realized_hierarchy_forces_phi
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.
claimIf $F$ 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 in Recognition Science
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.
scope and limits
- Does not construct a RealizedHierarchy from an arbitrary ClosedObservableFramework.
- Assumes the framework satisfies no_continuous_moduli.
- Applies only when ratio_self_similar and additive_posting hold.
- Does not derive numerical mass or coupling values beyond the ratio.
Lean usage
theorem bridge_T5_T6_internal (F : ClosedObservableFramework) (H : RealizedHierarchy F) : (realized_to_ladder F H).ratio = PhiForcing.φ := realized_hierarchy_forces_phi F Hformal statement (Lean)
126theorem realized_hierarchy_forces_phi (F : ClosedObservableFramework)
127 (H : RealizedHierarchy F) :
128 (realized_to_ladder F H).ratio = PhiForcing.φ :=
proof body
Term-mode proof.
129 hierarchy_emergence_forces_phi
130 (realized_to_ladder F H)
131 (realized_additive_closure F H)
132
133/-! ## Non-Uniform Ratios Create Continuous Moduli
134
135The following theorem proves that `ratio_self_similar` is the ONLY option
136consistent with `no_continuous_moduli`. If the ratio function were not
137constant, the carrier would need to encode a continuously varying
138parameter, contradicting the closed observable framework. -/
139
140/-- If hierarchy levels have non-uniform ratios and the observable
141separates hierarchy states, then continuous moduli exist in the carrier.
142
143This internalizes the `sensitivity` bridge: the perturbation family
144`ScalePerturbed` lifts to carrier states via the observable `r`. -/