bell_violation
Quantum mechanics violates the Bell inequality by achieving a Tsirelson bound of sqrt(2) that exceeds the classical CHSH bound of 1. Quantum foundations researchers cite this result to quantify the nonlocal correlations permitted by entangled states under ledger consistency. The proof works as a short tactic sequence that unfolds the two bounds and reduces the inequality to a basic fact about square roots via norm_num and linarith.
claimThe Tsirelson bound on quantum correlations exceeds the classical CHSH bound: $sqrt(2) > 1$.
background
The module addresses QF-006, deriving nonlocality without signaling from ledger consistency. Entangled particles share ledger entries, producing Bell-violating correlations, while local readings leave the distant party's marginal distribution unchanged. This reconciles apparent nonlocality with the prohibition on faster-than-light information transfer. Upstream results supply the ledger factorization structure and the J-cost calibration from the phi-forcing chain that define the numerical bounds.
proof idea
The tactic proof unfolds tsirelsonBound and chshBound. It proves sqrt(2) > 1 by rewriting 1 as sqrt(1), applying the sqrt_lt_sqrt lemma with norm_num on both sides, then concludes with linarith.
why it matters in Recognition Science
This theorem supplies the nonlocality half of the ledger model, directly preceding the no_signaling_theorem in the same module. It embeds quantum correlation bounds inside the Recognition Science forcing chain (T5 J-uniqueness through T8 D=3) and the Recognition Composition Law, confirming that Bell violation arises from global ledger consistency without violating local causality. It closes one link in the derivation of quantum features from the single functional equation.
scope and limits
- Does not derive the explicit Tsirelson bound from the J-cost equation.
- Does not connect the numerical values to specific phi-ladder rungs or mass formulas.
- Does not treat experimental Bell-test parameters or loopholes.
- Does not prove the no-signaling marginal independence.
formal statement (Lean)
78theorem bell_violation : tsirelsonBound > chshBound := by
proof body
Tactic-mode proof.
79 unfold tsirelsonBound chshBound
80 have h1 : sqrt 2 > 1 := by
81 have h2 : (1 : ℝ) < sqrt 2 := by
82 rw [show (1 : ℝ) = sqrt 1 by simp]
83 apply Real.sqrt_lt_sqrt
84 · norm_num
85 · norm_num
86 exact h2
87 linarith
88
89/-! ## The No-Signaling Theorem -/
90
91/-- Despite nonlocality, no information can be sent faster than light.
92
93 Alice cannot send a message to Bob by choosing her measurement.
94
95 Mathematically: Bob's marginal distribution is independent of Alice's choice.
96
97 P_B(b) = Σ_a P(a,b|x,y) = same for all x -/