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Numerical range polynomial bounds for simulation teams
Edition: 8 October 2026. Initial decision: Conditional engineering. The product interpretation and pricing are hypotheses; independent proof and buyer validation remain open.
Research finding
The complete Crouzeix statement bounds matrix-valued polynomial evaluation at an operator by twice its maximum norm over the numerical range. The constant is sharp and does not require a normal matrix.
Problem and buyer
A numerical-software team evaluating polynomial filters or iterative methods on nonnormal matrices needs more informative bounds than eigenvalues alone may provide.
What the finding could enable
A certificate toolkit could combine an outer enclosure of the numerical range with a rigorously bounded scalar or matrix-valued supremum. That may enable validated error estimates for narrowly defined polynomial computations.
Technical and commercial limits
Sampling the numerical-range boundary is not a certified enclosure. A norm inequality is not automatically a complete stability certificate for a nonlinear model, a discretization, or rational functions with poles in forbidden regions.
Minimal architecture
Matrix import -> validated numerical-range enclosure -> interval or exact polynomial bound -> operator-bound report. Compare the sharp constant with prior bounds and include enclosure and arithmetic errors.
Existing alternatives and differentiation
Established numerical linear algebra and hand-derived analysis are the baseline. A useful library must reduce conservatism or analyst time in a concrete application, with independent validation.
Monetization hypothesis
Hypothesis: AUD 5,000-15,000 for an engineering-method pilot, then a supported numerical-library license. Value comes from meaningful certificates, not simply plotting a numerical range.
Validation experiment
Test normal matrices, highly nonnormal matrices and the sharpness examples. Compare bounds with direct operator norms on manageable sizes and ensure numerical error never produces an invalid upper bound.
Conditions to reject or defer
Reject if computing a rigorous enclosure costs more than the direct calculation or if the tighter constant has negligible decision value.
Next concrete action
Identify one polynomial approximation workflow and derive the enclosure and supremum computation with explicit rounding control.
Source evidence
Repository sources are pinned to revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Selected scope notes and manuscript statements have been reviewed to the extent described above; these links do not represent successful kernel checks.
Family 325
Subject: The complete Crouzeix conjecture.