The explicit sources are valid (the witness of RCop.coprime_params)

The explicit sources are valid (the witness of RCop.coprime_params).

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

theorem NLogN.KFM.srcS_ok (d d' r : ℕ) (hd : 1 ≤ d) (hd' : d' ≤ d) (hr : 2 ∣ r)
    (hbig : 64 * d ^ 2 * (256 * d ^ 2).factorial ≤ r) :
    (∀ i : Fin d, Odd (srcS d d' r i)) ∧
    (∀ i j : Fin d, i ≠ j → Nat.Coprime (srcS d d' r i) (srcS d d' r j)) ∧
    ∀ i : Fin d, (4 * d - 2) * (if (i : ℕ) < d' then r / 2 else r) < 4 * d * srcS d d' r i ∧
      4 * d * srcS d d' r i ≤ (4 * d - 1) * (if (i : ℕ) < d' then r / 2 else r)

Context. The lemma belongs to the part KFM 14: the setup arithmetic (§9.2), in the integer form the machine computes.

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_Defs20.