Polynomial-bound evidence for nonnormal matrix workflows

Edition: 9 October 2026 Australia/Brisbane. Decision: Prototype bounded polynomial/model review; defer a general source-certified numerical library. Commercial confidence is low. No buyer demand, profitable delivery or independently accepted repository proof is established.

Research finding

Family 325 claims the sharp complete Crouzeix inequality: for any complex n-by-n matrix A, m-by-m coefficient matrices B_k and finite degree, ||sum A^k tensor B_k||_2 is at most twice max over z in W(A) of ||sum z^k B_k||_2. Both matrix dimensions must be positive; A need not be positive semidefinite or normal. W(A) is its entire Euclidean numerical range, including point and segment cases. Matrix coefficients require the specified tensor order, rather than a silent scalar-coefficient substitution.

The entire 623-line direct manuscript main.tex was read, including its coefficient/positive-kernel, amplified singular-pair, convex approximation, sharpness and conformal-collar arguments. Its referenced figure and bibliography were not reviewed. The 24-line family scope, 44-line Comparator, 15-line config, 41-line solution import entry, 119-line CompleteBound and 95-line Sharpness files were read. Full imported proof semantics, source compilation/kernel comparison and independent mathematical acceptance remain open. The companion structural manuscript is still individually pending. Comparator challenge placeholders are the intended proof interface.

The proof works through positive boundary kernels/Faber coefficients and an amplified top singular pair, then approximates arbitrary convex numerical ranges. It is a mathematical bound, not a ready numerical enclosure or maximum-search implementation. Its outer approximation calls on support values and analytical geometry; actual support computation, rounding and cost need engineering. Pinned direct argument, selected bound text.

A prior published theorem gives the complete constant 1+sqrt(2). The reference rounds it upward to rational 483/200=2.415, since (283/200)^2 exceeds two. This published prior result is separate from the repository's unverified sharper claim, and its proof was not executed here. For the same supremum upper bound, the ideal constant change alone is approximately 17.16%; enclosure conservatism can erase decision value. Crouzeix–Palencia published result, author abstract.

Problem and buyer

A scientific-computing team choosing polynomial filters or error polynomials for a nonnormal finite matrix needs a useful operator error bound. Eigenvalues alone can hide growth. For A=[[0,2],[0,0]] and p(z)=z, both eigenvalues are zero but ||p(A)||_2=2 and W(A) is the unit disk. That is a precise model failure which an auditable report can expose.

The commercial question is narrower than creating another numerical-range plot: does an engineer repeatedly need defensible polynomial error decisions where forming/evaluating a large matrix polynomial or obtaining a direct certified norm is expensive? No actual customer matrix, repeated workload or buying process was obtained. On all eight small generated cases, direct exact threshold evidence already suffices without the new theorem, so these examples establish no reason to purchase a standalone library.

What the finding could enable

Potential software is an error-budget review framework taking a matrix model, candidate scalar or matrix-valued polynomial and declared tolerance, then reporting numerical-range coverage, supremum control, arithmetic assumptions and theorem stage. A repeated-filter workflow could reuse a validated enclosure for many polynomials; this reuse/cache backend is not implemented here. A block-filter SDK could preserve matrix coefficients and tensor order. A regression integration could expose changes in enclosure, polynomial degree, tolerance and evidence status when simulation code changes.

These are application inferences from the inequality, not source implementations. A polynomial approximation error must first be represented by the actual error polynomial/operator model. A bound on an arbitrary polynomial is not automatically a nonlinear-system, discretization or physical-error certificate. Rational/analytic functions need pole/domain and approximation bridges beyond the implemented polynomial interface.

Technical and commercial limits

The new exact Fraction utility independently evaluates E=sum A^k tensor B_k and tests tolerance^2 I-E*E for positive semidefiniteness after exact realification. A complete true result proves the stated finite threshold under the ordinary algebraic criterion and trusted program implementation, without requiring Crouzeix. A complete false result disproves that threshold for this exact finite model. Formal correctness of this Python program was not proved.

Separately it forms twenty rational unit-direction Hermitian support matrices Re(conjugate(nu) A). Gershgorin/L1 initial upper bounds are tightened by exact PSD bisection; each final hI-H is checked PSD. Their halfspaces enclose the entire W(A). Axis directions give a closed bounding rectangle. A finite grid retains every cell potentially intersecting the enclosure, rejecting only cells wholly excluded by a halfspace. At each retained center it uses a Frobenius polynomial norm upper plus a coefficient-derived Lipschitz constant times cell radius. This covered-cell calculation is an upper bound, rather than a sampled boundary maximum.

All complex values use pairs of canonical rational strings. Square roots round upward with integer arithmetic on a 2^24 denominator. Base dimension is 1..8, coefficient dimension 1..4, degree 0..6, input rationals at most 128 bits, intermediate rationals at most 4096 bits, grid 1..32 and support bisections 0..16. Two million charges limit selected rational arithmetic, excluding parsing, native allocation, integer/gcd bit cost, support initialization and serialization. These are neither full bit-machine complexity nor wall-time/memory bounds.

If a budget is exhausted before direct completion, no direct threshold or theorem bound is supplied. If exhaustion occurs later, a completed direct threshold survives and the enclosure/supremum result remains unknown. An upper bound exceeding tolerance is inconclusive about the actual norm; it must not be labeled a failure of the finite matrix itself. Exact inputs still need a bridge to uncertain measured/floating inputs.

Validation experiment

857 finite controls include all 625 small complex Hermitian 2-by-2 principal-minor oracle cases, rank-one zero-pivot tests, rational rounding, eight generated cases, exact tensor indexing, finite Rayleigh points, thresholds, invalid models and budgets. 195 SciPy diagnostics compare floating SVD and Hermitian support eigenvalues in the preserved SciPy 1.18.1/NumPy 2.5.3 environment. No packages were altered. Floating agreement at 1e-11 is not rigorous rounding certification or a proof of either library.

On the sharp nilpotent case, grid 32 produces supremum upper 18365645/16777216, conditional new bound about 2.18936 and published-prior rounded bound about 2.64365. Only the conditional new upper fits a 2.2 tolerance. Yet a direct exact threshold already certifies 2.0, taking 231 charged arithmetic operations versus 84,404 across the full reference. The timing is one generated local run, not a scalable performance benchmark.

On a normal diagonal case with exact norm 2, the reference's numerical-range bounds are 4 and 4.83, illustrating global-constant conservatism. A nilpotent cancellation polynomial has actual evaluation zero while the covered-cell bounds remain positive. Specialized normal-matrix analysis, cancellation-aware polynomial bounds and direct norms are strong baselines. Existing SciPy already computes singular values and Hermitian eigenvalues; validated rounding/support coverage and useful workflow records would need to justify a new integration. SVD documentation, Hermitian eigenvalue documentation, saved eight-case comparison.

Existing alternatives and differentiation

Direct exact threshold checks, established SVD/Hermitian eigenvalue routines and normal-specific analysis are the practical baseline. The bounded prototype shows no standalone algorithmic advantage. Potential differentiation is reviewed entire-domain coverage, arithmetic/model evidence and repeated error-budget decisions; an actual repeated workflow must establish that value.

Minimal architecture

Exact or uncertainty-bounded matrix/model import -> independent direct threshold where practical -> validated support enclosure -> covered polynomial supremum -> separate prior/new theorem labels -> declared error-budget decision -> reproducible input/code/runtime and assumptions report. A large sparse implementation would require its own validated support solver and cost/error bridge. Do not turn floating support estimates into halfspace guarantees by rounding without a reviewed residual/error argument.

Monetization hypothesis

Hypothesis: AUD 9,000 for a narrowly scoped polynomial-bound engineering review within the shared evidence platform. At 35 assumed specialist hours and AUD 180/hour, direct labor is AUD 6,300, leaving AUD 2,700 before sales, support, proof review and overhead. At 55 hours labor is AUD 9,900. No quote, fee, license, willingness to pay or profit is established. These overlapping modules are not additional independent customers.

Conditions to reject or defer

Defer a supported numerical library until one real nonnormal workflow shows a repeated decision unavailable more cheaply from direct norms or established analysis. Reject the opportunity if enclosure conservatism, input uncertainty, exact arithmetic cost or support burden overwhelms the improved constant.

Next concrete action

The next local experiment should reuse one enclosure across a genuine polynomial-error family and compare decisions/work against direct and normal-specialized baselines, with source proof status explicit.