Sharp completion bounds for polynomial values of matrices

Henry Zweiman — September 9, 2026

A complete six-page manuscript with a full Markdown edition and reproducible exact checks. This is an internally checked proposed result, not an independently certified solution or a peer-reviewed publication.

For an algebraically closed field of characteristic zero, let E(f) be the maximum, over values lambda, of the minimum multiplicity of a root of f(t)-lambda. The manuscript proves that every prescription of r rows of an n by n matrix extends to f(X) exactly when n > E(f)(r-1). It classifies surjectivity and image linearity for sums A_i f_i(X_i) when the coefficient column spaces form a direct sum. For complementary A and B, AX^k+BY^l is surjective exactly when n > k(rank A-1) and n > l(rank B-1).

The broad source question is Gangwal--Mandal--Verma Question 1.1. The overlapping-coefficient problem, including Saini--Singh Question 6.1, remains unresolved here. The existing nilpotent obstruction and classical Kronecker decomposition are credited.

Run python3 check.py with Python 3.8 or later. The program uses only the standard library. Its finite checks support the written proof; they do not establish an infinite theorem by enumeration.