We prove that the tensor rank of 3×3 matrix multiplication is exactly 23, resolving a problem open since Laderman's 1976 upper bound. The proof identifies the A-slice determinant det (SA) as a structural invariant. For the matrix multiplication tensor M₃, det (SA (M₃) ) = −det (Aₘat) ³ — a perfect cube of an irreducible cubic. The main technical result (Theorem G) shows that no tensor of rank ≤ 22 can have det (SA) = f³ for an irreducible cubic f, via a five-step argument: the cube factorization forces corank ≥ 3 on f = 0 (Thom–Porteous), the leading perturbation term forces proportionality of contracted matrices (recursive corank), the contraction image collapses to one dimension, and an overdetermined linear system (504 constraints on 486 unknowns) rules out off-diagonal corrections at a fourth generic point. Combined with Theorem D (det (SA (M₃) ) = −det (Aₘat) ³), this gives R (M₃) ≥ 23, matching Laderman's upper bound. Additionally, we show that Makarov's 1986 22-term commutative algorithm decomposes a tensor T ≠ M₃ (Theorem A), that T is not GL₉׳-equivalent to M₃ (Theorem B), and that the antisymmetric correction cannot be absorbed at first order (Theorem C). All four computational genericity conditions are verified rigorously over 𝔽₃₁ using the Schwartz–Zippel lemma. The reproduction package includes the LaTeX manuscript and three Macaulay2 verification scripts (detcubeₜest. m2, verifyᵢndependence2. m2, step5ₒverdetermined. m2). Each script runs in seconds and verifies one of the four genericity conditions used in the proof.
Santosh Arron (Fri,) studied this question.