MMC

Fields covered — what each algebra means (and why ZT is not one)

Tag	Algebra	Coefficients	Notes
`F2` (= `Z2`)	GF(2)	{0, 1}, integers mod 2	Characteristic 2 — its own universe, no inclusion into the characteristic-0 chain. AlphaTensor's home field.
`F3` (= `Z3`)	GF(3)	{0, 1, 2}, integers mod 3	Ternary modular — three residues in a characteristic-3 field. Distinct from ZT below: same "three values" count, completely different arithmetic.
`Z`	integers ℤ	any integer	Characteristic 0. `Z ⊂ Q ⊂ R ⊂ C`.
↳ `ZT`	sub-class of Z — not a field	{−1, 0, +1} ⊂ ℤ	Ternary integer: a Z scheme whose every factor-matrix coefficient is −1, 0 or +1 (Perminov's "ternary" schemes). Carried as the per-scheme boolean `ZT`, present only when Z is covered. It is NOT F₂/Z₂ — that was a historical conflation; ZT is integer arithmetic, not mod-2.
`Q`	rationals ℚ	any rational	Characteristic 0; implies R, C.
`R`	reals ℝ	any real	Characteristic 0; implies C.
`C`	complex ℂ	any complex	Accepts every R/Q/Z scheme; plus C-only schemes (e.g. AlphaEvolve's complex-coefficient originals — its ⟨4,4,4⟩=48 rank is no longer C-only since Dumas–Pernet–Sedoglavic 2025 rationalised it to Q/R).

"Ternary" is overloaded: F3/Z3 is ternary modular (GF(3)), whereas ZT is ternary integer (coefficients in {−1,0,1} over ℤ). They are unrelated algebras that happen to share the word.

⟨∞,∞,∞⟩ — asymptotic exponent ω over time

The matrix-multiplication exponent ω — smallest constant such that ⟨n,n,n⟩ can be done in O(n^ω) operations as n → ∞. Lower bound is the information-theoretic ω ≥ 2; whether ω = 2 is open. Data from docs/omega-history.json. The plot also overlays the commutative exponent ω_c (data from docs/omega-history-commutative.json) — each green point is a single-shape upper bound log_n(R_c(n)); commutative algorithms do NOT lift to recursive matrix matmul, so the commutative series is not an aggregated asymptotic like NC ω.

log-scale y-axis (expands the band near ω = 2) y-axis is bounded [2, max(ω) + 0.05]; linear by default.

kind	year	ω ≤	method	source	notes

ω heatmap — per-format effective exponent

Each cell is a format ⟨n,m,p⟩ coloured by its effective exponent ω = 3·ln(r)/ln(n·m·p) (the best non-commutative rank we hold; lower = closer to optimal, 3 = naïve). Rows are the largest axis maxDim = p (2…32), labelled on the right. Each row is right-anchored on the cube axis: the rightmost cell of every row is the cube ⟨p,p,p⟩ (outlined), so the cubes line up in a single vertical column on the right. Moving left within a row visits the smaller shapes ⟨n,m,p⟩ (n≤m≤p), so the left edge is ragged — the lone ⟨2,2,2⟩ cell up top, the full ⟨…,…,32⟩ row at the bottom. Prime maxDim rows are flagged ◆ — they are the smoothness floor where serendipity and cube-projection have least to exploit. Block width tapers with distance from the cube so the cube and near-cube columns stay legible while the long tail stays compact. Hover a cell for the exact rank and ω.

Colour by: Column order: — same cells (best held NC rank per shape), recoloured / reordered.

Colour is self-calibrated to the best/worst ω actually held (not a fixed 2…3 window): best → worst no scheme ω < Strassen (log₂7 ≈ 2.807) — better-than-Strassen shape. Heatmap respects the Field filter.

MMC — Matrix Multiplication Catalog

⟨∞,∞,∞⟩ — asymptotic exponent ω over time

ω heatmap — per-format effective exponent

References