The source tetralemma

For every setting pair, the two measurable functions in (D) take exactly one value in $\{\pm1\}$ at each source state; their pushforward gives the joint outcome law with $\lambda$. Almost-sure qualifications can be imposed simultaneously for the four setting pairs by taking a finite union of null sets. A branching final vector is not either of these scalar outcome functions. Replacing (D) by deterministic evolution of that vector would change the premise.

Put $\alpha_x=\E[A\mid x]$ and $\beta_y=\E[B\mid y]$. Positivity gives $2P(A\ne B\mid xy)\ge|\alpha_x-\beta_y|$ and $2P(A=B\mid xy)\ge|\alpha_x+\beta_y|$. Let $L=P(A\ne B\mid00)+P(A\ne B\mid01)+P(A\ne B\mid10)+P(A=B\mid11)$, so $S=4-2L$. Writing $u=\alpha_0-\beta_0$, $v=\alpha_0-\beta_1$, $w=\alpha_1-\beta_0$, $z=\alpha_1+\beta_1$ gives $2L\ge|u|+|v|+|w|+|z|$. The combinations $u+v-w+z=2\alpha_0$, $-u+v+w+z=2\alpha_1$, $-u+v-w+z=2\beta_0$ and $u-v-w+z=2\beta_1$ bound all four biases by $L$. Thus the marginal guessing probability is at most $(1+L)/2=3/2-S/4$. Mixing the all-plus local box with a PR box attains this value. This is also the bound in [22, Eq. (14)].

(MI) makes the law of $T$ independent of the settings, so $P=\int P_t\,d\Prb(t)$ and $S=\int S_t\,d\Prb(t)$. By (NS) the Alice marginal given $x,t$ is independent of $y$. Lemma 2.3 and averaging prove (1). For fixed $x,y$, the random variable $F_A(\lambda,x,y)$ is a coordinate of $Z$. Hence, pointwise in $t$, the largest atom of $Z\mid t$ is no larger than the largest atom of that coordinate. (MI) identifies this coordinate's conditional law with the observed law at $x,y,t$. Averaging gives (2); Shannon entropy dominates average conditional min-entropy, by its pointwise version and Jensen's inequality.

If $\lambda=G(T)$ almost surely, then $Z$ is a function of $T$ and its conditional entropy is zero, a contradiction. Regular conditional source laws are concentrated on $\tau^{-1}(t)$ almost everywhere. A nondegenerate finite conditional response distribution therefore supplies different responses in those fibres, as in Theorem 6.1.

For (QT), the single-box bound $G_Q$ is imported from [22, Eq. (6)]. Extend it by $1$ for $s\le2$ on the quantum CHSH domain. The extension is concave: at $s=2$ its right derivative is $-1/4$, no larger than the left derivative $0$. Jensen gives $\E G_Q(S_t)\le G_Q(S)$, and the preceding coordinate argument applies unchanged.

Concavity of binary entropy on each half of $[0,1]$ gives $h_2(p)\ge2\min(p,1-p)$. Lemma 2.3 implies, at each $t$,

Since $u\mapsto(u-2)_+$ is convex, averaging and the deterministic-coordinate inequality give $H(Z\mid T)\ge(S-2)/2$. The comparison with $\fNS$ follows because a convex function lies below its endpoint chord: $\fNS(2)=0$, $\fNS(4)=1$.

For sharpness let $r=(S-2)/2$, let $C_0$ be a Bernoulli flag of mean $r$, and let $R_0$ be zero when $C_0=0$ and a fair bit when $C_0=1$. Take $\lambda=(C_0,R_0)$, $T=C_0$, independent settings, and all-plus responses in the first case. In the second case use $A=(-1)^{R_0}$ and $B=(-1)^{R_0+xy}$. The conditional boxes are local deterministic and PR, respectively, hence both no-signalling. Their average CHSH value is $2+2r$. The response tables in the PR branch are distinct and equiprobable, so $H(Z\mid T)=r$ and $p_{\rm guess}(Z\mid T)=1-r/2=3/2-S/4$. This is an extremal mathematical construction, not a physical PR-box claim or quantum-class sharpness assertion.

The exclusion is Theorem 2.4. The following four models all take the readout to be the complete specified initial state.

(C1) Branching: outside (SO) and the stated (D). Let the source be a unit ray of the two-qubit-plus-apparatus Hilbert space, prepared in $\Phi^+\otimes\ket{\mathrm{ready}}$. Independent settings select unitary measurement interactions with final vector

Use $A_0^{\rm op}=\sigma_z$, $A_1^{\rm op}=\sigma_x$, $B_0^{\rm op}=(\sigma_z+\sigma_x)/\sqrt2$, $B_1^{\rm op}=(\sigma_z-\sigma_x)/\sqrt2$. Orthogonal pointer branches have squared norms $P_Q(a,b\mid x,y)$, a no-signalling table. The complete vector is a deterministic function of source and settings, but no single actual pair $A,B$ has been selected. This is complete-state determinism, not (D). The branch-weight reading is the relative-state alternative [30, 31, 32]; derivation of subjective probabilities from branch weights is not a premise or result here.

(C2) Fundamental chance: violates only (D). Take $\Lambda$ to be pure rays with $\nu=\delta_{\Phi^+}$, $\tau=\mathrm{id}$, and independent settings. Sample one pair from $P_Q$ using fundamental chance not encoded in the initial source state. Then (SO), (MI) and (NS) hold, while (D) fails. Completeness here is completeness of the initial physical state, not a hidden initial encoding of subsequent chance. Pure-ray chance ontologies supply the relevant contrast [2, 33].

(C3) Measurement dependence: violates only (MI). Let $\lambda=(a_0,a_1,b_0,b_1)\in\{\pm1\}^4$, $A=a_x$, $B=b_y$, and take any positive setting law $q(x,y)$ with

The two unused entries are uniform. Summing over them reproduces $P_Q$ and normalizes each conditional source law. All its entries are positive, so each setting pair remains in the support at every $\lambda$. Conditional on $T=\lambda$, the responses are local deterministic and satisfy (NS). They satisfy (SO) and (D), but the source law depends on the settings. This is a measurement-dependent construction of the Brans type [35].

(C4) Accessible nonlocality: violates only (NS). Take $\lambda=(u,v)$ uniform on $[0,1]^2$, independently of the settings, with $T=\lambda$, and define

assigning $\mathrm{sgn}(0)=+1$ on the irrelevant threshold sets. Its averaged joint probabilities are $P_Q$. It satisfies (SO), (D) and (MI). For $y=1$, on the positive-measure interval between $(1-1/\sqrt2)/2$ and $(1+1/\sqrt2)/2$, Bob's definite output changes with $x$. Its box conditional on the readable $\lambda$ therefore violates (NS), although its unconditioned marginals do not signal. No claim that such complete access is physically available is required for the countermodel.

(C3) and (C4) are single-outcome, source-deterministic and source-complete. Neither is forced into the proposed surrogate branch. With a single preparation, convex-linearity imposes no nontrivial requirement; random mixtures of admitted preparation laws can moreover be represented by their mixtures. A density-matrix-only stochastic surrogate is already in the chance branch for the nondegenerate Tsirelson marginals. None of the arguments of Theorem 2.4 uses convex-linearity. These observations refute the proposed universal three-way claim; they do not classify every possible surrogate.

Apply Corollary 8.4 and Theorem 8.3. The Beltrametti–Bugajski model distinguishes proper-mixture laws while retaining rays as individual states [2].