realizedClosedScale_canonical_base_ratio_phi
plain-language theorem explainer
A realized closed-scale model, once assembled into its direct multilevel composition, has canonical base ratio exactly φ. Hierarchy and scale-normal-form arguments in the forcing chain cite this identity. The proof is a short term application of the uniform-seed base-ratio criterion, supplying the three canonical uniformity, growth, and seed-size facts for realized closed scales.
Claim. Let $F$ be a closed observable framework and let $H$ be a realized closed-scale model on $F$. Then the canonical base ratio of the multilevel composition built from $H$ equals $\varphi$.
background
The ambient module is the Unified Forcing Chain: T-1 through T8 are derived as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the step that forces φ as the self-similar fixed point of the discrete ledger.
A closed observable framework supplies a state space, a transition, and a positive ratio observable with nontrivial range and no external input. A realized closed-scale model packages an orbit of such a framework that realizes a closed geometric scale sequence (base state, positive amplitude, and a closed scale ladder).
The canonical base ratio extracts the multiplicative step of a multilevel composition. Upstream, the shifted cost $H(x)=J(x)+1$ rewrites the composition law as d'Alembert form; geometric scales are powers of φ. The companion lemma canonicalBaseRatio_eq_phi_of_uniform_seed states that any multilevel composition that is uniform, grows canonically, and has the right seed size has base ratio φ.
proof idea
Term-mode one-shot application of the uniform-seed criterion. Instantiate that lemma on the multilevel composition of the realized closed-scale model, then discharge its three hypotheses by the three specialized facts already proved for realized closed scales: canonical uniformity, canonical growth, and canonical seed size. No further algebraic work.
why it matters
Pins the base ratio of every realized closed-scale multilevel composition to φ, which is the T6 landmark of the forcing chain (φ as the unique self-similar fixed point). Downstream it is consumed by the canonical normal-form equivalence certificate for realized closed-scale models, which packages uniformity and growth into a level-equivalence between a realized model and its φ-uniform normal form. Without this identity the normal-form side of the hierarchy realization story would not close at the orbit-level composition.
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