ : ℕ, ∀ q, q.Prime → q₀ < q → ∃ K : ℝ, 0 ≤ K ∧
      ∀ (u : ℕ) (Pre G Suf : Shape), 1 ≤ u →
      (∀ a ∈ Pre, 1 ≤ a) → (∀ a ∈ G, 1 ≤ a) → (∀ a ∈ Suf, 1 ≤ a) →
      ∀ (I : List Sym) (t : IState HS) (xs : List (ZMod 2)) (R : List Item),
      t.G 0 = ints (enc2 xs) ++ R →
      xs.length = vol (binShape Pre G Suf (2 ^ u)) →
      ∃ c : ℕ, IRuns I l12prog (swI q (vol Pre) (vol G) (vol Suf) u) t
        (t.upd 0 (ints (AModel.pull (binShape Pre G Suf (2 ^ u)) (binShape Pre G Suf (2 ^ u))
          (swapPos Pre.length (Pre.length + 1 + G.length)) 0 (enc2 xs)) ++ R)) c ∧
        (c : ℝ) ≤ K * ((vol (binShape Pre G Suf (2 ^ u)) : ℝ) + 1) *
          (u : ℝ) ^ (1 - (2 : ℝ)⁻¹ ^ 50)

Context. The lemma belongs to the part KFL §4: the Lemma 12 swap wrapper: swI q vP vG vS u pushes the four Lemma 12 parameters (kP, kG, kS, aU), runs the fine Lemma 12 program l12F, and pops them. swI_spec: on any state whose stack 0 is enc2 xs ++ R (arbitrary tail R, no other hypothesis) it interchanges the two [2^u] fields of binShape Pre G Suf (2^u) at cost ≤ K (V + 1) u^τ, τ = 1 - 2^-50.

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.