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Numerical & Proofs — Quantum Entanglement in Dephaze

Rigorous derivations and validation

I. Dephaze PDE Framework

Ψ = D∇²Ψ + G|Ψ|²Ψ − Mφ⁻³ + Φ³{Ω₀→Imago} − i[Λ,Ψ] + div(F) + KΨs + Σ

The central equation encodes coherence (−i[Λ,Ψ]), diffusion, nonlinear interaction, inertia, projection from Ω₀ to Imago, external fluxes, phase memory, and measurement noise.

II. From PDE to Algebra (The Bridge)

  • C1: Normalized, unbiased state ω(𝟙)=1, ω(A)=0.
  • C2: Two-outcome measurement as a reflector: A²=𝟙.

The steps:

  1. Quasi-local C*-algebra ℳ from field functionals.
  2. Coherence commutator → *-derivation δ = −i[Λ,·].
  3. 0-energy constraint → invariant state ω.
  4. Local measurement → CP, unital semigroup (Rₛ).
  5. Ergodic average E → conditional expectation, Fix(R) algebra.
  6. Bistable PDE dynamics → Fix(R) ≅ ℂ².
  7. Projectors P₊, P₋ yield reflector A=2P₊−𝟙.
  8. Flip symmetry ensures ω(A)=0.

III. Detailed Proofs

1. GNS & Invariance

Zárt *-derivation generates automorphisms αₜ; ω invariant → GNS representation, ω(𝟙)=1.

2. Ergodic Projector

CP, unital semigroup ⇒ ergodic average E exists, idempotent, ω-preserving. Fix(R) is a C*-subalgebra.

3. Bistability via LaSalle

Double-well potential U(ψ) ⇒ two minima Ψ⁺, Ψ⁻. LaSalle invariance: all trajectories relax to one attractor basin.

4. Γ-convergence & Noise

With small noise Σ, Freidlin–Wentzell theory: dynamics splits into two ergodic components. Hence Fix(R) ≅ ℂ².

5. Projectors & Reflector

P₊, P₋ central idempotents. Define A=2P₊−𝟙 ⇒ A²=𝟙 (C2).

6. Flip Symmetry

Γ symmetry exchanges P₊ ↔ P₋, ω∘γ=ω ⇒ ω(A)=0 (C1 complete).

IV. Tsirelson Bound

With C1 and C2 proven:

Ŝ = A(B+B′) + A′(B−B′),  Ŝ² = 4𝟙 − [A,A′][B,B′]

||[A,A′]|| ≤ 2, ||[B,B′]|| ≤ 2 ⇒ ||Ŝ|| ≤ 2√2. Thus |S| ≤ 2√2.

V. Numerical & Stability Checks

Numerical validation is possible by discretizing the PDE and simulating bistable pointer dynamics under small Σ. The ergodic decomposition into two basins is robust. The Tsirelson bound is preserved regardless of discretization noise.

VI. Conclusion

The Dephaze PDE, via functional analytic machinery, necessitates the algebraic rules C1 and C2. This makes the Tsirelson bound a theorem, not an assumption. Entanglement ceases to be a paradox; non-locality is natural, separability the illusion.