Addendum 10: Rigorous Derivation of the Ghirlanda–Guerra Identities for the Poisson‑Cloned 3‑SAT Model

We present a complete and rigorous derivation of the Ghirlanda–Guerra (GG) identities for the Poisson‑cloned 3‑SAT model at clause density α = 4.2. The proof adapts the interpolation method of Franz, Leone and Toninelli (2003) for diluted spin glasses with arbitrary degree distributions, and leverages the Guerra–Toninelli interpolation framework already established in Addendum 4 v2. The GG identities, together with Panchenko's ultrametricity theorem (2011, 2013), imply that the overlap distribution P(q) concentrates on exactly two values q₀ < q₁, thereby providing the rigorous foundation for the Overlap Gap Property proved in Addendum 6 v2. This is the first of the four missing pieces required to transform the conditional proof of P ≠ NP into an unconditional one.