quantum_violation
The theorem establishes that the CHSH correlator at optimal angles equals the Tsirelson bound of 2√2, exceeding the classical limit of 2. Quantum foundations researchers would cite this for the Recognition Science ledger derivation of Bell violation. The proof is a term-mode reduction that rewrites to the optimal value then applies absolute-value and positivity lemmas to reach the equality.
claimFor optimal measurement angles the absolute value of the CHSH combination satisfies $|S| = 2√2$.
background
Recognition Science treats entanglement as arising from shared ledger entries created when two particles are produced together. The CHSH combination is defined as $S = E(a,b) - E(a,b') + E(a',b) + E(a',b')$, where each $E$ is a correlation function. Classical local realism requires $|S| ≤ 2$, while the Tsirelson bound reachable by quantum mechanics is $2√2$. The module derives the violation directly from the ledger structure without invoking hidden variables or faster-than-light signaling.
proof idea
The proof is a one-line term-mode wrapper. It rewrites the optimal CHSH value, applies abs_neg and abs_of_pos to handle the sign, then invokes mul_pos together with Real.sqrt_pos to confirm the positive square-root expression equals $2√2$.
why it matters in Recognition Science
This result completes the QF-005 derivation of Bell violation inside the Recognition Science framework. It supplies the concrete numerical prediction that feeds the ledger-based account of nonlocality and supports the planned PRL paper on quantum nonlocality from ledger structure. The equality anchors the quantum side of the CHSH comparison and closes the gap between the classical bound of 2 and the observed violation.
scope and limits
- Does not derive the Tsirelson bound from the J-function or phi-ladder.
- Does not address experimental loopholes or detector efficiency.
- Does not extend the result to multipartite or higher-dimensional systems.
formal statement (Lean)
163theorem quantum_violation :
164 |optimalCHSH| = tsirelsonBound := by
proof body
Term-mode proof.
165 rw [optimal_chsh_value, abs_neg, abs_of_pos]
166 exact mul_pos (by norm_num : (2 : ℝ) > 0) (Real.sqrt_pos.mpr (by norm_num))
167
168/-! ## The Ledger Explanation -/
169
170/-- In RS, Bell violation comes from **shared ledger entries**:
171
172 1. When entangled particles are created, they share a ledger entry
173 2. This entry encodes their correlation (not individual values)
174 3. Measurement "actualizes" the shared entry for both particles
175 4. The actualization is consistent (respects correlation) but not predetermined
176
177 This explains nonlocality without hidden variables or FTL signaling. -/