hierarchy_forces_phi
plain-language theorem explainer
Minimal closure of a geometric scale sequence forces its ratio to be the golden ratio phi. Researchers deriving self-similarity from ledger primitives in the Recognition Science framework cite this when advancing the forcing chain. The proof is a one-line wrapper applying the closed-ratio uniqueness theorem to the scales and closure predicate of the hierarchy.
Claim. If $H$ is a minimal discrete hierarchy consisting of a geometric scale sequence closed under the first non-trivial composition step, then the ratio of the scales equals $phi = (1 + sqrt(5))/2$.
background
The module isolates the minimal algebraic hierarchy data needed for the B1 closure step: a discrete geometric ledger together with the smallest closure condition scale 0 + scale 1 = scale 2. A MinimalHierarchy pairs a GeometricScaleSequence with the predicate that it is closed, which the doc-comment states is exactly the Fibonacci relation. This theorem rests on the upstream result that the unique positive closed ratio is phi, obtained by combining closure forcing the golden equation with positivity to select the single solution greater than zero.
proof idea
The proof is a one-line wrapper that applies closed_ratio_is_phi to the scales component and the minimalClosure field of the given MinimalHierarchy.
why it matters
This supplies the direct link from minimal hierarchy closure to the golden ratio and feeds the combined emergence theorem ledger_forces_phi, which derives the MinimalHierarchy package from uniform scale ladders plus additive closure and concludes phi. It fills the T6 step where phi is forced as the self-similar fixed point. The module doc-comment emphasizes that the first closure step is exactly the Fibonacci relation.
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