Foundation / Methods / Materials Combinator
Materials Hierarchy · M06

φCoherent materials combinator.

An interactive demonstrator for the five-branch Materials Hierarchy. Choose a φ-coherent element library and a target atomic number; the combinator enumerates every set of lower-Z elements whose atomic numbers sum exactly to the target, ranked by library purity, combinatorial order, and proximity to each library's functional terminus.

What this is A combinatorial map over the periodic table — not a claim about bulk material properties. Five φ-coherent branches derive from two root identities, φ²=φ+1 and d/dx(eˣ)=eˣ. Four are synthesis libraries that feed the subset-sum combinator (Σ Zᵢ = Ztarget): I Fibonacci (F), II Lucas (L), III Synthesis (S = F+L), and IV Product (P = F×L = F(2n)). The fifth, V Decay (D = round(eⁿ)), is not a synthesis library — it derives the Z ≥ 84 nuclear-stability boundary. Branch crossing draws candidates from all four synthesis libraries at once, generating cross-branch alloy candidates. The same classification underlies method M06 — Materials hierarchy.
Valid phi-coherent formulations