Results on tensors, characters, polynomial multiplication and states: rank and flattenings, interpolation, finite separation, the entropy inequality, concavity and tripling of the profile, diagonal growth, detecting states and characters, and the theorem, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Matrix multiplication with exponent at most 9/4 (OpenAI) — Theorem A. For every eps > 0 there is C_eps such that two n x n complex matrices can be multiplied with at most C_eps n^(9/4+eps) arithmetic operations; thus omega <= 9/4. Proved in the last lesson.
• Tensors of bilinear maps — Lemma 1.3. The tensors of a bilinear map in two choices of bases are restrictions of each other; the tensor of a tensor product of bilinear maps is the tensor product of their tensors.
• Flattening ranks — Lemma 1.5. Each flattening rank is monotone under restriction, additive on direct sums, multiplicative on tensor products, equal to 1 at the unit tensor and at most the rank.
• Matrix multiplication tensors — Lemma 1.6. T_n (x) T_m is isomorphic to T_nm; the three flattening ranks of T_n are n^2, so n^2 <= R(T_n) <= n^3; and T_n restricts to the sum of n unit tensors.
• Polynomial multiplication — Lemma 1.8. The tensor C(a,b) of multiplication of polynomials of degrees < a and < b has rank a+b-1 and flattening ranks a, b, a+b-1; exchanging its input legs gives C(b,a).
• From rank to algorithms — Proposition 1.10. If nu <= tau, then for every eps > 0 two n x n complex matrices can be multiplied with O(n^(tau+eps)) arithmetic operations, by recursion on a rank decomposition of some T_u.
• Characters on matrix multiplication — Lemma 2.5. Every character satisfies lambda(B_X(m)) = m^(p_X), and similarly for Y and Z, and lambda(T_m) = m^(3t) with t = (p_X+p_Y+p_Z)/3 in [1/3, 1].
• Interpolation — Lemma 2.9. If A' is a degeneration of A (a restriction followed by integer weights and the part of least weight), then lambda(A) >= lambda(A') for every character lambda.
• Permuted characters — Lemma 2.11. Composing a character with a permutation of the legs gives a character with the same three exponents in some order; the sum of p_X over the six permutations is 6t.
• Finite separation (OpenAI) — Proposition 3.2. If A = A_1 + ... + A_M has a shared X-leg and matched sectors, then 5M copies of A degenerate to the direct sum of the tensors A_h (x) B_X(M), each with its own X-variables.
• Counting words — Lemma 3.3. The multinomial coefficient N!/prod (N q_i)! lies between (N+1)^(-s) e^(N H(q)) and e^(N H(q)).
• Shared-leg entropy inequality (OpenAI) — Corollary 3.4. For a sum T of nonzero tensors T_i with a shared X-leg and matched sectors, every character lambda and every probability vector q, lambda(T) >= e^(p_X H(q)) prod lambda(T_i)^(q_i).
• The symmetrized profile — Lemma 4.1. P(a,b), the 6t-th root of the product of lambda(pi C(a,b)) over the six leg permutations pi, is symmetric, satisfies P(1,b) = b and P(a,b) <= (a+b-1)^(1/t).
• Discrete concavity (OpenAI) — Lemma 4.3. 2P(a,b) >= P(a,b+1) + P(a,b-1) for a >= 1 and b >= 2, from a degeneration of C(a,b) (x) B_X(2) built on the exact sequence 0 -> D V_(e-1) -> V_e (x) W -> V_(e+1) -> 0 with D = uw - vs.
• Shifted tripling (OpenAI) — Lemma 4.4. P(a, 3h+a-1) >= 3P(a,h) for all a, h >= 1, from a degeneration of C(a, 3h+a-1) into three sectors with a shared X-leg.
• Diagonal growth (OpenAI) — Lemma 5.1. A symmetric positive function on pairs of positive integers with P(1,b) = b, discrete concavity and shifted tripling satisfies P(a,a) >= a^(4/3).
• The bound on characters (OpenAI) — Theorem 5.2. Every character satisfies t(lambda) <= 3/4, that is, lambda(T_d) <= d^(9/4) for every d.
• Compactness of the state space — Lemma 6.3. For the countable field F, the functions 0 <= mu <= R on S form a compact subset of a normed space on which norm and pointwise convergence agree; the detecting states form a compact convex subset.
• Finitely generated cones — Lemma 6.5. A finitely generated cone in a finite-dimensional real vector space is closed, and every point outside it is separated from it by a linear functional that is nonnegative on the cone.
• Rational solutions — Lemma 6.6. A linear system with rational coefficients that has a nonnegative real solution has a nonnegative rational solution.
• Detecting states (OpenAI) — Proposition 6.7. If k < d^nu, there is a state mu with mu(T_d U) >= k mu(U) for all U; otherwise a catalytic comparison E + kU >= E + m + T_d U would force k >= d^nu.
• A multiplicative state — Lemma 7.2. A nonempty compact convex set of states that is mapped into itself by every normalized product mu -> mu(z .)/mu(z) contains a character; the proof uses Schauder's theorem and nested compact sets.
• Detecting characters (OpenAI, after Strassen) — Theorem 7.3. For integers d >= 2 and 0 <= k < d^nu there is a character lambda with lambda(T_d) >= k.
• The exponent of rank — Theorem 7.4. nu <= 9/4 over the field generated by the roots of unity, and hence over every field containing it.

Reading order

Read the lessons in order. Lessons 1 to 5 prove that every character satisfies lambda(T_d) <= d^(9/4); Lessons 6 and 7, which construct characters that are large on T_d, need only Lessons 1 and 2.