A family refuting Linden–Winter Conjecture 6

A six-atom classical family refutes Conjecture 6 of Linden and Winter [1]: no finite positive coefficient triple makes their proposed linear relaxation valid for all four-party quantum states. Their constrained theorem remains valid.

Conjecture

Linden and Winter proved a four-party entropy inequality under three linear entropy constraints. Conjecture 6 asks whether positive constants $k_1,k_2,k_3$ can make

hold for every four-party quantum state. Here $I(X;Y\mid Z)=H(XZ)+H(YZ)-H(Z)-H(XYZ)$ denotes conditional mutual information. Classical distributions embed as diagonal quantum states, so a classical family suffices.

Six-atom family

For an integer $L\geq2$, normalize these weights in $(A,B,C,D)$ order:

Given $D=0$, $A$ is constant; given $D=1$, $B$ is constant. Thus $I(A;B\mid D)=0$. For $B=0$, $A$ is constant; for $B=1$, the identity

establishes conditional independence of $A$ and $C$, hence $I(A;C\mid B)=0$. Writing $y_L=I(C;B\mid A)$ and $e_L=I(C;D)-I(C;AB)$ leaves $f_k=k_2y_L+e_L$. The values of $k_1$ and $k_3$ have no effect on this family.

Unbounded ratio

Set $\delta=10^{-L}$. The equivalent weights are $2,27(1-\delta),2/1000,27/1000,2\delta^3,27\delta^3$, with sum $Z=29029/1000-27\delta+29\delta^3$. Using natural logarithms, define

Conditional-entropy grouping gives

All arguments of $\Phi$ stay positive near zero, making these expressions analytic there. Their Taylor expansions are

The symbolic derivation receipt [3] records the lower-order cancellations and exact coefficients; they can also be checked by differentiating the displayed expressions. Therefore

For every finite positive $k_2$, choose $L$ with $R_L>k_2$ to obtain $f_k<0$. Analyticity and the leading coefficients establish the universal statement; the finite certificates below verify particular members of the family.

Verification and scope

The public note and standard-library interval verifier [2] check both exact structural identities, entropy signs, ratio bounds for $L=2,\ldots,16$, and explicit coefficient examples. The default finite certificates replayed successfully on October 2, 2026. The advertised arbitrary-coefficient invocation had a Python conversion defect; its merged correction [4] was replayed against the published code and certifies the supplied coefficient triple.

The family has $I(C;B\mid A)>0$, so it refutes the linear relaxation while preserving the original constrained theorem. It does not settle the existence of other unconstrained quantum entropy inequalities. The retained record includes an operator-supplied copy of correspondence reporting Andreas Winter’s confirmation; the original mailbox was not independently inspected. No peer review or publication acceptance is asserted.

Attribution and retained attempt

The preserved constructor and off-diagonal child assignments supplied this family as cited prior knowledge [5]. The saved findings certify and analyze it; they do not identify its first discovery. The journal records a director and five child attempts, all five ending down. Four Pi session files survive, including short or incomplete sessions. The director configuration declares OpenCode with GLM-5.2; child configurations include Pi and Claude Code. The Pi sessions declare GLM-5.2, but their retained assistant records end in errors, leaving the served model unverified.

The director transcript is missing, and native-session identifiers have not been joined reliably to the recorded agents. Complete attribution of each calculation is therefore unavailable. A recorded “winner” label does not establish that each child finished, that the proof was independent, or that the archive is complete.