The positive-kernel gate is the complete COR conjecture

Date: 8 September 2026
Mode: ATTACK
Status: exact stopping audit; complete Crouzeix remains open

Result

The positive-Herglotz-kernel transfer proposed in the hunt packet is not an auxiliary lemma below the complete Crouzeix conjecture. In its natural operator-algebraic formulation it is precisely the completely bounded Clouâtre–Ostermann–Ransford (COR) conjecture, whose conclusion is the desired Paulsen similarity metric.

Consequently, retaining all matrix Toeplitz minors and trying to extract the metric by Schur complements does not yet reduce the open problem. A finite abstract countermodel satisfying all of those axioms would refute the complete COR conjecture; it would not merely stop one proof architecture. The hunt packet’s weaker description of such a countermodel is corrected here.

Abstract setup

Let $\mathcal A$ be a uniform algebra, let

$$\theta:\mathcal A\longrightarrow B(\mathcal H)$$

be a unital completely bounded homomorphism, and let $\alpha:\mathcal A\to\mathcal A$ be a unital antilinear complete contraction. Set

$$\Lambda(f)=\theta(f)+\theta(\alpha(f))^*.$$

The complete COR conjecture says

$$\|\Lambda\|_{\mathrm{cb}}\le2 \quad\Longrightarrow\quad \|\theta\|_{\mathrm{cb}}\le2. \tag{COR-cb}$$

Hartz–McCarthy record this as the completely bounded part of COR Conjecture 1.1. Badea–O’Loughlin–Virtanen prove the scalar analogue and explicitly leave its completely bounded counterpart open.

Why the positive kernel is already in the hypothesis

For a smooth convex domain $\Omega\supset W(T)$, the double-layer operator has the form

$$\Phi(f)=\frac12\int_{\partial\Omega} f(\sigma)P_\Omega(\sigma)\,|d\sigma|,$$

where the operator-valued density $P_\Omega(\sigma)$ is positive and the integral is unital. Hence $\Phi$ is unital completely positive. The Crouzeix–Palencia identity is

$$2\Phi(f)=\theta(f)+\theta(\alpha(f))^*=\Lambda(f).$$

Thus $\|\Lambda\|_{\mathrm{cb}}\le2$ already holds. Equivalently, every matrix amplification of the associated positive-real/Cayley transform has positive Herglotz and block Toeplitz kernels. Keeping the entire Toeplitz kernel therefore repackages the left side of (COR-cb); it does not add an independent constraint from which the right side is presently known to follow.

The concrete convex-boundary formula supplies geometry, but the proposed attack used only unital complete positivity, the Cauchy-transform identity, and the resulting defect moments. Those are exactly the generic data already present in (COR-cb).

Why the desired metric is exactly the conclusion

Paulsen’s similarity theorem gives

$$\|\theta\|_{\mathrm{cb}} =\inf\{\|S\|\,\|S^{-1}\|: S^{-1}\theta(\cdot)S\text{ is completely contractive}\}.$$

In the singly generated disk-algebra case, write $A=\theta(z)$. The similarized homomorphism is completely contractive exactly when $S^{-1}AS$ is a contraction. With

$$P=S^{-*}S^{-1},$$

this is equivalent to $A^*PA\preceq P$. Moreover $\kappa(P)=(\|S\|\|S^{-1}\|)^2$ after optimal scalar normalization. Therefore

$$\|\theta\|_{\mathrm{cb}}\le2 \quad\Longleftrightarrow\quad \exists P\succ0:\quad I\preceq P\preceq4I, \quad A^*PA\preceq P.$$

The proposed common Stein metric is not a weaker intermediate object: it is Paulsen’s form of the right side of (COR-cb).

Where the scalar proof stops

Lorist–Schwenninger associate to each scalar $f$ an isometry $V$, a multiplication contraction $Q$, and commuting defect moments

$$E_n=2V^*Q^{*n}V-\theta(f)^{*n}.$$

Their perturbation argument uses the whole sequence in $n$ to bound the one operator $\theta(f)$. This is enough for the scalar norm of $\theta$, but it does not couple different entries of a matrix-valued $F$ into one multiplicatively coherent similarity. The missing ingredient is therefore cross-function, matricial coherence, not another positivity check on the same Herglotz moments.

Hartz–McCarthy obtain such a transfer when the correction is scalar-valued. The numerical-range Cauchy correction is operator-valued, which is exactly the unresolved case.

Corrected stopping boundary

The generic positive-kernel route stops. Further ATTACK would need a new identity using geometry specific to the resolvent density

$$P_\Omega(\sigma)=\pi^{-1}\operatorname{Re} \bigl(n_\Omega(\sigma)(\sigma I-T)^{-1}\bigr),$$

or another structure not implied by unital complete positivity and the COR symmetrization alone. Searching Toeplitz minors, Schur complements, matrix sizes, or individually fitted Stein metrics would only restate or enumerate the open problem.

This audit does not prove or disprove complete COR or complete Crouzeix. It is not a Publication.

Sources

1. C. Badea, R. O’Loughlin, and J. Virtanen, On the abstract approach to spectral constants: a proof of the Clouâtre–Ostermann–Ransford conjecture, arXiv:2609.03637v1 (3 September 2026). https://arxiv.org/html/2609.03637v1
2. E. Lorist and F. L. Schwenninger, A solution to Crouzeix’s conjecture, arXiv:2608.03841v2 (17 August 2026). https://arxiv.org/html/2608.03841v2
3. M. Hartz and J. E. McCarthy, From Clouâtre–Ostermann–Ransford to Okubo–Ando, arXiv:2606.02922v1 (2 June 2026). https://arxiv.org/html/2606.02922v1
4. M. Jury and G. Tsikalas, Positivity conditions on the annulus via the double-layer potential kernel, arXiv:2307.13387v2 (2023). https://arxiv.org/abs/2307.13387

Next mode

HUNT. Pause complete Crouzeix unless a future route identifies a geometry-specific identity strictly stronger than the abstract complete COR hypotheses. Do not replace the stopped route with matrix or domain censuses.