Bounded numerical-range polynomial comparison
Eight generated exact finite cases compare independent norm-threshold evidence with a separate covered numerical-range bound. Every listed direct threshold is complete and true. The source constant-two upper is conditional on the unverified repository theorem. The prior upper uses 483/200, rounding the published 1+sqrt(2) constant upward. SciPy norms below are floating diagnostics, not certified bounds. Exact inputs/results, SciPy/runtime diagnostics, 857 finite controls, 195 floating controls.
| Generated case | Tolerance | Floating norm diagnostic | Supremum upper (approx.) | Conditional source upper | Rounded prior upper | Conditional/prior upper fits | Direct/full arithmetic charges |
|---|---|---|---|---|---|---|---|
| nilpotent_sharp | 11/5 | 2 | 1.09468 | 2.18936 | 2.64365 | True/False | 231/84404 |
| normal_segment | 2 | 2 | 2 | 4 | 4.83 | False/False | 222/2633 |
| complex_point | 1 | 1 | 1 | 2 | 2.415 | False/False | 52/562 |
| constant_matrix | 2 | 2 | 2 | 4 | 4.83 | False/False | 815/16669 |
| matrix_coefficients | 10 | 2.75574 | 3.38113 | 6.76226 | 8.16543 | True/True | 1301/21037 |
| zero | 1/100 | 0 | 0 | 0 | 0 | True/True | 1109/2722 |
| imaginary_hermitian | 2 | 2 | 2.01562 | 4.03125 | 4.86773 | False/False | 297/10773 |
| nilpotent_cancellation | 1/100 | 0 | 4.67806 | 9.35612 | 11.2975 | False/False | 327/16427 |
The exact reports contain canonical rationals; table decimals are display only. The nilpotent sharp case has eigenvalues zero and direct norm two. A conditional 2.18936 upper fits tolerance 2.2 where the prior 2.64365 does not, but its direct exact check already proves 2.0 using 231 charges. The covered support/grid workflow uses 84,404 charges on the same tiny case. Direct and full counts have the same selected arithmetic convention, excluding integer/gcd bit cost and native setup/allocation; no general speed conclusion follows.
Normal/point/constant cases expose global-constant looseness. The nilpotent-cancellation polynomial evaluates to zero but a positive numerical-range upper remains. The matrix-coefficient case preserves ordered base-first Kronecker evaluation and independently checked indexing. Floating SciPy SVD/eigvalsh agree within 1e-11 on the generated diagnostics. Exact PSD permits zero pivots and treats a nonzero row at a zero pivot as indefinite.
The enclosure has twenty rational unit-direction support halfspaces with exact PSD support tests. Closed cells cover all possible intersection with the numerical range; only wholly excluded cells are removed. Frobenius center bounds plus coefficient Lipschitz/radius correction bound every retained cell. No sampled maximum is promoted to a certificate. Budgets retain only completed phases, and exact inputs still need uncertainty/physical-model bridges.
No customer workload, general sparse backend, source proof execution, formal Python proof, repeated-enclosure integration, broad stability claim or business advantage is established. Single-run elapsed timings are diagnostic only. Business/source decision.