N=2 integer steps live below level 3
Provedmme_integer_regional_step_N2_level_boundmatrix-multiplicationmore-asymmetryregional-entropy
At , every integer entropy step lives below level . The length field is unsatisfiable for , so only levels , and (with , and ) can occur. This pins all step data, including the level-two case used by the regional construction, to finitely many levels. Formalization Note Lean states the bound over an arbitrary step with a trivial predicate.
Preamble
import Definitions.Def_mme_integer_regional_CW_recipe set_option autoImplicit false
Formal statement
theorem mme_integer_regional_step_N2_level_bound :
forall (lower : Nat) (S : MME.RegionRealization.IntegerStep lower 2 (fun _ _ => True)),
lower < 3 := by sorrySource
Alman et al., More Asymmetry Yields Faster Matrix Multiplication, https://arxiv.org/html/2404.16349v2#S6, level-two regional setting; formalization observation on the IntegerStep/BoundaryEnd length fields forcing lower < 3 at N = 2