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Phys-33-A1 · revision r1 · on Phys-33

OpenAI claim summary for Phys-33

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The constant-two numerical-range theorem addresses the Crouzeix matrix-function subquestion. Spectral completeness, inverse spectral problems, instability, and the other approximation clauses in the same entry remain separate.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Non-self-adjoint spectral theory: completeness, reconstruction, and matrix-function bounds
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Non-self-adjoint spectral theory: completeness, reconstruction, and matrix-function bounds

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 325; 2026-09-26), the asserted result is that finite matrices obey the sharp constant-two numerical-range bound for matrix-valued polynomials in every coefficient size, including degenerate numerical ranges; In manuscript 2 (family 325; 2026-09-23), the asserted result is that the sharp complete constant-two numerical-range estimate holds for bounded operators without restrictions on Hilbert-space dimension or separability, also for the specified holomorphic and rational calculus.

Mathematical statement of the claimed result

Manuscript 1.
For integers $n,m\ge1$, $d\ge0$, a complex matrix $A\in M_n(\mathbb C)$ and coefficients $B_k\in M_m(\mathbb C)$, let \[ P(z)=\sum_{k=0}^d B_kz^k,\qquad P[A]=\sum_{k=0}^d A^k\otimes B_k, \quad W(A)=\{x^*Ax:x\in\mathbb C^n,\ \|x\|_2=1\}. \] With Hilbert-space operator norms, the claimed complete inequality is \[ \|P[A]\|\le2\max_{z\in W(A)}\|P(z)\|. \] The factor $2$ is asserted sharp uniformly over all three integer parameters. Matrix-valued coefficients are allowed; the scalar polynomial case follows with $m=1$.
Manuscript 2.
For $n,m\ge1$, $A\in M_n(\mathbb C)$, $d\ge0$ and arbitrary $B_0,\ldots,B_d\in M_m(\mathbb C)$, write $P(z)=\sum_jB_jz^j$ and $W(A)=\{x^*Ax:\|x\|_2=1\}$. The claimed complete polynomial inequality is \[ \left\|\sum_{j=0}^dA^j\otimes B_j\right\|\le2\max_{z\in W(A)}\|P(z)\|. \] Hilbert-space operator norms are used, and the constant two is claimed sharp independently of matrix sizes and polynomial degree. The scalar case takes $m=1$.

Outline of the argument

For manuscript 1 (family 325; 2026-09-26), exterior conformal coordinates produce Faber coefficient estimates and a contractive correction kernel. Parseval separation of positive and negative modes controls amplified singular vectors, and enclosing-domain limits include point and segment numerical ranges.

For manuscript 2 (family 325; 2026-09-23), for finite matrices, an optimal positive metric makes the conformal disk coordinate contractive. Endpoint extremal data yield a common continuous boundary density whose ordered comparison forces condition number at most two; domain approximation and finite compressions extend the bound to general operators.

Formal verification

For manuscript 1 (family 325; 2026-09-26), OpenAI lists Lean statements for this assertion: finite matrices obey the sharp constant-two numerical-range bound for matrix-valued polynomials in every coefficient size, including degenerate numerical ranges.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 2 (family 325; 2026-09-23), OpenAI lists Lean statements for this assertion: the sharp complete constant-two numerical-range estimate holds for bounded operators without restrictions on Hilbert-space dimension or separability, also for the specified holomorphic and rational calculus.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are CompleteCrouzeix.lean, CrouzeixHilbert.lean, DirectCrouzeix.lean, HilbertCrouzeix.lean, StructuralCrouzeix.lean.

Their named declarations include OAI.CompleteCrouzeix.main, OAI.CrouzeixHilbert.hilbert, OAI.DirectCrouzeix.complete_crouzeix, OAI.DirectCrouzeix.uniform_sharpness, OAI.HilbertCrouzeix.exists_finite_compression, OAI.HilbertCrouzeix.numericalClosure_geometry, OAI.HilbertCrouzeix.rational_denominator_isUnit, OAI.HilbertCrouzeix.zeroConclusion, OAI.StructuralCrouzeixReference.challenge.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources