slug: krishna-2022-cstar-schoenberg
bibkey: krishna2022cstarschoenberg
doi: 10.48550/arXiv.2206.06653
url: https://arxiv.org/abs/2206.06653v1
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Algebra/CStarSchoenberg.result

The C*-algebraic Schoenberg Conjecture is false

Problem

K. Mahesh Krishna, “C*-algebraic Schoenberg Conjecture”, arXiv:2206.06653v1 (14 June 2022; single version), Section 2. The source text is quoted verbatim:

Conjecture 2.1 (C-algebraic Schoenberg Conjecture).* Let $\mathcal{A}$ be a C*-algebra. Let $d\in \mathbb{N}\setminus \{1\}$, $P(z):=(z-a_1)(z-a_2)\cdots (z-a_d)$ be a polynomial over $\mathcal{A}$ with $a_1,a_2,\dots ,a_d\in \mathcal{A}$. If $P^{\prime }$ can be written as $P^{\prime }(z)=d(z-b_1)(z-b_2)\cdots (z-b_{d-1})$ on $\mathcal{A}$ with $b_1,b_2,\dots ,b_{d-1}\in \mathcal{A}$, then $$\sum _{k=1}^{d-1}{b}_k{b}_k^{\ast }\le \frac{1}{d^2}\left[\sum _{j=1}^d{a}_j\right]{\left[\sum _{j=1}^d{a}_j\right]}^{\ast }+\frac{d-2}{d}\sum _{j=1}^d{a}_j{a}_j^{\ast }$$ and $$\sum _{k=1}^{d-1}{b}_k^{\ast }{b}_k\le \frac{1}{d^2}{\left[\sum _{j=1}^d{a}_j\right]}^{\ast }\left[\sum _{j=1}^d{a}_j\right]+\frac{d-2}{d}\sum _{j=1}^d{a}_j^{\ast }{a}_j.$$

The ordered derivative is defined in the source immediately before the conjecture. The encoding uses zero-based Fin indices, complex scalar coefficients, star, and the C*-order. It restricts the source to unital C*-algebras in Type with PartialOrder and StarOrderedRing, and d >= 2; this subclass is enough for a counterexample to the source’s universal assertion.

Motivation

The same cubic matrix polynomial settles this conjecture and the two linked conjectures in the de Bruin-Sharma dossier (krishna-2022-cstar-de-bruin-sharma.md) and the Kushel-Tyaglov dossier (krishna-2022-cstar-kushel-tyaglov.md). Each dossier records one named OpenProblemResolutionClaim on its own result node.

Gap

The published source states the conjecture and the bounded literature check recorded in #14845 did not identify a prior settlement in that searched scope. Exhaustive literature coverage and publication priority remain ASSUMED-UNVERIFIED.

Route

In M_2(C) let x=E_12, y=E_11+E_12, and $$a_1=(2x+y)/3,a_2=(-x+y)/3,a_3=(-x-2y)/3,u=(x+y)/3,(b_1,b_2)=(u,-u).$$ Then xy=0 != yx, sum_j a_j=0, and the ordered derivative equals 3(z-u)(z+u) for every matrix z. The first inequality has right side minus left side D_2.1^(1)=-2t^2 E_11/9. Its first diagonal entry is negative, so the C*-order inequality fails. The fixed Lean result is D5/S3/Quantum/Algebra/CStarSchoenberg.result.

Falsifier

The encoded statement retains the full-algebra factorization hypothesis, both inequalities, the source coefficients and the noncommutative product order. A failure of factorization, a sign error in the displayed defect, or a changed source clause would invalidate this route. The Lean result verifies the fixed witness and the negation of the stated claim.

Evidence

The settling module is D5/S3/Quantum/Algebra/CStarSchoenberg.lean. The witness lemmas sum_a, factorization and negative_e11_not_posSemidef supply the zero-sum, whole-algebra factorization and positivity obstruction. The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9). The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}.

Experiment: exact symbolic matrix check at https://github.com/the-omega-institute/trureturing-experiments/tree/51f6d69f317a872443b4d3360ae4b28808dee017/docs/reports/krishna-2022-cstar-schoenberg-critical-points, entry check.py, SHA-256 702b68b36e2ae270381994118ce00660b5310f00ad5c514705b95334a7b0f4dd. Command: python3 docs/reports/krishna-2022-cstar-schoenberg-critical-points/check.py in the experiments repository; exit 0. The tested scope is symbolic real t,c>0 and an arbitrary symbolic 2x2 matrix z, with x=tE_12, y=t(cE_11+E_12) and the displayed definitions of a,u,b. The factorization residual and sum_j a_j are zero matrices.

The first defects are $$D_2.1^{(}1)=-2t^2{E}_{1}1/9,D_2.3^{(}1)=2t^4(c^2-7)E_{1}1/81,D_2.4^{(}1)=-t^4(43c^2+158)E_{1}1/81.$$ The second defects are $$D_2.1^{(}2)=t^2\left(\begin{array}{cc}0& -c\\ -c& -2\end{array}\right)/9,D_2.3^{(}2)=t^4\left(\begin{array}{cc}-5c^2& -c(c^2+7)\\ -c(c^2+7)& -5c^2-14\end{array}\right)/81,$$ $$D_2.4^{(}2)=t^4\left(\begin{array}{cc}-44c^2& -c(c^2+79)\\ -c(c^2+79)& -5c^2-158\end{array}\right)/81.$$ Their least eigenvalues are respectively negative for t,c>0, with exact formulas in the experiment entry. Conjecture 2.1 fails throughout t,c>0. The fixed Lean result uses t=c=1 and the first inequality.

Triage

What the settlement shows

Mechanism — proved. In a noncommutative ring with ${\sum }_j{a}_j=0$, $$P^{\prime }(z)-3(z-u)(z+u)=z(2a_1+a_2-3u)-(2a_1+a_2-3u)z+a_1{a}_2-(a_1+a_2{)}^2+3u^2.$$ With $2a_1+a_2=x+y=3u$, the remainder is $xy/3$. Hence a one-sided zero divisor $xy=0\ne yx$ satisfies the factorization while $x{y}^{\ast }+y{x}^{\ast }\ne 0$ changes the energies. The fixed instance is kernel-checked by CStarSchoenberg.factorization, imported by the latter two modules where applicable.

Witness family — computed. The experiment entry above verifies the symbolic family for $t,c>0$, all six defect matrices and the least-eigenvalue formulas. The first inequality of 2.3 fails for $0<c<\sqrt{7}$, and its second inequality fails for every $c>0$. The other two conjectures’ first and second inequalities fail for every $t,c>0$. Uniform extension beyond this tested symbolic scope is not claimed by computation. The fixed first-inequality refutation for this dossier is kernel-checked.

Corner embedding — proved in prose, not formalized. For every $n\ge 2$, embed $x,y$ in the upper-left corner of $M_n(\mathbb{C})$. The identities $xy=0$, $2a_1+a_2=3u$ and ${\sum }_j{a}_j=0$ persist, and the displayed algebraic identity applies to every $z\in M_n(\mathbb{C})$. The negative corner defect therefore persists. This embeds the present refutation in every matrix size $n\ge 2$.

Commutative case — proved (literature) for 2.1 and 2.3, not formalized. In a commutative unital C*-algebra, Gelfand evaluation sends the factorization and C*-order to pointwise scalar statements. The classical results apply: Malamud–Pereira for 2.1 and Cheung–Ng for 2.3. For these two conjectures, commutativity removes the one-sided-zero-divisor mechanism. This literature consequence is not formalized here.

Conjecture 2.4 already fails in $\mathbb{C}$ — computed. The scalar inequality that the source quotes as the Kushel–Tyaglov theorem (Section 1), which Conjecture 2.4 lifts term by term, is false as printed. Take $d=3$ and $a=(1+i,1-i,1)$. Then $P(z)=(z-1{)}^3+(z-1)$, so $P^{\prime }(z)=3(z-b_1)(z-b_2)$ with $b_{1,2}=1\pm i/\sqrt{3}$. The left side is $32/9$ and the right side is $28/9$, so the inequality fails by $4/9$. The same data refute Conjecture 2.4 in the commutative C*-algebra $\mathbb{C}$, so commutativity does not restore 2.4 as transcribed. The exact check is pinned at scalar-kushel-tyaglov.py; command python3 scalar-kushel-tyaglov.py, exit 0, script SHA-256 19281dba6dfa200069d585239213a1151175da8fd53d207b8756d842701412f4. Whether Kushel and Tyaglov’s published theorem differs from the source’s quotation was not checked against their paper: ASSUMED-UNVERIFIED.

Degree two — proved (source). Krishna’s Theorem 2.2 proves Conjecture 2.1 for $d=2$ with equality, and the source states the degree-two case of Conjecture 2.3. The present counterexample has $d=3$ and leaves those degree-two statements intact.

Trace versions — proved in prose, not formalized. $M_2(\mathbb{C})$ is a factor: a matrix commuting with every matrix unit is scalar. Its normalized trace $\tau =\mathrm{Tr}/2$ is faithful, since $\tau (v^{\ast }v)=\frac{1}{2}{\sum }_{i,j}|v_{ij}{|}^2$ vanishes only for $v=0$. At $t=c=1$, applying $\tau$ to the first-inequality defects $-\frac{2}{9}{E}_{11}$, $-\frac{4}{27}{E}_{11}$ and $-\frac{67}{27}{E}_{11}$ gives $-\frac{1}{9}$, $-\frac{2}{27}$ and $-\frac{67}{54}$, respectively. Each is negative, so the same witness refutes all three inequalities taken in trace.

Open. Whether Conjectures 2.1 and 2.3 hold for $d\ge 3$ under extra hypotheses excluding one-sided zero divisors among the relevant products, short of full commutativity, remains open here. Conjecture 2.4 as transcribed has no such repair, because it fails in $\mathbb{C}$. Krishna’s companion C*-algebraic Gauss–Lucas/Sendov, Casas–Alvero and Smale mean-value conjectures are outside this delivery. Any source result relying on one of the three conjectures as an assumption loses that unconditional implication; the source’s independent definitions, scalar results and degree-two theorem are unaffected.

ASSUMED-UNVERIFIED

The literature check is bounded by #14845 and the cited source; exhaustive absence of another settlement is ASSUMED-UNVERIFIED. The symbolic experiment is evidence only for its stated scope. The unfinished escape audit for the non-result public witness declarations is tracked in #14861; no declared_validated claim is made.