realized_ratio_eq_base
plain-language theorem explainer
In a realized hierarchy inside a closed observable framework, every adjacent level ratio equals the base ratio from levels 0 to 1. Researchers deriving uniform scaling from self-similar dynamics cite this to confirm constant ratios across the hierarchy. The argument applies induction on the level index, reducing each successor step with the self-similar ratio property and the inductive hypothesis.
Claim. Let $F$ be a closed observable framework and $H$ a realized hierarchy in $F$. For every natural number $k$, the ratio of successive level observables satisfies $$frac{ell(k+1)}{ell(k)} = frac{ell(1)}{ell(0)},$$ where $ell(k)$ is the value of the positive observable at the $k$-th iterate of the dynamics on the base state.
background
A ClosedObservableFramework consists of a state space $S$, dynamics map $T:S to S$, and positive observable $r:S to mathbb{R}$ with the nontriviality condition that at least two states yield distinct $r$ values. The RealizedHierarchy structure augments this by fixing a base state, defining levels via $r$ composed with iterated $T$, and adding the ratio_self_similar field (consecutive ratios equal) together with additive posting and growth conditions. This module replaces external bridge hypotheses on sensitivity and additive composition with these native fields of the framework, as the module documentation states: the self-similar dynamics follows from J-cost invariance under the scale factor. Upstream, the shifted cost $H(x) = J(x) + 1$ reparametrizes the recognition composition law into d'Alembert form, though it is not invoked directly in this declaration.
proof idea
Proof by induction on $k$. The base case at zero holds by reflexivity. In the successor case, the ratio_self_similar property at the current index equates the next ratio to the prior ratio, which the inductive hypothesis then reduces to the base ratio.
why it matters
This theorem is invoked by realized_uniform_ratios to establish that all adjacent ratios coincide. It advances the hierarchy realization that replaces bridge hypotheses in the T5 to T6 transition, where J-uniqueness and the phi fixed point force uniform scaling from self-similar dynamics. It touches the open question of whether the closed framework alone suffices to derive the self-similar property without additional structure.
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