Lemma 16 (layout)

Lemma 16 (layout). The layout L is reached from the initial slot order by |Lst| ≤ D(b+1) single-slot moves followed by at most Dq - 1 exchanges of K-bit chunks; running these primitives on the records is the pull by relabel init tgt.

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

theorem Synth.paper_layout (hK : 0 < K) (hℓ : 1 ≤ ℓ) :
    ∃ ops : List LOp,
      (ops.filter LOp.isMv).length = (lstL D ℓ K long).length ∧
      (lstL D ℓ K long).length ≤ D * (bb ℓ K + 1) ∧
      (ops.filter fun o => !o.isMv).length ≤ D * qq ℓ K - 1 ∧
      (∀ o ∈ ops, o.WF (initL D ℓ long).length) ∧
      applyN ops (initL D ℓ long) = tgtL D ℓ K long ∧
      ∀ {α : Type} (d : α) (xs : List α), xs.length = 2 ^ (initL D ℓ long).length →
        runArr (initL D ℓ long).length d ops xs =
          pull (binSh (initL D ℓ long).length) (binSh (initL D ℓ long).length)
            (relabel (initL D ℓ long) (tgtL D ℓ K long)) d xs

Context. The lemma belongs to the part §6.2 Lemma 16: the paper's layout L.

Formalization Note. This lemma belongs to the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: the array model of tape data (mixed-radix arrays and pull maps), the interchange networks of §3–4 (Propositions 7, 8, 11, Lemmas 9–12), the butterfly layers of §5 (Lemma 13, Proposition 14), the synthesis of §6–7 (Lemma 16, Proposition 18), the resampling and parameter choices of §8–9, and the final glue IntMul.OpenAI.Glue.kappaBound_of_machine, which derives the platform bound KappaBound (1/2^182) (from Def_IntMul_MultitapeModel) for any correct multitape machine meeting the paper's time recurrence. All names refer to the definition bundles IntMul_Kappa_K_Defs1–IntMul_Kappa_K_Defs9.