Realizability of the correlators of boxes violating chained monogamy

W. Kłobus, M. Oszmaniec, R. Augusiak, A. Grudka, Communication strength of correlations violating monogamy relations, Found. Phys. 46 (2016) 620–634, arXiv:1408.1223v2, Section 5:

Although we cannot prove it as in the case $M=2$, we conjecture that all possible values of the correlators in $S_{A\to BE}^i$ and $S_{B\to AE}^i$ that satisfy inequalities (\ref{ElPrat}) can always be realized with some signaling probability distribution $\vec{p}$ for which $R_M(\vec{p})=2M+\Delta$.

A box is a family $p(a,b,e|A_i,B_j)$ of distributions on $a,b,e=\pm1$, $i,j=0,\dots,M-1$, with one setting $E$ for the third party, all one- and three-party expectation values zero, and a common value of $\langle B_0E\rangle_{A_i}$. $R_M=I^M_{AB}+2\langle B_0E\rangle$ with the chained Bell expression $I^M_{AB}=\sum_k(\langle A_kB_k\rangle+\langle A_{k+1}B_k\rangle)$, $A_M=-A_0$, and $\Delta\in[0,2]$. The coordinates are $x_A^i=\langle B_iE\rangle_{A_i}$, $y_A^i=\langle B_iE\rangle_{A_{i+1}}$, $x_B^i=\langle A_iE\rangle_{B_{i-1}}$, $y_B^i=\langle A_iE\rangle_{B_i}$ ($1\le i\le M-1$), $x_B^0=\langle A_0E\rangle_{B_0}$, $y_B^0=\langle A_0E\rangle_{B_{M-1}}$, and (ElPrat) is $$\sum_{i=1}^{M-1}(-1)^{a_i}(x_A^i-y_B^i)+\sum_{i=1}^{M-2}(-1)^{b_i} (x_B^{i+1}-y_A^i)+(-1)^c(y_A^{M-1}+y_B^0)+x_B^1+x_B^0\ge\Delta$$ for all $a_i,b_i,c\in{0,1}$.

The paper derives the monogamy relation $|I^M_{AB}|+2|\langle B_0E\rangle|\le2M$ of nonsignaling boxes from three-variable inequalities and reads a violation by $\Delta>0$ as forced signaling: the correlators in at least one pair $S^i_{A\to BE}$, $S^i_{B\to AE}$ must differ, and each pair defines a binary classical channel. The communication strength of the violation is the least over violating boxes of the largest capacity of these channels. The conjecture identifies the set over which this minimization runs: the correlator vectors cut out by (ElPrat). The frozen declaration D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.result proves it.

Issue #12401 classifies the conjecture as Tier 1 and records the checks made before any Lean:

• arXiv:1408.1223 v2 is the latest version and states the conjecture; Crossref, OpenAlex and INSPIRE report no citing work;
• results on monogamy of nonsignaling correlations (Toner, quant-ph/0601172; Augusiak–Demianowicz–Pawłowski–Tura–Acín, arXiv:1307.6390; Ramanathan–Horodecki, arXiv:1402.4453; Augusiak, arXiv:1611.00651) and on communication cost (Montina–Wolf, arXiv:1312.6290) fix aggregate Bell values of nonsignaling boxes, not every coordinate of a signaling box; a later account by W. Kłobus (2023 summary of professional accomplishments, §4.3.3) gives a lower bound on the communication strength and no proof of the conjecture.

These are seat-reported and orchestrator-reported readings, not-found-in-searched-scope; they do not establish exhaustive worldwide novelty. The realizable correlator triples of three $\pm1$ variables with zero odd moments form a tetrahedron; this is the $n=3$ case of Huber–Marić, arXiv:1311.2002v2, Theorem 3, and of the cut polytope of $K_3$ (arXiv:1706.06182v2, §4), and is used in step 1 of the Route.

1. For $|v|,|w|\le1$ and $-1+|v+w|\le u\le1-|v-w|$, the function $(1+ab,u+ae,v+be,w)/8$ is a probability distribution with correlators $\langle AB\rangle=u$, $\langle AE\rangle=v$, $\langle BE\rangle=w$ and vanishing one- and three-party means.
2. Let $T=\sum_{i=1}^{M-1}|x_A^i-y_B^i|+\sum_{i=1}^{M-2}|x_B^{i+1}-y_A^i| +|y_A^{M-1}+y_B^0|$. Choosing every sign in (ElPrat) against its term gives $x_B^0+x_B^1-T\ge\Delta$.
3. Put $t=(2+\Delta+T)/(2+x_B^0+x_B^1)\in(0,1]$. At $(A_0,B_0)$ and $(A_1,B_0)$ take $(u,v,w)=(t,x_B^j,x_B^j,t)$; at the other $(A_i,B_0)$ take $(0,0,t)$; at $(A_i,B_i)$ and $(A_{i+1},B_i)$, $i\ge1$, the prescribed $(v,w)$ with $u=1-|v-w|$; at $(A_0,B_{M-1})$ the prescribed $(v,w)=(y_B^0,y_A^{M-1})$ with $u=-1+|v+w|$; and $(0,0,0)$ elsewhere.
4. The box realizes the coordinates, $\langle B_0E\rangle_{A_i}=t$ for all $i$, and $R_M=(2M-2-T)+t(x_B^0+x_B^1+2)=2M+\Delta$.

The kernel-checked result states, for every $M\ge2$, every $\Delta\in[0,2]$ and all coordinate sequences with values in $[-1,1]$ on their index ranges that satisfy (ElPrat), that some box (indexed by natural numbers, a probability distribution on Bool × Bool × Bool at every setting pair with $i,j<M$) has vanishing one- and three-party means, a common $\langle B_0E\rangle_{A_i}$, $R_M=2M+\Delta$, and the prescribed coordinates. It reads $y_A^{M-1}=\langle B_{M-1}E\rangle_{A_M}$ at the setting pair $(A_0,B_{M-1})$: $A_M=-A_0$ relabels the outcome of $A_0$, which does not change a correlator of $B$ and $E$. The case $M\ge3$ is the conjecture; for $M=2$ (ElPrat) differs from the paper's printed list (see Triage).

Exact Fraction arithmetic (issue #12401): the box of Route step 3 realizes 1440 random admissible coordinate vectors with entries in $\tfrac1{12}\mathbb Z$ for $M=2,\dots,7$ and $\Delta\in{0,\tfrac13,\tfrac12,1,\tfrac32,2}$, checking nonnegativity, normalization, the common $\langle B_0E\rangle$, all $4M-2$ correlators and $R_M$; a box with one entry changed by $1/100$ is rejected by the same verifier. A second construction puts $\langle B_0E\rangle=1$ and every $u$ at its extreme, which gives the largest $R_M$ for the given coordinates, $2M+x_B^0+x_B^1-T$; it realizes 1000 further admissible samples by mixing with the box for $\langle B_0E\rangle=-1$, and on 15827 non-admissible samples its value of $R_M$ is below $2M+\Delta$, as the converse direction predicts. (The box of Route step 3 is defined by the formula for $t$ only on admissible data: outside them $t$ can exceed $1$ and a probability becomes negative.)

The canonical source is D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.lean. Its public declarations are IsBox, OddMomentsVanish, corrAB, corrAE, corrBE, chainR, ElPrat, CoordinateBounds, Realizes, claim and result; the outcome sign is the frozen D5/S3/Combinatorics/IsingUniquenessSets.sgn. The frozen module state has statement identity sha256:457daf112b0653b47d2b60cd8593fed7c8f4c0b80f3970f0369a70f48cea1665. The result declaration has statement identity sha256:189745a8f3608dc5f2cf3c74b97968e0de7b0d9ee8126daa88b6ff6a30963cba. The Freeze event is sha256:29c7ef363071c78eeee2044226eeba96ec3dc9f0cc8cc7cb0aeffd286a53bf75; its project-level frozen prerequisite is the Freeze event of D5/S3/Combinatorics/IsingUniquenessSets. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Tier 1 published conjecture; resolution Proved by D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.result. proof_shape: content; admission_basis: open-problem-resolution (issue #12401). Utility kind none (a statement for every $M$, every $\Delta$ and every admissible coordinate vector).

What the settlement shows

• Proved by result: every coordinate vector in $[-1,1]^{4M-2}$ satisfying (ElPrat) is realized by a box of the paper's class with a common $\langle B_0E\rangle$ and $R_M=2M+\Delta$, for all $M\ge2$ and $\Delta\in[0,2]$.
• Proved inside the proof of result: (ElPrat) implies the single inequality $x_B^0+x_B^1-T\ge\Delta$; the realizing box can be taken with $\langle B_0E\rangle_{A_i}=t=(2+\Delta+T)/(2+x_B^0+x_B^1)$.
• Mechanism (read off the proof; not stated in Lean): the setting pairs decouple. Each pair only has to realize a point of the correlator tetrahedron, and the only link between pairs is the common $\langle B_0E\rangle$. For fixed coordinates the largest $R_M$ is $2M+x_B^0+x_B^1-T$, attained with $\langle B_0E\rangle=1$, so (ElPrat) is exactly the condition that this maximum reaches $2M+\Delta$; since $x_B^0+x_B^1\le2$, also $t\ge\tfrac12$.
• Follows from it (not stated in Lean): with the paper's derivation of (ElPrat) as a necessary condition (l. 704–716), the set of coordinate vectors of $\mathcal P^M_\Delta$ is exactly the (ElPrat) polytope inside $[-1,1]^{4M-2}$, so the communication strength $C^M_\Delta$ is a minimization over this explicit polytope. At $\Delta=0$ the realizing box need not signal; for $\Delta>0$ every realizing box signals.
• Checked by reading (orchestrator, issue #12401): the paper's printed $M=2$ list (l. 468–471, labels niert1–niert4) has the signs of $x_A^1,y_A^1$ reversed in niert2 and niert3; the point $(x_A^1,y_A^1,x_B^0,y_B^0,x_B^1,y_B^1)=(1,1,1,1,1,-1)$ satisfies all four printed inequalities at $\Delta=2$ but has $x_B^0+x_B^1-T=-2$, so it is not realizable. (ElPrat) at $M=2$ is the corrected list.
• Open here: the value of $C^M_\Delta$ for $M\ge3$ (the minimization of the largest channel capacity over the polytope); whether the paper's reported values of $C_\Delta$ for $M=2$ were computed with the printed or with the correct inequalities (not checked); and the analogous realizability question when the common-value assumption on $\langle B_0E\rangle_{A_i}$ is dropped, as the paper does for $M=2$ at the end of Section 4.

The literature checks do not establish worldwide priority; the 2023 summary by W. Kłobus and the Huber–Marić results are seat-reported readings. The Lean kernel verifies the encoded statement and its axiom closure; its correspondence to the paper, including the reading of the class $\mathcal P$, of $R_M$ and of the setting $A_M=-A_0$, is checked by reading the source and the definitions.