Minimum-Rank and Mortality Bounds for Finite Real Matrix Monoids

For any family of real $n\times n$ matrices generating a finite entire product monoid of minimum rank $s$, some word attains rank $s$ within $$B(n,s)=n2^{n-s}-\frac{n(n+1)}2+\frac{s(s-1)}2$$ letters. The same holds over $\mathbb Q$, without requiring a finite generating alphabet. For mortality, $s=0$, giving a bound of order $n2^n$. Over $\mathbb Q$, this improves Kiefer–Ryzhikov's (2026) $3^{n^2}$ bounds for mortality and minimum-rank diameter, and Almeida–Steinberg's (2009) mortality bound $(2n-1)^{n^2}-1$ for $n>1$. The proof uses rank-dependent sandwich descent. Further results give the sharp planar threshold four, additive invariant-flag bounds with sharp small-block examples, and cubic-size compressed witness existence with correct evaluation. Boundedness and individually finite-power generators alone admit no uniform planar mortality bound. General optimality and polynomial-time synthesis are not claimed.

For any family of real $n\times n$ matrices generating a finite entire product monoid of minimum rank $s$, some word attains rank $s$ within $$B(n,s)=n2^{n-s}-\frac{n(n+1)}2+\frac{s(s-1)}2$$ letters. The same holds over $\mathbb Q$, without requiring a finite generating alphabet. For mortality, $s=0$, giving a bound of order $n2^n$. Over $\mathbb Q$, this improves Kiefer–Ryzhikov's (2026) $3^{n^2}$ bounds for mortality and minimum-rank diameter, and Almeida–Steinberg's (2009) mortality bound $(2n-1)^{n^2}-1$ for $n>1$. The proof uses rank-dependent sandwich descent. Further results give the sharp planar threshold four, additive invariant-flag bounds with sharp small-block examples, and cubic-size compressed witness existence with correct evaluation. Boundedness and individually finite-power generators alone admit no uniform planar mortality bound. General optimality and polynomial-time synthesis are not claimed.