zsz_le_B (KFV §35: phase 1 of the layout (the tops of the long axes) as a fixed IL loop)

The following holds (Lean statement, in the scope of module Solutions.KFV32LayLoop1):

theorem NLogN.KFV.zsz_le_B {x : ℤ} {B : ℕ} (h1 : -(B : ℤ) ≤ x) (h2 : x ≤ B) : zsz x ≤ B

Context. The lemma belongs to the part KFV §35: phase 1 of the layout (the tops of the long axes) as a fixed IL loop: The long-axis flags are the bits of the mask register gMk (mkOf). Iteration k < D of a unary loop moves the bit at position rank + flag·k(ℓ-1) to position rank (mvB), then rank += flag, k += 1 (registers gRk, gI). ph1_ok: the loop computes runArr n 0 (ops1 …).

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.