The Tsirelson Bound in Absolute Frame Theory: Two Convergent Routes to the Quantum Correlation Ceiling

A no-signaling theory may, in principle, be more nonlocal than quantum mechanics: the Popescu--Rohrlich box reaches the algebraic Clauser--Horne--Shimony--Holt value $4$, whereas nature stops at the Tsirelson value $2\sqrt2$. Absolute Frame Theory (AFT) produces nonlocal correlations---two systems entangled in $\mathcal{M}$ share a single point of the substratum $\mathcal{A}$ (Axiom II)---and forbids superluminal signaling (the channel is Bremermann-bounded). We ask where those correlations fall, and we establish, by two independent routes, that they cannot exceed the Tsirelson bound: AFT admits no super-quantum correlations. The operator route proves, inside the reconstructed Hilbert theory, the Tsirelson operator identity, with the local commutativity $[A_i,B_j]=0$ itself derived from AFT structure---the tensor factorization that local tomography supplies and the field microcausality of the constrained embedding---rather than assumed. The information route reduces the bound to a physical principle: AFT's shared resource is read out through a completely positive, trace-preserving channel of finite classical capacity (a theorem of the channel, founded on action quantization and the trace over the inaccessible fiber), this channel obeys the data-processing inequality, and the combination places AFT in the class that satisfies Information Causality, which singles out $2\sqrt2$ among the no-signaling theories. We state plainly what this does and does not achieve. It does not close the question of why the bound is $2\sqrt2$ rather than some other value: both routes, chased to their foundations, terminate at the same posited fact---the $\mathcal{M}$--$\mathcal{A}$ signature transition (the Wick rotation) that installs the complex-Hilbert structure---and finite capacity alone, stripped of that structure, does not exclude a Popescu--Rohrlich box. The two routes are thus the operator-algebraic and information-theoretic faces of one ceiling, and they reduce the selection of $2\sqrt2$ to a single substratum fact, with the residue an instance of the deepest open question of the programme rather than a problem specific to nonlocal correlations.