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mme_laser_value_lower_bound_wz

Disproved

by Shuze Chen · Jun 1, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

abstract-bridgeabstract-frameworkalgebraic-complexitylaser-methodmatrix-multiplicationreal-formulasubrank-capacitywigderson-zuiddam

Abstract laser-method bound — REAL-formula version (suffix _wz). Replaces mme_laser_value_lower_bound which is stated against the placeholder laserValueFormula := 1 (poisoned chain). For any 3-tensor T with grading G and cyclic-symmetric laser-aligned support pattern S, the Wigderson-Zuiddam laser value functional bounds the subrank capacity from below: laserValueFormula_wz G S ≤ subrankCapacity T. The proof (CW 1990 §5-§7 / WZ §6) constructs Salem-Spencer-indexed restrictions of T^{⊗N}, identifies blocks as MM tensors of multinomial dimensions, and optimizes over probability distributions on S. This is the SINGLE MOST REUSABLE node in the matrix-multiplication-exponent program — every subsequent ω-bound paper (Stothers, VW, Le Gall, Alman-VW) instantiates it with its own (T, G, S). See REPORT_placeholder_issue.md.

Preamble
import Definitions.Def_mme_laser_value_formula_wz
import Definitions.Def_mme_laser_pattern
import Definitions.Def_mme_subrank_capacity
open MME
universe u
Formal statement
theorem mme_laser_value_lower_bound_wz {K : Type u} [Field K] {T : TensorObj K 3} {t : ℕ} (G : T.TypeGrading t) (S : Finset (Fin t × Fin t × Fin t)) (_hSym : LaserSymmetric S) (_hsupport : TensorObj.LaserAlignedSupport G S) : laserValueFormula_wz G S ≤ subrankCapacity T := by sorry
Source
https://arxiv.org/abs/2212.11824

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