The network acts as the sequence of its steps

The network acts as the sequence of its steps.

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

theorem NLogN.KFN.roleOpF_net {st : Fin m × κ → ℕ} {n : ℕ} (h : SLaws st n) (T : Butterfly.Op κ) :
    ∀ (l : List (RStep (Fin W) m)) (X : Fin W → List M), (∀ w, (X w).length = n) →
      roleOpF st n (denListR T l) X = runSteps st n T l X

Context. The lemma belongs to the part KFN: the role streams of a network invocation: A network state on W roles is stored role-major: role w is a list X w of n records, and record j of a role carries the group address fbit st j : Addr m κ (selected bits with record strides st, KFNSel). roleOpF st n A X is the action of a network operator A : ROp (Fin W) m κ on these streams.

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.