Absence of critical Bernoulli site percolation on $\mathbb Z^d$ for $d \ge 3$

Claims $\theta(p_c)=0$ for nearest-neighbour Bernoulli site percolation on $\mathbb Z^d$ for every $d\ge3$ (theorem KNAll.Site.site_no_percolation_at_critical), by proving the Kozma-Nitzan gluing inequality for independent hyperedges, representing site percolation by incidence hyperedges, and running the lattice exploration of the bond proof. The intermediate dimensions were open. The planar case is classical and outside the formal statement. The same repository proves Kozma-Nitzan Conjectures 1 to 4 and 6, answers Questions 5, 7 and 9, and refutes one reading of Question 8. OpenAI's release (24 September 2026) later gave critical site nonpercolation on $\mathbb Z^3$ only, crediting this work; its quasi-transitive theorem is bond only, so the site form of Benjamini-Schramm Conjecture 4 beyond $\mathbb Z^d$ stays open.

Claims $\theta(p_c)=0$ for nearest-neighbour Bernoulli site percolation on $\mathbb Z^d$ for every $d\ge3$ (theorem KNAll.Site.site_no_percolation_at_critical), by proving the Kozma-Nitzan gluing inequality for independent hyperedges, representing site percolation by incidence hyperedges, and running the lattice exploration of the bond proof. The intermediate dimensions were open. The planar case is classical and outside the formal statement. The same repository proves Kozma-Nitzan Conjectures 1 to 4 and 6, answers Questions 5, 7 and 9, and refutes one reading of Question 8. OpenAI's release (24 September 2026) later gave critical site nonpercolation on $\mathbb Z^3$ only, crediting this work; its quasi-transitive theorem is bond only, so the site form of Benjamini-Schramm Conjecture 4 beyond $\mathbb Z^d$ stays open.