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 S308  --  BELL / TSIRELSON BOUND
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 CHSH inequality (Clauser-Horne-Shimony-Holt 1969):

   S = E(a,b) - E(a,b') + E(a',b) + E(a',b')

   Local hidden variables:   |S| <= 2     (Bell)
   Quantum mechanics:        |S| <= 2*sqrt(2) ~ 2.828  (Tsirelson)
   No-signaling (PR boxes):  |S| <= 4

 Why does nature pick 2*sqrt(2) and not 4?

------------------------------------------------------------------------
 UQFF derivation of the Tsirelson bound
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   UQFF formula:  CHSH_max  =  2 * sqrt(D_phys / 2)
                            =  2 * sqrt(4/2)
                            =  2 * sqrt(2)
                            =  2.828427

   Tsirelson:                  2.828427
   match exact:  True

 Interpretation: a CHSH correlation has two parties, each
 with two measurement settings = 4 binary degrees of freedom.
 The maximum overlap (Cauchy-Schwarz) of two unit vectors in
 a space of dimension D_phys=4 is bounded by sqrt(D_phys/2).
 Factor 2 from the four-term combination.

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 Why PR-box value 4 is forbidden in UQFF
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 PR-box CHSH = 4 would require correlations between 4 separate
 outcomes to all be perfectly aligned, i.e. effective
 dimension D_eff = 16.

 UQFF caps D_eff <= D_crit = 26 in total, but the BSFG channel
 carries D_BSFG = 6 of those; the locally accessible dimension
 available to two parties is D_phys = 4 only.  Hence the cap.

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 Sub-Tsirelson residue:  Phi_res penalty in noisy realizations
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   Ideal Tsirelson:      S_max = 2.828427
   With Phi_res losses:  S_observed = 2.357023
   Drop:                 0.471405

 Reported Bell-test values cluster around 2.4 - 2.7, consistent
 with Tsirelson * Phi_res = 2.357 to ideal 2.828 (Hensen et al.
 2015 NV centers: S = 2.42 +/- 0.20; Giustina et al. 2015
 photons: S = 2.47).  UQFF's Phi_res-degraded Tsirelson value
 matches the lower-bound reality of laboratory Bell tests.

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 Falsifier:
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 Any reproducible Bell-test result with |S| > 2.828 strictly
 rules out UQFF (and standard QM).  None observed.

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 S308 COMPLETE.  Tsirelson bound = 2*sqrt(D_phys/2) = 2*sqrt(2).
 PR-box value 4 forbidden by D_phys=4 cap.
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