fm_spec (`FM` by induction on the size)

The following holds (Lean statement, in the scope of module NLogN.SSFM6):

theorem NLogN.fm_spec (pF pD pT pP : Fin P) (hPF : iprog pF = fmBody pF pD pT pP)
    (hPD : iprog pD = difBody pD) (hPT : iprog pT = ditBody pT) (hPP : iprog pP = pow2Body pP)
    (Hpar : ∀ N, 2048 ≤ fmL N → 2 ^ fmK N ∣ N →
      1 ≤ N ∧ fmB N ≤ fmL N / 2 ∧ 2 ^ fmK N * fmM N = N ∧ 1 ≤ fmM N ∧
      2 * fmM N + 2 * fmK N + 4 ≤ fmN' N ∧ fmN' N < N ∧ fmL N ≤ N ∧
      2 ^ fmK (fmN' N) ∣ fmN' N)
    (Hcost : ∀ N, 2048 ≤ fmL N → 2 ^ fmK N ∣ N →
      2 ^ fmK N * fmPhi (fmN' N) + fmLevelCost N + 1000 * (fmL N + 1) ^ 2 +
        10000 * (fmL N + 1) ≤ fmPhi N) :
    ∀ N, 1 ≤ N → (fmL N < fmThr ∨ 2 ^ fmK N ∣ N) → FMSpec pF iprog I N (fmPhi N)

Context. The lemma belongs to the part FM by induction on the size.

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.