The per-axis base data exist under the base hypotheses on every axis

The per-axis base data exist under the base hypotheses on every axis.

The following holds (Lean statement, in the scope of module Solutions.KFMR19):

theorem NLogN.KFM.axisMaps_exists (p α : ℕ) (s t : Fin d → ℕ) [∀ i, NeZero (s i)] [∀ i, NeZero (t i)]
    (hst : ∀ i, s i < t i) (hcop : ∀ i, Nat.Coprime (s i) (t i)) (hα : 2 ≤ α)
    (hαp : α ^ 2 ≤ p) (hx : ∀ i, s i ≤ α ^ 2 * (t i - s i))
    (hp : ∀ i, 100 * (IntMul.HvdH.jIter p (s i) (t i) α *
      (10 * IntMul.HvdH.eWidth p α + 12) + 1) ≤ 2 ^ p)
    (hpA : 4 * (6 * IntMul.HvdH.sWidth p α + 4) ≤ 2 ^ p) :
    Nonempty (AxisMaps (d := d) p α s t)

Context. The lemma belongs to the part Item 12 (math side): Lemma 19/20 with the base witnesses and a permutation-free B̃₀: The resampling identities of Resample19/ResampleTensor/ResampleFinal are existential in the maps A, B₀. The machine computes the base thread's fixed-point maps Sapprox and Bapprox. This file pins the witnesses to those maps.

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.