realized_hierarchy_forces_phi
plain-language theorem explainer
A realized hierarchy on a closed observable framework forces the scale ratio of its derived ladder to equal the golden ratio φ. Researchers closing the T5-T6 bridge in Recognition Science cite this to replace external sensitivity and additive-composition hypotheses. The proof is a one-line wrapper that feeds the realized ladder and its additive closure into the emergence theorem.
Claim. If $F$ is a closed observable framework and $H$ is a realized hierarchy on $F$ (with self-similar ratios and additive posting derived from observable $r$ and dynamics $T$), then the ratio of the ladder obtained from $F$ and $H$ equals the golden ratio $φ$.
background
The module internalizes hierarchies inside ClosedObservableFramework, replacing free-floating level interfaces with carrier states observed by $r$ and generated by $T$. A RealizedHierarchy structure supplies baseState, levels via iterated $T$, the self-similarity field ratio_self_similar, and the additive_posting field that yields Fibonacci closure. Upstream CostAlgebra defines the shifted cost $H(x) = J(x) + 1$ that converts the Recognition Composition Law into the d'Alembert equation, while HierarchyEmergence supplies the forcing result under uniform ratios and additive closure.
proof idea
The proof is a one-line wrapper that applies hierarchy_emergence_forces_phi to the ladder produced by realized_to_ladder and the additive closure produced by realized_additive_closure.
why it matters
This supplies the internal step for bridge_T5_T6_internal, which states the full RS-internal T5-T6 bridge without external hypotheses. It realizes the forcing of φ as the self-similar fixed point (T6) from J-uniqueness (T5) via the Recognition Composition Law and the eight-tick octave structure. The result closes the earlier gap that required separate sensitivity and HasAdditiveComposition assumptions on ZeroParameterComparisonLedger.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.