mme_omega_le_of_subrank_capacity

The abstract ω-bound from subrank capacity (Strassen–Schönhage / Wigderson–Zuiddam).

For any order-3 tensor T : TensorObj K 3, real upper bound R ≥ 1 on the asymptotic rank, and real lower bound V > 1 on the subrank capacity,

Proof sketch (Hölder + Schönhage's τ + asymptotic limit; deferred to a sub-decomposition):

1. From $V \leq \widetilde V(T)$, for each $\varepsilon > 0$ extract infinitely many $N$ with $\bigoplus_i \langle a_i, b_i, c_i\rangle \leq T^{\otimes N}$ and $\sum_i (a_i b_i c_i)^{1/3} \geq V^N(1-\varepsilon)$, where the number of summands $k_N$ is subexponential in $N$.

2. Apply mme_asymptotic_sum_inequality (Schönhage's τ-theorem, already Proved) to get $\sum_i (a_i b_i c_i)^{\omega/3} \leq \widetilde R(T^{\otimes N}) \leq R^N$.

3. Use Hölder's inequality with conjugate exponents $(\omega, \omega/(\omega-1))$:

4. Taking $N$-th roots and using $k_N^{1/N} \to 1$ (subexponential growth), the limit gives $V^{\omega} \leq R$, i.e. $\omega \leq \log R / \log V$.

Reusability — first-class principle. This theorem carries zero content specific to any particular tensor or paper. Every subsequent ω-bound improvement instantiates this exact bridge with its own tensor's $(R, V)$ pair; only the lower bound on subrank capacity differs paper-to-paper. The abstract framework here is the load-bearing piece of the entire matrix-multiplication-exponent program.