hierarchy_forces_golden_equation
plain-language theorem explainer
A minimal discrete hierarchy with the smallest closure condition forces its scale ratio to satisfy the golden equation r squared equals r plus one. Researchers tracing the Recognition Science forcing chain from T0 to T6 would cite this as the algebraic source of the self-similar fixed point. The proof is a direct one-line application of the general closure theorem to the hierarchy's scale sequence and closure datum.
Claim. Let $H$ be a minimal hierarchy consisting of a geometric scale sequence closed under the first additive composition. Then the ratio $r$ of consecutive scales satisfies $r^2 = r + 1$.
background
The HierarchyMinimality module isolates the minimal algebraic data for the B1 closure step: a discrete geometric ledger together with the smallest closure condition scale 0 plus scale 1 equals scale 2. MinimalHierarchy is the structure carrying a GeometricScaleSequence together with the predicate that the sequence is closed under the first non-trivial composition. Upstream, the theorem closure_forces_golden_equation states that if a geometric scale sequence is closed under additive composition then the ratio r must satisfy r squared equals r plus one.
proof idea
One-line wrapper that applies closure_forces_golden_equation to the scales field and the minimalClosure field of the supplied hierarchy.
why it matters
This declaration supplies the first closure step inside the hierarchy minimality module and is explicitly described as the Fibonacci relation. It feeds the forcing of phi as the unique positive self-similar fixed point in the T0-to-T8 chain. The result sits at the algebraic root of the Recognition Composition Law and the subsequent derivation of the eight-tick octave and three spatial dimensions.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.