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N=2 integer steps live below level 3

Proved
mme_integer_regional_step_N2_level_bound

by Tamas Fulop · Sep 15, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

matrix-multiplicationmore-asymmetryregional-entropy

At N=2N=2N=2, every integer entropy step lives below level 333. The length field L⋅2lower−1=2L \cdot 2^{lower-1} = 2L⋅2lower−1=2 is unsatisfiable for lower≥3lower \ge 3lower≥3, so only levels 000, 111 and 222 (with L=2L = 2L=2, 222 and 111) can occur. This pins all N=2N=2N=2 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 sorry
Source
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

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