bridge_T5_T6_internal
plain-language theorem explainer
In a closed observable framework equipped with a realized hierarchy of states generated by iterating the dynamics, the extracted ladder ratio equals the golden ratio φ. Researchers deriving the Recognition Science forcing chain from T5 J-uniqueness to T6 self-similarity cite this as the internal derivation that avoids external composition axioms. The proof is a direct one-line application of the realized hierarchy forces phi lemma.
Claim. Let $F$ be a closed observable framework and let $H$ be a realized hierarchy on $F$. Then the ratio of the ladder obtained from $F$ and $H$ equals $φ$, the golden ratio.
background
A ClosedObservableFramework consists of a state space $S$, dynamics map $T$, and positive observable $r$ satisfying nontrivial variation and closure with no external input. A RealizedHierarchy on $F$ supplies a base state together with levels obtained by iterating $T$, enforcing self-similar scaling and additive posting with growth factor strictly greater than 1. The module HierarchyDynamics closes the structural gap between T5 (J-uniqueness via the cost functional) and T6 (φ forced by self-similarity) by deriving the Fibonacci recurrence from zero-parameter ledger composition without invoking external bridge hypotheses.
proof idea
One-line wrapper that applies the lemma realized_hierarchy_forces_phi to the given ClosedObservableFramework and RealizedHierarchy, which internally routes through ratio_self_similar, additive_posting, and the golden equation.
why it matters
This supplies the core RS-internal T5→T6 bridge, feeding directly into bridge_T5_T6_from_realized_closed_scale and HasLocalComposition. It realizes the module goal of deriving the golden equation from multilevel composition and minimality, completing the step from J-uniqueness to the eight-tick octave and D=3 without external sensitivity or additive-composition axioms.
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