# Numerical range polynomial certificate toolkit

Family 325: The complete Crouzeix conjecture. First review, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Simulation teams can bound polynomial operator evaluations for nonnormal matrices using the sharp constant-two inequality.

## Applicability and commercial boundary

The numerical-range enclosure and scalar supremum must themselves be certified; sampled plots alone provide no rigorous upper bound.

## Initial business decision

Conditional engineering. Commercial demand and profitability remain hypotheses.

## Next verification action

Develop one rounded-arithmetic enclosure workflow and compare with direct operator norms.

## Evidence scope

The catalog statement was individually reviewed. Main-paper and selected formal-scope passages were additionally inspected for families 090, 093, 094, 097, 325, 328 and 332; this record does not claim a full proof audit or independent Lean verification. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).