realizedHierarchy_canonical_base_ratio_phi
plain-language theorem explainer
Any realized hierarchy over a closed observable framework has multilevel composition whose canonical base ratio equals φ. Hierarchy and forcing-chain arguments cite this to pin self-similar scale to the golden ratio. The proof is a short term application of the uniform-seed base-ratio lemma, feeding uniformity, growth, and seed-size facts for the realized hierarchy.
Claim. Let $F$ be a closed observable framework and $H$ a realized hierarchy on $F$. Form the multilevel composition of $H$. Then its canonical base ratio (level $1$ over level $0$) equals $\varphi$, the unique positive self-similar fixed point forced by the Recognition cost structure.
background
The module UnifiedForcingChain aims at a complete inevitability chain T-1 through T8 from the Recognition Composition Law plus normalization and calibration. T6 is the step that forces $\varphi$ as the self-similar fixed point of the discrete ledger.
A closed observable framework supplies a state space, dynamics $T$, and a positive ratio observable $r$ with nontrivial range and no external input. A realized hierarchy on $F$ is an RS-native hierarchy: a base state, levels $k \mapsto r(T^{[k]}(\mathrm{base}))$, positivity, and growth $1 < \mathrm{levels},1/\mathrm{levels},0$. The multilevel composition packages those levels as a nontrivial multilevel composition.
The canonical base ratio of such a composition is simply $\mathrm{levels},1/\mathrm{levels},0$. Upstream, $\varphi$ is already forced in the PhiForcing layer; the present result identifies that abstract scale with the concrete base ratio of any realized hierarchy once uniformity, growth, and seed size are in hand.
proof idea
Term-mode one-shot application of canonicalBaseRatio_eq_phi_of_uniform_seed to the multilevel composition of $H$. The three side hypotheses are discharged by the sibling facts that the realized hierarchy is canonically uniform, has canonical growth, and has the required canonical seed size. No extra algebraic work sits in this declaration; it is the glue that routes those three properties into the $\varphi$-identification lemma.
why it matters
This pins the geometric scale of every realized hierarchy to $\varphi$, the T6 landmark of the forcing chain (self-similarity in the discrete ledger). Downstream it feeds canonical_realized_hierarchy_normal_form_equivalence, which packages the certificate that a realized hierarchy is level-equivalent to its $\varphi$-uniform normal form (uniformity and growth fields of that structure). Without base ratio $=\varphi$, the normal-form equivalence cannot close. In the broader RS picture this is the bridge from abstract hierarchy data to the $\varphi$-ladder used for mass rungs and constant derivations further up the chain.
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