Why the Tsirelson bound is 2√2
The Tsirelson bound is 2√2 because it is the algebraic maximum 4 brought down by the price of reading: 4 correlations, times the constant ½, times the diagonal √2 of the square. Bell's classical bound, 2, follows the same formula with the side of the square, 1, in place of the diagonal.
What physics establishes
FactTwo entangled particles, measured far apart, give results more strongly correlated than any local hidden-variable theory allows: this is Bell's theorem (1964), confirmed by experiment from Aspect (1982) to the loophole-free tests of 2015.
FactThe violation itself has a limit. Boris Tsirelson proved in 1980 that the CHSH combination of correlations never exceeds 2√2 ≈ 2.828. Standard physics obtains this number through operator algebra in Hilbert space, without saying why it is this number.
The ARC reading
TheoremA protocol with no wave function and no operator recovers the exact singlet correlation, E(a, b) = −cos θ. The pair shares a state λ on the sphere; Alice reads the sign of her projection; the state updates; Bob reads his projection against a threshold μ that belongs to his reading device. The cosine follows in two lines from the linear response of the reading.
TheoremWithout the threshold μ, reading the sign alone, the same protocol gives 4: the Popescu-Rohrlich box, the algebraic maximum. The threshold is what brings 4 down to 2√2. "Reading the sign alone is a free reading; nature grants no free reading."
TheoremThe bound decomposes into three factors: 2√2 = 4 × ½ × √2. The 4 counts the four CHSH correlations, which answer to the four pairs of the grid (1, 8), (2, 7), (3, 6), (4, 5). The ½ is the constant k, the barycentre 4.5 over 9. The √2 is the diagonal of the square in the founding figure. With the side 1 instead of the diagonal, one recovers the classical bound: 4 × ½ × 1 = 2.
ReadingThe step from classical to quantum is therefore exactly √2, the ratio of diagonal to side: the view that sees objects reads the side, the view that sees relations reads the diagonal.
What this reading does not claim
The model is not local in Bell's sense, and Bell's theorem is not disputed: the state update is a shared relational memory, not a signal. The version of record is Tsirelson Without Hilbert v4.1 (July 2026); version 4.0, which read the pure sign, is withdrawn and its error is published.
Further reading
- Volume 7, The Physics Reread (English summary and table of statements)
- Chapter 1: the Tsirelson bound without Hilbert space (in French)
- Volume 7, full PDF (in French)